Generating functions for Sequences of interest to Arvind Ayyer and Yuval Roi\ chamn By Shalosh B. Ekhad Let n be a positive integer, and let Lambda and Mu be two partitions of n. L\ et F[Lambda,Mu][k]:= The inner product of the k-th power of the Lambda-column o\ f the Character Table of Sn, and the Mu-column and let f(Lamda,Mu](x):=Sum(F[Lambda,Mu][k]*x^k,k=0..infinity Here is everything up to n=, 9 ------------------------------------------ Doing the symmetric group on, 1, elements There are, 1, partitions of, 1, here there are in the usual order Regarding Lambda=, [1] 1 F[[1], [1]](x) = - ------ -1 + x and in Maple notation F[[1],[1]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 2, elements There are, 2, partitions of, 2, here there are in the usual order Regarding Lambda=, [2] 1 F[[2], [2]](x) = - ------ -1 + x and in Maple notation F[[2],[2]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[2], [1, 1]](x) = 0 and in Maple notation F[[2],[1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [1, 1] 1 F[[1, 1], [2]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1],[2]](x) = -1/(1+x)/(-1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 x F[[1, 1], [1, 1]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1],[1, 1]](x) = -x/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 3, elements There are, 3, partitions of, 3, here there are in the usual order Regarding Lambda=, [3] 1 F[[3], [3]](x) = - ------ -1 + x and in Maple notation F[[3],[3]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[3], [2, 1]](x) = 0 and in Maple notation F[[3],[2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[3], [1, 1, 1]](x) = 0 and in Maple notation F[[3],[1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [2, 1] 2 x + x - 1 F[[2, 1], [3]](x) = ------------------ (1 + x) (-1 + 2 x) and in Maple notation F[[2, 1],[3]](x) = (x^2+x-1)/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 x F[[2, 1], [2, 1]](x) = - ------------------ (1 + x) (-1 + 2 x) and in Maple notation F[[2, 1],[2, 1]](x) = -x/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 2 x F[[2, 1], [1, 1, 1]](x) = - ------------------ (1 + x) (-1 + 2 x) and in Maple notation F[[2, 1],[1, 1, 1]](x) = -x^2/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 Regarding Lambda=, [1, 1, 1] 1 F[[1, 1, 1], [3]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1],[3]](x) = -1/(1+x)/(-1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1], [2, 1]](x) = 0 and in Maple notation F[[1, 1, 1],[2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 x F[[1, 1, 1], [1, 1, 1]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1],[1, 1, 1]](x) = -x/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 4, elements There are, 5, partitions of, 4, here there are in the usual order Regarding Lambda=, [4] 1 F[[4], [4]](x) = - ------ -1 + x and in Maple notation F[[4],[4]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[4], [3, 1]](x) = 0 and in Maple notation F[[4],[3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[4], [2, 2]](x) = 0 and in Maple notation F[[4],[2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[4], [2, 1, 1]](x) = 0 and in Maple notation F[[4],[2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[4], [1, 1, 1, 1]](x) = 0 and in Maple notation F[[4],[1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [3, 1] 3 x - 3 x + 1 F[[3, 1], [4]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[3, 1],[4]](x) = (x^3-3*x+1)/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 x (-1 + 2 x) F[[3, 1], [3, 1]](x) = - --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[3, 1],[3, 1]](x) = -x*(-1+2*x)/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684, 435848050 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 2 x F[[3, 1], [2, 2]](x) = - ------------------ (1 + x) (-1 + 3 x) and in Maple notation F[[3, 1],[2, 2]](x) = -x^2/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581, 1195742, 3587227, 10761680, 32285041, 96855122, 290565367 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 2 x F[[3, 1], [2, 1, 1]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[3, 1],[2, 1, 1]](x) = x^2/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 1, 3, 10, 30, 91, 273, 820, 2460, 7381, 22143, 66430, 199290, 597871, 1793613, 5380840, 16142520, 48427561, 145282683, 435848050 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 3 x F[[3, 1], [1, 1, 1, 1]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[3, 1],[1, 1, 1, 1]](x) = x^3/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 0, 1, 3, 10, 30, 91, 273, 820, 2460, 7381, 22143, 66430, 199290, 597871, 1793613, 5380840, 16142520, 48427561, 145282683 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 Regarding Lambda=, [2, 2] 2 x + x - 1 F[[2, 2], [4]](x) = ------------------ (1 + x) (-1 + 2 x) and in Maple notation F[[2, 2],[4]](x) = (x^2+x-1)/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 F[[2, 2], [3, 1]](x) = 0 and in Maple notation F[[2, 2],[3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 x F[[2, 2], [2, 2]](x) = - ------------------ (1 + x) (-1 + 2 x) and in Maple notation F[[2, 2],[2, 2]](x) = -x/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 F[[2, 2], [2, 1, 1]](x) = 0 and in Maple notation F[[2, 2],[2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 2 x F[[2, 2], [1, 1, 1, 1]](x) = - ------------------ (1 + x) (-1 + 2 x) and in Maple notation F[[2, 2],[1, 1, 1, 1]](x) = -x^2/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763 ---------------------------------- Their sum is 1 - ------------------ (1 + x) (-1 + 2 x) and in Maple notation -1/(1+x)/(-1+2*x) The first 20 term , starting with k=1 are 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051 Regarding Lambda=, [2, 1, 1] 3 x - 3 x + 1 F[[2, 1, 1], [4]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[2, 1, 1],[4]](x) = (x^3-3*x+1)/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 2 x F[[2, 1, 1], [3, 1]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[2, 1, 1],[3, 1]](x) = x^2/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 1, 3, 10, 30, 91, 273, 820, 2460, 7381, 22143, 66430, 199290, 597871, 1793613, 5380840, 16142520, 48427561, 145282683, 435848050 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 2 x F[[2, 1, 1], [2, 2]](x) = - ------------------ (1 + x) (-1 + 3 x) and in Maple notation F[[2, 1, 1],[2, 2]](x) = -x^2/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581, 1195742, 3587227, 10761680, 32285041, 96855122, 290565367 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 x (-1 + 2 x) F[[2, 1, 1], [2, 1, 1]](x) = - --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[2, 1, 1],[2, 1, 1]](x) = -x*(-1+2*x)/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684, 435848050 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 3 x F[[2, 1, 1], [1, 1, 1, 1]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 3 x) and in Maple notation F[[2, 1, 1],[1, 1, 1, 1]](x) = x^3/(1+x)/(-1+x)/(-1+3*x) The first 20 term , starting with k=1 are 0, 0, 1, 3, 10, 30, 91, 273, 820, 2460, 7381, 22143, 66430, 199290, 597871, 1793613, 5380840, 16142520, 48427561, 145282683 ---------------------------------- Their sum is 2 x + x - 1 ------------------ (1 + x) (-1 + 3 x) and in Maple notation (x^2+x-1)/(1+x)/(-1+3*x) The first 20 term , starting with k=1 are 1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834 Regarding Lambda=, [1, 1, 1, 1] 1 F[[1, 1, 1, 1], [4]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1],[4]](x) = -1/(1+x)/(-1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1], [3, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1],[3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1], [2, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1],[2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1], [2, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1],[2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 x F[[1, 1, 1, 1], [1, 1, 1, 1]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1],[1, 1, 1, 1]](x) = -x/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 5, elements There are, 7, partitions of, 5, here there are in the usual order Regarding Lambda=, [5] 1 F[[5], [5]](x) = - ------ -1 + x and in Maple notation F[[5],[5]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[5], [4, 1]](x) = 0 and in Maple notation F[[5],[4, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[5], [3, 2]](x) = 0 and in Maple notation F[[5],[3, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[5], [3, 1, 1]](x) = 0 and in Maple notation F[[5],[3, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[5], [2, 2, 1]](x) = 0 and in Maple notation F[[5],[2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[5], [2, 1, 1, 1]](x) = 0 and in Maple notation F[[5],[2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[5], [1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[5],[1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [4, 1] 4 3 2 3 x - x - 8 x + 6 x - 1 F[[4, 1], [5]](x) = -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) and in Maple notation F[[4, 1],[5]](x) = (3*x^4-x^3-8*x^2+6*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 1, 1, 4, 11, 40, 147, 568, 2227, 8824, 35123, 140152, 559923, 2238328, 8950579, 35796856, 143176499, 572684152, 2290692915, 9162684280 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 2 x (5 x - 5 x + 1) F[[4, 1], [4, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) and in Maple notation F[[4, 1],[4, 1]](x) = -x*(5*x^2-5*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 1, 4, 11, 40, 147, 568, 2227, 8824, 35123, 140152, 559923, 2238328, 8950579, 35796856, 143176499, 572684152, 2290692915, 9162684280, 36650562355 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 2 2 x (x - 3 x + 1) F[[4, 1], [3, 2]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) and in Maple notation F[[4, 1],[3, 2]](x) = -x^2*(x^2-3*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 1, 3, 12, 45, 176, 693, 2752, 10965, 43776, 174933, 699392, 2796885, 11186176, 44741973, 178962432, 715838805, 2863333376, 11453289813, 45813071872 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 2 x F[[4, 1], [3, 1, 1]](x) = - ------------------ (1 + x) (-1 + 4 x) and in Maple notation F[[4, 1],[3, 1, 1]](x) = -x^2/(1+x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861, 3355443, 13421773, 53687091, 214748365, 858993459, 3435973837, 13743895347, 54975581389 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 3 x (3 x - 2) F[[4, 1], [2, 2, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) and in Maple notation F[[4, 1],[2, 2, 1]](x) = x^3*(3*x-2)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 0, 2, 9, 40, 165, 672, 2709, 10880, 43605, 174592, 698709, 2795520, 11183445, 44736512, 178951509, 715816960, 2863289685, 11453202432, 45812897109 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 3 x F[[4, 1], [2, 1, 1, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) and in Maple notation F[[4, 1],[2, 1, 1, 1]](x) = -x^3/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 0, 1, 6, 29, 126, 525, 2142, 8653, 34782, 139469, 558558, 2235597, 8945118, 35785933, 143154654, 572640461, 2290605534, 9162509517, 36650212830 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 4 x F[[4, 1], [1, 1, 1, 1, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) and in Maple notation F[[4, 1],[1, 1, 1, 1, 1]](x) = -x^4/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 0, 0, 1, 6, 29, 126, 525, 2142, 8653, 34782, 139469, 558558, 2235597, 8945118, 35785933, 143154654, 572640461, 2290605534, 9162509517 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 Regarding Lambda=, [3, 2] 3 x - 5 x + 1 F[[3, 2], [5]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[3, 2],[5]](x) = (x^3-5*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 1, 6, 26, 131, 651, 3256, 16276, 81381, 406901, 2034506, 10172526, 50862631, 254313151, 1271565756, 6357828776, 31789143881, 158945719401, 794728597006 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[3, 2], [4, 1]](x) = - ------------------ (1 + x) (-1 + 5 x) and in Maple notation F[[3, 2],[4, 1]](x) = -x^2/(1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 x (-1 + 4 x) F[[3, 2], [3, 2]](x) = - --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[3, 2],[3, 2]](x) = -x*(-1+4*x)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 1, 6, 26, 131, 651, 3256, 16276, 81381, 406901, 2034506, 10172526, 50862631, 254313151, 1271565756, 6357828776, 31789143881, 158945719401, 794728597006, 3973642985026 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[3, 2], [3, 1, 1]](x) = ------------------- (-1 + x) (-1 + 5 x) and in Maple notation F[[3, 2],[3, 1, 1]](x) = x^2/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 6, 31, 156, 781, 3906, 19531, 97656, 488281, 2441406, 12207031, 61035156, 305175781, 1525878906, 7629394531, 38146972656, 190734863281, 953674316406, 4768371582031 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[3, 2], [2, 2, 1]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[3, 2],[2, 2, 1]](x) = x^2/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 5, 26, 130, 651, 3255, 16276, 81380, 406901, 2034505, 10172526, 50862630, 254313151, 1271565755, 6357828776, 31789143880, 158945719401, 794728597005, 3973642985026 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[3, 2], [2, 1, 1, 1]](x) = - ------------------ (1 + x) (-1 + 5 x) and in Maple notation F[[3, 2],[2, 1, 1, 1]](x) = -x^2/(1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 3 x F[[3, 2], [1, 1, 1, 1, 1]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[3, 2],[1, 1, 1, 1, 1]](x) = x^3/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 5, 26, 130, 651, 3255, 16276, 81380, 406901, 2034505, 10172526, 50862630, 254313151, 1271565755, 6357828776, 31789143880, 158945719401, 794728597005 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 Regarding Lambda=, [3, 1, 1] 3 2 8 x - 7 x - 5 x + 1 F[[3, 1, 1], [5]](x) = ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation F[[3, 1, 1],[5]](x) = (8*x^3-7*x^2-5*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 1, 13, 61, 397, 2317, 14029, 83917, 504013, 3023053, 18140365, 108838093, 653036749, 3918204109, 23509257421, 141055478989, 846333005005, 5077997767885, 30467987131597 ---------------------------------- Their sum is 3 2 4 x - 4 x - 4 x + 1 ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation (4*x^3-4*x^2-4*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 9, 45, 285, 1677, 10125, 60621, 363981, 2183373, 13101261, 78605517, 471637197, 2829814989, 16978906317, 101873405133, 611240496333, 3667442846925, 22004657343693, 132027943537869, 792167662275789 2 x F[[3, 1, 1], [4, 1]](x) = ------------------- (-1 + x) (-1 + 6 x) and in Maple notation F[[3, 1, 1],[4, 1]](x) = x^2/(-1+x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 7, 43, 259, 1555, 9331, 55987, 335923, 2015539, 12093235, 72559411, 435356467, 2612138803, 15672832819, 94036996915, 564221981491, 3385331888947, 20311991333683, 121871948002099 ---------------------------------- Their sum is 3 2 4 x - 4 x - 4 x + 1 ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation (4*x^3-4*x^2-4*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 9, 45, 285, 1677, 10125, 60621, 363981, 2183373, 13101261, 78605517, 471637197, 2829814989, 16978906317, 101873405133, 611240496333, 3667442846925, 22004657343693, 132027943537869, 792167662275789 2 2 x F[[3, 1, 1], [3, 2]](x) = - -------------------- (1 + 2 x) (-1 + 6 x) and in Maple notation F[[3, 1, 1],[3, 2]](x) = -2*x^2/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 2, 8, 56, 320, 1952, 11648, 70016, 419840, 2519552, 15116288, 90699776, 544194560, 3265175552, 19591036928, 117546254336, 705277460480, 4231664893952, 25389989101568, 152339935133696 ---------------------------------- Their sum is 3 2 4 x - 4 x - 4 x + 1 ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation (4*x^3-4*x^2-4*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 9, 45, 285, 1677, 10125, 60621, 363981, 2183373, 13101261, 78605517, 471637197, 2829814989, 16978906317, 101873405133, 611240496333, 3667442846925, 22004657343693, 132027943537869, 792167662275789 x (-1 + 4 x) F[[3, 1, 1], [3, 1, 1]](x) = - ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation F[[3, 1, 1],[3, 1, 1]](x) = -x*(-1+4*x)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 1, 13, 61, 397, 2317, 14029, 83917, 504013, 3023053, 18140365, 108838093, 653036749, 3918204109, 23509257421, 141055478989, 846333005005, 5077997767885, 30467987131597, 182807921741005 ---------------------------------- Their sum is 3 2 4 x - 4 x - 4 x + 1 ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation (4*x^3-4*x^2-4*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 9, 45, 285, 1677, 10125, 60621, 363981, 2183373, 13101261, 78605517, 471637197, 2829814989, 16978906317, 101873405133, 611240496333, 3667442846925, 22004657343693, 132027943537869, 792167662275789 2 2 x F[[3, 1, 1], [2, 2, 1]](x) = - -------------------- (1 + 2 x) (-1 + 6 x) and in Maple notation F[[3, 1, 1],[2, 2, 1]](x) = -2*x^2/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 2, 8, 56, 320, 1952, 11648, 70016, 419840, 2519552, 15116288, 90699776, 544194560, 3265175552, 19591036928, 117546254336, 705277460480, 4231664893952, 25389989101568, 152339935133696 ---------------------------------- Their sum is 3 2 4 x - 4 x - 4 x + 1 ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation (4*x^3-4*x^2-4*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 9, 45, 285, 1677, 10125, 60621, 363981, 2183373, 13101261, 78605517, 471637197, 2829814989, 16978906317, 101873405133, 611240496333, 3667442846925, 22004657343693, 132027943537869, 792167662275789 2 x F[[3, 1, 1], [2, 1, 1, 1]](x) = ------------------- (-1 + x) (-1 + 6 x) and in Maple notation F[[3, 1, 1],[2, 1, 1, 1]](x) = x^2/(-1+x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 7, 43, 259, 1555, 9331, 55987, 335923, 2015539, 12093235, 72559411, 435356467, 2612138803, 15672832819, 94036996915, 564221981491, 3385331888947, 20311991333683, 121871948002099 ---------------------------------- Their sum is 3 2 4 x - 4 x - 4 x + 1 ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation (4*x^3-4*x^2-4*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 9, 45, 285, 1677, 10125, 60621, 363981, 2183373, 13101261, 78605517, 471637197, 2829814989, 16978906317, 101873405133, 611240496333, 3667442846925, 22004657343693, 132027943537869, 792167662275789 2 x (-1 + 4 x) F[[3, 1, 1], [1, 1, 1, 1, 1]](x) = - ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation F[[3, 1, 1],[1, 1, 1, 1, 1]](x) = -x^2*(-1+4*x)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 1, 13, 61, 397, 2317, 14029, 83917, 504013, 3023053, 18140365, 108838093, 653036749, 3918204109, 23509257421, 141055478989, 846333005005, 5077997767885, 30467987131597 ---------------------------------- Their sum is 3 2 4 x - 4 x - 4 x + 1 ----------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) and in Maple notation (4*x^3-4*x^2-4*x+1)/(-1+x)/(1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 9, 45, 285, 1677, 10125, 60621, 363981, 2183373, 13101261, 78605517, 471637197, 2829814989, 16978906317, 101873405133, 611240496333, 3667442846925, 22004657343693, 132027943537869, 792167662275789 Regarding Lambda=, [2, 2, 1] 3 x - 5 x + 1 F[[2, 2, 1], [5]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[2, 2, 1],[5]](x) = (x^3-5*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 1, 6, 26, 131, 651, 3256, 16276, 81381, 406901, 2034506, 10172526, 50862631, 254313151, 1271565756, 6357828776, 31789143881, 158945719401, 794728597006 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[2, 2, 1], [4, 1]](x) = - ------------------ (1 + x) (-1 + 5 x) and in Maple notation F[[2, 2, 1],[4, 1]](x) = -x^2/(1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[2, 2, 1], [3, 2]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[2, 2, 1],[3, 2]](x) = x^2/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 5, 26, 130, 651, 3255, 16276, 81380, 406901, 2034505, 10172526, 50862630, 254313151, 1271565755, 6357828776, 31789143880, 158945719401, 794728597005, 3973642985026 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[2, 2, 1], [3, 1, 1]](x) = ------------------- (-1 + x) (-1 + 5 x) and in Maple notation F[[2, 2, 1],[3, 1, 1]](x) = x^2/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 6, 31, 156, 781, 3906, 19531, 97656, 488281, 2441406, 12207031, 61035156, 305175781, 1525878906, 7629394531, 38146972656, 190734863281, 953674316406, 4768371582031 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 x (-1 + 4 x) F[[2, 2, 1], [2, 2, 1]](x) = - --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[2, 2, 1],[2, 2, 1]](x) = -x*(-1+4*x)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 1, 6, 26, 131, 651, 3256, 16276, 81381, 406901, 2034506, 10172526, 50862631, 254313151, 1271565756, 6357828776, 31789143881, 158945719401, 794728597006, 3973642985026 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 2 x F[[2, 2, 1], [2, 1, 1, 1]](x) = - ------------------ (1 + x) (-1 + 5 x) and in Maple notation F[[2, 2, 1],[2, 1, 1, 1]](x) = -x^2/(1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 4, 21, 104, 521, 2604, 13021, 65104, 325521, 1627604, 8138021, 40690104, 203450521, 1017252604, 5086263021, 25431315104, 127156575521, 635782877604, 3178914388021 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 3 x F[[2, 2, 1], [1, 1, 1, 1, 1]](x) = --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation F[[2, 2, 1],[1, 1, 1, 1, 1]](x) = x^3/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 5, 26, 130, 651, 3255, 16276, 81380, 406901, 2034505, 10172526, 50862630, 254313151, 1271565755, 6357828776, 31789143880, 158945719401, 794728597005 ---------------------------------- Their sum is 3 x - 4 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (x^3-4*x+1)/(1+x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 6, 27, 136, 677, 3386, 16927, 84636, 423177, 2115886, 10579427, 52897136, 264485677, 1322428386, 6612141927, 33060709636, 165303548177, 826517740886, 4132588704427, 20662943522136 Regarding Lambda=, [2, 1, 1, 1] 4 2 3 x - 8 x - 2 x + 1 F[[2, 1, 1, 1], [5]](x) = ------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) and in Maple notation F[[2, 1, 1, 1],[5]](x) = (3*x^4-8*x^2-2*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 1, 0, 4, 6, 40, 126, 568, 2142, 8824, 34782, 140152, 558558, 2238328, 8945118, 35796856, 143154654, 572684152, 2290605534, 9162684280 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 2 x F[[2, 1, 1, 1], [4, 1]](x) = - ---------------------------- (1 + x) (1 + 2 x) (-1 + 4 x) and in Maple notation F[[2, 1, 1, 1],[4, 1]](x) = -x^2/(1+x)/(1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 1, 1, 11, 29, 147, 525, 2227, 8653, 35123, 139469, 559923, 2235597, 8950579, 35785933, 143176499, 572640461, 2290692915, 9162509517, 36650562355 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 2 x F[[2, 1, 1, 1], [3, 2]](x) = - -------------------- (1 + 2 x) (-1 + 4 x) and in Maple notation F[[2, 1, 1, 1],[3, 2]](x) = -x^2/(1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 1, 2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 2 x F[[2, 1, 1, 1], [3, 1, 1]](x) = - ------------------ (1 + x) (-1 + 4 x) and in Maple notation F[[2, 1, 1, 1],[3, 1, 1]](x) = -x^2/(1+x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861, 3355443, 13421773, 53687091, 214748365, 858993459, 3435973837, 13743895347, 54975581389 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 3 3 x F[[2, 1, 1, 1], [2, 2, 1]](x) = ----------------------------- (-1 + x) (1 + 2 x) (-1 + 4 x) and in Maple notation F[[2, 1, 1, 1],[2, 2, 1]](x) = 3*x^3/(-1+x)/(1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 0, 3, 9, 45, 165, 693, 2709, 10965, 43605, 174933, 698709, 2796885, 11183445, 44741973, 178951509, 715838805, 2863289685, 11453289813, 45812897109 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 2 x (5 x + 2 x - 1) F[[2, 1, 1, 1], [2, 1, 1, 1]](x) = - ------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) and in Maple notation F[[2, 1, 1, 1],[2, 1, 1, 1]](x) = -x*(5*x^2+2*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x ) The first 20 term , starting with k=1 are 1, 0, 4, 6, 40, 126, 568, 2142, 8824, 34782, 140152, 558558, 2238328, 8945118, 35796856, 143154654, 572684152, 2290605534, 9162684280, 36650212830 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 3 x F[[2, 1, 1, 1], [1, 1, 1, 1, 1]](x) = - ---------------------------- (1 + x) (1 + 2 x) (-1 + 4 x) and in Maple notation F[[2, 1, 1, 1],[1, 1, 1, 1, 1]](x) = -x^3/(1+x)/(1+2*x)/(-1+4*x) The first 20 term , starting with k=1 are 0, 0, 1, 1, 11, 29, 147, 525, 2227, 8653, 35123, 139469, 559923, 2235597, 8950579, 35785933, 143176499, 572640461, 2290692915, 9162509517 ---------------------------------- Their sum is 3 2 x - x - 3 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 4 x) and in Maple notation (x^3-x^2-3*x+1)/(1+x)/(-1+x)/(-1+4*x) The first 20 term , starting with k=1 are 1, 4, 14, 56, 222, 888, 3550, 14200, 56798, 227192, 908766, 3635064, 14540254, 58161016, 232644062, 930576248, 3722304990, 14889219960, 59556879838, 238227519352 Regarding Lambda=, [1, 1, 1, 1, 1] 1 F[[1, 1, 1, 1, 1], [5]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1],[5]](x) = -1/(1+x)/(-1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1], [4, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1],[4, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1], [3, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1],[3, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1], [3, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1],[3, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1], [2, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1],[2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1], [2, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1],[2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 x F[[1, 1, 1, 1, 1], [1, 1, 1, 1, 1]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1],[1, 1, 1, 1, 1]](x) = -x/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 6, elements There are, 11, partitions of, 6, here there are in the usual order Regarding Lambda=, [6] 1 F[[6], [6]](x) = - ------ -1 + x and in Maple notation F[[6],[6]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [5, 1]](x) = 0 and in Maple notation F[[6],[5, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [4, 2]](x) = 0 and in Maple notation F[[6],[4, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [4, 1, 1]](x) = 0 and in Maple notation F[[6],[4, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [3, 3]](x) = 0 and in Maple notation F[[6],[3, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [3, 2, 1]](x) = 0 and in Maple notation F[[6],[3, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [3, 1, 1, 1]](x) = 0 and in Maple notation F[[6],[3, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [2, 2, 2]](x) = 0 and in Maple notation F[[6],[2, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [2, 2, 1, 1]](x) = 0 and in Maple notation F[[6],[2, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [2, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[6],[2, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[6], [1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[6],[1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [5, 1] 5 4 3 2 11 x - 7 x - 29 x + 31 x - 10 x + 1 F[[5, 1], [6]](x) = ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[6]](x) = (11*x^5-7*x^4-29*x^3+31*x^2-10*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-\ 1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 1, 4, 11, 41, 161, 694, 3151, 14851, 71621, 350384, 1729091, 8577661, 42686281, 212828074, 1062335831, 5306276471, 26515196141, 132527465764 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x (19 x - 24 x + 9 x - 1) F[[5, 1], [5, 1]](x) = - ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[5, 1]](x) = -x*(19*x^3-24*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/( -1+5*x) The first 20 term , starting with k=1 are 1, 1, 4, 11, 41, 161, 694, 3151, 14851, 71621, 350384, 1729091, 8577661, 42686281, 212828074, 1062335831, 5306276471, 26515196141, 132527465764, 662491871371 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 2 x (3 x - 5 x + 1) F[[5, 1], [4, 2]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[4, 2]](x) = -x^2*(3*x^2-5*x+1)/(1+x)/(-1+x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 54, 241, 1113, 5293, 25644, 125761, 621423, 3084973, 15358434, 76592881, 382366533, 1910039053, 9544814424, 47707929601, 238491220443, 1192310819533 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 3 2 x (5 x - 13 x + 7 x - 1) F[[5, 1], [4, 1, 1]](x) = - ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[4, 1, 1]](x) = -x^2*(5*x^3-13*x^2+7*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3* x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 251, 1183, 5713, 27975, 138151, 685663, 3413213, 17021095, 84971251, 424454943, 2121073513, 10601769415, 52998063551, 264957989023, 1324693002613 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x (5 x - 5 x + 1) F[[5, 1], [3, 3]](x) = ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[3, 3]](x) = x^3*(5*x^2-5*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 5, 25, 120, 581, 2835, 13945, 68990, 342661, 1706265, 8509865, 42484260, 212224741, 1060531295, 5300873785, 26499009930, 132478950821, 662346413925 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x F[[5, 1], [3, 2, 1]](x) = ----------------------------- (1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation F[[5, 1],[3, 2, 1]](x) = 2*x^3/(1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 2, 12, 66, 340, 1722, 8652, 43346, 216900, 1084842, 5424892, 27125826, 135631860, 678164762, 3390834732, 16954195506, 84771021220, 423855193482, 2119276142172 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x (5 x - 4 x + 1) F[[5, 1], [3, 1, 1, 1]](x) = ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[3, 1, 1, 1]](x) = x^3*(5*x^2-4*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-\ 1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 190, 1001, 5166, 26335, 133230, 670901, 3368926, 16888235, 84572670, 423259201, 2117486286, 10591007735, 52965778510, 264861133901, 1324402437246 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 4 x (5 x - 2) F[[5, 1], [2, 2, 2]](x) = - ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[2, 2, 2]](x) = -x^4*(5*x-2)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 2, 15, 90, 490, 2562, 13125, 66530, 335280, 1684122, 8443435, 42284970, 211626870, 1058737682, 5295492945, 26482867410, 132430523260, 662201131242 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 4 3 x F[[5, 1], [2, 2, 1, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[2, 2, 1, 1]](x) = -3*x^4/(1+x)/(-1+x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 24, 150, 840, 4473, 23184, 118380, 599280, 3018543, 15159144, 75995010, 380572920, 1904658213, 9528671904, 47659502040, 238345937760, 1191874971483 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[5, 1], [2, 1, 1, 1, 1]](x) = 4 x ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[2, 1, 1, 1, 1]](x) = x^4/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 1, 10, 70, 420, 2331, 12390, 64240, 328240, 1662661, 8378370, 42088410, 211034460, 1056954991, 5290133950, 26466768580, 132382183080, 662056023321 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[5, 1], [1, 1, 1, 1, 1, 1]](x) = 5 x ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[5, 1],[1, 1, 1, 1, 1, 1]](x) = x^5/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 1, 10, 70, 420, 2331, 12390, 64240, 328240, 1662661, 8378370, 42088410, 211034460, 1056954991, 5290133950, 26466768580, 132382183080 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 Regarding Lambda=, [4, 2] 4 3 2 12 x - 2 x - 27 x + 12 x - 1 F[[4, 2], [6]](x) = -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[6]](x) = (12*x^4-2*x^3-27*x^2+12*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 2, 13, 92, 769, 6734, 60061, 538904, 4845217, 43592186, 392285389, 3530435636, 31773522145, 285960503558, 2573640944797, 23162757741488, 208464787388353, 1876182989640050, 16885646616195085 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 2 x (3 x - 6 x + 1) F[[4, 2], [5, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[5, 1]](x) = -x^2*(3*x^2-6*x+1)/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 6, 49, 420, 3721, 33306, 299209, 2691240, 24216241, 217931406, 1961338369, 17651912460, 158866813561, 1429800126306, 12868197549529, 115813767184080, 1042323872371681, 9380914754490006, 84428232499844689 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x (15 x - 10 x + 1) F[[4, 2], [4, 2]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[4, 2]](x) = -x*(15*x^2-10*x+1)/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 2, 13, 92, 769, 6734, 60061, 538904, 4845217, 43592186, 392285389, 3530435636, 31773522145, 285960503558, 2573640944797, 23162757741488, 208464787388353, 1876182989640050, 16885646616195085, 151970818674059660 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x F[[4, 2], [4, 1, 1]](x) = ------------------- (-1 + x) (-1 + 9 x) and in Maple notation F[[4, 2],[4, 1, 1]](x) = x^2/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 3 2 x (3 x - 2) F[[4, 2], [3, 3]](x) = -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[3, 3]](x) = 2*x^3*(3*x-2)/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 0, 4, 42, 400, 3660, 33124, 298662, 2689600, 24211320, 217916644, 1961294082, 17651779600, 158866414980, 1429798930564, 12868193962302, 115813756422400, 1042323840086640, 9380914657634884, 84428232209279322 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 2 x F[[4, 2], [3, 2, 1]](x) = - ------------------ (1 + x) (-1 + 9 x) and in Maple notation F[[4, 2],[3, 2, 1]](x) = -2*x^2/(1+x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 2, 16, 146, 1312, 11810, 106288, 956594, 8609344, 77484098, 697356880, 6276211922, 56485907296, 508373165666, 4575358490992, 41178226418930, 370604037770368, 3335436339933314, 30018927059399824, 270170343534598418 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x F[[4, 2], [3, 1, 1, 1]](x) = ------------------- (-1 + x) (-1 + 9 x) and in Maple notation F[[4, 2],[3, 1, 1, 1]](x) = x^2/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 2 x (3 x - 6 x + 1) F[[4, 2], [2, 2, 2]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[2, 2, 2]](x) = -x^2*(3*x^2-6*x+1)/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 6, 49, 420, 3721, 33306, 299209, 2691240, 24216241, 217931406, 1961338369, 17651912460, 158866813561, 1429800126306, 12868197549529, 115813767184080, 1042323872371681, 9380914754490006, 84428232499844689 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 3 6 x F[[4, 2], [2, 2, 1, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[2, 2, 1, 1]](x) = -6*x^3/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 0, 6, 72, 708, 6552, 59514, 537264, 4840296, 43577424, 392241102, 3530302776, 31773123564, 285959307816, 2573637357570, 23162746979808, 208464755103312, 1876182892784928, 16885646325629718, 151970817802363560 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 3 2 x (3 x - 2) F[[4, 2], [2, 1, 1, 1, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[2, 1, 1, 1, 1]](x) = 2*x^3*(3*x-2)/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 0, 4, 42, 400, 3660, 33124, 298662, 2689600, 24211320, 217916644, 1961294082, 17651779600, 158866414980, 1429798930564, 12868193962302, 115813756422400, 1042323840086640, 9380914657634884, 84428232209279322 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 4 6 x F[[4, 2], [1, 1, 1, 1, 1, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 9 x) and in Maple notation F[[4, 2],[1, 1, 1, 1, 1, 1]](x) = -6*x^4/(1+x)/(-1+x)/(-1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 0, 0, 6, 72, 708, 6552, 59514, 537264, 4840296, 43577424, 392241102, 3530302776, 31773123564, 285959307816, 2573637357570, 23162746979808, 208464755103312, 1876182892784928, 16885646325629718 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 Regarding Lambda=, [4, 1, 1] 5 4 3 2 18 x - 5 x - 41 x + 4 x + 10 x - 1 F[[4, 1, 1], [6]](x) = ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[6]](x) = (18*x^5-5*x^4-41*x^3+4*x^2+10*x-1)/(-1+x)/(1+x)/(1+2*x)/( -1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 1, 1, 16, 137, 1396, 13881, 138916, 1388857, 13888996, 138888761, 1388889316, 13888888377, 138888890596, 1388888886841, 13888888895716, 138888888880697, 1388888888916196, 13888888888856121, 138888888888998116 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 x (-1 + 4 x) F[[4, 1, 1], [5, 1]](x) = ----------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[5, 1]](x) = x^2*(-1+4*x)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 1, 7, 71, 695, 6951, 69447, 694471, 6944455, 69444551, 694444487, 6944444871, 69444444615, 694444446151, 6944444445127, 69444444451271, 694444444447175, 6944444444471751, 69444444444455367, 694444444444553671 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x (-1 + 4 x) F[[4, 1, 1], [4, 2]](x) = - -------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[4, 2]](x) = -2*x^2*(-1+4*x)/(1+2*x)/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 2, 12, 128, 1248, 12512, 124992, 1250048, 12499968, 125000192, 1249999872, 12500000768, 124999999488, 1250000003072, 12499999997952, 125000000012288, 1249999999991808, 12500000000049152, 124999999999967232, 1250000000000196608 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[4, 1, 1], [4, 1, 1]](x) = 3 2 x (22 x + x - 9 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[4, 1, 1]](x) = -x*(22*x^3+x^2-9*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x) /(-1+10*x) The first 20 term , starting with k=1 are 1, 1, 16, 137, 1396, 13881, 138916, 1388857, 13888996, 138888761, 1388889316, 13888888377, 138888890596, 1388888886841, 13888888895716, 138888888880697, 1388888888916196, 13888888888856121, 138888888888998116, 1388888888888757817 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x (5 x - 5 x + 1) F[[4, 1, 1], [3, 3]](x) = - --------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[3, 3]](x) = -x^2*(5*x^2-5*x+1)/(-1+x)/(1+x)/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 1, 7, 70, 695, 6946, 69447, 694450, 6944455, 69444466, 694444487, 6944444530, 69444444615, 694444444786, 6944444445127, 69444444445810, 694444444447175, 6944444444449906, 69444444444455367, 694444444444466290 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x F[[4, 1, 1], [3, 2, 1]](x) = -------------------- (-1 + x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[3, 2, 1]](x) = 2*x^2/(-1+x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 2, 22, 222, 2222, 22222, 222222, 2222222, 22222222, 222222222, 2222222222, 22222222222, 222222222222, 2222222222222, 22222222222222, 222222222222222, 2222222222222222, 22222222222222222, 222222222222222222, 2222222222222222222 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[4, 1, 1], [3, 1, 1, 1]](x) = 2 2 x (8 x - 4 x - 1) ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[3, 1, 1, 1]](x) = x^2*(8*x^2-4*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/ (-1+10*x) The first 20 term , starting with k=1 are 0, 1, 14, 137, 1390, 13881, 138894, 1388857, 13888910, 138888761, 1388888974, 13888888377, 138888889230, 1388888886841, 13888888890254, 138888888880697, 1388888888894350, 13888888888856121, 138888888888910734, 1388888888888757817 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[4, 1, 1], [2, 2, 2]](x) = 2 2 x (6 x - 4 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[2, 2, 2]](x) = -x^2*(6*x^2-4*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-\ 1+10*x) The first 20 term , starting with k=1 are 0, 1, 6, 71, 690, 6951, 69426, 694471, 6944370, 69444551, 694444146, 6944444871, 69444443250, 694444446151, 6944444439666, 69444444451271, 694444444425330, 6944444444471751, 69444444444367986, 694444444444553671 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 x F[[4, 1, 1], [2, 2, 1, 1]](x) = ---------------------- (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[2, 2, 1, 1]](x) = x^2/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 1, 12, 124, 1248, 12496, 124992, 1249984, 12499968, 124999936, 1249999872, 12499999744, 124999999488, 1249999998976, 12499999997952, 124999999995904, 1249999999991808, 12499999999983616, 124999999999967232, 1249999999999934464 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x (10 x - 5 x + 1) F[[4, 1, 1], [2, 1, 1, 1, 1]](x) = - ----------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[2, 1, 1, 1, 1]](x) = -x^2*(10*x^2-5*x+1)/(-1+x)/(1+2*x)/(-1+2*x)/( -1+10*x) The first 20 term , starting with k=1 are 0, 1, 6, 70, 690, 6946, 69426, 694450, 6944370, 69444466, 694444146, 6944444530, 69444443250, 694444444786, 6944444439666, 69444444445810, 694444444425330, 6944444444449906, 69444444444367986, 694444444444466290 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[4, 1, 1], [1, 1, 1, 1, 1, 1]](x) = 3 2 x (8 x - 4 x - 1) ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[4, 1, 1],[1, 1, 1, 1, 1, 1]](x) = x^3*(8*x^2-4*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1 +2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 0, 1, 14, 137, 1390, 13881, 138894, 1388857, 13888910, 138888761, 1388888974, 13888888377, 138888889230, 1388888886841, 13888888890254, 138888888880697, 1388888888894350, 13888888888856121, 138888888888910734 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 Regarding Lambda=, [3, 3] 5 4 3 2 11 x - 3 x - 30 x + 11 x + 4 x - 1 F[[3, 3], [6]](x) = ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[6]](x) = (11*x^5-3*x^4-30*x^3+11*x^2+4*x-1)/(1+x)/(-1+2*x)/(-1+x)/(1+ 3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 0, 4, 1, 41, 70, 694, 2331, 14851, 64240, 350384, 1662661, 8577661, 42088410, 212828074, 1056954991, 5306276471, 26466768580, 132527465764 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x (5 x - 2 x + 1) F[[3, 3], [5, 1]](x) = - ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[5, 1]](x) = -x^3*(5*x^2-2*x+1)/(1+x)/(-1+2*x)/(-1+x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 2, 25, 90, 581, 2562, 13945, 66530, 342661, 1684122, 8509865, 42284970, 212224741, 1058737682, 5300873785, 26482867410, 132478950821, 662201131242 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 x (-1 + 3 x) F[[3, 3], [4, 2]](x) = - ----------------------------- (1 + 3 x) (-1 + x) (-1 + 5 x) and in Maple notation F[[3, 3],[4, 2]](x) = -x^2*(-1+3*x)/(1+3*x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 0, 13, 24, 241, 840, 5293, 23184, 125761, 599280, 3084973, 15159144, 76592881, 380572920, 1910039053, 9528671904, 47707929601, 238345937760, 1192310819533 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x (5 x + 6 x - 3) F[[3, 3], [4, 1, 1]](x) = ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[4, 1, 1]](x) = x^3*(5*x^2+6*x-3)/(1+x)/(-1+2*x)/(-1+x)/(1+3*x)/(-1+5* x) The first 20 term , starting with k=1 are 0, 0, 3, 6, 55, 190, 1183, 5166, 27975, 133230, 685663, 3368926, 17021095, 84572670, 424454943, 2117486286, 10601769415, 52965778510, 264957989023, 1324402437246 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x (19 x - 8 x - 4 x + 1) F[[3, 3], [3, 3]](x) = - ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[3, 3]](x) = -x*(19*x^3-8*x^2-4*x+1)/(1+x)/(-1+2*x)/(-1+x)/(1+3*x)/(-1 +5*x) The first 20 term , starting with k=1 are 1, 0, 4, 1, 41, 70, 694, 2331, 14851, 64240, 350384, 1662661, 8577661, 42088410, 212828074, 1056954991, 5306276471, 26466768580, 132527465764, 662056023321 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x F[[3, 3], [3, 2, 1]](x) = ----------------------------- (1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation F[[3, 3],[3, 2, 1]](x) = 2*x^3/(1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 2, 12, 66, 340, 1722, 8652, 43346, 216900, 1084842, 5424892, 27125826, 135631860, 678164762, 3390834732, 16954195506, 84771021220, 423855193482, 2119276142172 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 2 x (5 x + 2 x - 1) F[[3, 3], [3, 1, 1, 1]](x) = - --------------------------------------- (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[3, 1, 1, 1]](x) = -x^2*(5*x^2+2*x-1)/(1+x)/(-1+2*x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 1, 13, 35, 251, 1001, 5713, 26335, 138151, 670901, 3413213, 16888235, 84971251, 423259201, 2121073513, 10591007735, 52998063551, 264861133901, 1324693002613 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 2 x (x + 4 x - 1) F[[3, 3], [2, 2, 2]](x) = ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[2, 2, 2]](x) = x^2*(x^2+4*x-1)/(1+x)/(-1+2*x)/(-1+x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 0, 11, 10, 161, 420, 3151, 12390, 71621, 328240, 1729091, 8378370, 42686281, 211034460, 1062335831, 5290133950, 26515196141, 132382183080, 662491871371 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 3 x F[[3, 3], [2, 2, 1, 1]](x) = - ---------------------------- (1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[2, 2, 1, 1]](x) = -3*x^3/(1+x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 3, 3, 54, 150, 1113, 4473, 25644, 118380, 621423, 3018543, 15358434, 75995010, 382366533, 1904658213, 9544814424, 47659502040, 238491220443, 1191874971483 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 4 5 x F[[3, 3], [2, 1, 1, 1, 1]](x) = --------------------------------------- (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[2, 1, 1, 1, 1]](x) = 5*x^4/(1+x)/(-1+2*x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 5, 15, 120, 490, 2835, 13125, 68990, 335280, 1706265, 8443435, 42484260, 211626870, 1060531295, 5295492945, 26499009930, 132430523260, 662346413925 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[3, 3], [1, 1, 1, 1, 1, 1]](x) = 3 2 x (x + 4 x - 1) ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[3, 3],[1, 1, 1, 1, 1, 1]](x) = x^3*(x^2+4*x-1)/(1+x)/(-1+2*x)/(-1+x)/(1+3*x) /(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 0, 11, 10, 161, 420, 3151, 12390, 71621, 328240, 1729091, 8378370, 42686281, 211034460, 1062335831, 5290133950, 26515196141, 132382183080 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 Regarding Lambda=, [3, 2, 1] 3 2 22 x - 17 x - 15 x + 1 F[[3, 2, 1], [6]](x) = ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[6]](x) = (22*x^3-17*x^2-15*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 1, 5, 93, 1453, 23309, 372813, 5965261, 95443661, 1527099597, 24433591501, 390937468109, 6254999481549, 100079991721165, 1601279867505869, 25620477880159437, 409927646082419917, 6558842337318980813, 104941477397103168717, 1679063638353651748045 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 2 x F[[3, 2, 1], [5, 1]](x) = - --------------------- (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[5, 1]](x) = -2*x^2/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 2, 28, 456, 7280, 116512, 1864128, 29826176, 477218560, 7635497472, 122167958528, 1954687338496, 31274997411840, 500399958597632, 8006399337545728, 128102389400764416, 2049638230412165120, 32794211686594772992, 524707386985516105728, 8395318191768258215936 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 3 x F[[3, 2, 1], [4, 2]](x) = -------------------- (-1 + x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[4, 2]](x) = 3*x^2/(-1+x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 3, 51, 819, 13107, 209715, 3355443, 53687091, 858993459, 13743895347, 219902325555, 3518437208883, 56294995342131, 900719925474099, 14411518807585587, 230584300921369395, 3689348814741910323, 59029581035870565171, 944473296573929042739, 15111572745182864683827 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 4 x F[[3, 2, 1], [4, 1, 1]](x) = - --------------------- (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[4, 1, 1]](x) = -4*x^2/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 4, 56, 912, 14560, 233024, 3728256, 59652352, 954437120, 15270994944, 244335917056, 3909374676992, 62549994823680, 1000799917195264, 16012798675091456, 256204778801528832, 4099276460824330240, 65588423373189545984, 1049414773971032211456, 16790636383536516431872 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 2 x F[[3, 2, 1], [3, 3]](x) = - --------------------- (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[3, 3]](x) = -2*x^2/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 2, 28, 456, 7280, 116512, 1864128, 29826176, 477218560, 7635497472, 122167958528, 1954687338496, 31274997411840, 500399958597632, 8006399337545728, 128102389400764416, 2049638230412165120, 32794211686594772992, 524707386985516105728, 8395318191768258215936 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 x (-1 + 10 x) F[[3, 2, 1], [3, 2, 1]](x) = - ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[3, 2, 1]](x) = -x*(-1+10*x)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 5, 93, 1453, 23309, 372813, 5965261, 95443661, 1527099597, 24433591501, 390937468109, 6254999481549, 100079991721165, 1601279867505869, 25620477880159437, 409927646082419917, 6558842337318980813, 104941477397103168717, 1679063638353651748045, 26865018213658425871565 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 4 x F[[3, 2, 1], [3, 1, 1, 1]](x) = - --------------------- (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[3, 1, 1, 1]](x) = -4*x^2/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 4, 56, 912, 14560, 233024, 3728256, 59652352, 954437120, 15270994944, 244335917056, 3909374676992, 62549994823680, 1000799917195264, 16012798675091456, 256204778801528832, 4099276460824330240, 65588423373189545984, 1049414773971032211456, 16790636383536516431872 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 2 x F[[3, 2, 1], [2, 2, 2]](x) = - --------------------- (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[2, 2, 2]](x) = -2*x^2/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 2, 28, 456, 7280, 116512, 1864128, 29826176, 477218560, 7635497472, 122167958528, 1954687338496, 31274997411840, 500399958597632, 8006399337545728, 128102389400764416, 2049638230412165120, 32794211686594772992, 524707386985516105728, 8395318191768258215936 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 3 x F[[3, 2, 1], [2, 2, 1, 1]](x) = -------------------- (-1 + x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[2, 2, 1, 1]](x) = 3*x^2/(-1+x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 3, 51, 819, 13107, 209715, 3355443, 53687091, 858993459, 13743895347, 219902325555, 3518437208883, 56294995342131, 900719925474099, 14411518807585587, 230584300921369395, 3689348814741910323, 59029581035870565171, 944473296573929042739, 15111572745182864683827 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 2 x F[[3, 2, 1], [2, 1, 1, 1, 1]](x) = - --------------------- (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[2, 1, 1, 1, 1]](x) = -2*x^2/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 2, 28, 456, 7280, 116512, 1864128, 29826176, 477218560, 7635497472, 122167958528, 1954687338496, 31274997411840, 500399958597632, 8006399337545728, 128102389400764416, 2049638230412165120, 32794211686594772992, 524707386985516105728, 8395318191768258215936 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 2 x (-1 + 10 x) F[[3, 2, 1], [1, 1, 1, 1, 1, 1]](x) = - ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation F[[3, 2, 1],[1, 1, 1, 1, 1, 1]](x) = -x^2*(-1+10*x)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 0, 1, 5, 93, 1453, 23309, 372813, 5965261, 95443661, 1527099597, 24433591501, 390937468109, 6254999481549, 100079991721165, 1601279867505869, 25620477880159437, 409927646082419917, 6558842337318980813, 104941477397103168717, 1679063638353651748045 ---------------------------------- Their sum is 3 2 8 x - 4 x - 14 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 16 x) and in Maple notation (8*x^3-4*x^2-14*x+1)/(-1+x)/(1+2*x)/(-1+16*x) The first 20 term , starting with k=1 are 1, 29, 429, 6925, 110669, 1770957, 28334797, 453357773, 7253722317, 116059561165, 1856952970445, 29711247543501, 475379960663245, 7606079370677453, 121697269930708173, 1947156318891592909, 31154501102264962253, 498472017636240444621, 7975552282179845016781, 127608836514877524462797 Regarding Lambda=, [3, 1, 1, 1] 5 4 3 2 18 x - 5 x - 41 x + 4 x + 10 x - 1 F[[3, 1, 1, 1], [6]](x) = ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[6]](x) = (18*x^5-5*x^4-41*x^3+4*x^2+10*x-1)/(-1+x)/(1+x)/(1+2*x )/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 1, 1, 16, 137, 1396, 13881, 138916, 1388857, 13888996, 138888761, 1388889316, 13888888377, 138888890596, 1388888886841, 13888888895716, 138888888880697, 1388888888916196, 13888888888856121, 138888888888998116 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[3, 1, 1, 1], [5, 1]](x) = 2 2 x (6 x - 4 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[5, 1]](x) = -x^2*(6*x^2-4*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-\ 1+10*x) The first 20 term , starting with k=1 are 0, 1, 6, 71, 690, 6951, 69426, 694471, 6944370, 69444551, 694444146, 6944444871, 69444443250, 694444446151, 6944444439666, 69444444451271, 694444444425330, 6944444444471751, 69444444444367986, 694444444444553671 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x (-1 + 4 x) F[[3, 1, 1, 1], [4, 2]](x) = - -------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[4, 2]](x) = -2*x^2*(-1+4*x)/(1+2*x)/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 2, 12, 128, 1248, 12512, 124992, 1250048, 12499968, 125000192, 1249999872, 12500000768, 124999999488, 1250000003072, 12499999997952, 125000000012288, 1249999999991808, 12500000000049152, 124999999999967232, 1250000000000196608 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[3, 1, 1, 1], [4, 1, 1]](x) = 2 2 x (8 x - 4 x - 1) ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[4, 1, 1]](x) = x^2*(8*x^2-4*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/ (-1+10*x) The first 20 term , starting with k=1 are 0, 1, 14, 137, 1390, 13881, 138894, 1388857, 13888910, 138888761, 1388888974, 13888888377, 138888889230, 1388888886841, 13888888890254, 138888888880697, 1388888888894350, 13888888888856121, 138888888888910734, 1388888888888757817 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x (10 x - 5 x + 1) F[[3, 1, 1, 1], [3, 3]](x) = - ----------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[3, 3]](x) = -x^2*(10*x^2-5*x+1)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+10* x) The first 20 term , starting with k=1 are 0, 1, 6, 70, 690, 6946, 69426, 694450, 6944370, 69444466, 694444146, 6944444530, 69444443250, 694444444786, 6944444439666, 69444444445810, 694444444425330, 6944444444449906, 69444444444367986, 694444444444466290 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x F[[3, 1, 1, 1], [3, 2, 1]](x) = -------------------- (-1 + x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[3, 2, 1]](x) = 2*x^2/(-1+x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 2, 22, 222, 2222, 22222, 222222, 2222222, 22222222, 222222222, 2222222222, 22222222222, 222222222222, 2222222222222, 22222222222222, 222222222222222, 2222222222222222, 22222222222222222, 222222222222222222, 2222222222222222222 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[3, 1, 1, 1], [3, 1, 1, 1]](x) = 3 2 x (22 x + x - 9 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[3, 1, 1, 1]](x) = -x*(22*x^3+x^2-9*x+1)/(-1+x)/(1+x)/(1+2*x)/(-\ 1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 1, 16, 137, 1396, 13881, 138916, 1388857, 13888996, 138888761, 1388889316, 13888888377, 138888890596, 1388888886841, 13888888895716, 138888888880697, 1388888888916196, 13888888888856121, 138888888888998116, 1388888888888757817 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 x (-1 + 4 x) F[[3, 1, 1, 1], [2, 2, 2]](x) = ----------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[2, 2, 2]](x) = x^2*(-1+4*x)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 1, 7, 71, 695, 6951, 69447, 694471, 6944455, 69444551, 694444487, 6944444871, 69444444615, 694444446151, 6944444445127, 69444444451271, 694444444447175, 6944444444471751, 69444444444455367, 694444444444553671 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 x F[[3, 1, 1, 1], [2, 2, 1, 1]](x) = ---------------------- (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[2, 2, 1, 1]](x) = x^2/(-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 1, 12, 124, 1248, 12496, 124992, 1249984, 12499968, 124999936, 1249999872, 12499999744, 124999999488, 1249999998976, 12499999997952, 124999999995904, 1249999999991808, 12499999999983616, 124999999999967232, 1249999999999934464 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 2 2 x (5 x - 5 x + 1) F[[3, 1, 1, 1], [2, 1, 1, 1, 1]](x) = - --------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[2, 1, 1, 1, 1]](x) = -x^2*(5*x^2-5*x+1)/(-1+x)/(1+x)/(-1+2*x)/( -1+10*x) The first 20 term , starting with k=1 are 0, 1, 7, 70, 695, 6946, 69447, 694450, 6944455, 69444466, 694444487, 6944444530, 69444444615, 694444444786, 6944444445127, 69444444445810, 694444444447175, 6944444444449906, 69444444444455367, 694444444444466290 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 F[[3, 1, 1, 1], [1, 1, 1, 1, 1, 1]](x) = 3 2 x (8 x - 4 x - 1) ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) and in Maple notation F[[3, 1, 1, 1],[1, 1, 1, 1, 1, 1]](x) = x^3*(8*x^2-4*x-1)/(-1+x)/(1+x)/(1+2*x)/ (-1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 0, 0, 1, 14, 137, 1390, 13881, 138894, 1388857, 13888910, 138888761, 1388888974, 13888888377, 138888889230, 1388888886841, 13888888890254, 138888888880697, 1388888888894350, 13888888888856121, 138888888888910734 ---------------------------------- Their sum is 3 2 4 x - 9 x - 8 x + 1 ------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) and in Maple notation (4*x^3-9*x^2-8*x+1)/(-1+x)/(1+2*x)/(-1+10*x) The first 20 term , starting with k=1 are 1, 12, 104, 1060, 10548, 105572, 1055524, 10555620, 105555428, 1055555812, 10555555044, 105555556580, 1055555553508, 10555555559652, 105555555547364, 1055555555571940, 10555555555522788, 105555555555621092, 1055555555555424484, 10555555555555817700 Regarding Lambda=, [2, 2, 2] 5 4 3 2 11 x - 7 x - 29 x + 31 x - 10 x + 1 F[[2, 2, 2], [6]](x) = ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[6]](x) = (11*x^5-7*x^4-29*x^3+31*x^2-10*x+1)/(1+x)/(-1+x)/(-1+2*x) /(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 1, 4, 11, 41, 161, 694, 3151, 14851, 71621, 350384, 1729091, 8577661, 42686281, 212828074, 1062335831, 5306276471, 26515196141, 132527465764 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 4 x (5 x - 2) F[[2, 2, 2], [5, 1]](x) = - ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[5, 1]](x) = -x^4*(5*x-2)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 2, 15, 90, 490, 2562, 13125, 66530, 335280, 1684122, 8443435, 42284970, 211626870, 1058737682, 5295492945, 26482867410, 132430523260, 662201131242 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 2 x (3 x - 5 x + 1) F[[2, 2, 2], [4, 2]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[4, 2]](x) = -x^2*(3*x^2-5*x+1)/(1+x)/(-1+x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 54, 241, 1113, 5293, 25644, 125761, 621423, 3084973, 15358434, 76592881, 382366533, 1910039053, 9544814424, 47707929601, 238491220443, 1192310819533 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x (5 x - 4 x + 1) F[[2, 2, 2], [4, 1, 1]](x) = ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[4, 1, 1]](x) = x^3*(5*x^2-4*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-\ 1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 190, 1001, 5166, 26335, 133230, 670901, 3368926, 16888235, 84572670, 423259201, 2117486286, 10591007735, 52965778510, 264861133901, 1324402437246 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 4 x F[[2, 2, 2], [3, 3]](x) = ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[3, 3]](x) = x^4/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 1, 10, 70, 420, 2331, 12390, 64240, 328240, 1662661, 8378370, 42088410, 211034460, 1056954991, 5290133950, 26466768580, 132382183080, 662056023321 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x F[[2, 2, 2], [3, 2, 1]](x) = ----------------------------- (1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[3, 2, 1]](x) = 2*x^3/(1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 2, 12, 66, 340, 1722, 8652, 43346, 216900, 1084842, 5424892, 27125826, 135631860, 678164762, 3390834732, 16954195506, 84771021220, 423855193482, 2119276142172 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 2, 2], [3, 1, 1, 1]](x) = 2 3 2 x (5 x - 13 x + 7 x - 1) - ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[3, 1, 1, 1]](x) = -x^2*(5*x^3-13*x^2+7*x-1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 251, 1183, 5713, 27975, 138151, 685663, 3413213, 17021095, 84971251, 424454943, 2121073513, 10601769415, 52998063551, 264957989023, 1324693002613 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 2, 2], [2, 2, 2]](x) = 3 2 x (19 x - 24 x + 9 x - 1) - ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[2, 2, 2]](x) = -x*(19*x^3-24*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 3*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 1, 4, 11, 41, 161, 694, 3151, 14851, 71621, 350384, 1729091, 8577661, 42686281, 212828074, 1062335831, 5306276471, 26515196141, 132527465764, 662491871371 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 4 3 x F[[2, 2, 2], [2, 2, 1, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[2, 2, 1, 1]](x) = -3*x^4/(1+x)/(-1+x)/(-1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 24, 150, 840, 4473, 23184, 118380, 599280, 3018543, 15159144, 75995010, 380572920, 1904658213, 9528671904, 47659502040, 238345937760, 1191874971483 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 2, 2], [2, 1, 1, 1, 1]](x) = 3 2 x (5 x - 5 x + 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[2, 1, 1, 1, 1]](x) = x^3*(5*x^2-5*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3 *x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 5, 25, 120, 581, 2835, 13945, 68990, 342661, 1706265, 8509865, 42484260, 212224741, 1060531295, 5300873785, 26499009930, 132478950821, 662346413925 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 2, 2], [1, 1, 1, 1, 1, 1]](x) = 5 x ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 2, 2],[1, 1, 1, 1, 1, 1]](x) = x^5/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+5*x ) The first 20 term , starting with k=1 are 0, 0, 0, 0, 1, 10, 70, 420, 2331, 12390, 64240, 328240, 1662661, 8378370, 42088410, 211034460, 1056954991, 5290133950, 26466768580, 132382183080 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 Regarding Lambda=, [2, 2, 1, 1] 4 2 12 x - 27 x - 6 x + 1 F[[2, 2, 1, 1], [6]](x) = ------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[6]](x) = (12*x^4-27*x^2-6*x+1)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 0, 13, 72, 769, 6552, 60061, 537264, 4845217, 43577424, 392285389, 3530302776, 31773522145, 285959307816, 2573640944797, 23162746979808, 208464787388353, 1876182892784928, 16885646616195085 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x (-1 + 3 x) F[[2, 2, 1, 1], [5, 1]](x) = - ----------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[5, 1]](x) = -x^2*(-1+3*x)/(-1+x)/(1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 4, 49, 400, 3721, 33124, 299209, 2689600, 24216241, 217916644, 1961338369, 17651779600, 158866813561, 1429798930564, 12868197549529, 115813756422400, 1042323872371681, 9380914657634884, 84428232499844689 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 2 x (-1 + 3 x) F[[2, 2, 1, 1], [4, 2]](x) = - ------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[4, 2]](x) = -2*x^2*(-1+3*x)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 2, 6, 92, 708, 6734, 59514, 538904, 4840296, 43592186, 392241102, 3530435636, 31773123564, 285960503558, 2573637357570, 23162757741488, 208464755103312, 1876182989640050, 16885646325629718, 151970818674059660 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x F[[2, 2, 1, 1], [4, 1, 1]](x) = ------------------- (-1 + x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[4, 1, 1]](x) = x^2/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 3 6 x F[[2, 2, 1, 1], [3, 3]](x) = ----------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[3, 3]](x) = 6*x^3/(-1+x)/(1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 0, 6, 42, 420, 3660, 33306, 298662, 2691240, 24211320, 217931406, 1961294082, 17651912460, 158866414980, 1429800126306, 12868193962302, 115813767184080, 1042323840086640, 9380914754490006, 84428232209279322 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 2 x F[[2, 2, 1, 1], [3, 2, 1]](x) = - ------------------ (1 + x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[3, 2, 1]](x) = -2*x^2/(1+x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 2, 16, 146, 1312, 11810, 106288, 956594, 8609344, 77484098, 697356880, 6276211922, 56485907296, 508373165666, 4575358490992, 41178226418930, 370604037770368, 3335436339933314, 30018927059399824, 270170343534598418 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x F[[2, 2, 1, 1], [3, 1, 1, 1]](x) = ------------------- (-1 + x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[3, 1, 1, 1]](x) = x^2/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x (-1 + 3 x) F[[2, 2, 1, 1], [2, 2, 2]](x) = - ----------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[2, 2, 2]](x) = -x^2*(-1+3*x)/(-1+x)/(1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 1, 4, 49, 400, 3721, 33124, 299209, 2689600, 24216241, 217916644, 1961338369, 17651779600, 158866813561, 1429798930564, 12868197549529, 115813756422400, 1042323872371681, 9380914657634884, 84428232499844689 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 2 x (15 x + 6 x - 1) F[[2, 2, 1, 1], [2, 2, 1, 1]](x) = - ------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[2, 2, 1, 1]](x) = -x*(15*x^2+6*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+9* x) The first 20 term , starting with k=1 are 1, 0, 13, 72, 769, 6552, 60061, 537264, 4845217, 43577424, 392285389, 3530302776, 31773522145, 285959307816, 2573640944797, 23162746979808, 208464787388353, 1876182892784928, 16885646616195085, 151970817802363560 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 3 6 x F[[2, 2, 1, 1], [2, 1, 1, 1, 1]](x) = ----------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[2, 1, 1, 1, 1]](x) = 6*x^3/(-1+x)/(1+3*x)/(-1+9*x) The first 20 term , starting with k=1 are 0, 0, 6, 42, 420, 3660, 33306, 298662, 2691240, 24211320, 217931406, 1961294082, 17651912460, 158866414980, 1429800126306, 12868193962302, 115813767184080, 1042323840086640, 9380914754490006, 84428232209279322 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 F[[2, 2, 1, 1], [1, 1, 1, 1, 1, 1]](x) = 3 2 x (-1 + 3 x) - ------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) and in Maple notation F[[2, 2, 1, 1],[1, 1, 1, 1, 1, 1]](x) = -2*x^3*(-1+3*x)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+9*x) The first 20 term , starting with k=1 are 0, 0, 2, 6, 92, 708, 6734, 59514, 538904, 4840296, 43592186, 392241102, 3530435636, 31773123564, 285960503558, 2573637357570, 23162757741488, 208464755103312, 1876182989640050, 16885646325629718 ---------------------------------- Their sum is 3 2 4 x - x - 8 x + 1 --------------------------- (1 + x) (-1 + x) (-1 + 9 x) and in Maple notation (4*x^3-x^2-8*x+1)/(1+x)/(-1+x)/(-1+9*x) The first 20 term , starting with k=1 are 1, 9, 77, 693, 6233, 56097, 504869, 4543821, 40894385, 368049465, 3312445181, 29812006629, 268308059657, 2414772536913, 21732952832213, 195596575489917, 1760369179409249, 15843322614683241, 142589903532149165, 1283309131789342485 Regarding Lambda=, [2, 1, 1, 1, 1] 5 4 3 2 11 x - 3 x - 30 x + 11 x + 4 x - 1 F[[2, 1, 1, 1, 1], [6]](x) = ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[6]](x) = (11*x^5-3*x^4-30*x^3+11*x^2+4*x-1)/(1+x)/(-1+2*x)/( -1+x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 0, 4, 1, 41, 70, 694, 2331, 14851, 64240, 350384, 1662661, 8577661, 42088410, 212828074, 1056954991, 5306276471, 26466768580, 132527465764 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 1, 1, 1, 1], [5, 1]](x) = 2 2 x (x + 4 x - 1) ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[5, 1]](x) = x^2*(x^2+4*x-1)/(1+x)/(-1+2*x)/(-1+x)/(1+3*x)/(-\ 1+5*x) The first 20 term , starting with k=1 are 0, 1, 0, 11, 10, 161, 420, 3151, 12390, 71621, 328240, 1729091, 8378370, 42686281, 211034460, 1062335831, 5290133950, 26515196141, 132382183080, 662491871371 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 x (-1 + 3 x) F[[2, 1, 1, 1, 1], [4, 2]](x) = - ----------------------------- (1 + 3 x) (-1 + x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[4, 2]](x) = -x^2*(-1+3*x)/(1+3*x)/(-1+x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 1, 0, 13, 24, 241, 840, 5293, 23184, 125761, 599280, 3084973, 15159144, 76592881, 380572920, 1910039053, 9528671904, 47707929601, 238345937760, 1192310819533 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 2 2 x (5 x + 2 x - 1) F[[2, 1, 1, 1, 1], [4, 1, 1]](x) = - --------------------------------------- (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[4, 1, 1]](x) = -x^2*(5*x^2+2*x-1)/(1+x)/(-1+2*x)/(1+3*x)/(-1 +5*x) The first 20 term , starting with k=1 are 0, 1, 1, 13, 35, 251, 1001, 5713, 26335, 138151, 670901, 3413213, 16888235, 84971251, 423259201, 2121073513, 10591007735, 52998063551, 264861133901, 1324693002613 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 4 5 x F[[2, 1, 1, 1, 1], [3, 3]](x) = --------------------------------------- (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[3, 3]](x) = 5*x^4/(1+x)/(-1+2*x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 0, 5, 15, 120, 490, 2835, 13125, 68990, 335280, 1706265, 8443435, 42484260, 211626870, 1060531295, 5295492945, 26499009930, 132430523260, 662346413925 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 2 x F[[2, 1, 1, 1, 1], [3, 2, 1]](x) = ----------------------------- (1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[3, 2, 1]](x) = 2*x^3/(1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 2, 12, 66, 340, 1722, 8652, 43346, 216900, 1084842, 5424892, 27125826, 135631860, 678164762, 3390834732, 16954195506, 84771021220, 423855193482, 2119276142172 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 1, 1, 1, 1], [3, 1, 1, 1]](x) = 3 2 x (5 x + 6 x - 3) ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[3, 1, 1, 1]](x) = x^3*(5*x^2+6*x-3)/(1+x)/(-1+2*x)/(-1+x)/(1 +3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 3, 6, 55, 190, 1183, 5166, 27975, 133230, 685663, 3368926, 17021095, 84572670, 424454943, 2117486286, 10601769415, 52965778510, 264957989023, 1324402437246 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 1, 1, 1, 1], [2, 2, 2]](x) = 3 2 x (5 x - 2 x + 1) - ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[2, 2, 2]](x) = -x^3*(5*x^2-2*x+1)/(1+x)/(-1+2*x)/(-1+x)/(1+3 *x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 2, 25, 90, 581, 2562, 13945, 66530, 342661, 1684122, 8509865, 42284970, 212224741, 1058737682, 5300873785, 26482867410, 132478950821, 662201131242 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 3 3 x F[[2, 1, 1, 1, 1], [2, 2, 1, 1]](x) = - ---------------------------- (1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[2, 2, 1, 1]](x) = -3*x^3/(1+x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 3, 3, 54, 150, 1113, 4473, 25644, 118380, 621423, 3018543, 15358434, 75995010, 382366533, 1904658213, 9544814424, 47659502040, 238491220443, 1191874971483 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 1, 1, 1, 1], [2, 1, 1, 1, 1]](x) = 3 2 x (19 x - 8 x - 4 x + 1) - ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[2, 1, 1, 1, 1]](x) = -x*(19*x^3-8*x^2-4*x+1)/(1+x)/(-1+2*x)/ (-1+x)/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 0, 4, 1, 41, 70, 694, 2331, 14851, 64240, 350384, 1662661, 8577661, 42088410, 212828074, 1056954991, 5306276471, 26466768580, 132527465764, 662056023321 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 F[[2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]](x) = 3 2 x (x + 4 x - 1) ------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 3 x) (-1 + 5 x) and in Maple notation F[[2, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1]](x) = x^3*(x^2+4*x-1)/(1+x)/(-1+2*x)/(-1+x )/(1+3*x)/(-1+5*x) The first 20 term , starting with k=1 are 0, 0, 1, 0, 11, 10, 161, 420, 3151, 12390, 71621, 328240, 1729091, 8378370, 42686281, 211034460, 1062335831, 5290133950, 26515196141, 132382183080 ---------------------------------- Their sum is 2 2 (2 x - 5 x + 1) (x + x - 1) -------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) and in Maple notation (2*x^2-5*x+1)*(x^2+x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x) The first 20 term , starting with k=1 are 1, 4, 15, 70, 337, 1664, 8275, 41290, 206277, 1031044, 5154535, 25771310, 128853817, 644263624, 3221307195, 16106514130, 80532526957, 402662547404, 2013312562255, 10066562461750 Regarding Lambda=, [1, 1, 1, 1, 1, 1] 1 F[[1, 1, 1, 1, 1, 1], [6]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1],[6]](x) = -1/(1+x)/(-1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [5, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[5, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [4, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[4, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [4, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[4, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [3, 3]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[3, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [3, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[3, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [3, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[3, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [2, 2, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[2, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [2, 2, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[2, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 x F[[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1]](x) = -x/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 7, elements There are, 15, partitions of, 7, here there are in the usual order Regarding Lambda=, [7] 1 F[[7], [7]](x) = - ------ -1 + x and in Maple notation F[[7],[7]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [6, 1]](x) = 0 and in Maple notation F[[7],[6, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [5, 2]](x) = 0 and in Maple notation F[[7],[5, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [5, 1, 1]](x) = 0 and in Maple notation F[[7],[5, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [4, 3]](x) = 0 and in Maple notation F[[7],[4, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [4, 2, 1]](x) = 0 and in Maple notation F[[7],[4, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [4, 1, 1, 1]](x) = 0 and in Maple notation F[[7],[4, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [3, 3, 1]](x) = 0 and in Maple notation F[[7],[3, 3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [3, 2, 2]](x) = 0 and in Maple notation F[[7],[3, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [3, 2, 1, 1]](x) = 0 and in Maple notation F[[7],[3, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [3, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[7],[3, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [2, 2, 2, 1]](x) = 0 and in Maple notation F[[7],[2, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [2, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[7],[2, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [2, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[7],[2, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[7], [1, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[7],[1, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [6, 1] F[[6, 1], [7]](x) = 6 5 4 3 2 53 x - 45 x - 132 x + 179 x - 80 x + 15 x - 1 ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[7]](x) = (53*x^6-45*x^5-132*x^4+179*x^3-80*x^2+15*x-1)/(1+x)/(-1+x)/( -1+2*x)/(-1+3*x)/(-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 1, 4, 11, 41, 162, 714, 3397, 17251, 92048, 509444, 2893683, 16734381, 97965934, 578241694, 3431848769, 20442513431, 122066210220, 730058053464 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [6, 1]](x) = 4 3 2 x (91 x - 135 x + 68 x - 14 x + 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[6, 1]](x) = -x*(91*x^4-135*x^3+68*x^2-14*x+1)/(1+x)/(-1+x)/(-1+2*x)/( -1+3*x)/(-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 1, 4, 11, 41, 162, 714, 3397, 17251, 92048, 509444, 2893683, 16734381, 97965934, 578241694, 3431848769, 20442513431, 122066210220, 730058053464, 4371040179055 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [5, 2]](x) = 2 2 2 x (13 x - 8 x + 1) (2 x - 4 x + 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[5, 2]](x) = -x^2*(13*x^2-8*x+1)*(2*x^2-4*x+1)/(1+x)/(-1+x)/(-1+2*x)/( -1+3*x)/(-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 255, 1253, 6497, 35115, 195859, 1117633, 6480981, 38002055, 224522663, 1333312893, 7944989065, 47451587875, 283839750267, 1699570199033, 10183580356349 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [5, 1, 1]](x) = 2 4 3 2 x (27 x - 68 x + 47 x - 12 x + 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[5, 1, 1]](x) = -x^2*(27*x^4-68*x^3+47*x^2-12*x+1)/(1+x)/(-1+x)/(-1+2* x)/(-1+3*x)/(-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 256, 1268, 6643, 36285, 204286, 1174558, 6850273, 40335815, 239018716, 1422328248, 8487433903, 50740260145, 303709579546, 1819345716338, 10904475725533 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [4, 3]](x) = 3 3 2 x (16 x - 23 x + 9 x - 1) ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[4, 3]](x) = x^3*(16*x^3-23*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/ (-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 1, 6, 34, 185, 1015, 5656, 32068, 184575, 1075129, 6318686, 37375702, 222085045, 1323764443, 7907398596, 47303030536, 283250923595, 1697231078557, 10174272389386 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [4, 2, 1]](x) = 3 3 2 x (37 x - 48 x + 18 x - 2) ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[4, 2, 1]](x) = x^3*(37*x^3-48*x^2+18*x-2)/(1+x)/(-1+x)/(-1+2*x)/(-1+3 *x)/(-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 2, 12, 70, 395, 2247, 12922, 75140, 440865, 2604217, 15458432, 92074710, 549720535, 3287341787, 19679917542, 117902358280, 706703735405, 4237375308957, 25412847166252 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 3 x (-1 + 4 x) F[[6, 1], [4, 1, 1, 1]](x) = ---------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation F[[6, 1],[4, 1, 1, 1]](x) = x^3*(-1+4*x)/(1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 200, 1161, 6826, 40495, 241500, 1444421, 8652446, 51871755, 311100400, 1866209281, 11196070866, 67172859815, 403026440900, 2418126447741, 14508662006086 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [3, 3, 1]](x) = 4 2 x (27 x - 19 x + 3) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[3, 3, 1]](x) = -x^4*(27*x^2-19*x+3)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-\ 1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 26, 180, 1141, 6993, 42252, 253830, 1521707, 9117603, 54632578, 327436200, 1962979473, 11770725333, 70593946904, 423436669290, 2540095802839, 15238429494183 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [3, 2, 2]](x) = 4 2 x (9 x - 10 x + 2) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[3, 2, 2]](x) = -x^4*(9*x^2-10*x+2)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 2, 20, 151, 1015, 6468, 40110, 245177, 1486925, 8978134, 54074020, 325200603, 1954034355, 11734939400, 70450792250, 422864028829, 2537805197305, 15229266984666 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [3, 2, 1, 1]](x) = 4 2 x (17 x - 15 x + 3) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[3, 2, 1, 1]](x) = -x^4*(17*x^2-15*x+3)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x) /(-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 30, 230, 1575, 10213, 64260, 397260, 2429625, 14759723, 89279190, 538537090, 3242605275, 19500966033, 117186541320, 703840445720, 4225922106525, 25367034269143 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [3, 1, 1, 1, 1]](x) = 4 2 x (9 x - 5 x + 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[3, 1, 1, 1, 1]](x) = -x^4*(9*x^2-5*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x )/(-1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 1, 10, 80, 575, 3891, 25320, 160510, 999625, 6150881, 37538930, 227832540, 1377586275, 8308471471, 50024421340, 300846246170, 1807892426525, 10858662653661 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [2, 2, 2, 1]](x) = 5 x (16 x - 5) ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[2, 2, 2, 1]](x) = x^5*(16*x-5)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x )/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 59, 490, 3514, 23415, 149793, 935660, 5760128, 35140105, 213139927, 1287978510, 7764243942, 46730390075, 280960318061, 1688068569040, 10137622176556 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [2, 2, 1, 1, 1]](x) = 5 2 x (5 x - 2) ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[2, 2, 1, 1, 1]](x) = 2*x^5*(5*x-2)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 4, 50, 434, 3220, 22008, 143430, 907918, 5642120, 34646612, 211100890, 1279625802, 7730240700, 46592594416, 280403776430, 1685826303686, 10128604774960 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [2, 1, 1, 1, 1, 1]](x) = 5 x - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[2, 1, 1, 1, 1, 1]](x) = -x^5/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/ (-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 1, 15, 146, 1170, 8427, 56925, 369292, 2333760, 14496053, 89015355, 542444838, 3288672270, 19869829279, 119775517305, 720895369184, 4334389616700 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[6, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 6 x - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) and in Maple notation F[[6, 1],[1, 1, 1, 1, 1, 1, 1]](x) = -x^6/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4* x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 1, 15, 146, 1170, 8427, 56925, 369292, 2333760, 14496053, 89015355, 542444838, 3288672270, 19869829279, 119775517305, 720895369184 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 Regarding Lambda=, [5, 2] 5 4 3 2 80 x - 30 x - 166 x + 124 x - 22 x + 1 F[[5, 2], [7]](x) = -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[7]](x) = (80*x^5-30*x^4-166*x^3+124*x^2-22*x+1)/(1+x)/(-1+x)/(-1+2*x) /(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 2, 15, 142, 1695, 22094, 299839, 4141454, 57643839, 804997006, 11257862463, 157537510286, 2205089777983, 30868644733838, 432145353402687, 6049940910564238, 84698608525764927, 1185777134028514190, 16600859564407253311 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 3 2 x (10 x - 37 x + 15 x - 1) F[[5, 2], [6, 1]](x) = - -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[6, 1]](x) = -x^2*(10*x^3-37*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+6*x )/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 7, 68, 771, 9744, 130163, 1784896, 24764467, 345358592, 4826957619, 67529032704, 945116218163, 13229885624320, 185207950177075, 2592848611131392, 36299504407819059, 508190804821475328, 7114657726172967731, 99605126918455951360 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 3 2 x (88 x - 94 x + 20 x - 1) F[[5, 2], [5, 2]](x) = - -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[5, 2]](x) = -x*(88*x^3-94*x^2+20*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+6*x)/ (-1+14*x) The first 20 term , starting with k=1 are 1, 2, 15, 142, 1695, 22094, 299839, 4141454, 57643839, 804997006, 11257862463, 157537510286, 2205089777983, 30868644733838, 432145353402687, 6049940910564238, 84698608525764927, 1185777134028514190, 16600859564407253311, 232411912029752320910 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 x (-1 + 10 x) F[[5, 2], [5, 1, 1]](x) = --------------------------------- (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[5, 1, 1]](x) = x^2*(-1+10*x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 12, 140, 1760, 23376, 319552, 4427200, 61701120, 862136576, 12059835392, 168777231360, 2362518446080, 33073081470976, 463010079916032, 6482062754693120, 90748408380784640, 1270474896221208576, 17786631620437737472, 249012741126172180480 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 3 2 x (14 x + 9 x - 11 x + 1) F[[5, 2], [4, 3]](x) = -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[4, 3]](x) = x^2*(14*x^3+9*x^2-11*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+6*x)/ (-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 128, 1623, 21692, 297479, 4127404, 57559751, 804492908, 11254838727, 157519369580, 2204980937159, 30867991695724, 432141435187655, 6049917401301356, 84698467470242247, 1185776287695487340, 16600854486409310663, 232411881561765101932 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (14 x - 19 x + 2) F[[5, 2], [4, 2, 1]](x) = - ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[4, 2, 1]](x) = -x^2*(14*x^2-19*x+2)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 2, 23, 293, 3895, 53257, 737863, 10283513, 143689415, 2009972537, 28129538503, 393753074233, 5512180244935, 77168346652217, 1080343792447943, 15124734730128953, 211745816036864455, 2964438603406282297, 41502123521028682183, 581029627734445231673 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 x (-1 + 4 x) F[[5, 2], [4, 1, 1, 1]](x) = - ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation F[[5, 2],[4, 1, 1, 1]](x) = -x^2*(-1+4*x)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 153, 2135, 29881, 418311, 5856313, 81988295, 1147835961, 16069703111, 224975842873, 3149661798855, 44095265181241, 617333712531911, 8642671975435833, 120997407656079815, 1693963707185073721, 23715491900590944711, 332016886608273051193 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 3 2 x (28 x - 24 x + 10 x - 1) F[[5, 2], [3, 3, 1]](x) = - -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[3, 3, 1]](x) = -x^2*(28*x^3-24*x^2+10*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 12, 165, 2272, 31565, 440384, 6156109, 86129664, 1205479629, 16874699776, 236233704653, 3307199307776, 46300354956493, 648202357260288, 9074817328827597, 127047348566622208, 1778662315710794957, 24901269034619371520, 348617746172680129741 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 3 2 x (14 x - 37 x + 11 x - 1) F[[5, 2], [3, 2, 2]](x) = - -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[3, 2, 2]](x) = -x^2*(14*x^3-37*x^2+11*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 156, 2211, 31184, 438067, 6142144, 86045747, 1204975872, 16871676723, 236215565312, 3307090469683, 46299701923840, 648198439056179, 9074793819586560, 127047207511143219, 1778661469377855488, 24901263956621603635, 348617715704693260288 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 3 14 x (-1 + 4 x) F[[5, 2], [3, 2, 1, 1]](x) = ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[3, 2, 1, 1]](x) = 14*x^3*(-1+4*x)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 14, 238, 3570, 51310, 726194, 10213518, 143269490, 2007453070, 28114421874, 393662374798, 5511636049010, 77165081478030, 1080324201405554, 15124617183880078, 211745110759382130, 2964434371741410190, 41502098131039493234, 581029475394510185358 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 3 4 x F[[5, 2], [3, 1, 1, 1, 1]](x) = - --------------------------------- (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[3, 1, 1, 1, 1]](x) = -4*x^3/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 4, 88, 1440, 21440, 307904, 4357248, 61281280, 859617280, 12044719104, 168686532608, 2361974251520, 33069816299520, 462990488879104, 6481945208455168, 90747703103324160, 1270470664556380160, 17786606230448635904, 249012588786237308928 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[5, 2], [2, 2, 2, 1]](x) = 3 2 x (14 x + 3 x - 4) - -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[2, 2, 2, 1]](x) = -x^3*(14*x^2+3*x-4)/(1+x)/(-1+x)/(-1+2*x)/(-1+6*x)/ (-1+14*x) The first 20 term , starting with k=1 are 0, 0, 4, 85, 1364, 20137, 288148, 4071417, 57223828, 802477369, 11242745492, 157446810169, 2204545580692, 30865379556921, 432125762354836, 6049823364304441, 84697903248260756, 1185772902363598393, 16600834174417976980, 232411759689817099833 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[5, 2], [2, 2, 1, 1, 1]](x) = 3 x (10 x + 3) -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[2, 2, 1, 1, 1]](x) = x^3*(10*x+3)/(1+x)/(-1+x)/(-1+2*x)/(-1+6*x)/(-1+ 14*x) The first 20 term , starting with k=1 are 0, 0, 3, 76, 1303, 19756, 285831, 4057452, 57139911, 801973612, 11239722439, 157428670828, 2204436742599, 30864726524268, 432121844150727, 6049799855063404, 84697762192781767, 1185772056030658924, 16600829096420209095, 232411729221830230380 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[5, 2], [2, 1, 1, 1, 1, 1]](x) = 4 x (38 x - 25) - -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[2, 1, 1, 1, 1, 1]](x) = -x^4*(38*x-25)/(1+x)/(-1+x)/(-1+2*x)/(-1+6*x) /(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 25, 512, 8189, 120832, 1728909, 24428544, 343343053, 4814864384, 67456473293, 944680861696, 13227273485517, 185192277344256, 2592754574134477, 36298940185837568, 508187419489586381, 7114637414181634048, 99605005046507949261 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[5, 2], [1, 1, 1, 1, 1, 1, 1]](x) = 4 x (10 x + 3) -------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[5, 2],[1, 1, 1, 1, 1, 1, 1]](x) = x^4*(10*x+3)/(1+x)/(-1+x)/(-1+2*x)/(-1+6*x )/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 76, 1303, 19756, 285831, 4057452, 57139911, 801973612, 11239722439, 157428670828, 2204436742599, 30864726524268, 432121844150727, 6049799855063404, 84697762192781767, 1185772056030658924, 16600829096420209095 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 Regarding Lambda=, [5, 1, 1] F[[5, 1, 1], [7]](x) = 6 5 4 3 2 285 x - 73 x - 680 x + 181 x + 66 x - 20 x + 1 ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[7]](x) = (285*x^6-73*x^5-680*x^4+181*x^3+66*x^2-20*x+1)/(-1+x)/(1+ x)/(-1+3*x)/(1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 1, 16, 162, 2351, 34211, 510366, 7635652, 114457501, 1716423621, 25744356716, 386154890342, 5792272824651, 86883835465831, 1303256263411066, 19548837570020232, 293232531788063801, 4398487817665384841, 65977316470494313416 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 2 x (47 x - 16 x + 1) F[[5, 1, 1], [6, 1]](x) = -------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[6, 1]](x) = x^2*(47*x^2-16*x+1)/(-1+x)/(1+x)/(-1+3*x)/(-1+5*x)/(-1 +15*x) The first 20 term , starting with k=1 are 0, 1, 7, 74, 966, 13851, 204797, 3057844, 45799276, 686653781, 10298149587, 154464017214, 2316919302386, 34753585288111, 521302759677577, 7819536301726184, 117293019073054296, 1759395158874668841, 26390926747143444767, 395863898027656152754 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 3 2 2 x (75 x + 13 x - 13 x + 1) F[[5, 1, 1], [5, 2]](x) = - --------------------------------------------------- (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[5, 2]](x) = -2*x^2*(75*x^3+13*x^2-13*x+1)/(1+x)/(-1+3*x)/(1+3*x)/( -1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 2, 12, 162, 2184, 32082, 476532, 7129362, 106835664, 1602054882, 24028317852, 360412634562, 5406127907544, 81091614101682, 1216372680464772, 18245582583555762, 273683700559729824, 4105255317714892482, 61578828811629365292, 923682427406553172962 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [5, 1, 1]](x) = 4 3 2 x (390 x - 107 x - 61 x + 19 x - 1) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[5, 1, 1]](x) = -x*(390*x^4-107*x^3-61*x^2+19*x-1)/(-1+x)/(1+x)/(-1 +3*x)/(1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 1, 16, 162, 2351, 34211, 510366, 7635652, 114457501, 1716423621, 25744356716, 386154890342, 5792272824651, 86883835465831, 1303256263411066, 19548837570020232, 293232531788063801, 4398487817665384841, 65977316470494313416, 989659743081883041322 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 2 x (20 x - 11 x + 1) F[[5, 1, 1], [4, 3]](x) = - ----------------------------------------- (1 + x) (-1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[4, 3]](x) = -x^2*(20*x^2-11*x+1)/(1+x)/(-1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 11, 150, 2158, 31891, 475881, 7125560, 106819388, 1601968581, 24027910951, 360410555770, 5406117735018, 81091562840471, 1216372426151621, 18245581308402780, 273683694201901048, 4105255285893463561, 61578828652683645891, 923682426611534010590 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [4, 2, 1]](x) = 2 4 3 2 x (75 x + 55 x - 5 x - 15 x + 2) ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[4, 2, 1]](x) = x^2*(75*x^4+55*x^3-5*x^2-15*x+2)/(-1+x)/(1+x)/(-1+3 *x)/(1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 2, 25, 365, 5330, 79432, 1188075, 17806035, 267007780, 4004720462, 60068760125, 901021325305, 13515268906230, 202728780143892, 3040930429596175, 45613950093886175, 684209219610180680, 10263138135276941722, 153947071234344816225, 2309206064542158816645 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 x (-1 + 5 x) F[[5, 1, 1], [4, 1, 1, 1]](x) = ------------------------------- (-1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[4, 1, 1, 1]](x) = x^2*(-1+5*x)/(-1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 14, 203, 3020, 45221, 678074, 10170383, 152553560, 2288296841, 34324432934, 514866434963, 7722996347300, 115844944678061, 1737674168576594, 26065112523865943, 390976687843640240, 5864650317611556881, 87969754764044213054, 1319546321460275775323 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [3, 3, 1]](x) = 3 3 2 x (135 x + 57 x - 95 x + 15) ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[3, 3, 1]](x) = x^3*(135*x^3+57*x^2-95*x+15)/(-1+x)/(1+x)/(-1+3*x)/ (1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 15, 205, 3182, 47450, 712285, 10679655, 160189212, 2402744500, 36040856555, 540610703105, 8109151237642, 121637216705550, 1824558004042425, 27368368780102555, 410525525413660472, 6157882849335050600, 92368242581709597895, 1385523637930188958005 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [3, 2, 2]](x) = 2 3 2 x (159 x - 55 x + 9 x - 1) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[3, 2, 2]](x) = -x^2*(159*x^3-55*x^2+9*x-1)/(-1+x)/(1+x)/(-1+3*x)/( 1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 11, 210, 3126, 47411, 711361, 10677220, 160170476, 2402670501, 36040427511, 540608735030, 8109140865826, 121637166440791, 1824557747935661, 27368367513917640, 410525519039689176, 6157882817594334281, 92368242422618595811, 1385523637135896209050 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [3, 2, 1, 1]](x) = 2 4 3 2 x (150 x - 55 x + 19 x - x - 1) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[3, 2, 1, 1]](x) = -x^2*(150*x^4-55*x^3+19*x^2-x-1)/(-1+x)/(1+x)/(-\ 1+3*x)/(1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 21, 336, 5210, 78751, 1184911, 17789486, 266927220, 4004311101, 60066733001, 901011130636, 13515218110030, 202728525631451, 3040929158628291, 45613943734263786, 684209187826417640, 10263137976315079801, 153947070439664646781, 2309206060568370548936 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [3, 1, 1, 1, 1]](x) = 2 3 2 x (75 x - 71 x + 13 x - 1) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[3, 1, 1, 1, 1]](x) = -x^2*(75*x^3-71*x^2+13*x-1)/(-1+x)/(1+x)/(-1+ 3*x)/(1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 7, 146, 2190, 33651, 506837, 7620196, 114373660, 1716024101, 25742300067, 386144784246, 5792221762730, 86883581750551, 1303254990051697, 19548831217572296, 293232499982777400, 4398487658768093001, 65977315675620433727, 989659739108675904346 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 3 x (25 x - 7) F[[5, 1, 1], [2, 2, 2, 1]](x) = ----------------------------------------- (1 + x) (-1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[2, 2, 2, 1]](x) = x^3*(25*x-7)/(1+x)/(-1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 7, 129, 2054, 31370, 473277, 7112539, 106754284, 1601643060, 24026283347, 360402417749, 5406077044914, 81091359389950, 1216371408899017, 18245576222139759, 273683668770585944, 4105255158736888040, 61578828016900768287, 923682423432619622569 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [2, 2, 1, 1, 1]](x) = 3 2 4 x (15 x - 7 x + 2) - --------------------------------------------------- (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[2, 2, 1, 1, 1]](x) = -4*x^3*(15*x^2-7*x+2)/(1+x)/(-1+3*x)/(1+3*x)/ (-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 8, 124, 2048, 31240, 472808, 7109284, 106739648, 1601561680, 24025891208, 360400383244, 5406067005248, 81091308527320, 1216371155781608, 18245574950574004, 273683662423518848, 4105255126947744160, 61578827858051904008, 923682422637891025564 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [2, 1, 1, 1, 1, 1]](x) = 3 2 x (45 x - 16 x + 3) -------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[2, 1, 1, 1, 1, 1]](x) = x^3*(45*x^2-16*x+3)/(-1+x)/(1+x)/(-1+3*x)/ (-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 3, 53, 862, 13330, 202193, 3044823, 45734172, 686328260, 10296521983, 154455879193, 2316878612282, 34753381837590, 521301742424973, 7819531215463163, 117292993641739192, 1759395031718093320, 26390926111360567163, 395863894848741764733 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[5, 1, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (75 x - 71 x + 13 x - 1) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 15 x) and in Maple notation F[[5, 1, 1],[1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(75*x^3-71*x^2+13*x-1)/(-1+x)/(1+x )/(-1+3*x)/(1+3*x)/(-1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 1, 7, 146, 2190, 33651, 506837, 7620196, 114373660, 1716024101, 25742300067, 386144784246, 5792221762730, 86883581750551, 1303254990051697, 19548831217572296, 293232499982777400, 4398487658768093001, 65977315675620433727 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 Regarding Lambda=, [4, 3] F[[4, 3], [7]](x) = 6 5 4 3 2 102 x - 17 x - 232 x + 73 x + 52 x - 18 x + 1 ------------------------------------------------------------ (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[7]](x) = (102*x^6-17*x^5-232*x^4+73*x^3+52*x^2-18*x+1)/(1+x)/(-1+x)/( 1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 1, 11, 112, 1519, 20988, 293119, 4100524, 57396287, 803502700, 11248862527, 157483369836, 2204764379455, 30866690101612, 432133616675135, 6049870454391148, 84698185645615423, 1185774596174889324, 16600844334995073343 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (32 x - 14 x + 1) F[[4, 3], [6, 1]](x) = - ------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[6, 1]](x) = -x^2*(32*x^2-14*x+1)/(1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 4, 52, 656, 9040, 125760, 1758016, 24600832, 344368384, 4820980736, 67493032960, 944899657728, 13228584030208, 185200131653632, 2592801664221184, 36299222583148544, 508189113300877312, 7114647574758555648, 99605066000807231488 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (14 x - 10 x + 1) F[[4, 3], [5, 2]](x) = - ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[5, 2]](x) = -x^2*(14*x^2-10*x+1)/(1+x)/(-1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 9, 113, 1519, 21017, 293223, 4101049, 57398343, 803511353, 11248896967, 157483509305, 2204764936647, 30866692337209, 432133625614791, 6049870490177081, 84698185788748231, 1185774596747529785, 16600844337285591495, 232411820653278891577 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (10 x - 9 x + 1) F[[4, 3], [5, 1, 1]](x) = - ------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[5, 1, 1]](x) = -x^2*(10*x^2-9*x+1)/(1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x ) The first 20 term , starting with k=1 are 0, 1, 9, 120, 1620, 22496, 314064, 4393600, 61496640, 860898816, 12052364544, 168732231680, 2362247746560, 33071454478336, 463000306765824, 6482004071055360, 90748056099962880, 1270472781820461056, 17786618931169787904, 249012664979111280640 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[4, 3], [4, 3]](x) = 4 3 2 x (122 x - 55 x - 44 x + 17 x - 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[4, 3]](x) = -x*(122*x^4-55*x^3-44*x^2+17*x-1)/(1+x)/(-1+x)/(1+2*x)/(-\ 1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 1, 11, 112, 1519, 20988, 293119, 4100524, 57396287, 803502700, 11248862527, 157483369836, 2204764379455, 30866690101612, 432133616675135, 6049870454391148, 84698185645615423, 1185774596174889324, 16600844334995073343, 232411820644116382060 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[4, 3], [4, 2, 1]](x) = 2 4 3 2 x (70 x + 5 x - 39 x + 2 x + 1) ------------------------------------------------------------ (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[4, 2, 1]](x) = x^2*(70*x^4+5*x^3-39*x^2+2*x+1)/(1+x)/(-1+x)/(1+2*x)/( -1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 20, 270, 3755, 52366, 732375, 10249870, 143484935, 2008734606, 28122067655, 393708073870, 5511909545415, 77166719656846, 1080334019297735, 15124676046480270, 211745463756042695, 2964436489005491086, 41502110831760732615, 581029551587384157070 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 x (-1 + 4 x) F[[4, 3], [4, 1, 1, 1]](x) = - ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation F[[4, 3],[4, 1, 1, 1]](x) = -x^2*(-1+4*x)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 153, 2135, 29881, 418311, 5856313, 81988295, 1147835961, 16069703111, 224975842873, 3149661798855, 44095265181241, 617333712531911, 8642671975435833, 120997407656079815, 1693963707185073721, 23715491900590944711, 332016886608273051193 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 3 2 x (42 x - 9 x - 5 x + 1) F[[4, 3], [3, 3, 1]](x) = - --------------------------------------------------- (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[3, 3, 1]](x) = -x^2*(42*x^3-9*x^2-5*x+1)/(1+x)/(1+2*x)/(-1+2*x)/(-1+4 *x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 12, 161, 2247, 31389, 439299, 6149389, 86088819, 1205232077, 16873205811, 236224704717, 3307145168691, 46300029557965, 648200402633523, 9074805592100045, 127047278110470963, 1778661892830645453, 24901266496765834035, 348617730943267949773 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (14 x - 7 x + 1) F[[4, 3], [3, 2, 2]](x) = - ------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[3, 2, 2]](x) = -x^2*(14*x^2-7*x+1)/(1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x ) The first 20 term , starting with k=1 are 0, 1, 11, 160, 2236, 31360, 439152, 6148864, 86086592, 1205223424, 16873170688, 236224565248, 3307144608768, 46300027322368, 648200393682944, 9074805556314112, 127047277967294464, 1778661892258004992, 24901266494475141120, 348617730934105440256 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[4, 3], [3, 2, 1, 1]](x) = 2 4 3 2 x (14 x + 31 x - 6 x + x - 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[3, 2, 1, 1]](x) = -x^2*(14*x^4+31*x^3-6*x^2+x-1)/(1+x)/(-1+x)/(1+2*x) /(-1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 17, 261, 3710, 52201, 731682, 10247161, 143473970, 2008691001, 28121892722, 393707375161, 5511906748530, 77166708473401, 1080333974555762, 15124675867528761, 211745463040203890, 2964436486142201401, 41502110820307442802, 581029551541571259961 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 3 x (18 x - 7) F[[4, 3], [3, 1, 1, 1, 1]](x) = ------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[3, 1, 1, 1, 1]](x) = x^3*(18*x-7)/(1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 7, 108, 1580, 22320, 313392, 4390848, 61485760, 860855040, 12052189952, 168731532288, 2362244951040, 33071443292160, 463000262029312, 6482003892092928, 90748055384145920, 1270472778957127680, 17786618919716585472, 249012664933298208768 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[4, 3], [2, 2, 2, 1]](x) = 2 3 2 x (18 x + 31 x - 11 x + 1) ------------------------------------------------------------ (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[2, 2, 2, 1]](x) = x^2*(18*x^3+31*x^2-11*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1 +2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 7, 106, 1479, 20862, 292551, 4098382, 57387463, 803467918, 11248722375, 157482811278, 2204762141127, 30866681156494, 432133580878279, 6049870311236494, 84698185072931271, 1185774593884283790, 16600844325832389063, 232411820607466169230 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 3 2 x (7 x - 3) F[[4, 3], [2, 2, 1, 1, 1]](x) = ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[2, 2, 1, 1, 1]](x) = 2*x^3*(7*x-3)/(1+x)/(-1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 6, 100, 1468, 20812, 292404, 4097772, 57385236, 803458924, 11248687252, 157482670444, 2204761581204, 30866678915436, 432133571927700, 6049870275428716, 84698184929754772, 1185774593311555948, 16600844323541696148, 232411820598303310188 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[4, 3], [2, 1, 1, 1, 1, 1]](x) = 3 2 x (32 x - 10 x + 3) - --------------------------------------------------- (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[2, 1, 1, 1, 1, 1]](x) = -x^3*(32*x^2-10*x+3)/(1+x)/(1+2*x)/(-1+2*x)/( -1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 41, 627, 8893, 125235, 1755789, 24592179, 344333261, 4820841267, 67492473037, 944897422131, 13228575079629, 185200095867699, 2592801521044685, 36299222010508083, 508189111010184397, 7114647565596046131, 99605065964156669133 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[4, 3], [1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (18 x + 31 x - 11 x + 1) ------------------------------------------------------------ (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 3],[1, 1, 1, 1, 1, 1, 1]](x) = x^3*(18*x^3+31*x^2-11*x+1)/(1+x)/(-1+x)/(1 +2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 1, 7, 106, 1479, 20862, 292551, 4098382, 57387463, 803467918, 11248722375, 157482811278, 2204762141127, 30866681156494, 432133580878279, 6049870311236494, 84698185072931271, 1185774593884283790, 16600844325832389063 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 Regarding Lambda=, [4, 2, 1] 4 3 2 60 x - 9 x - 175 x + 40 x - 1 F[[4, 2, 1], [7]](x) = --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[7]](x) = (60*x^4-9*x^3-175*x^2+40*x-1)/(-1+x)/(1+x)/(-1+5*x)/(-1+ 35*x) The first 20 term , starting with k=1 are 0, 1, 9, 301, 10434, 364801, 12766059, 446802301, 15638031684, 547330864801, 19156579047309, 670480260552301, 23466809088812934, 821338317955864801, 28746841127692328559, 1006139439465416802301, 35214880381270514594184, 1232520813344372643364801, 43138228467052565680609809, 1509837996346837414635552301 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (25 x - 2) F[[4, 2, 1], [6, 1]](x) = - ------------------------------ (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[6, 1]](x) = -x^2*(25*x-2)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 53, 1797, 62578, 2188672, 76595703, 2680810547, 93828173828, 3283985107422, 114939473876953, 4022881561279297, 140800854522705078, 4928029907684326172, 172481046765899658203, 6036836636791229248047, 211289282287616729736328, 7395124880066204071044922, 258829370802315235137939453, 9059027978081023693084716797 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (19 x - 2) F[[4, 2, 1], [5, 2]](x) = --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[5, 2]](x) = 2*x^2*(19*x-2)/(-1+x)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 4, 122, 4184, 145972, 5106684, 178722222, 6255219184, 218932378472, 7662631781684, 268192105034722, 9386723639594184, 328535327202690972, 11498736451178656684, 402455775786675347222, 14085952152510748969184, 493008325337761773003472, 17255291386821089850531684, 603935198538735283745659722, 21137731948855720625983344184 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (25 x + 30 x - 4) F[[4, 2, 1], [5, 1, 1]](x) = --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[5, 1, 1]](x) = x^2*(25*x^2+30*x-4)/(-1+x)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 4, 130, 4479, 156380, 5471354, 191487630, 6702018229, 234570393880, 8209962565104, 287348683675130, 10057203898111979, 352002136281331380, 12320074769083658854, 431202616914113362630, 15092091591974894205729, 528223205719025929768880, 18487812200165430704752604, 647073427005787690480550130, 22647569945202557245890299479 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (35 x - 4) F[[4, 2, 1], [4, 3]](x) = - ------------------------------ (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[4, 3]](x) = -x^2*(35*x-4)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 4, 121, 4179, 145946, 5106554, 178721571, 6255215929, 218932362196, 7662631700304, 268192104627821, 9386723637559679, 328535327192518446, 11498736451127794054, 402455775786421034071, 14085952152509477403429, 493008325337755415174696, 17255291386821058061387804, 603935198538735124799940321, 21137731948855719831254747179 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (115 x - 31 x + 1) F[[4, 2, 1], [4, 2, 1]](x) = - --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[4, 2, 1]](x) = -x*(115*x^2-31*x+1)/(-1+x)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 9, 301, 10434, 364801, 12766059, 446802301, 15638031684, 547330864801, 19156579047309, 670480260552301, 23466809088812934, 821338317955864801, 28746841127692328559, 1006139439465416802301, 35214880381270514594184, 1232520813344372643364801, 43138228467052565680609809, 1509837996346837414635552301, 52844329872139297591315375434 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 5 x F[[4, 2, 1], [4, 1, 1, 1]](x) = - ------------------- (1 + x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[4, 1, 1, 1]](x) = -5*x^2/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 5, 170, 5955, 208420, 7294705, 255314670, 8936013455, 312760470920, 10946616482205, 383131576877170, 13409605190700955, 469336181674533420, 16426766358608669705, 574936822551303439670, 20122788789295620388455, 704297607625346713595920, 24650416266887134975857205, 862764569341049724155002170, 30196759926936740345425075955 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (35 x + 21 x - 5) F[[4, 2, 1], [3, 3, 1]](x) = --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[3, 3, 1]](x) = x^2*(35*x^2+21*x-5)/(-1+x)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 5, 179, 6255, 218854, 7659505, 268080729, 9382815755, 328398502604, 11493947347005, 402288155924479, 14080085451253255, 492802990763346354, 17248104676564534505, 603683663678995768229, 21128928228761037190755, 739512488006617228190104, 25882937080231507619222005, 905902797808102289835611979, 31706597923283577760060628255 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (22 x - 5) F[[4, 2, 1], [3, 2, 2]](x) = --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[3, 2, 2]](x) = x^2*(22*x-5)/(-1+x)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 5, 178, 6250, 218828, 7659375, 268080078, 9382812500, 328398486328, 11493947265625, 402288155517578, 14080085449218750, 492802990753173828, 17248104676513671875, 603683663678741455078, 21128928228759765625000, 739512488006610870361328, 25882937080231475830078125, 905902797808102130889892578, 31706597923283576965332031250 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (25 x - 8) F[[4, 2, 1], [3, 2, 1, 1]](x) = --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[3, 2, 1, 1]](x) = x^2*(25*x-8)/(-1+x)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 8, 295, 10408, 364670, 12765408, 446799045, 15638015408, 547330783420, 19156578640408, 670480258517795, 23466809078640408, 821338317905002170, 28746841127438015408, 1006139439464145236545, 35214880381264156765408, 1232520813344340854220920, 43138228467052406734890408, 1509837996346836619906955295, 52844329872139293617672390408 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (25 x - 5 x - 3) F[[4, 2, 1], [3, 1, 1, 1, 1]](x) = --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[3, 1, 1, 1, 1]](x) = x^2*(25*x^2-5*x-3)/(-1+x)/(1+x)/(-1+5*x)/(-1+ 35*x) The first 20 term , starting with k=1 are 0, 3, 125, 4453, 156250, 5470703, 191484375, 6702001953, 234570312500, 8209962158203, 287348681640625, 10057203887939453, 352002136230468750, 12320074768829345703, 431202616912841796875, 15092091591968536376953, 528223205718994140625000, 18487812200165271759033203, 647073427005786895751953125, 22647569945202553272247314453 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 3 x F[[4, 2, 1], [2, 2, 2, 1]](x) = ------------------------------ (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[2, 2, 2, 1]](x) = 3*x^2/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 3, 117, 4158, 145842, 5106033, 178718967, 6255202908, 218932297092, 7662631374783, 268192103000217, 9386723629421658, 328535327151828342, 11498736450924343533, 402455775785403781467, 14085952152504391140408, 493008325337729983859592, 17255291386820930904812283, 603935198538734489017062717, 21137731948855716652340359158 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (35 x - 4 x + 3) F[[4, 2, 1], [2, 2, 1, 1, 1]](x) = - --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[2, 2, 1, 1, 1]](x) = -x^2*(35*x^2-4*x+3)/(-1+x)/(1+x)/(-1+5*x)/(-1 +35*x) The first 20 term , starting with k=1 are 0, 3, 116, 4153, 145816, 5105903, 178718316, 6255199653, 218932280816, 7662631293403, 268192102593316, 9386723627387153, 328535327141655816, 11498736450873480903, 402455775785149468316, 14085952152503119574653, 493008325337723626030816, 17255291386820899115668403, 603935198538734330071343316, 21137731948855715857611762153 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (10 x + 1) F[[4, 2, 1], [2, 1, 1, 1, 1, 1]](x) = ------------------------------ (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[2, 1, 1, 1, 1, 1]](x) = x^2*(10*x+1)/(1+x)/(-1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 49, 1776, 62474, 2188151, 76593099, 2680797526, 93828108724, 3283984781901, 114939472249349, 4022881553141276, 140800854482014974, 4928029907480875651, 172481046764882405599, 6036836636786142985026, 211289282287591298421224, 7395124880066076914469401, 258829370802314599355061849, 9059027978081020514170328776 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 F[[4, 2, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 3 x (25 x - 8) --------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 2, 1],[1, 1, 1, 1, 1, 1, 1]](x) = x^3*(25*x-8)/(-1+x)/(1+x)/(-1+5*x)/(-1+ 35*x) The first 20 term , starting with k=1 are 0, 0, 8, 295, 10408, 364670, 12765408, 446799045, 15638015408, 547330783420, 19156578640408, 670480258517795, 23466809078640408, 821338317905002170, 28746841127438015408, 1006139439464145236545, 35214880381264156765408, 1232520813344340854220920, 43138228467052406734890408, 1509837996346836619906955295 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 Regarding Lambda=, [4, 1, 1, 1] 4 3 2 116 x + 96 x - 65 x - 17 x + 1 F[[4, 1, 1, 1], [7]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[7]](x) = (116*x^4+96*x^3-65*x^2-17*x+1)/(1+x)/(-1+2*x)/(1+4*x)/ (-1+20*x) The first 20 term , starting with k=1 are 0, 1, 1, 39, 617, 12791, 253641, 5080759, 101581897, 2031767991, 40634833481, 812698762679, 16253966856777, 325079370673591, 6501587279221321, 130031746121231799, 2600634920277021257, 52012698414130097591, 1040253968248241689161, 20805079365102271688119 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 x (8 x - 1) F[[4, 1, 1, 1], [6, 1]](x) = ----------------------------- (1 + x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[6, 1]](x) = x^2*(8*x-1)/(1+x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 7, 201, 3767, 76361, 1523127, 30478921, 609512887, 12190519881, 243809349047, 4876191175241, 97523806727607, 1950476201661001, 39009523764784567, 780190476369433161, 15603809523093695927, 312076190479053787721, 6241523809512356277687, 124830476190522003460681 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x (8 x - 1) F[[4, 1, 1, 1], [5, 2]](x) = - -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[5, 2]](x) = -2*x^2*(8*x-1)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 20, 456, 8848, 177952, 3554880, 71113856, 1422211328, 28444488192, 568888714240, 11377778477056, 227555552759808, 4551111122296832, 91022222177484800, 1820444444623405056, 36408888888173068288, 728177777780641103872, 14563555555544102338560, 291271111111156924153856 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 x (10 x + 1) F[[4, 1, 1, 1], [5, 1, 1]](x) = - ----------------------------- (1 + x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[5, 1, 1]](x) = -x^2*(10*x+1)/(1+x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 25, 471, 9545, 190391, 3809865, 76189111, 1523814985, 30476168631, 609523896905, 12190475840951, 243809525207625, 4876190470598071, 97523809546179145, 1950476190386711991, 39009523809881723465, 780190476189044534711, 15603809523815250432585, 312076190476167569698231 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x (8 x - 1) F[[4, 1, 1, 1], [4, 3]](x) = - -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[4, 3]](x) = -2*x^2*(8*x-1)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 20, 456, 8848, 177952, 3554880, 71113856, 1422211328, 28444488192, 568888714240, 11377778477056, 227555552759808, 4551111122296832, 91022222177484800, 1820444444623405056, 36408888888173068288, 728177777780641103872, 14563555555544102338560, 291271111111156924153856 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x (10 x + 1) F[[4, 1, 1, 1], [4, 2, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[4, 2, 1]](x) = 2*x^2*(10*x+1)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 56, 1104, 22240, 444352, 8889216, 177776384, 3555560960, 71111089152, 1422222309376, 28444444094464, 568888890286080, 11377777772183552, 227555555577921536, 4551111111021625344, 91022222222580121600, 1820444444443012759552, 36408888888894615453696, 728177777777754871169024 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 x (44 x + 16 x - 1) F[[4, 1, 1, 1], [4, 1, 1, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[4, 1, 1, 1]](x) = -x*(44*x^2+16*x-1)/(1+x)/(-1+2*x)/(1+4*x)/(-1 +20*x) The first 20 term , starting with k=1 are 1, 1, 39, 617, 12791, 253641, 5080759, 101581897, 2031767991, 40634833481, 812698762679, 16253966856777, 325079370673591, 6501587279221321, 130031746121231799, 2600634920277021257, 52012698414130097591, 1040253968248241689161, 20805079365102271688119, 416101587301495675851337 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x F[[4, 1, 1, 1], [3, 3, 1]](x) = - --------------------- (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[3, 3, 1]](x) = -2*x^2/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 32, 672, 13312, 266752, 5332992, 106668032, 2133327872, 42666688512, 853333245952, 17066667016192, 341333331935232, 6826666672259072, 136533333310963712, 2730666666756145152, 54613333332975419392, 1092266666668098322432, 21845333333327606710272, 436906666666689573158912 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x F[[4, 1, 1, 1], [3, 2, 2]](x) = - --------------------- (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[3, 2, 2]](x) = -2*x^2/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 32, 672, 13312, 266752, 5332992, 106668032, 2133327872, 42666688512, 853333245952, 17066667016192, 341333331935232, 6826666672259072, 136533333310963712, 2730666666756145152, 54613333332975419392, 1092266666668098322432, 21845333333327606710272, 436906666666689573158912 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x (10 x + 1) F[[4, 1, 1, 1], [3, 2, 1, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[3, 2, 1, 1]](x) = 2*x^2*(10*x+1)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 56, 1104, 22240, 444352, 8889216, 177776384, 3555560960, 71111089152, 1422222309376, 28444444094464, 568888890286080, 11377777772183552, 227555555577921536, 4551111111021625344, 91022222222580121600, 1820444444443012759552, 36408888888894615453696, 728177777777754871169024 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 x (10 x + 1) F[[4, 1, 1, 1], [3, 1, 1, 1, 1]](x) = - ----------------------------- (1 + x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[3, 1, 1, 1, 1]](x) = -x^2*(10*x+1)/(1+x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 25, 471, 9545, 190391, 3809865, 76189111, 1523814985, 30476168631, 609523896905, 12190475840951, 243809525207625, 4876190470598071, 97523809546179145, 1950476190386711991, 39009523809881723465, 780190476189044534711, 15603809523815250432585, 312076190476167569698231 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x (8 x - 1) F[[4, 1, 1, 1], [2, 2, 2, 1]](x) = - -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[2, 2, 2, 1]](x) = -2*x^2*(8*x-1)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 20, 456, 8848, 177952, 3554880, 71113856, 1422211328, 28444488192, 568888714240, 11377778477056, 227555552759808, 4551111122296832, 91022222177484800, 1820444444623405056, 36408888888173068288, 728177777780641103872, 14563555555544102338560, 291271111111156924153856 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 2 x (8 x - 1) F[[4, 1, 1, 1], [2, 2, 1, 1, 1]](x) = - -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[2, 2, 1, 1, 1]](x) = -2*x^2*(8*x-1)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 20, 456, 8848, 177952, 3554880, 71113856, 1422211328, 28444488192, 568888714240, 11377778477056, 227555552759808, 4551111122296832, 91022222177484800, 1820444444623405056, 36408888888173068288, 728177777780641103872, 14563555555544102338560, 291271111111156924153856 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 2 x (8 x - 1) F[[4, 1, 1, 1], [2, 1, 1, 1, 1, 1]](x) = ----------------------------- (1 + x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[2, 1, 1, 1, 1, 1]](x) = x^2*(8*x-1)/(1+x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 7, 201, 3767, 76361, 1523127, 30478921, 609512887, 12190519881, 243809349047, 4876191175241, 97523806727607, 1950476201661001, 39009523764784567, 780190476369433161, 15603809523093695927, 312076190479053787721, 6241523809512356277687, 124830476190522003460681 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 F[[4, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 2 2 x (44 x + 16 x - 1) - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation F[[4, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1]](x) = -x^2*(44*x^2+16*x-1)/(1+x)/(-1+2*x)/ (1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 1, 39, 617, 12791, 253641, 5080759, 101581897, 2031767991, 40634833481, 812698762679, 16253966856777, 325079370673591, 6501587279221321, 130031746121231799, 2600634920277021257, 52012698414130097591, 1040253968248241689161, 20805079365102271688119 ---------------------------------- Their sum is 4 3 2 32 x + 16 x - 60 x - 16 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 20 x) and in Maple notation (32*x^4+16*x^3-60*x^2-16*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 23, 361, 7415, 147145, 2946743, 58917961, 1178423735, 23568210505, 471365254583, 9427300889161, 188546034544055, 3770920623739465, 75418412743159223, 1508368253789311561, 30167365080080936375, 603347301584438334025, 12066946031757485108663, 241338920634874822169161, 4826778412698595950816695 Regarding Lambda=, [3, 3, 1] 4 3 2 23 x + x - 63 x - 18 x + 1 F[[3, 3, 1], [7]](x) = -------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[7]](x) = (23*x^4+x^3-63*x^2-18*x+1)/(-1+x)/(1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 1, 42, 802, 17043, 357283, 7504764, 157594564, 3309502245, 69499497925, 1459489604046, 30649281242086, 643634907412407, 13516333051674727, 283842994097126688, 5960702876003788168, 125174760396187168329, 2628669968319607684489, 55202069334712729925490 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (-1 + 9 x) F[[3, 3, 1], [6, 1]](x) = - ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[6, 1]](x) = -x^2*(-1+9*x)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 10, 235, 4852, 102133, 2144062, 45027487, 945570664, 19857003625, 416997017074, 8756937535699, 183895687718236, 3861809443677277, 81097998312439846, 1703057964575585671, 35764217256044252368, 751048562377058439889, 15772019809917839817178, 331212416008275798422203 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 2 x (7 x - 1) F[[3, 3, 1], [5, 2]](x) = - ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[5, 2]](x) = -2*x^2*(7*x-1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 24, 546, 11328, 238290, 5002872, 105063954, 2206332096, 46333006818, 972993044760, 20432854235202, 429089938053504, 9010888701780786, 189228662729424888, 3973801917341837490, 83449840264106842752, 1752446645546458931394, 36801379556474991858456, 772828970685976766130018 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (9 x + 1) F[[3, 3, 1], [5, 1, 1]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[5, 1, 1]](x) = x^2*(9*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 28, 577, 12160, 255241, 5360428, 112567897, 2363929120, 49642501681, 1042492564828, 21892343772817, 459739219494880, 9654523608595321, 202744995782893228, 4257644911433583337, 89410543140126773440, 1877621405942597672161, 39430049524794744825628, 828031040020689060207457 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (7 x + 1) F[[3, 3, 1], [4, 3]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[4, 3]](x) = x^2*(7*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 26, 539, 11348, 238229, 5003054, 105063407, 2206333736, 46333001897, 972993059522, 20432854190915, 429089938186364, 9010888701382205, 189228662730620630, 3973801917338250263, 83449840264117604432, 1752446645546426646353, 36801379556475088713578, 772828970685976475564651 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (7 x + 3) F[[3, 3, 1], [4, 2, 1]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[4, 2, 1]](x) = x^2*(7*x+3)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 3, 64, 1351, 28360, 595603, 12507544, 262658791, 5515833520, 115832507203, 2432482641424, 51082135499431, 1072724845399480, 22527221753654803, 473071656825953704, 9934504793347419271, 208624600660288630240, 4381116613866082758403, 92003448891187673356384, 1932072426714941334194311 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 2 x F[[3, 3, 1], [4, 1, 1, 1]](x) = - --------------------- (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[4, 1, 1, 1]](x) = -2*x^2/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 36, 774, 16200, 340362, 7147116, 150090894, 3151904400, 66190005522, 1389990076596, 29189791726614, 612985625904600, 12872698145059482, 270326661043060476, 5676859881913835934, 119214057520161856800, 2503495207923485086242, 52573399366392928530756, 1104041386694252273986854 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (40 x + 17 x - 1) F[[3, 3, 1], [3, 3, 1]](x) = - -------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[3, 3, 1]](x) = -x*(40*x^2+17*x-1)/(-1+x)/(1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 1, 42, 802, 17043, 357283, 7504764, 157594564, 3309502245, 69499497925, 1459489604046, 30649281242086, 643634907412407, 13516333051674727, 283842994097126688, 5960702876003788168, 125174760396187168329, 2628669968319607684489, 55202069334712729925490, 1159243456028964422781610 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 2 x F[[3, 3, 1], [3, 2, 2]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[3, 2, 2]](x) = 2*x^2/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 38, 812, 17012, 357374, 7504490, 157595384, 3309499784, 69499505306, 1459489581902, 30649281308516, 643634907213116, 13516333052272598, 283842994095333074, 5960702876009169008, 125174760396171025808, 2628669968319656112050, 55202069334712584642806, 1159243456028964858629660 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (14 x + 3) F[[3, 3, 1], [3, 2, 1, 1]](x) = - ----------------------------- (1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[3, 2, 1, 1]](x) = -x^2*(14*x+3)/(1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 3, 65, 1348, 28370, 595573, 12507635, 262658518, 5515834340, 115832504743, 2432482648805, 51082135477288, 1072724845465910, 22527221753455513, 473071656826551575, 9934504793345625658, 208624600660294011080, 4381116613866066615883, 92003448891187721783945, 1932072426714941188911628 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (9 x - 2) F[[3, 3, 1], [3, 1, 1, 1, 1]](x) = ----------------------------- (1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[3, 1, 1, 1, 1]](x) = x^2*(9*x-2)/(1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 25, 587, 12130, 255332, 5360155, 112568717, 2363926660, 49642509062, 1042492542685, 21892343839247, 459739219295590, 9654523609193192, 202744995781099615, 4257644911438964177, 89410543140110630920, 1877621405942646099722, 39430049524794599542945, 828031040020689496055507 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (7 x + 1) F[[3, 3, 1], [2, 2, 2, 1]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[2, 2, 2, 1]](x) = x^2*(7*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 26, 539, 11348, 238229, 5003054, 105063407, 2206333736, 46333001897, 972993059522, 20432854190915, 429089938186364, 9010888701382205, 189228662730620630, 3973801917338250263, 83449840264117604432, 1752446645546426646353, 36801379556475088713578, 772828970685976475564651 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (7 x + 1) F[[3, 3, 1], [2, 2, 1, 1, 1]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[2, 2, 1, 1, 1]](x) = x^2*(7*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 26, 539, 11348, 238229, 5003054, 105063407, 2206333736, 46333001897, 972993059522, 20432854190915, 429089938186364, 9010888701382205, 189228662730620630, 3973801917338250263, 83449840264117604432, 1752446645546426646353, 36801379556475088713578, 772828970685976475564651 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (-1 + 9 x) F[[3, 3, 1], [2, 1, 1, 1, 1, 1]](x) = - ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[2, 1, 1, 1, 1, 1]](x) = -x^2*(-1+9*x)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 10, 235, 4852, 102133, 2144062, 45027487, 945570664, 19857003625, 416997017074, 8756937535699, 183895687718236, 3861809443677277, 81097998312439846, 1703057964575585671, 35764217256044252368, 751048562377058439889, 15772019809917839817178, 331212416008275798422203 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 3 2 x F[[3, 3, 1], [1, 1, 1, 1, 1, 1, 1]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 3, 1],[1, 1, 1, 1, 1, 1, 1]](x) = 2*x^3/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 2, 38, 812, 17012, 357374, 7504490, 157595384, 3309499784, 69499505306, 1459489581902, 30649281308516, 643634907213116, 13516333052272598, 283842994095333074, 5960702876009169008, 125174760396171025808, 2628669968319656112050, 55202069334712584642806 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 Regarding Lambda=, [3, 2, 2] 5 4 3 2 69 x + x - 191 x + 9 x + 21 x - 1 F[[3, 2, 2], [7]](x) = ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[7]](x) = (69*x^5+x^4-191*x^3+9*x^2+21*x-1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 2, 42, 812, 17043, 357374, 7504764, 157595384, 3309502245, 69499505306, 1459489604046, 30649281308516, 643634907412407, 13516333052272598, 283842994097126688, 5960702876009169008, 125174760396187168329, 2628669968319656112050, 55202069334712729925490 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (-1 + 9 x) F[[3, 2, 2], [6, 1]](x) = - ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[6, 1]](x) = -x^2*(-1+9*x)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 10, 235, 4852, 102133, 2144062, 45027487, 945570664, 19857003625, 416997017074, 8756937535699, 183895687718236, 3861809443677277, 81097998312439846, 1703057964575585671, 35764217256044252368, 751048562377058439889, 15772019809917839817178, 331212416008275798422203 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 3 2 2 x (21 x - 10 x - 8 x + 1) F[[3, 2, 2], [5, 2]](x) = - ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[5, 2]](x) = -2*x^2*(21*x^3-10*x^2-8*x+1)/(1+x)/(-1+x)/(1+3*x)/(-1+ 3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 26, 546, 11348, 238290, 5003054, 105063954, 2206333736, 46333006818, 972993059522, 20432854235202, 429089938186364, 9010888701780786, 189228662730620630, 3973801917341837490, 83449840264117604432, 1752446645546458931394, 36801379556475088713578, 772828970685976766130018 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 3 2 x (27 x - 42 x - 4 x - 1) F[[3, 2, 2], [5, 1, 1]](x) = ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[5, 1, 1]](x) = x^2*(27*x^3-42*x^2-4*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+ 3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 25, 577, 12130, 255241, 5360155, 112567897, 2363926660, 49642501681, 1042492542685, 21892343772817, 459739219295590, 9654523608595321, 202744995781099615, 4257644911433583337, 89410543140110630920, 1877621405942597672161, 39430049524794599542945, 828031040020689060207457 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (7 x + 1) F[[3, 2, 2], [4, 3]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[4, 3]](x) = x^2*(7*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 26, 539, 11348, 238229, 5003054, 105063407, 2206333736, 46333001897, 972993059522, 20432854190915, 429089938186364, 9010888701382205, 189228662730620630, 3973801917338250263, 83449840264117604432, 1752446645546426646353, 36801379556475088713578, 772828970685976475564651 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 3 2 x (21 x + 44 x - 2 x - 3) F[[3, 2, 2], [4, 2, 1]](x) = ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[4, 2, 1]](x) = x^2*(21*x^3+44*x^2-2*x-3)/(1+x)/(-1+x)/(1+3*x)/(-1+ 3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 3, 65, 1351, 28370, 595603, 12507635, 262658791, 5515834340, 115832507203, 2432482648805, 51082135499431, 1072724845465910, 22527221753654803, 473071656826551575, 9934504793347419271, 208624600660294011080, 4381116613866082758403, 92003448891187721783945, 1932072426714941334194311 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 2 x F[[3, 2, 2], [4, 1, 1, 1]](x) = - --------------------- (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[4, 1, 1, 1]](x) = -2*x^2/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 36, 774, 16200, 340362, 7147116, 150090894, 3151904400, 66190005522, 1389990076596, 29189791726614, 612985625904600, 12872698145059482, 270326661043060476, 5676859881913835934, 119214057520161856800, 2503495207923485086242, 52573399366392928530756, 1104041386694252273986854 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 2 x (6 x - 17 x - 1) F[[3, 2, 2], [3, 3, 1]](x) = ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[3, 3, 1]](x) = x^2*(6*x^2-17*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+3*x)/(-\ 1+21*x) The first 20 term , starting with k=1 are 0, 1, 38, 802, 17012, 357283, 7504490, 157594564, 3309499784, 69499497925, 1459489581902, 30649281242086, 643634907213116, 13516333051674727, 283842994095333074, 5960702876003788168, 125174760396171025808, 2628669968319607684489, 55202069334712584642806, 1159243456028964422781610 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 F[[3, 2, 2], [3, 2, 2]](x) = 3 2 x (120 x - 10 x - 19 x + 1) - ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[3, 2, 2]](x) = -x*(120*x^3-10*x^2-19*x+1)/(1+x)/(-1+x)/(1+3*x)/(-1 +3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 2, 42, 812, 17043, 357374, 7504764, 157595384, 3309502245, 69499505306, 1459489604046, 30649281308516, 643634907412407, 13516333052272598, 283842994097126688, 5960702876009169008, 125174760396187168329, 2628669968319656112050, 55202069334712729925490, 1159243456028964858629660 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 F[[3, 2, 2], [3, 2, 1, 1]](x) = 2 3 2 x (42 x - 26 x + x + 3) - ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[3, 2, 1, 1]](x) = -x^2*(42*x^3-26*x^2+x+3)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 3, 64, 1348, 28360, 595573, 12507544, 262658518, 5515833520, 115832504743, 2432482641424, 51082135477288, 1072724845399480, 22527221753455513, 473071656825953704, 9934504793345625658, 208624600660288630240, 4381116613866066615883, 92003448891187673356384, 1932072426714941188911628 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 F[[3, 2, 2], [3, 1, 1, 1, 1]](x) = 2 3 2 x (27 x + 21 x + 14 x - 2) ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[3, 1, 1, 1, 1]](x) = x^2*(27*x^3+21*x^2+14*x-2)/(1+x)/(-1+x)/(1+3* x)/(-1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 28, 587, 12160, 255332, 5360428, 112568717, 2363929120, 49642509062, 1042492564828, 21892343839247, 459739219494880, 9654523609193192, 202744995782893228, 4257644911438964177, 89410543140126773440, 1877621405942646099722, 39430049524794744825628, 828031040020689496055507 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (7 x + 1) F[[3, 2, 2], [2, 2, 2, 1]](x) = ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[2, 2, 2, 1]](x) = x^2*(7*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 26, 539, 11348, 238229, 5003054, 105063407, 2206333736, 46333001897, 972993059522, 20432854190915, 429089938186364, 9010888701382205, 189228662730620630, 3973801917338250263, 83449840264117604432, 1752446645546426646353, 36801379556475088713578, 772828970685976475564651 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 F[[3, 2, 2], [2, 2, 1, 1, 1]](x) = 2 3 2 x (21 x - 25 x - 3 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[2, 2, 1, 1, 1]](x) = x^2*(21*x^3-25*x^2-3*x-1)/(1+x)/(-1+x)/(1+3*x )/(-1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 24, 539, 11328, 238229, 5002872, 105063407, 2206332096, 46333001897, 972993044760, 20432854190915, 429089938053504, 9010888701382205, 189228662729424888, 3973801917338250263, 83449840264106842752, 1752446645546426646353, 36801379556474991858456, 772828970685976475564651 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 2 x (-1 + 9 x) F[[3, 2, 2], [2, 1, 1, 1, 1, 1]](x) = - ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[2, 1, 1, 1, 1, 1]](x) = -x^2*(-1+9*x)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 10, 235, 4852, 102133, 2144062, 45027487, 945570664, 19857003625, 416997017074, 8756937535699, 183895687718236, 3861809443677277, 81097998312439846, 1703057964575585671, 35764217256044252368, 751048562377058439889, 15772019809917839817178, 331212416008275798422203 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 F[[3, 2, 2], [1, 1, 1, 1, 1, 1, 1]](x) = 3 2 x (6 x - 17 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 2, 2],[1, 1, 1, 1, 1, 1, 1]](x) = x^3*(6*x^2-17*x-1)/(1+x)/(-1+x)/(1+3*x) /(-1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 1, 38, 802, 17012, 357283, 7504490, 157594564, 3309499784, 69499497925, 1459489581902, 30649281242086, 643634907213116, 13516333051674727, 283842994095333074, 5960702876003788168, 125174760396171025808, 2628669968319607684489, 55202069334712584642806 ---------------------------------- Their sum is 3 2 23 x - 42 x - 18 x + 1 ------------------------------ (-1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation (23*x^3-42*x^2-18*x+1)/(-1+x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 22, 423, 8964, 187965, 3948066, 82906947, 1741053168, 36562094649, 767804053230, 16123884920991, 338601583931292, 7110633260785653, 149323298481813114, 3135789268102132155, 65851574630192604936, 1382883067233901214577, 29040544411912355973318, 609851432650158184038039, 12806880085653325739003700 Regarding Lambda=, [3, 2, 1, 1] 4 3 2 60 x + 8 x - 175 x - 30 x + 1 F[[3, 2, 1, 1], [7]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[7]](x) = (60*x^4+8*x^3-175*x^2-30*x+1)/(1+x)/(-1+x)/(1+5*x)/(-1 +35*x) The first 20 term , starting with k=1 are 0, 1, 8, 301, 10408, 364801, 12765408, 446802301, 15638015408, 547330864801, 19156578640408, 670480260552301, 23466809078640408, 821338317955864801, 28746841127438015408, 1006139439465416802301, 35214880381264156765408, 1232520813344372643364801, 43138228467052406734890408, 1509837996346837414635552301 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (25 x + 11 x - 2) F[[3, 2, 1, 1], [6, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[6, 1]](x) = -x^2*(25*x^2+11*x-2)/(1+x)/(-1+x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 49, 1797, 62474, 2188672, 76593099, 2680810547, 93828108724, 3283985107422, 114939472249349, 4022881561279297, 140800854482014974, 4928029907684326172, 172481046764882405599, 6036836636791229248047, 211289282287591298421224, 7395124880066204071044922, 258829370802314599355061849, 9059027978081023693084716797 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 4 x F[[3, 2, 1, 1], [5, 2]](x) = - ----------------------------- (1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[5, 2]](x) = -4*x^2/(1+x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 4, 116, 4184, 145816, 5106684, 178718316, 6255219184, 218932280816, 7662631781684, 268192102593316, 9386723639594184, 328535327141655816, 11498736451178656684, 402455775785149468316, 14085952152510748969184, 493008325337723626030816, 17255291386821089850531684, 603935198538734330071343316, 21137731948855720625983344184 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (25 x + 5 x + 4) F[[3, 2, 1, 1], [5, 1, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[5, 1, 1]](x) = x^2*(25*x^2+5*x+4)/(1+x)/(-1+x)/(1+5*x)/(-1+35*x ) The first 20 term , starting with k=1 are 0, 4, 125, 4479, 156250, 5471354, 191484375, 6702018229, 234570312500, 8209962565104, 287348681640625, 10057203898111979, 352002136230468750, 12320074769083658854, 431202616912841796875, 15092091591974894205729, 528223205718994140625000, 18487812200165430704752604, 647073427005786895751953125, 22647569945202557245890299479 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (35 x + 3 x - 4) F[[3, 2, 1, 1], [4, 3]](x) = - -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[4, 3]](x) = -x^2*(35*x^2+3*x-4)/(1+x)/(-1+x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 4, 117, 4179, 145842, 5106554, 178718967, 6255215929, 218932297092, 7662631700304, 268192103000217, 9386723637559679, 328535327151828342, 11498736451127794054, 402455775785403781467, 14085952152509477403429, 493008325337729983859592, 17255291386821058061387804, 603935198538734489017062717, 21137731948855719831254747179 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (25 x + 9) F[[3, 2, 1, 1], [4, 2, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[4, 2, 1]](x) = x^2*(25*x+9)/(1+x)/(-1+x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 9, 295, 10434, 364670, 12766059, 446799045, 15638031684, 547330783420, 19156579047309, 670480258517795, 23466809088812934, 821338317905002170, 28746841127692328559, 1006139439464145236545, 35214880381270514594184, 1232520813344340854220920, 43138228467052565680609809, 1509837996346836619906955295, 52844329872139297591315375434 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 5 x F[[3, 2, 1, 1], [4, 1, 1, 1]](x) = - ------------------- (1 + x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[4, 1, 1, 1]](x) = -5*x^2/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 5, 170, 5955, 208420, 7294705, 255314670, 8936013455, 312760470920, 10946616482205, 383131576877170, 13409605190700955, 469336181674533420, 16426766358608669705, 574936822551303439670, 20122788789295620388455, 704297607625346713595920, 24650416266887134975857205, 862764569341049724155002170, 30196759926936740345425075955 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (35 x + 28 x + 5) F[[3, 2, 1, 1], [3, 3, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[3, 3, 1]](x) = x^2*(35*x^2+28*x+5)/(1+x)/(-1+x)/(1+5*x)/(-1+35* x) The first 20 term , starting with k=1 are 0, 5, 178, 6255, 218828, 7659505, 268080078, 9382815755, 328398486328, 11493947347005, 402288155517578, 14080085451253255, 492802990753173828, 17248104676564534505, 603683663678741455078, 21128928228761037190755, 739512488006610870361328, 25882937080231507619222005, 905902797808102130889892578, 31706597923283577760060628255 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (29 x + 5) F[[3, 2, 1, 1], [3, 2, 2]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[3, 2, 2]](x) = x^2*(29*x+5)/(1+x)/(-1+x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 5, 179, 6250, 218854, 7659375, 268080729, 9382812500, 328398502604, 11493947265625, 402288155924479, 14080085449218750, 492802990763346354, 17248104676513671875, 603683663678995768229, 21128928228759765625000, 739512488006617228190104, 25882937080231475830078125, 905902797808102289835611979, 31706597923283576965332031250 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (115 x + 22 x - 1) F[[3, 2, 1, 1], [3, 2, 1, 1]](x) = - -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[3, 2, 1, 1]](x) = -x*(115*x^2+22*x-1)/(1+x)/(-1+x)/(1+5*x)/(-1+ 35*x) The first 20 term , starting with k=1 are 1, 8, 301, 10408, 364801, 12765408, 446802301, 15638015408, 547330864801, 19156578640408, 670480260552301, 23466809078640408, 821338317955864801, 28746841127438015408, 1006139439465416802301, 35214880381264156765408, 1232520813344372643364801, 43138228467052406734890408, 1509837996346837414635552301, 52844329872139293617672390408 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (25 x + 40 x + 3) F[[3, 2, 1, 1], [3, 1, 1, 1, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[3, 1, 1, 1, 1]](x) = x^2*(25*x^2+40*x+3)/(1+x)/(-1+x)/(1+5*x)/( -1+35*x) The first 20 term , starting with k=1 are 0, 3, 130, 4453, 156380, 5470703, 191487630, 6702001953, 234570393880, 8209962158203, 287348683675130, 10057203887939453, 352002136281331380, 12320074768829345703, 431202616914113362630, 15092091591968536376953, 528223205719025929768880, 18487812200165271759033203, 647073427005787690480550130, 22647569945202553272247314453 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (31 x + 3) F[[3, 2, 1, 1], [2, 2, 2, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[2, 2, 2, 1]](x) = x^2*(31*x+3)/(1+x)/(-1+x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 3, 121, 4158, 145946, 5106033, 178721571, 6255202908, 218932362196, 7662631374783, 268192104627821, 9386723629421658, 328535327192518446, 11498736450924343533, 402455775786421034071, 14085952152504391140408, 493008325337755415174696, 17255291386820930904812283, 603935198538735124799940321, 21137731948855716652340359158 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 x (35 x + 3) F[[3, 2, 1, 1], [2, 2, 1, 1, 1]](x) = - ----------------------------- (1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[2, 2, 1, 1, 1]](x) = -x^2*(35*x+3)/(1+x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 3, 122, 4153, 145972, 5105903, 178722222, 6255199653, 218932378472, 7662631293403, 268192105034722, 9386723627387153, 328535327202690972, 11498736450873480903, 402455775786675347222, 14085952152503119574653, 493008325337761773003472, 17255291386820899115668403, 603935198538735283745659722, 21137731948855715857611762153 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 2 2 x (10 x + 23 x + 1) F[[3, 2, 1, 1], [2, 1, 1, 1, 1, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[2, 1, 1, 1, 1, 1]](x) = x^2*(10*x^2+23*x+1)/(1+x)/(-1+x)/(1+5*x )/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 53, 1776, 62578, 2188151, 76595703, 2680797526, 93828173828, 3283984781901, 114939473876953, 4022881553141276, 140800854522705078, 4928029907480875651, 172481046765899658203, 6036836636786142985026, 211289282287616729736328, 7395124880066076914469401, 258829370802315235137939453, 9059027978081020514170328776 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 F[[3, 2, 1, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 3 x (25 x + 9) -------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[3, 2, 1, 1],[1, 1, 1, 1, 1, 1, 1]](x) = x^3*(25*x+9)/(1+x)/(-1+x)/(1+5*x)/(-\ 1+35*x) The first 20 term , starting with k=1 are 0, 0, 9, 295, 10434, 364670, 12766059, 446799045, 15638031684, 547330783420, 19156579047309, 670480258517795, 23466809088812934, 821338317905002170, 28746841127692328559, 1006139439464145236545, 35214880381270514594184, 1232520813344340854220920, 43138228467052565680609809, 1509837996346836619906955295 ---------------------------------- Their sum is 2 12 x + 33 x - 1 ------------------- (1 + x) (-1 + 35 x) and in Maple notation (12*x^2+33*x-1)/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 57, 1973, 69077, 2417673, 84618577, 2961650173, 103657756077, 3628021462673, 126980751193577, 4444326291775173, 155551420212131077, 5444299707424587673, 190550489759860568577, 6669267141595119900173, 233424349955829196506077, 8169852248454021877712673, 285944828695890765719943577, 10008069004356176800198025173, 350282415152466188006930881077 Regarding Lambda=, [3, 1, 1, 1, 1] 5 4 3 2 95 x - 2 x - 226 x + 45 x + 13 x - 1 F[[3, 1, 1, 1, 1], [7]](x) = ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[7]](x) = (95*x^5-2*x^4-226*x^3+45*x^2+13*x-1)/(-1+x)/(1+x)/( -1+3*x)/(1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 1, 16, 146, 2351, 33651, 510366, 7620196, 114457501, 1716024101, 25744356716, 386144784246, 5792272824651, 86883581750551, 1303256263411066, 19548831217572296, 293232531788063801, 4398487658768093001, 65977316470494313416 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 x (11 x - 1) F[[3, 1, 1, 1, 1], [6, 1]](x) = ----------------------------------------- (-1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[6, 1]](x) = x^2*(11*x-1)/(-1+x)/(-1+3*x)/(1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 3, 74, 862, 13851, 202193, 3057844, 45734172, 686653781, 10296521983, 154464017214, 2316878612282, 34753585288111, 521301742424973, 7819536301726184, 117292993641739192, 1759395158874668841, 26390926111360567163, 395863898027656152754 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 2 2 x (25 x + 8 x - 1) F[[3, 1, 1, 1, 1], [5, 2]](x) = - ---------------------------------------- (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[5, 2]](x) = -2*x^2*(25*x^2+8*x-1)/(1+x)/(-1+3*x)/(1+5*x)/(-1 +15*x) The first 20 term , starting with k=1 are 0, 2, 8, 162, 2048, 32082, 472808, 7129362, 106739648, 1602054882, 24025891208, 360412634562, 5406067005248, 81091614101682, 1216371155781608, 18245582583555762, 273683662423518848, 4105255317714892482, 61578827858051904008, 923682427406553172962 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [5, 1, 1]](x) = 2 2 x (25 x - 6 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[5, 1, 1]](x) = -x^2*(25*x^2-6*x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+ 5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 7, 162, 2190, 34211, 506837, 7635652, 114373660, 1716423621, 25742300067, 386154890342, 5792221762730, 86883835465831, 1303254990051697, 19548837570020232, 293232499982777400, 4398487817665384841, 65977315675620433727, 989659743081883041322 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 2 x (20 x - 7 x + 1) F[[3, 1, 1, 1, 1], [4, 3]](x) = - ----------------------------------------- (-1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[4, 3]](x) = -x^2*(20*x^2-7*x+1)/(-1+x)/(-1+3*x)/(1+5*x)/(-1+ 15*x) The first 20 term , starting with k=1 are 0, 1, 7, 150, 2054, 31891, 473277, 7125560, 106754284, 1601968581, 24026283347, 360410555770, 5406077044914, 81091562840471, 1216371408899017, 18245581308402780, 273683668770585944, 4105255285893463561, 61578828016900768287, 923682426611534010590 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [4, 2, 1]](x) = 2 3 x (25 x + 5 x - 2) ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[4, 2, 1]](x) = x^2*(25*x^3+5*x-2)/(-1+x)/(1+x)/(-1+3*x)/(1+5 *x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 2, 21, 365, 5210, 79432, 1184911, 17806035, 266927220, 4004720462, 60066733001, 901021325305, 13515218110030, 202728780143892, 3040929158628291, 45613950093886175, 684209187826417640, 10263138135276941722, 153947070439664646781, 2309206064542158816645 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 2 x (-1 + 5 x) F[[3, 1, 1, 1, 1], [4, 1, 1, 1]](x) = ------------------------------- (-1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[4, 1, 1, 1]](x) = x^2*(-1+5*x)/(-1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 14, 203, 3020, 45221, 678074, 10170383, 152553560, 2288296841, 34324432934, 514866434963, 7722996347300, 115844944678061, 1737674168576594, 26065112523865943, 390976687843640240, 5864650317611556881, 87969754764044213054, 1319546321460275775323 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [3, 3, 1]](x) = 3 2 x (45 x - 62 x - 11) ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[3, 3, 1]](x) = x^3*(45*x^2-62*x-11)/(-1+x)/(1+x)/(-1+3*x)/(1 +5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 11, 205, 3126, 47450, 711361, 10679655, 160170476, 2402744500, 36040427511, 540610703105, 8109140865826, 121637216705550, 1824557747935661, 27368368780102555, 410525519039689176, 6157882849335050600, 92368242422618595811, 1385523637930188958005 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [3, 2, 2]](x) = 2 2 x (31 x - 2 x - 1) ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[3, 2, 2]](x) = x^2*(31*x^2-2*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+5 *x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 15, 210, 3182, 47411, 712285, 10677220, 160189212, 2402670501, 36040856555, 540608735030, 8109151237642, 121637166440791, 1824558004042425, 27368367513917640, 410525525413660472, 6157882817594334281, 92368242581709597895, 1385523637135896209050 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [3, 2, 1, 1]](x) = 2 3 2 x (50 x - 35 x + 12 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[3, 2, 1, 1]](x) = -x^2*(50*x^3-35*x^2+12*x+1)/(-1+x)/(1+x)/( -1+3*x)/(1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 1, 25, 336, 5330, 78751, 1188075, 17789486, 267007780, 4004311101, 60068760125, 901011130636, 13515268906230, 202728525631451, 3040930429596175, 45613943734263786, 684209219610180680, 10263137976315079801, 153947071234344816225, 2309206060568370548936 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [3, 1, 1, 1, 1]](x) = 3 2 x (130 x - 43 x - 12 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[3, 1, 1, 1, 1]](x) = -x*(130*x^3-43*x^2-12*x+1)/(-1+x)/(1+x) /(-1+3*x)/(1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 1, 16, 146, 2351, 33651, 510366, 7620196, 114457501, 1716024101, 25744356716, 386144784246, 5792272824651, 86883581750551, 1303256263411066, 19548831217572296, 293232531788063801, 4398487658768093001, 65977316470494313416, 989659739108675904346 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 3 x (25 x - 11) F[[3, 1, 1, 1, 1], [2, 2, 2, 1]](x) = ----------------------------------------- (-1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[2, 2, 2, 1]](x) = x^3*(25*x-11)/(-1+x)/(-1+3*x)/(1+5*x)/(-1+ 15*x) The first 20 term , starting with k=1 are 0, 0, 11, 129, 2158, 31370, 475881, 7112539, 106819388, 1601643060, 24027910951, 360402417749, 5406117735018, 81091359389950, 1216372426151621, 18245576222139759, 273683694201901048, 4105255158736888040, 61578828652683645891, 923682423432619622569 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [2, 2, 1, 1, 1]](x) = 3 4 x (5 x - 3) - ---------------------------------------- (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[2, 2, 1, 1, 1]](x) = -4*x^3*(5*x-3)/(1+x)/(-1+3*x)/(1+5*x)/( -1+15*x) The first 20 term , starting with k=1 are 0, 0, 12, 124, 2184, 31240, 476532, 7109284, 106835664, 1601561680, 24028317852, 360400383244, 5406127907544, 81091308527320, 1216372680464772, 18245574950574004, 273683700559729824, 4105255126947744160, 61578828811629365292, 923682422637891025564 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1]](x) = 3 x (45 x - 7) ----------------------------------------- (-1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1]](x) = x^3*(45*x-7)/(-1+x)/(-1+3*x)/(1+5*x) /(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 7, 53, 966, 13330, 204797, 3044823, 45799276, 686328260, 10298149587, 154455879193, 2316919302386, 34753381837590, 521302759677577, 7819531215463163, 117293019073054296, 1759395031718093320, 26390926747143444767, 395863894848741764733 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 F[[3, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 3 2 x (25 x - 6 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 15 x) and in Maple notation F[[3, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(25*x^2-6*x+1)/(-1+x)/(1+x)/ (-1+3*x)/(1+5*x)/(-1+15*x) The first 20 term , starting with k=1 are 0, 0, 1, 7, 162, 2190, 34211, 506837, 7635652, 114373660, 1716423621, 25742300067, 386154890342, 5792221762730, 86883835465831, 1303254990051697, 19548837570020232, 293232499982777400, 4398487817665384841, 65977315675620433727 ---------------------------------- Their sum is 4 3 2 19 x - 5 x - 38 x + 17 x - 1 --------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) and in Maple notation (19*x^4-5*x^3-38*x^2+17*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) The first 20 term , starting with k=1 are 1, 12, 159, 2342, 34989, 524432, 7865259, 117975242, 1769617689, 26544232532, 398163389559, 5972450548142, 89586757336389, 1343801357388632, 20157020352857859, 302355305268953042, 4535329578962551089, 68029943684223032732, 1020449155262699790159, 15306737328938559749942 Regarding Lambda=, [2, 2, 2, 1] 5 4 3 2 51 x + 2 x - 113 x + 36 x + 12 x - 1 F[[2, 2, 2, 1], [7]](x) = ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[7]](x) = (51*x^5+2*x^4-113*x^3+36*x^2+12*x-1)/(-1+x)/(1+x)/(-1+ 2*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 1, 11, 106, 1519, 20862, 293119, 4098382, 57396287, 803467918, 11248862527, 157482811278, 2204764379455, 30866681156494, 432133616675135, 6049870311236494, 84698185645615423, 1185774593884283790, 16600844334995073343 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [6, 1]](x) = 2 3 2 x (16 x - 21 x - 9 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[6, 1]](x) = -x^2*(16*x^3-21*x^2-9*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1 +4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 3, 52, 627, 9040, 125235, 1758016, 24592179, 344368384, 4820841267, 67493032960, 944897422131, 13228584030208, 185200095867699, 2592801664221184, 36299222010508083, 508189113300877312, 7114647565596046131, 99605066000807231488 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [5, 2]](x) = 2 3 2 x (14 x + 4 x - 6 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[5, 2]](x) = -x^2*(14*x^3+4*x^2-6*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+ 4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 6, 113, 1468, 21017, 292404, 4101049, 57385236, 803511353, 11248687252, 157483509305, 2204761581204, 30866692337209, 432133571927700, 6049870490177081, 84698184929754772, 1185774596747529785, 16600844323541696148, 232411820653278891577 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 x (-1 + 5 x) F[[2, 2, 2, 1], [5, 1, 1]](x) = - -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[5, 1, 1]](x) = -x^2*(-1+5*x)/(-1+2*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 7, 120, 1580, 22496, 313392, 4393600, 61485760, 860898816, 12052189952, 168732231680, 2362244951040, 33071454478336, 463000262029312, 6482004071055360, 90748055384145920, 1270472781820461056, 17786618919716585472, 249012664979111280640 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (9 x + 5 x - 1) F[[2, 2, 2, 1], [4, 3]](x) = ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[4, 3]](x) = x^2*(9*x^2+5*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+4*x)/(-1 +14*x) The first 20 term , starting with k=1 are 0, 1, 7, 112, 1479, 20988, 292551, 4100524, 57387463, 803502700, 11248722375, 157483369836, 2204762141127, 30866690101612, 432133580878279, 6049870454391148, 84698185072931271, 1185774596174889324, 16600844325832389063, 232411820644116382060 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (35 x + 6 x + 1) F[[2, 2, 2, 1], [4, 2, 1]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[4, 2, 1]](x) = x^2*(35*x^2+6*x+1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+14 *x) The first 20 term , starting with k=1 are 0, 1, 17, 270, 3710, 52366, 731682, 10249870, 143473970, 2008734606, 28121892722, 393708073870, 5511906748530, 77166719656846, 1080333974555762, 15124676046480270, 211745463040203890, 2964436489005491086, 41502110820307442802, 581029551587384157070 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 x (-1 + 4 x) F[[2, 2, 2, 1], [4, 1, 1, 1]](x) = - ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[4, 1, 1, 1]](x) = -x^2*(-1+4*x)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 153, 2135, 29881, 418311, 5856313, 81988295, 1147835961, 16069703111, 224975842873, 3149661798855, 44095265181241, 617333712531911, 8642671975435833, 120997407656079815, 1693963707185073721, 23715491900590944711, 332016886608273051193 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [3, 3, 1]](x) = 2 3 2 x (21 x - 8 x - x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[3, 3, 1]](x) = -x^2*(21*x^3-8*x^2-x+1)/(-1+x)/(1+x)/(-1+2*x)/(1 +4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 161, 2236, 31389, 439152, 6149389, 86086592, 1205232077, 16873170688, 236224704717, 3307144608768, 46300029557965, 648200393682944, 9074805592100045, 127047277967294464, 1778661892830645453, 24901266494475141120, 348617730943267949773 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [3, 2, 2]](x) = 2 3 2 x (7 x - 21 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[3, 2, 2]](x) = -x^2*(7*x^3-21*x^2+1)/(-1+x)/(1+x)/(-1+2*x)/(1+4 *x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 12, 160, 2247, 31360, 439299, 6148864, 86088819, 1205223424, 16873205811, 236224565248, 3307145168691, 46300027322368, 648200402633523, 9074805556314112, 127047278110470963, 1778661892258004992, 24901266496765834035, 348617730934105440256 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (7 x - 9 x - 1) F[[2, 2, 2, 1], [3, 2, 1, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[3, 2, 1, 1]](x) = -x^2*(7*x^2-9*x-1)/(1+x)/(-1+2*x)/(1+4*x)/(-1 +14*x) The first 20 term , starting with k=1 are 0, 1, 20, 261, 3755, 52201, 732375, 10247161, 143484935, 2008691001, 28122067655, 393707375161, 5511909545415, 77166708473401, 1080334019297735, 15124675867528761, 211745463756042695, 2964436486142201401, 41502110831760732615, 581029551541571259961 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 3 9 x F[[2, 2, 2, 1], [3, 1, 1, 1, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[3, 1, 1, 1, 1]](x) = 9*x^3/(-1+2*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 9, 108, 1620, 22320, 314064, 4390848, 61496640, 860855040, 12052364544, 168731532288, 2362247746560, 33071443292160, 463000306765824, 6482003892092928, 90748056099962880, 1270472778957127680, 17786618931169787904, 249012664933298208768 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [2, 2, 2, 1]](x) = 3 2 x (61 x - 38 x - 11 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[2, 2, 2, 1]](x) = -x*(61*x^3-38*x^2-11*x+1)/(-1+x)/(1+x)/(-1+2* x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 1, 11, 106, 1519, 20862, 293119, 4098382, 57396287, 803467918, 11248862527, 157482811278, 2204764379455, 30866681156494, 432133616675135, 6049870311236494, 84698185645615423, 1185774593884283790, 16600844334995073343, 232411820607466169230 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [2, 2, 1, 1, 1]](x) = 3 2 x (14 x + 8 x - 9) ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[2, 2, 1, 1, 1]](x) = x^3*(14*x^2+8*x-9)/(-1+x)/(1+x)/(-1+2*x)/( 1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 9, 100, 1519, 20812, 293223, 4097772, 57398343, 803458924, 11248896967, 157482670444, 2204764936647, 30866678915436, 432133625614791, 6049870275428716, 84698185788748231, 1185774593311555948, 16600844337285591495, 232411820598303310188 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [2, 1, 1, 1, 1, 1]](x) = 3 2 x (16 x - 7 x + 4) - ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[2, 1, 1, 1, 1, 1]](x) = -x^3*(16*x^2-7*x+4)/(-1+x)/(1+x)/(-1+2* x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 4, 41, 656, 8893, 125760, 1755789, 24600832, 344333261, 4820980736, 67492473037, 944899657728, 13228575079629, 185200131653632, 2592801521044685, 36299222583148544, 508189111010184397, 7114647574758555648, 99605065964156669133 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 2, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 3 2 x (9 x + 5 x - 1) ------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 1],[1, 1, 1, 1, 1, 1, 1]](x) = x^3*(9*x^2+5*x-1)/(-1+x)/(1+x)/(-1+2 *x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 1, 7, 112, 1479, 20988, 292551, 4100524, 57387463, 803502700, 11248722375, 157483369836, 2204762141127, 30866690101612, 432133580878279, 6049870454391148, 84698185072931271, 1185774596174889324, 16600844325832389063 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 Regarding Lambda=, [2, 2, 1, 1, 1] F[[2, 2, 1, 1, 1], [7]](x) = 6 5 4 3 2 160 x + 36 x - 350 x - 32 x + 88 x + 8 x - 1 ----------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[7]](x) = (160*x^6+36*x^5-350*x^4-32*x^3+88*x^2+8*x-1)/(-1+x) /(1+x)/(-1+2*x)/(1+2*x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 0, 15, 76, 1695, 19756, 299839, 4057452, 57643839, 801973612, 11257862463, 157428670828, 2205089777983, 30864726524268, 432145353402687, 6049799855063404, 84698608525764927, 1185772056030658924, 16600859564407253311 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 x (-1 + 10 x) F[[2, 2, 1, 1, 1], [6, 1]](x) = - -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[6, 1]](x) = -x^2*(-1+10*x)/(-1+2*x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 0, 68, 512, 9744, 120832, 1784896, 24428544, 345358592, 4814864384, 67529032704, 944680861696, 13229885624320, 185192277344256, 2592848611131392, 36298940185837568, 508190804821475328, 7114637414181634048, 99605126918455951360 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [5, 2]](x) = 2 3 2 x (20 x + 60 x + 13 x - 2) ----------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[5, 2]](x) = x^2*(20*x^3+60*x^2+13*x-2)/(-1+x)/(1+x)/(-1+2*x) /(1+2*x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 2, 3, 142, 1303, 22094, 285831, 4141454, 57139911, 804997006, 11239722439, 157537510286, 2204436742599, 30868644733838, 432121844150727, 6049940910564238, 84697762192781767, 1185777134028514190, 16600829096420209095, 232411912029752320910 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (20 x - 4 x + 1) F[[2, 2, 1, 1, 1], [5, 1, 1]](x) = ------------------------------------------ (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[5, 1, 1]](x) = x^2*(20*x^2-4*x+1)/(-1+2*x)/(1+2*x)/(1+6*x)/( -1+14*x) The first 20 term , starting with k=1 are 0, 1, 4, 140, 1440, 23376, 307904, 4427200, 61281280, 862136576, 12044719104, 168777231360, 2361974251520, 33073081470976, 462990488879104, 6482062754693120, 90747703103324160, 1270474896221208576, 17786606230448635904, 249012741126172180480 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (14 x - 5 x + 1) F[[2, 2, 1, 1, 1], [4, 3]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[4, 3]](x) = x^2*(14*x^2-5*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+14 *x) The first 20 term , starting with k=1 are 0, 1, 4, 128, 1364, 21692, 288148, 4127404, 57223828, 804492908, 11242745492, 157519369580, 2204545580692, 30867991695724, 432125762354836, 6049917401301356, 84697903248260756, 1185776287695487340, 16600834174417976980, 232411881561765101932 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [4, 2, 1]](x) = 2 4 3 2 x (28 x + 60 x + 3 x - 2 x + 2) - ----------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[4, 2, 1]](x) = -x^2*(28*x^4+60*x^3+3*x^2-2*x+2)/(-1+x)/(1+x) /(-1+2*x)/(1+2*x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 2, 14, 293, 3570, 53257, 726194, 10283513, 143269490, 2009972537, 28114421874, 393753074233, 5511636049010, 77168346652217, 1080324201405554, 15124734730128953, 211745110759382130, 2964438603406282297, 41502098131039493234, 581029627734445231673 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 x (-1 + 4 x) F[[2, 2, 1, 1, 1], [4, 1, 1, 1]](x) = - ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[4, 1, 1, 1]](x) = -x^2*(-1+4*x)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 153, 2135, 29881, 418311, 5856313, 81988295, 1147835961, 16069703111, 224975842873, 3149661798855, 44095265181241, 617333712531911, 8642671975435833, 120997407656079815, 1693963707185073721, 23715491900590944711, 332016886608273051193 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [3, 3, 1]](x) = 2 3 2 x (56 x + 8 x - 4 x - 1) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[3, 3, 1]](x) = -x^2*(56*x^3+8*x^2-4*x-1)/(1+x)/(-1+2*x)/(1+2 *x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 11, 165, 2211, 31565, 438067, 6156109, 86045747, 1205479629, 16871676723, 236233704653, 3307090469683, 46300354956493, 648198439056179, 9074817328827597, 127047207511143219, 1778662315710794957, 24901263956621603635, 348617746172680129741 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 2 2 x (28 x - 4 x - 1) F[[2, 2, 1, 1, 1], [3, 2, 2]](x) = - ------------------------------------------ (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[3, 2, 2]](x) = -x^2*(28*x^2-4*x-1)/(-1+2*x)/(1+2*x)/(1+6*x)/ (-1+14*x) The first 20 term , starting with k=1 are 0, 1, 12, 156, 2272, 31184, 440384, 6142144, 86129664, 1204975872, 16874699776, 236215565312, 3307199307776, 46299701923840, 648202357260288, 9074793819586560, 127047348566622208, 1778661469377855488, 24901269034619371520, 348617715704693260288 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [3, 2, 1, 1]](x) = 3 3 2 x (112 x + 56 x - 54 x - 23) ----------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[3, 2, 1, 1]](x) = x^3*(112*x^3+56*x^2-54*x-23)/(-1+x)/(1+x)/ (-1+2*x)/(1+2*x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 23, 238, 3895, 51310, 737863, 10213518, 143689415, 2007453070, 28129538503, 393662374798, 5512180244935, 77165081478030, 1080343792447943, 15124617183880078, 211745816036864455, 2964434371741410190, 41502123521028682183, 581029475394510185358 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [3, 1, 1, 1, 1]](x) = 3 4 x (2 x - 3) - ------------------------------------------ (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[3, 1, 1, 1, 1]](x) = -4*x^3*(2*x-3)/(-1+2*x)/(1+2*x)/(1+6*x) /(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 12, 88, 1760, 21440, 319552, 4357248, 61701120, 859617280, 12059835392, 168686532608, 2362518446080, 33069816299520, 463010079916032, 6481945208455168, 90748408380784640, 1270470664556380160, 17786631620437737472, 249012588786237308928 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [2, 2, 2, 1]](x) = 3 x (14 x - 11) - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[2, 2, 2, 1]](x) = -x^3*(14*x-11)/(1+x)/(-1+2*x)/(1+6*x)/(-1+ 14*x) The first 20 term , starting with k=1 are 0, 0, 11, 85, 1623, 20137, 297479, 4071417, 57559751, 802477369, 11254838727, 157446810169, 2204980937159, 30865379556921, 432141435187655, 6049823364304441, 84698467470242247, 1185772902363598393, 16600854486409310663, 232411759689817099833 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [2, 2, 1, 1, 1]](x) = 4 3 2 x (176 x - 4 x - 74 x - 8 x + 1) - ----------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[2, 2, 1, 1, 1]](x) = -x*(176*x^4-4*x^3-74*x^2-8*x+1)/(-1+x)/ (1+x)/(-1+2*x)/(1+2*x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 0, 15, 76, 1695, 19756, 299839, 4057452, 57643839, 801973612, 11257862463, 157428670828, 2205089777983, 30864726524268, 432145353402687, 6049799855063404, 84698608525764927, 1185772056030658924, 16600859564407253311, 232411729221830230380 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [2, 1, 1, 1, 1, 1]](x) = 3 x (38 x - 7) - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[2, 1, 1, 1, 1, 1]](x) = -x^3*(38*x-7)/(1+x)/(-1+2*x)/(1+6*x) /(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 7, 25, 771, 8189, 130163, 1728909, 24764467, 343343053, 4826957619, 67456473293, 945116218163, 13227273485517, 185207950177075, 2592754574134477, 36299504407819059, 508187419489586381, 7114657726172967731, 99605005046507949261 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 F[[2, 2, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (20 x + 60 x + 13 x - 2) ----------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1]](x) = x^3*(20*x^3+60*x^2+13*x-2)/(-1+x) /(1+x)/(-1+2*x)/(1+2*x)/(1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 2, 3, 142, 1303, 22094, 285831, 4141454, 57139911, 804997006, 11239722439, 157537510286, 2204436742599, 30868644733838, 432121844150727, 6049940910564238, 84697762192781767, 1185777134028514190, 16600829096420209095 ---------------------------------- Their sum is 3 2 4 x + 8 x - 14 x + 1 ------------------------------ (1 + x) (-1 + 2 x) (-1 + 14 x) and in Maple notation (4*x^3+8*x^2-14*x+1)/(1+x)/(-1+2*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 11, 129, 1775, 24769, 346623, 4852417, 67933247, 951064257, 13314897215, 186408556225, 2609719777599, 36536076867265, 511505076103487, 7161071065372353, 100254994915060031, 1403569928810534593, 19649979003346872639, 275099706046854993601, 3851395884655967463743 Regarding Lambda=, [2, 1, 1, 1, 1, 1] F[[2, 1, 1, 1, 1, 1], [7]](x) = 7 6 5 4 3 2 106 x - 29 x - 303 x + 75 x + 92 x - 22 x - 5 x + 1 --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[7]](x) = (106*x^7-29*x^6-303*x^5+75*x^4+92*x^3-22*x^2-5*x +1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 0, 4, 0, 41, 15, 714, 1170, 17251, 56925, 509444, 2333760, 16734381, 89015355, 578241694, 3288672270, 20442513431, 119775517305, 730058053464 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [6, 1]](x) = 2 4 3 2 x (2 x - 43 x + 12 x + 5 x - 1) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[6, 1]](x) = -x^2*(2*x^4-43*x^3+12*x^2+5*x-1)/(1+x)/(-1+x) /(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 0, 11, 1, 162, 146, 3397, 8427, 92048, 369292, 2893683, 14496053, 97965934, 542444838, 3431848769, 19869829279, 122066210220, 720895369184, 4371040179055 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [5, 2]](x) = 2 3 2 x (26 x - 2 x - 6 x + 1) - -------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[5, 2]](x) = -x^2*(26*x^3-2*x^2-6*x+1)/(1+x)/(-1+3*x)/(-1+ 2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 0, 13, 4, 255, 434, 6497, 22008, 195859, 907918, 6480981, 34646612, 224522663, 1279625802, 7944989065, 46592594416, 283839750267, 1685826303686, 10183580356349 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [5, 1, 1]](x) = 2 5 4 3 2 x (54 x - x - 42 x + 10 x + 5 x - 1) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[5, 1, 1]](x) = -x^2*(54*x^5-x^4-42*x^3+10*x^2+5*x-1)/(1+x )/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 1, 0, 13, 10, 256, 575, 6643, 25320, 204286, 999625, 6850273, 37538930, 239018716, 1377586275, 8487433903, 50024421340, 303709579546, 1807892426525, 10904475725533 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [4, 3]](x) = 4 3 2 x (32 x + 22 x - 25 x + 6) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[4, 3]](x) = x^4*(32*x^3+22*x^2-25*x+6)/(1+x)/(-1+x)/(1+2* x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 6, 5, 185, 490, 5656, 23415, 184575, 935660, 6318686, 35140105, 222085045, 1287978510, 7907398596, 46730390075, 283250923595, 1688068569040, 10174272389386 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [4, 2, 1]](x) = 4 3 2 x (74 x - 31 x - 30 x + 12) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[4, 2, 1]](x) = x^4*(74*x^3-31*x^2-30*x+12)/(1+x)/(-1+x)/( 1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 12, 30, 395, 1575, 12922, 64260, 440865, 2429625, 15458432, 89279190, 549720535, 3242605275, 19679917542, 117186541320, 706703735405, 4225922106525, 25412847166252 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [4, 1, 1, 1]](x) = 3 x (-1 + 4 x) ---------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[4, 1, 1, 1]](x) = x^3*(-1+4*x)/(1+x)/(-1+3*x)/(-1+2*x)/(-\ 1+6*x) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 200, 1161, 6826, 40495, 241500, 1444421, 8652446, 51871755, 311100400, 1866209281, 11196070866, 67172859815, 403026440900, 2418126447741, 14508662006086 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [3, 3, 1]](x) = 4 3 2 x (54 x - 11 x - 5 x - 3) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[3, 3, 1]](x) = -x^4*(54*x^3-11*x^2-5*x-3)/(1+x)/(-1+x)/(1 +2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 20, 180, 1015, 6993, 40110, 253830, 1486925, 9117603, 54074020, 327436200, 1954034355, 11770725333, 70450792250, 423436669290, 2537805197305, 15238429494183 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [3, 2, 2]](x) = 4 3 2 x (18 x + 25 x - 16 x - 2) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[3, 2, 2]](x) = -x^4*(18*x^3+25*x^2-16*x-2)/(1+x)/(-1+x)/( 1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 0, 2, 26, 151, 1141, 6468, 42252, 245177, 1521707, 8978134, 54632578, 325200603, 1962979473, 11734939400, 70593946904, 422864028829, 2540095802839, 15229266984666 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [3, 2, 1, 1]](x) = 3 4 3 2 x (34 x - 5 x - 9 x + 7 x - 2) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[3, 2, 1, 1]](x) = -x^3*(34*x^4-5*x^3-9*x^2+7*x-2)/(1+x)/( -1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 2, 3, 70, 230, 2247, 10213, 75140, 397260, 2604217, 14759723, 92074710, 538537090, 3287341787, 19500966033, 117902358280, 703840445720, 4237375308957, 25367034269143 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1]](x) = 3 4 3 2 x (18 x - 73 x + 19 x + 14 x - 3) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1]](x) = -x^3*(18*x^4-73*x^3+19*x^2+14*x-3)/( 1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 3, 1, 55, 80, 1268, 3891, 36285, 160510, 1174558, 6150881, 40335815, 227832540, 1422328248, 8308471471, 50740260145, 300846246170, 1819345716338, 10858662653661 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [2, 2, 2, 1]](x) = 3 4 3 2 x (32 x - 14 x + 11 x - 5 x + 1) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[2, 2, 2, 1]](x) = x^3*(32*x^4-14*x^3+11*x^2-5*x+1)/(1+x)/ (-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 1, 0, 34, 59, 1015, 3514, 32068, 149793, 1075129, 5760128, 37375702, 213139927, 1323764443, 7764243942, 47303030536, 280960318061, 1697231078557, 10137622176556 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1]](x) = 3 2 x (-1 + 5 x) (2 x + 6 x - 3) ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1]](x) = x^3*(-1+5*x)*(2*x^2+6*x-3)/(1+x)/(-1 +x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 3, 0, 55, 50, 1253, 3220, 35115, 143430, 1117633, 5642120, 38002055, 211100890, 1333312893, 7730240700, 47451587875, 280403776430, 1699570199033, 10128604774960 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1]](x) = 5 4 3 2 x (182 x - 43 x - 77 x + 19 x + 5 x - 1) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1]](x) = -x*(182*x^5-43*x^4-77*x^3+19*x^2+ 5*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 0, 4, 0, 41, 15, 714, 1170, 17251, 56925, 509444, 2333760, 16734381, 89015355, 578241694, 3288672270, 20442513431, 119775517305, 730058053464, 4334389616700 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 F[[2, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (2 x - 43 x + 12 x + 5 x - 1) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) and in Maple notation F[[2, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(2*x^4-43*x^3+12*x^2+5*x-\ 1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x) The first 20 term , starting with k=1 are 0, 0, 1, 0, 11, 1, 162, 146, 3397, 8427, 92048, 369292, 2893683, 14496053, 97965934, 542444838, 3431848769, 19869829279, 122066210220, 720895369184 ---------------------------------- Their sum is 2 3 2 (x + x - 1) (8 x - 20 x + 9 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) and in Maple notation (x^2+x-1)*(8*x^3-20*x^2+9*x-1)/(1+x)/(-1+x)/(-1+3*x)/(-1+2*x)/(-1+6*x) The first 20 term , starting with k=1 are 1, 4, 15, 74, 397, 2260, 13211, 78270, 466713, 2791736, 16725127, 100275586, 601429349, 3607906332, 21645434163, 129866604422, 779181646705, 4675035984448, 28050054306719, 168299841215178 Regarding Lambda=, [1, 1, 1, 1, 1, 1, 1] 1 F[[1, 1, 1, 1, 1, 1, 1], [7]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[7]](x) = -1/(1+x)/(-1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [6, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[6, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [5, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[5, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [5, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[5, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [4, 3]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[4, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [4, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[4, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [4, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[4, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [3, 3, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[3, 3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [3, 2, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[3, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [3, 2, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[3, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[2, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 x F[[1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1]](x) = -x/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 8, elements There are, 22, partitions of, 8, here there are in the usual order Regarding Lambda=, [8] 1 F[[8], [8]](x) = - ------ -1 + x and in Maple notation F[[8],[8]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [7, 1]](x) = 0 and in Maple notation F[[8],[7, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [6, 2]](x) = 0 and in Maple notation F[[8],[6, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [6, 1, 1]](x) = 0 and in Maple notation F[[8],[6, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [5, 3]](x) = 0 and in Maple notation F[[8],[5, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [5, 2, 1]](x) = 0 and in Maple notation F[[8],[5, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [5, 1, 1, 1]](x) = 0 and in Maple notation F[[8],[5, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [4, 4]](x) = 0 and in Maple notation F[[8],[4, 4]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [4, 3, 1]](x) = 0 and in Maple notation F[[8],[4, 3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [4, 2, 2]](x) = 0 and in Maple notation F[[8],[4, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [4, 2, 1, 1]](x) = 0 and in Maple notation F[[8],[4, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [4, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[8],[4, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [3, 3, 2]](x) = 0 and in Maple notation F[[8],[3, 3, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [3, 3, 1, 1]](x) = 0 and in Maple notation F[[8],[3, 3, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [3, 2, 2, 1]](x) = 0 and in Maple notation F[[8],[3, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [3, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[8],[3, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [3, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[8],[3, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [2, 2, 2, 2]](x) = 0 and in Maple notation F[[8],[2, 2, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [2, 2, 2, 1, 1]](x) = 0 and in Maple notation F[[8],[2, 2, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [2, 2, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[8],[2, 2, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [2, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[8],[2, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[8], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[8],[1, 1, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [7, 1] F[[7, 1], [8]](x) = 7 6 5 4 3 2 309 x - 317 x - 724 x + 1180 x - 650 x + 169 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[8]](x) = (309*x^7-317*x^6-724*x^5+1180*x^4-650*x^3+169*x^2-21*x+1)/(1 +x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 1, 1, 4, 11, 41, 162, 715, 3424, 17686, 97493, 567986, 3462537, 21880951, 142148644, 942800317, 6349172750, 43233294236, 296737912815, 2048310985708 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [7, 1]](x) = 5 4 3 2 x (531 x - 881 x + 535 x - 151 x + 20 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[7, 1]](x) = -x*(531*x^5-881*x^4+535*x^3-151*x^2+20*x-1)/(1+x)/(-1+x)/ (-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 1, 4, 11, 41, 162, 715, 3424, 17686, 97493, 567986, 3462537, 21880951, 142148644, 942800317, 6349172750, 43233294236, 296737912815, 2048310985708, 14196341292463 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [6, 2]](x) = 2 5 4 3 2 x (155 x - 424 x + 344 x - 118 x + 18 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[6, 2]](x) = -x^2*(155*x^5-424*x^4+344*x^3-118*x^2+18*x-1)/(1+x)/(-1+x )/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 256, 1273, 6763, 38055, 224986, 1385593, 8816413, 57535855, 382733716, 2582380713, 17605674463, 120934304455, 835206380446, 5790534838633, 40257187308913 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [6, 1, 1]](x) = 2 5 4 3 2 x (154 x - 424 x + 344 x - 118 x + 18 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[6, 1, 1]](x) = -x^2*(154*x^5-424*x^4+344*x^3-118*x^2+18*x-1)/(1+x)/(-\ 1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 256, 1274, 6784, 38328, 227821, 1411465, 9034015, 59269301, 396037006, 2681843076, 18335646706, 126221141374, 873133398211, 6060757555307, 42172970828857 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [5, 3]](x) = 3 3 2 x (35 x - 41 x + 12 x - 1) ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[5, 3]](x) = x^3*(35*x^3-41*x^2+12*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x) /(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 199, 1155, 6888, 42295, 266853, 1723799, 11353870, 75947235, 514118787, 3511826683, 24149990052, 166894875455, 1157524606801, 8049117729807, 56076624029034 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [5, 2, 1]](x) = 3 3 2 2 x (34 x - 41 x + 12 x - 1) ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[5, 2, 1]](x) = 2*x^3*(34*x^3-41*x^2+12*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1 +4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 2, 12, 70, 400, 2346, 14196, 88622, 568440, 3727570, 24869020, 168100374, 1147342560, 7886823674, 54493820484, 377922764926, 2627932540360, 18308466199458, 127726245780588 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [5, 1, 1, 1]](x) = 3 4 3 2 x (105 x - 157 x + 77 x - 15 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[5, 1, 1, 1]](x) = x^3*(105*x^4-157*x^3+77*x^2-15*x+1)/(1+x)/(-1+x)/(-\ 1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 200, 1176, 7161, 45130, 292725, 1941401, 13087316, 89250525, 613581150, 4241798926, 29436826971, 204821893220, 1427747323475, 9964901249751, 69610473450126 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [4, 4]](x) = 4 3 2 x (49 x - 44 x + 12 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[4, 4]](x) = -x^4*(49*x^3-44*x^2+12*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x )/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 1, 9, 65, 434, 2835, 18501, 121675, 808698, 5432009, 36829793, 251652765, 1730097642, 11950688023, 82844692485, 575822844935, 4010155964066, 27967467026877 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [4, 3, 1]](x) = 4 3 2 x (105 x - 109 x + 33 x - 3) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[4, 3, 1]](x) = -x^4*(105*x^3-109*x^2+33*x-3)/(1+x)/(-1+x)/(-1+2*x)/(-\ 1+3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 30, 235, 1680, 11613, 79380, 541845, 3708210, 25475923, 175705530, 1215978855, 8438861340, 58694702433, 408927969480, 2852631357265, 19918349297070, 139175134003143 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [4, 2, 2]](x) = 4 2 2 x (14 x - 8 x + 1) - ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[4, 2, 2]](x) = -2*x^4*(14*x^2-8*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x)/( -1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 2, 20, 160, 1176, 8358, 58548, 407780, 2835272, 19715674, 137213076, 955947720, 6666675288, 46532979950, 325026322004, 2271518378380, 15881726564424, 111075284793186 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 4 3 x F[[7, 1], [4, 2, 1, 1]](x) = - ------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (7 x - 1) and in Maple notation F[[7, 1],[4, 2, 1, 1]](x) = -3*x^4/(1+x)/(-1+x)/(-1+3*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 30, 240, 1770, 12663, 89460, 628680, 4408140, 30879123, 216220290, 1513741320, 10596787110, 74179303383, 519260504520, 3634839674160, 25443926146680, 178107628309443 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [4, 1, 1, 1, 1]](x) = 4 3 2 x (35 x - 38 x + 11 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[4, 1, 1, 1, 1]](x) = -x^4*(35*x^3-38*x^2+11*x-1)/(1+x)/(-1+x)/(-1+2*x )/(-1+3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 1, 10, 80, 595, 4326, 31185, 223735, 1598740, 11381051, 80740660, 571096890, 4029574185, 28376295676, 199521019435, 1401248313545, 9832422298930, 68948127036201 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [3, 3, 2]](x) = 5 x (14 x - 5) ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation F[[7, 1],[3, 3, 2]](x) = x^5*(14*x-5)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/( 7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 66, 616, 4998, 37905, 277452, 1991792, 14144196, 99824725, 702059358, 4927632528, 34546505994, 242038474865, 1695122892984, 11869279994824, 83098655256792 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [3, 3, 1, 1]](x) = 5 2 x (14 x - 3) ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[3, 3, 1, 1]](x) = 2*x^5*(14*x-3)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x)/(-1+5 *x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 6, 80, 756, 6216, 47754, 353640, 2564232, 18359792, 130432302, 922041120, 6497136828, 45685276728, 320787784050, 2250325644920, 15775762809744, 110545465845024 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [3, 2, 2, 1]](x) = 5 2 5 x (7 x - 6 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[3, 2, 2, 1]](x) = 5*x^5*(7*x^2-6*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/ (-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 75, 770, 6720, 53865, 411075, 3044690, 22129140, 158883725, 1131605475, 8016200010, 56579009760, 398342342585, 2799681721275, 19653536590730, 137850876848580 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [3, 2, 1, 1, 1]](x) = 5 4 x (-1 + 3 x) ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[3, 2, 1, 1, 1]](x) = 4*x^5*(-1+3*x)/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x)/(-\ 1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 4, 60, 624, 5544, 45276, 351540, 2642728, 19444128, 140974548, 1011710700, 7208658912, 51102985752, 360968569420, 2543161519140, 17884611005976, 125606969638416 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [3, 1, 1, 1, 1, 1]](x) = 5 2 x (14 x - 6 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[3, 1, 1, 1, 1, 1]](x) = x^5*(14*x^2-6*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 3*x)/(-1+4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 1, 15, 161, 1491, 12684, 102060, 790042, 5949042, 43910867, 319444125, 2299476543, 16425607653, 116676326950, 825425468610, 5822266334864, 40980660009324 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [2, 2, 2, 2]](x) = 6 x (21 x - 5) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[2, 2, 2, 2]](x) = -x^6*(21*x-5)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4* x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 5, 84, 924, 8442, 69825, 544698, 4097588, 30114084, 217942725, 1561151592, 11104767492, 78611513526, 554646210305, 3904240549566, 27437793186636 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [2, 2, 2, 1, 1]](x) = 6 x (35 x - 9) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[2, 2, 2, 1, 1]](x) = -x^6*(35*x-9)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 9, 154, 1722, 15960, 133623, 1052898, 7984944, 59059000, 429546117, 3088567482, 22032503766, 156303867720, 1104558828291, 7784256595906, 54752221591788 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [2, 2, 1, 1, 1, 1]](x) = 6 5 x - ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[2, 2, 1, 1, 1, 1]](x) = -5*x^6/(1+x)/(-1+x)/(-1+2*x)/(-1+4*x)/(-1+5*x )/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 5, 90, 1050, 10080, 86835, 699930, 5403200, 40514760, 297762465, 2157925770, 15484600950, 110332535040, 782208316895, 5525576849610, 38932494306300 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 6 x ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[2, 1, 1, 1, 1, 1, 1]](x) = x^6/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x )/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 1, 21, 273, 2835, 25872, 217602, 1733446, 13303290, 99462363, 729972243, 5286836919, 37927017765, 270222716674, 1915783519944, 13533849421092 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[7, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 7 x ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (7 x - 1) and in Maple notation F[[7, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^7/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+ 4*x)/(-1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 1, 21, 273, 2835, 25872, 217602, 1733446, 13303290, 99462363, 729972243, 5286836919, 37927017765, 270222716674, 1915783519944 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 Regarding Lambda=, [6, 2] F[[6, 2], [8]](x) = 7 6 5 4 3 2 3350 x - 2660 x - 7314 x + 8733 x - 3378 x + 568 x - 41 x + 1 ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[8]](x) = (3350*x^7-2660*x^6-7314*x^5+8733*x^4-3378*x^3+568*x^2-41*x+1 )/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 2, 16, 167, 2361, 39042, 705976, 13400357, 260948071, 5148978032, 102282555486, 2038694391747, 40704384131581, 813392950304222, 16260913155825796, 325148811777310337, 6502281757165011891, 130038690531319893612, 2600704365352864600906 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [7, 1]](x) = 2 5 4 3 2 x (750 x - 2070 x + 1412 x - 353 x + 34 x - 1) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[7, 1]](x) = -x^2*(750*x^5-2070*x^4+1412*x^3-353*x^2+34*x-1)/(1+x)/(-1 +x)/(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 7, 73, 951, 14811, 258027, 4796863, 92386231, 1812622171, 35903374947, 714586264503, 14256958881711, 284791737978931, 5692361464176667, 113812501751345743, 2275902786478580391, 45514583376613646091, 910256944659951215187, 18204791667740621364583 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [6, 2]](x) = 5 4 3 2 x (4650 x - 6140 x + 2694 x - 501 x + 39 x - 1) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[6, 2]](x) = -x*(4650*x^5-6140*x^4+2694*x^3-501*x^2+39*x-1)/(1+x)/(-1+ x)/(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 1, 2, 16, 167, 2361, 39042, 705976, 13400357, 260948071, 5148978032, 102282555486, 2038694391747, 40704384131581, 813392950304222, 16260913155825796, 325148811777310337, 6502281757165011891, 130038690531319893612, 2600704365352864600906, 52013392858501764652127 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [6, 1, 1]](x) = 2 4 3 2 x (750 x - 810 x + 260 x - 30 x + 1) ----------------------------------------------------------------- (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[6, 1, 1]](x) = x^2*(750*x^4-810*x^3+260*x^2-30*x+1)/(-1+x)/(-1+2*x)/( -1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 12, 155, 2340, 39831, 730422, 13964665, 272948520, 5395993331, 107292469482, 2139587358525, 42729186836700, 853958434359031, 17072917172495742, 341395835865277985, 6827291679337580880, 136539583396732655931, 2730729166983842261202, 54613958334919927183045 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [5, 3]](x) = 2 4 3 2 x (330 x - 578 x + 218 x - 28 x + 1) ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[5, 3]](x) = x^2*(330*x^4-578*x^3+218*x^2-28*x+1)/(1+x)/(-1+x)/(-1+2*x )/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 13, 184, 2934, 51471, 958713, 18473974, 362508124, 7180593541, 142916845863, 2851389741564, 56958337422714, 1138472241971611, 22762500095953813, 455180556024145954, 9102916668964891704, 182051388900201081681, 3640958333389178518563, 72818472222498584935144 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 2 2 x (195 x - 45 x + 2) F[[6, 2], [5, 2, 1]](x) = --------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[5, 2, 1]](x) = x^2*(195*x^2-45*x+2)/(1+x)/(-1+x)/(-1+5*x)/(-1+10*x)/( -1+20*x) The first 20 term , starting with k=1 are 0, 2, 25, 372, 6225, 112872, 2143725, 41750372, 823831225, 16365187872, 326191018725, 6512701125372, 130142870706225, 2601746099562872, 52023809862893725, 1040365081060500372, 20806190484667581225, 416112698455083937872, 8322142857354784768725, 166441746032805669875372 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [5, 1, 1, 1]](x) = 2 5 4 3 2 x (1750 x - 2450 x + 1250 x - 294 x + 30 x - 1) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[5, 1, 1, 1]](x) = -x^2*(1750*x^5-2450*x^4+1250*x^3-294*x^2+30*x-1)/(1 +x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 11, 178, 3150, 59171, 1146621, 22573678, 447939020, 8923727641, 178125601731, 3559030862828, 71145849064290, 1422569524322311, 28447917070948841, 568923613152094378, 11378125010283207960, 227559027829506988181, 4551145833593232075951, 91022569445748948936328 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [4, 4]](x) = 2 5 4 3 2 x (1100 x - 2660 x + 1611 x - 369 x + 34 x - 1) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[4, 4]](x) = -x^2*(1100*x^5-2660*x^4+1611*x^3-369*x^2+34*x-1)/(1+x)/(-\ 1+x)/(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 7, 89, 1408, 25066, 472437, 9167299, 180556978, 3583333556, 71388862567, 1424999758909, 28472220579348, 569166656703646, 11380555498747897, 227583333021328919, 4551388887217010518, 91024999991193199336, 1820472222176398616427, 36409166666430390293329 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [4, 3, 1]](x) = 2 5 4 3 2 x (500 x - 250 x + 255 x - 127 x + 22 x - 1) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[4, 3, 1]](x) = -x^2*(500*x^5-250*x^4+255*x^3-127*x^2+22*x-1)/(1+x)/(-\ 1+x)/(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 19, 339, 6210, 117856, 2290659, 45133789, 895807540, 17847091986, 356249343549, 7118052251839, 142291650061470, 2845138805514916, 56895832915068439, 1137847220125311489, 22756249989493089000, 455118055502931546646, 9102291666403188757329, 182045138887570067784739 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [4, 2, 2]](x) = 2 4 3 2 x (1140 x - 900 x + 242 x - 27 x + 1) ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[4, 2, 2]](x) = x^2*(1140*x^4-900*x^3+242*x^2-27*x+1)/(1+x)/(-1+x)/(-1 +2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 14, 249, 4710, 91581, 1805034, 35830369, 713872670, 14249913381, 284721767154, 5691664303689, 113805543390630, 2275833271109581, 45513888572175674, 910249998394060209, 18204722214103036590, 364091666625712808181, 7281805555349354574594, 145635833332296601739929 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 3 15 x F[[6, 2], [4, 2, 1, 1]](x) = ------------------------------ (1 + x) (-1 + 4 x) (-1 + 20 x) and in Maple notation F[[6, 2],[4, 2, 1, 1]](x) = 15*x^3/(1+x)/(-1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 15, 345, 7095, 142665, 2856375, 57139785, 1142844855, 22857093705, 457142660535, 9142856356425, 182857139711415, 3657142844559945, 73142857092525495, 1462857142655816265, 29257142856337550775, 585142857139635917385, 11702857142844257955255, 234057142857091317535305 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [4, 1, 1, 1, 1]](x) = 3 3 2 x (5 x - 2) (50 x - 90 x + 23 x - 2) ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[4, 1, 1, 1, 1]](x) = x^3*(5*x-2)*(50*x^3-90*x^2+23*x-2)/(1+x)/(-1+x)/ (-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 4, 108, 2455, 52225, 1077174, 21879228, 440994565, 8854283175, 177431157244, 3552086418298, 71076404619675, 1421875079877525, 28440972626503714, 568854168707648568, 11377430565838760785, 227552083385062538275, 4551076389148787620584, 91021875001304504470038 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 3 x (25 x - 6) F[[6, 2], [3, 3, 2]](x) = - ------------------------------------------ (-1 + x) (-1 + 5 x) (-1 + 4 x) (-1 + 20 x) and in Maple notation F[[6, 2],[3, 3, 2]](x) = -x^3*(25*x-6)/(-1+x)/(-1+5*x)/(-1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 6, 155, 3276, 66385, 1331946, 26659815, 533299416, 10666498445, 213332497686, 4266662510275, 85333312638756, 1706666563543305, 34133332819114626, 682666664101165535, 13653333320528197296, 273066666602730464965, 5461333333014010238766, 109226666665071482849595 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [3, 3, 1, 1]](x) = 3 4 3 2 4 x (400 x - 515 x + 204 x - 34 x + 2) ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[3, 3, 1, 1]](x) = 4*x^3*(400*x^4-515*x^3+204*x^2-34*x+2)/(1+x)/(-1+x) /(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 8, 192, 4152, 86020, 1749468, 35274792, 708317072, 14194357740, 284166211428, 5686108747792, 113749987834392, 2275277715552660, 45508333016617388, 910194442838499192, 18204166658547470112, 364086111070157230780, 7281749999793798975348, 145635277776741046096992 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [3, 2, 2, 1]](x) = 3 4 3 2 x (500 x - 650 x + 55 x + 44 x - 6) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[3, 2, 2, 1]](x) = -x^3*(500*x^4-650*x^3+55*x^2+44*x-6)/(1+x)/(-1+x)/( -1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 6, 202, 4825, 103975, 2151786, 43744932, 881918715, 17708203225, 354860454916, 7104163363462, 142152761173605, 2843749916628075, 56881944026183646, 1137708331236430792, 22754861100604216495, 455104166614042690525, 9102152777514299933976, 182043749998681179026922 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [3, 2, 1, 1, 1]](x) = 3 2 x (200 x - 45 x - 3) - --------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[3, 2, 1, 1, 1]](x) = -x^3*(200*x^2-45*x-3)/(1+x)/(-1+x)/(-1+5*x)/(-1+ 10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 3, 150, 4003, 90650, 1921503, 39528150, 801609003, 16142965650, 323968796503, 6490478903150, 129920648484003, 2599523877340650, 52001587640671503, 1040142858838278150, 20803968262445359003, 416090476232861715650, 8321920635132562546503, 166439523810583447653150 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [3, 1, 1, 1, 1, 1]](x) = 4 2 x (350 x - 210 x + 31) ----------------------------------------------------------------- (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[3, 1, 1, 1, 1, 1]](x) = x^4*(350*x^2-210*x+31)/(-1+x)/(-1+2*x)/(-1+5* x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 0, 31, 1092, 27335, 605430, 12714681, 260448552, 5270993395, 106042469610, 2127087358781, 42604186837212, 852708434360055, 17060417172497790, 341270835865282081, 6826041679337589072, 136527083396732672315, 2730604166983842293970, 54612708334919927248581 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [2, 2, 2, 2]](x) = 3 4 3 2 x (500 x - 540 x + 105 x - 9 x + 1) ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[2, 2, 2, 2]](x) = x^3*(500*x^4-540*x^3+105*x^2-9*x+1)/(1+x)/(-1+x)/(-\ 1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 1, 32, 850, 19505, 416871, 8611722, 175001380, 3527777915, 70833306841, 1419444203012, 28416665023110, 568611101146725, 11374999943189611, 227527777465767902, 4550833331661444040, 91019444435637621935, 1820416666620843017181, 36408611110874834650392 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [2, 2, 2, 1, 1]](x) = 4 2 x (730 x - 378 x + 47) ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[2, 2, 2, 1, 1]](x) = x^4*(730*x^2-378*x+47)/(1+x)/(-1+x)/(-1+2*x)/(-1 +5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 0, 47, 1549, 37590, 819840, 17085117, 348619299, 7041704780, 141527957230, 2837500853187, 56819448534849, 1137083353084770, 22748611207069020, 455041667135265257, 9101527780076019199, 182037500011312225560, 3640819444500289695210, 72817083333609696177327 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [2, 2, 1, 1, 1, 1]](x) = 4 2 5 x (50 x + 12 x - 5) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[2, 2, 1, 1, 1, 1]](x) = -5*x^4*(50*x^2+12*x-5)/(1+x)/(-1+x)/(-1+2*x)/ (-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 0, 25, 965, 25140, 567060, 12011415, 247059075, 5010088930, 100893666170, 2024805502005, 40565495240985, 812004061411920, 16247024266930080, 325009922888407795, 6500892868276095695, 130024801642430950110, 2600565476463975602790, 52012003969612875544785 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [2, 1, 1, 1, 1, 1, 1]](x) = 4 3 2 x (1250 x - 930 x + 133 x + 3) ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[2, 1, 1, 1, 1, 1, 1]](x) = x^4*(1250*x^3-930*x^2+133*x+3)/(1+x)/(-1+x )/(-1+2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 256, 7865, 188580, 4102413, 85441776, 1743177705, 35208930460, 707641819973, 14187514437096, 284097293534145, 5685417019731540, 113743057306899933, 2275208342034133216, 45507638932169196185, 910187500215506759820, 18204097223296176898293 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[6, 2], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 5 2 5 x (50 x + 12 x - 5) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 4 x) (-1 + 10 x) (-1 + 20 x) and in Maple notation F[[6, 2],[1, 1, 1, 1, 1, 1, 1, 1]](x) = -5*x^5*(50*x^2+12*x-5)/(1+x)/(-1+x)/(-1 +2*x)/(-1+5*x)/(-1+4*x)/(-1+10*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 25, 965, 25140, 567060, 12011415, 247059075, 5010088930, 100893666170, 2024805502005, 40565495240985, 812004061411920, 16247024266930080, 325009922888407795, 6500892868276095695, 130024801642430950110, 2600565476463975602790 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (1 + x) (-1 + x) (-1 + 5 x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(1+x)/(-1+x)/(-1+5*x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 Regarding Lambda=, [6, 1, 1] F[[6, 1, 1], [8]](x) = ( 8 7 6 5 4 3 2 9720 x + 747 x - 21798 x - 3503 x + 4960 x + 89 x - 288 x + 34 x - 1 )/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[8]](x) = (9720*x^8+747*x^7-21798*x^6-3503*x^5+4960*x^4+89*x^3-288* x^2+34*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 1, 16, 162, 2646, 48758, 972791, 19996369, 416278291, 8710237635, 182638389066, 3832961621996, 80470467858236, 1689685917505132, 35481668918957641, 745099487895371043, 15646949564027419281, 328584685824089507249, 6900267119818388995316 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [7, 1]](x) = 2 5 4 3 2 x (567 x + 1467 x - 81 x - 181 x + 29 x - 1) ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[7, 1]](x) = x^2*(567*x^5+1467*x^4-81*x^3-181*x^2+29*x-1)/(-1+x)/(1 +x)/(-1+3*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 7, 74, 1006, 17282, 332458, 6734855, 139353487, 2908593713, 60925063309, 1278058252736, 26827092214468, 563260848047444, 11827511545029460, 248369085278903117, 5215673111465160349, 109528437637039212875, 2300090919507736085311, 48301852922613828900398 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 F[[6, 1, 1], [6, 2]](x) = - x 6 5 4 3 2 (4050 x - 576 x - 3213 x + 262 x + 331 x - 56 x + 2)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[6, 2]](x) = -x^2*(4050*x^6-576*x^5-3213*x^4+262*x^3+331*x^2-56*x+2 )/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 12, 165, 2540, 46922, 930472, 19083955, 396818280, 8298807192, 173971721132, 3650715541145, 76641036471220, 1609247223371062, 33792268960156992, 709620392619162735, 14901873238873625360, 312937944724865662532, 6571684310177143758052, 138005257815794036684725 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [6, 1, 1]](x) = - x 6 5 4 3 2 (10692 x + 2817 x - 3555 x - 151 x + 269 x - 33 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[6, 1, 1]](x) = -x*(10692*x^6+2817*x^5-3555*x^4-151*x^3+269*x^2-33* x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 1, 1, 16, 162, 2646, 48758, 972791, 19996369, 416278291, 8710237635, 182638389066, 3832961621996, 80470467858236, 1689685917505132, 35481668918957641, 745099487895371043, 15646949564027419281, 328584685824089507249, 6900267119818388995316, 144905508049965664027510 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 5 4 3 2 F[[6, 1, 1], [5, 3]](x) = - x (1089 x - 468 x + 104 x + 97 x - 23 x + 1)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[5, 3]](x) = -x^2*(1089*x^5-468*x^4+104*x^3+97*x^2-23*x+1)/(-1+x)/( 1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 11, 184, 3259, 63375, 1284675, 26569418, 554301143, 11607631849, 243467190439, 5110180907052, 107290172616627, 2252881259605523, 47308596787753403, 993463355366438686, 20862575926805213311, 438112703992571201997, 9200354271726103807167, 193207327109882444949920 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [5, 2, 1]](x) = 2 3 2 x (27 x - 232 x + 47 x - 2) - -------------------------------------------------- (-1 + x) (1 + x) (-1 + 6 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[5, 2, 1]](x) = -x^2*(27*x^3-232*x^2+47*x-2)/(-1+x)/(1+x)/(-1+6*x)/ (-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 25, 396, 7225, 142560, 2915125, 60533244, 1265178625, 26515415448, 556348742125, 11679078977172, 245222638452025, 5149334315266416, 108132957320906725, 2270764573223931180, 47685808498159056625, 1001399752016003659464, 21029374762791377804125, 441616689803496620819268 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 F[[6, 1, 1], [5, 1, 1, 1]](x) = x 6 5 4 3 2 (3402 x + 3816 x + 504 x - 721 x - 20 x + 20 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[5, 1, 1, 1]](x) = x^2*(3402*x^6+3816*x^5+504*x^4-721*x^3-20*x^2+20 *x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 14, 209, 3865, 76936, 1584009, 33005064, 690965030, 14492009321, 304174337104, 6386277821419, 134099613051495, 2815983240839406, 59134678533852299, 1241819572447792274, 26078133224513951260, 547640099305738043191, 11500435810319182257594, 241509095604263774005629 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 3 5 4 3 2 F[[6, 1, 1], [4, 4]](x) = x (2268 x + 9 x - 1341 x + 53 x + 55 x - 4)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[4, 4]](x) = x^3*(2268*x^5+9*x^4-1341*x^3+53*x^2+55*x-4)/(-1+x)/(1+ x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 4, 81, 1553, 31120, 637812, 13247801, 276839161, 5801142540, 121710287600, 2554885251721, 53643266671449, 1126424416622960, 23654153454522868, 496730379109015041, 10431276311495996417, 219056247340733475580, 4600176195232558199616, 96603655096883993452961 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [4, 3, 1]](x) = 3 4 3 2 x (3402 x - 2034 x - 621 x + 271 x - 18) ------------------------------------------------------------------------- (-1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[4, 3, 1]](x) = x^3*(3402*x^4-2034*x^3-621*x^2+271*x-18)/(-1+x)/(-1 +3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 18, 359, 7390, 151890, 3156188, 65939839, 1381508580, 28981496630, 608333543158, 12772464920919, 268198681775570, 5631963216303970, 118269337475471928, 2483639027347536599, 52156265743739680360, 1095280194379795049910, 23000871595248278558498, 483018191056187467594879 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [4, 2, 2]](x) = 2 4 3 2 x (189 x + 606 x - 176 x + 22 x - 1) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[4, 2, 2]](x) = -x^2*(189*x^4+606*x^3-176*x^2+22*x-1)/(-1+x)/(1+x)/ (-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 11, 279, 5735, 120123, 2511551, 52633071, 1104129695, 23175518715, 486579655991, 10217187472383, 214551884602055, 4505507026956027, 94614898059847631, 1986906074604751215, 41724966269480561615, 876223738574338756059, 18400693523832682291271, 386414519073657439077567 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 2 x (9 x - 3 x - 1) F[[6, 1, 1], [4, 2, 1, 1]](x) = - -------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[4, 2, 1, 1]](x) = -x^2*(9*x^2-3*x-1)/(-1+x)/(1+x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 0, 1, 21, 433, 9120, 191431, 4020321, 84425923, 1772946840, 37231876261, 781869423621, 16419257829613, 344804414621160, 7240892706446491, 152058746837169921, 3193233683575187503, 67057907355095080080, 1408216054456948254121, 29572537143596058619221, 621023280015516795155593 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 F[[6, 1, 1], [4, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (1134 x + 2772 x - 1206 x - 290 x + 216 x - 27 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[4, 1, 1, 1, 1]](x) = -x^2*(1134*x^6+2772*x^5-1206*x^4-290*x^3+216* x^2-27*x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 7, 167, 3435, 73276, 1550612, 32706402, 688272970, 14467798001, 303956398317, 6384316527337, 134081961072605, 2815824374424426, 59133248733128122, 1241806704253829972, 26078017410741386340, 547639056981897956551, 11500426429404379340027, 241509011176031564726307 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [3, 3, 2]](x) = 3 2 x (120 x + 49 x + 6) - ----------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[3, 3, 2]](x) = -x^3*(120*x^2+49*x+6)/(-1+x)/(1+x)/(1+2*x)/(-1+6*x) /(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 6, 181, 4126, 88556, 1871492, 39370737, 827207352, 17373867412, 364866352378, 7662284036693, 160908509157578, 3379081956900468, 70960740687713064, 1490175671982902449, 31293689816934445204, 657167490387240034124, 13800517323522174663950, 289810863946305168102405 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [3, 3, 1, 1]](x) = 3 3 2 x (567 x + 21 x + 61 x - 9) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[3, 3, 1, 1]](x) = -x^3*(567*x^3+21*x^2+61*x-9)/(-1+x)/(1+x)/(-1+6* x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 9, 236, 5427, 117110, 2485161, 52393376, 1101978999, 23156142770, 486405331533, 10215618375116, 214537763258091, 4505379933266030, 94613754221420625, 1986895780044559256, 41724873618481880703, 876222904715221487690, 18400686019101014296437, 386414451531071264862596 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [3, 2, 2, 1]](x) = 3 4 3 2 x (1134 x - 414 x + 252 x + 37 x - 9) ------------------------------------------------------------------------- (-1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[3, 2, 2, 1]](x) = x^3*(1134*x^4-414*x^3+252*x^2+37*x-9)/(-1+x)/(-1 +3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 9, 278, 6580, 144600, 3089849, 65342788, 1376128560, 28933076450, 607897702489, 12768542354898, 268163378149940, 5631645483673300, 118266477877012929, 2483613290961405608, 52156034116221454720, 1095278109732131019150, 23000852833418914861169, 483018022199723194318918 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [3, 2, 1, 1, 1]](x) = 3 2 x (225 x - 74 x + 9) -------------------------------------------------- (-1 + x) (1 + x) (-1 + 6 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[3, 2, 1, 1, 1]](x) = x^3*(225*x^2-74*x+9)/(-1+x)/(1+x)/(-1+6*x)/(-\ 1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 9, 250, 5913, 130750, 2808837, 59576650, 1256569281, 26437931350, 555651385245, 11672802765250, 245166152544729, 5148825942100750, 108128381962415733, 2270723394997512250, 47685437894121286257, 1001396416579663726150, 21029344743864318404301, 441616419633153086220850 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 3 4 3 2 F[[6, 1, 1], [3, 1, 1, 1, 1, 1]](x) = x (648 x - 1089 x - 18 x + 22 x - 3)/ ((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[3, 1, 1, 1, 1, 1]](x) = x^3*(648*x^4-1089*x^3-18*x^2+22*x-3)/(-1+x )/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 3, 80, 1877, 42115, 912730, 19458285, 411433074, 8666652830, 182246103677, 3829431252790, 80438694336091, 1689399957599445, 35479095278012844, 745076325143010395, 15646741099240030928, 328582809641148294760, 6900250234171772800231, 144905356079147425815900 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 3 4 3 2 F[[6, 1, 1], [2, 2, 2, 2]](x) = x (873 x + 72 x + 110 x - 14 x - 1)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[2, 2, 2, 2]](x) = x^3*(873*x^4+72*x^3+110*x^2-14*x-1)/(-1+x)/(1+x) /(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 1, 48, 1235, 28198, 611331, 13008926, 274687645, 5781773976, 121535955761, 2553316220884, 53629145261055, 1126297323530834, 23653009615497991, 496720084554203922, 10431183660491934665, 219055413481664634772, 4600168690500841777221, 96603587554298255086040 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 3 F[[6, 1, 1], [2, 2, 2, 1, 1]](x) = - x 5 4 3 2 (2268 x - 531 x - 1089 x + 158 x - 9 x + 3)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[2, 2, 2, 1, 1]](x) = -x^3*(2268*x^5-531*x^4-1089*x^3+158*x^2-9*x+3 )/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 3, 93, 2459, 55994, 1218427, 25971547, 548921943, 11559204288, 243031357151, 5106258274601, 107254869057427, 2252563526376982, 47305737189892275, 993437618974926855, 20862344299292368511, 438110619344858743676, 9200335509896788537399, 193207158253417735825909 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 3 F[[6, 1, 1], [2, 2, 1, 1, 1, 1]](x) = x 5 4 3 2 (486 x - 1152 x - 117 x - 16 x + x - 2)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[2, 2, 1, 1, 1, 1]](x) = x^3*(486*x^5-1152*x^4-117*x^3-16*x^2+x-2)/ (-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 2, 67, 1720, 39480, 864042, 18485537, 391437440, 8250374710, 173535873082, 3646792864407, 76605732779160, 1608929489743940, 33789409361100122, 709594656224063677, 14901641611350018880, 312935860077120919170, 6571665548347731633162, 138005088959329036995347 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[6, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (1701 x - 441 x + 153 x - 14 x + 1) - ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x) and in Maple notation F[[6, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -x^3*(1701*x^4-441*x^3+153*x^2-14*x+1)/ (-1+x)/(1+x)/(-1+3*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 1, 22, 586, 13531, 299152, 6435373, 136662247, 2884375012, 60707131903, 1276096892224, 26809440302008, 563101981034593, 11826081744903154, 248356217079559975, 5215557297697976269, 109527395313150698674, 2300081538592981595305, 48301768494381183773026 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 4 F[[6, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = x 4 3 2 (648 x - 1089 x - 18 x + 22 x - 3)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 9 x) (-1 + 21 x)) and in Maple notation F[[6, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^4*(648*x^4-1089*x^3-18*x^2+22*x-3) /(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+9*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 80, 1877, 42115, 912730, 19458285, 411433074, 8666652830, 182246103677, 3829431252790, 80438694336091, 1689399957599445, 35479095278012844, 745076325143010395, 15646741099240030928, 328582809641148294760, 6900250234171772800231 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+3*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 Regarding Lambda=, [5, 3] 8 7 6 5 4 3 F[[5, 3], [8]](x) = (8272 x - 2548 x - 18990 x + 2626 x + 5549 x - 761 x 2 - 260 x + 38 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x)) and in Maple notation F[[5, 3],[8]](x) = (8272*x^8-2548*x^7-18990*x^6+2626*x^5+5549*x^4-761*x^3-260*x ^2+38*x-1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 1, 26, 497, 12684, 341625, 9440068, 263057593, 7353102692, 205761654329, 5760076150500, 161269628708409, 4515424601055460, 126430638773587513, 3540045385615775972, 99121145796346777145, 2775390832296994191588, 77710930804301520145977, 2175905937520431111878884 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [7, 1]](x) = 2 4 3 2 x (1680 x - 644 x - 102 x + 30 x - 1) - ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[7, 1]](x) = -x^2*(1680*x^4-644*x^3-102*x^2+30*x-1)/(1+2*x)/(1+4*x)/(-\ 1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 8, 146, 3352, 87192, 2377536, 65938720, 1840014464, 51457785728, 1440192692224, 40319143463424, 1128873512073216, 31607833307305984, 885013082526236672, 24780303810242551808, 693847881685538603008, 19427734437187207004160, 543976501741221752340480, 15231341423754083080732672 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [6, 2]](x) = 2 4 3 2 2 x (1280 x - 424 x - 110 x + 29 x - 1) - ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[6, 2]](x) = -2*x^2*(1280*x^4-424*x^3-110*x^2+29*x-1)/(1+2*x)/(1+4*x)/ (-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 18, 384, 9240, 246112, 6762528, 188098304, 5254197120, 146992476672, 4114538469888, 115194576461824, 3225323126937600, 90307797543084032, 2528605830982606848, 70800838267334098944, 1982422221481775431680, 55507809701486849032192, 1554218546641574505873408, 43518118055964040352497664 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [6, 1, 1]](x) = 2 4 3 2 x (112 x - 524 x + 42 x + 20 x - 1) - ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[6, 1, 1]](x) = -x^2*(112*x^4-524*x^3+42*x^2+20*x-1)/(1+2*x)/(1+4*x)/( -1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 18, 382, 9600, 257224, 7090272, 197396704, 5515865856, 154331646592, 4320161235456, 120953263023616, 3386578866757632, 94823083244062720, 2655035080867307520, 74340869763882016768, 2081543228389233328128, 58283199144892090974208, 1631929463556987137163264, 45694023854595536762306560 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [5, 3]](x) = - x 6 5 4 3 2 (9648 x + 116 x - 4466 x + 566 x + 247 x - 37 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x)) and in Maple notation F[[5, 3],[5, 3]](x) = -x*(9648*x^6+116*x^5-4466*x^4+566*x^3+247*x^2-37*x+1)/(1+ x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 1, 26, 497, 12684, 341625, 9440068, 263057593, 7353102692, 205761654329, 5760076150500, 161269628708409, 4515424601055460, 126430638773587513, 3540045385615775972, 99121145796346777145, 2775390832296994191588, 77710930804301520145977, 2175905937520431111878884, 60925365000571842065043001 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 2 2 x (68 x + 14 x - 1) F[[5, 3], [5, 2, 1]](x) = ------------------------------------------ (-1 + x) (1 + 2 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[5, 2, 1]](x) = 2*x^2*(68*x^2+14*x-1)/(-1+x)/(1+2*x)/(-1+10*x)/(-1+28* x) The first 20 term , starting with k=1 are 0, 2, 46, 1086, 28430, 776014, 21528462, 600796814, 16802311054, 470264709006, 13165411853198, 368611531887502, 10320922892854158, 288983840999908238, 8091527547997447054, 226562571343928484750, 6343749997629997638542, 177624979933639933748110, 4973499238141918145209230, 139257976667973708065334158 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[5, 3], [5, 1, 1, 1]](x) = - x 5 4 3 2 (3920 x - 1268 x - 286 x + 82 x - 19 x + 1)/((-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x)) and in Maple notation F[[5, 3],[5, 1, 1, 1]](x) = -x^2*(3920*x^5-1268*x^4-286*x^3+82*x^2-19*x+1)/(-1+ x)/(1+2*x)/(1+4*x)/(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 20, 564, 15250, 421706, 11746770, 328299154, 9186150770, 257149951986, 7199574048370, 201581827028594, 5644228663090290, 158037777625265266, 4425051523608083570, 123901380161965931634, 3469238019536656669810, 97138658297041419410546, 2719882369817185513184370, 76156705729881434888412274 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 3 2 x (336 x + 76 x - 26 x + 1) F[[5, 3], [4, 4]](x) = - ------------------------------------------------------ (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[4, 4]](x) = -x^2*(336*x^3+76*x^2-26*x+1)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1 +10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 10, 248, 6248, 170208, 4712736, 131460736, 3675851392, 102873904640, 2879968543232, 80634120259584, 2257705354684416, 63215249947942912, 1770021998341070848, 49560565953818427392, 1387695346703694725120, 38855464707707747303424, 1087952961815765384822784, 30462682430841499494645760 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[5, 3], [4, 3, 1]](x) = - x 6 5 4 3 2 (1120 x - 3864 x + 756 x + 894 x - 93 x - 30 x + 2)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x)) and in Maple notation F[[5, 3],[4, 3, 1]](x) = -x^2*(1120*x^6-3864*x^5+756*x^4+894*x^3-93*x^2-30*x+2) /(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 46, 1137, 30590, 843577, 23494926, 656601017, 18372323470, 514299947577, 14399148446606, 403163654755897, 11288457331774350, 316075555261713977, 8850103047305651086, 247802760324110814777, 6938476039074745017230, 194277316594085702110777, 5439764739634393932948366, 152313411459762915589721657 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [4, 2, 2]](x) = 2 3 2 2 x (224 x + 24 x - 18 x + 1) - ------------------------------------------------------ (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[4, 2, 2]](x) = -2*x^2*(224*x^3+24*x^2-18*x+1)/(1+2*x)/(1+4*x)/(-1+2*x )/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 32, 888, 24160, 672160, 18767616, 525004160, 14695072256, 411412197888, 11519040839680, 322528146307072, 9030738085404672, 252860166436069376, 7080079660030951424, 198242180481582465024, 5550780553481445572608, 155421850497491929268224, 4351811763929728205979648, 121850728890032573019324416 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 x (20 x + 1) F[[5, 3], [4, 2, 1, 1]](x) = -------------------------------- (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation F[[5, 3],[4, 2, 1, 1]](x) = x^2*(20*x+1)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 48, 1360, 38400, 1075456, 30117888, 843304960, 23612620800, 661153447936, 18512297852928, 518344340930560, 14513641567027200, 406381963893538816, 11378694989354631168, 318603459702198108160, 8920896871666915737600, 249785112406677935620096, 6993983147387068096708608, 195831528126837975427317760 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [4, 1, 1, 1, 1]](x) = 3 3 2 2 x (280 x - 286 x - 19 x + 7) ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[4, 1, 1, 1, 1]](x) = 2*x^3*(280*x^3-286*x^2-19*x+7)/(1+2*x)/(1+4*x)/( -1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 14, 494, 14560, 414760, 11677344, 327604704, 9179206400, 257080507520, 7198879604224, 201574882584064, 5644159218647040, 158037083180820480, 4425044579163643904, 123901310717521485824, 3469237325092212244480, 97138651352596974960640, 2719882300372741068816384, 76156705035436990443945984 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 2 x (56 x + 2 x - 1) F[[5, 3], [3, 3, 2]](x) = - ------------------------------------------ (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation F[[5, 3],[3, 3, 2]](x) = -x^2*(56*x^2+2*x-1)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 24, 640, 17952, 501952, 14055552, 393543424, 11019231744, 308538293248, 8639072471040, 241894026047488, 6773032733515776, 189644916488126464, 5310057661734617088, 148681614527764037632, 4163085206778466664448, 116566385789784181964800, 3263858802113974274359296, 91388046459191073524678656 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [3, 3, 1, 1]](x) = 2 5 4 3 2 x (448 x - 752 x - 164 x + 70 x - 8 x + 1) ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[3, 3, 1, 1]](x) = x^2*(448*x^5-752*x^4-164*x^3+70*x^2-8*x+1)/(1+x)/(-\ 1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 26, 831, 23602, 666599, 18712050, 524448583, 14689516658, 411356642247, 11518485283954, 322522590751175, 9030682529848434, 252859610880512455, 7080074104475393138, 198242124926026904007, 5550779997925890006130, 155421844941936373690823, 4351811708374172650380402, 121850728334477017463681479 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [3, 2, 2, 1]](x) = 2 4 3 2 x (1120 x - 280 x - 44 x + 6 x - 1) - ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[3, 2, 2, 1]](x) = -x^2*(1120*x^4-280*x^3-44*x^2+6*x-1)/(1+2*x)/(1+4*x )/(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 32, 1000, 29200, 829696, 23356032, 655212160, 18358434560, 514161058816, 14397759557632, 403149765867520, 11288318442885120, 316074166372827136, 8850089158416760832, 247802621435221934080, 6938474650185856122880, 194277302705196813254656, 5439764600745505044037632, 152313410070874026700963840 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 3 24 x F[[5, 3], [3, 2, 1, 1, 1]](x) = --------------------------------- (1 + 2 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[3, 2, 1, 1, 1]](x) = 24*x^3/(1+2*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 24, 864, 26208, 753792, 21306240, 598574592, 16780088832, 470042486784, 13163189630976, 368589309665280, 10320700670631936, 288981618777686016, 8091505325775224832, 226562349121706262528, 6343747775407775416320, 177624957711417711525888, 4973499015919695922987008, 139257974445751485843111936 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [3, 1, 1, 1, 1, 1]](x) = 3 3 2 6 x (168 x - 18 x - 5 x - 1) - ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[3, 1, 1, 1, 1, 1]](x) = -6*x^3*(168*x^3-18*x^2-5*x-1)/(1+2*x)/(1+4*x) /(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 6, 258, 8352, 244728, 6965280, 196146720, 5503365888, 154206646656, 4318911235584, 120940763023872, 3386453866758144, 94821833244063744, 2655022580867309568, 74340744763882020864, 2081541978389233336320, 58283186644892090990592, 1631929338556987137196032, 45694022604595536762372096 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [2, 2, 2, 2]](x) = 3 3 2 x (1456 x - 252 x + 6 x + 5) --------------------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[2, 2, 2, 2]](x) = x^3*(1456*x^3-252*x^2+6*x+5)/(-1+x)/(1+2*x)/(1+4*x) /(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 5, 191, 5695, 164647, 4657191, 130905159, 3670295879, 102818348999, 2879412987847, 80628564703687, 2257649799129543, 63214694392385991, 1770016442785518023, 49560510398262866375, 1387694791148139180487, 38855459152152191726023, 1087952906260209829310919, 30462681875285943939002823 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [2, 2, 2, 1, 1]](x) = 3 2 2 x (104 x - 10 x + 5) ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[2, 2, 2, 1, 1]](x) = 2*x^3*(104*x^2-10*x+5)/(1+2*x)/(1+4*x)/(-1+2*x)/ (-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 10, 360, 11288, 327744, 9301152, 261668736, 7339213696, 205622765568, 5758687261184, 161255739820032, 4515285712164864, 126429249884700672, 3540031496726880256, 99121006907457896448, 2775389443408105275392, 77710916915412631289856, 2175905798631542222880768, 60925363611682953176285184 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 3 F[[5, 3], [2, 2, 1, 1, 1, 1]](x) = - x 5 4 3 2 (2560 x - 1744 x + 388 x - 46 x + 52 x + 5)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x)) and in Maple notation F[[5, 3],[2, 2, 1, 1, 1, 1]](x) = -x^3*(2560*x^5-1744*x^4+388*x^3-46*x^2+52*x+5 )/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 5, 242, 7855, 232210, 6623655, 186709362, 5240308295, 146853587570, 4113149581255, 115180687572082, 3225184238049735, 90306408654191730, 2528591942093722055, 70800699378445196402, 1982420832592886559175, 55507795812597960088690, 1554218407752685617050055, 43518116667075151463390322 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 3 F[[5, 3], [2, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (2800 x - 1756 x + 238 x - 28 x - 38 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x)) and in Maple notation F[[5, 3],[2, 1, 1, 1, 1, 1, 1]](x) = x^3*(2800*x^5-1756*x^4+238*x^3-28*x^2-38*x -1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 76, 2657, 80246, 2308089, 65244270, 1833070009, 51388341262, 1439498247737, 40312199018894, 1128804067628601, 31607138862861198, 885006138081791545, 24780234365798105998, 693847187241094155833, 19427727492742762554254, 543976432296777307885113, 15231340729309638636266382 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[5, 3], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 4 2 2 x (104 x - 10 x + 5) ----------------------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 2 x) (-1 + 4 x) (-1 + 10 x) (-1 + 28 x) and in Maple notation F[[5, 3],[1, 1, 1, 1, 1, 1, 1, 1]](x) = 2*x^4*(104*x^2-10*x+5)/(1+2*x)/(1+4*x)/ (-1+2*x)/(-1+4*x)/(-1+10*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 0, 10, 360, 11288, 327744, 9301152, 261668736, 7339213696, 205622765568, 5758687261184, 161255739820032, 4515285712164864, 126429249884700672, 3540031496726880256, 99121006907457896448, 2775389443408105275392, 77710916915412631289856, 2175905798631542222880768 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(-1+28* x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 Regarding Lambda=, [5, 2, 1] F[[5, 2, 1], [8]](x) = 6 5 4 3 2 4672 x - 32 x - 8467 x - 1399 x + 1176 x - 82 x + 1 ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[8]](x) = (4672*x^6-32*x^5-8467*x^4-1399*x^3+1176*x^2-82*x+1)/(-1+x )/(1+x)/(1+2*x)/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 9, 464, 27359, 1716020, 109264983, 6984009028, 446833399863, 28595046901124, 1830046351103927, 117122380063762052, 7495822941578728887, 479732518141039929988, 30702878759106709425591, 1964984202152112404232836, 125758988322843724031061431, 8048575242823734829146913412, 515108815383306812958292929975, 32966964182013040571754459970180 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 2 2 x (288 x - 52 x + 1) F[[5, 2, 1], [7, 1]](x) = - -------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[7, 1]](x) = -2*x^2*(288*x^2-52*x+1)/(1+2*x)/(-1+4*x)/(-1+16*x)/(-1 +64*x) The first 20 term , starting with k=1 are 0, 2, 60, 3144, 190064, 11988768, 764481984, 48882097280, 3127738353408, 200163801199104, 12810300024101888, 819856269508724736, 52470754336051064832, 3358127526907285544960, 214920149712467089539072, 13754889389444308913651712, 880312917849978421991899136, 56340026693207301466136641536, 3605761707578206213308656910336, 230768749272412220363918405009408 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 3 2 x (768 x + 616 x - 173 x + 4) F[[5, 2, 1], [6, 2]](x) = ----------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[6, 2]](x) = x^2*(768*x^3+616*x^2-173*x+4)/(-1+x)/(1+2*x)/(-1+4*x)/ (-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 4, 159, 8781, 539905, 34203609, 2183435049, 139650351241, 8936190768585, 571893302479689, 36600804853948489, 2342445646587197001, 149916427556574382665, 9594649862420828791369, 614057567175734806278729, 39299683914939858504831561, 2515179764407236249493443145, 160971504823680484893479768649, 10302176307141428872168890077769, 659339283631865493243275126542921 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [6, 1, 1]](x) = 2 3 2 x (96 x - 832 x + 173 x - 4) - --------------------------------------------------- (-1 + x) (1 + x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[6, 1, 1]](x) = -x^2*(96*x^3-832*x^2+173*x-4)/(-1+x)/(1+x)/(-1+4*x) /(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 4, 163, 9152, 565811, 35896320, 2292327219, 146628395008, 9382928724787, 600486822281216, 38430826771460915, 2459567635713490944, 157412244243153630003, 10074382280481877000192, 644760444333561648198451, 41264668091471493028904960, 2640938752320152327442084659, 169020080059945377385307701248, 10817285122419794207730079839027, 692306247812199470176675934765056 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 2 x (768 x - 200 x + 5) F[[5, 2, 1], [5, 3]](x) = - -------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[5, 3]](x) = -x^2*(768*x^2-200*x+5)/(1+2*x)/(-1+4*x)/(-1+16*x)/(-1+ 64*x) The first 20 term , starting with k=1 are 0, 5, 210, 12108, 752936, 47838384, 3056063136, 195498560192, 12510476184192, 800647569255168, 51241077928274432, 3279423123346861056, 209882986069204051968, 13432509607229172396032, 859680590843468975874048, 55019557429674846074880000, 3521251669350275456839614464, 225360106740034994175088459776, 14423046829788117466237658333184, 923074997081253563263859548749824 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [5, 2, 1]](x) = 4 3 2 x (3520 x + 1440 x - 901 x + 73 x - 1) - ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[5, 2, 1]](x) = -x*(3520*x^4+1440*x^3-901*x^2+73*x-1)/(-1+x)/(1+x)/ (1+2*x)/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 1, 9, 464, 27359, 1716020, 109264983, 6984009028, 446833399863, 28595046901124, 1830046351103927, 117122380063762052, 7495822941578728887, 479732518141039929988, 30702878759106709425591, 1964984202152112404232836, 125758988322843724031061431, 8048575242823734829146913412, 515108815383306812958292929975, 32966964182013040571754459970180, 2109885707608537069271614541229495 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 2 x (832 x - 168 x + 5) F[[5, 2, 1], [5, 1, 1, 1]](x) = - -------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[5, 1, 1, 1]](x) = -x^2*(832*x^2-168*x+5)/(1+2*x)/(-1+4*x)/(-1+16*x )/(-1+64*x) The first 20 term , starting with k=1 are 0, 5, 242, 14796, 935720, 59710640, 3818680992, 244350831296, 15637737319040, 1000803734956800, 64051255784417792, 4099277438168247296, 262353709130257704960, 16790636633736499343360, 1074600732549536727867392, 68774446691016765587767296, 4401564585150615648419348480, 281700133400448083954630328320, 18028808536841616292560799137792, 1153843746345270465436009695543296 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [4, 4]](x) = 2 4 3 2 x (1408 x + 360 x - 7 x - 62 x + 2) ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[4, 4]](x) = x^2*(1408*x^4+360*x^3-7*x^2-62*x+2)/(-1+x)/(1+x)/(1+2* x)/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 2, 102, 6007, 375736, 23907523, 1527845088, 97746297203, 6255190369152, 400323021073459, 25620526747323904, 1639711366204626739, 104941489907102005248, 6716254753574589219635, 429840294621094549708800, 27509778702027184079467315, 1760625834470173905307009024, 112680053366738075918598419251, 7211523414841587994419132235776, 461537498539787249812748367246131 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 2 x (640 x - 256 x + 9) F[[5, 2, 1], [4, 3, 1]](x) = - -------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[4, 3, 1]](x) = -x^2*(640*x^2-256*x+9)/(1+2*x)/(-1+4*x)/(-1+16*x)/( -1+64*x) The first 20 term , starting with k=1 are 0, 9, 482, 29580, 1871400, 119421104, 7637361312, 488701659840, 31275474627200, 2001607469869824, 128102511568660992, 8198554876335795200, 524707418260512614400, 33581273267472987500544, 2149201465099073410998272, 137548893382033530996572160, 8803129170301231296122880000, 563400266800896167906397323264, 36057617073683232585110145073152, 2307687492690540930871973578014720 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [4, 2, 2]](x) = 2 4 3 2 x (512 x + 1520 x - 220 x - 117 x + 6) ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[4, 2, 2]](x) = x^2*(512*x^4+1520*x^3-220*x^2-117*x+6)/(-1+x)/(1+x) /(1+2*x)/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 6, 375, 23480, 1494211, 95490272, 6109143411, 390949397376, 25020188814387, 1601282921696768, 102481960387745587, 6558843119193700352, 419765922098411127603, 26865018413818406559744, 1719361168876698993783603, 110039114654385869036945408, 7042503335421129744733451059, 450720213427599249650479923200, 28846093658736703113293909668659, 1846149994149074617420871559020544 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 9 x F[[5, 2, 1], [4, 2, 1, 1]](x) = -------------------- (-1 + x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[4, 2, 1, 1]](x) = 9*x^2/(-1+x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 9, 585, 37449, 2396745, 153391689, 9817068105, 628292358729, 40210710958665, 2573485501354569, 164703072086692425, 10540996613548315209, 674623783267092173385, 43175922129093899096649, 2763259016262009542185545, 176848577040768610699874889, 11318308930609191084791992905, 724371771558988229426687545929, 46359793379775246683308002939465, 2967026776305615787731712188125769 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [4, 1, 1, 1, 1]](x) = 2 2 x (320 x - 32 x + 3) - -------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[4, 1, 1, 1, 1]](x) = -x^2*(320*x^2-32*x+3)/(1+2*x)/(-1+4*x)/(-1+16 *x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 3, 214, 14340, 928440, 59594128, 3816816864, 244321005120, 15637260100480, 1000796099459328, 64051133616459264, 4099275483480908800, 262353677855260293120, 16790636133336540745728, 1074600724543137390321664, 68774446562914376187002880, 4401564583100977418007183360, 281700133367653872268035555328, 18028808536316908905575283032064, 1153843746336875147244241437327360 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 4 x F[[5, 2, 1], [3, 3, 2]](x) = ---------------------- (-1 + 4 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[3, 3, 2]](x) = 4*x^2/(-1+4*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 4, 272, 17472, 1118464, 71582720, 4581298176, 293203099648, 18764998443008, 1200959900614656, 76861433640386560, 4919131752988934144, 314824432191308562432, 20148763660243815104512, 1289520874255604435124224, 82529335952358684921692160, 5281877500950955839283265536, 338040160060861173731308863488, 21634570243895115118872486739968, 1384612495609287367608114029264896 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [3, 3, 1, 1]](x) = 2 3 2 x (768 x - 264 x + 68 x - 5) - --------------------------------------------------- (-1 + x) (1 + x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[3, 3, 1, 1]](x) = -x^2*(768*x^3-264*x^2+68*x-5)/(-1+x)/(1+x)/(-1+4 *x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 5, 352, 23117, 1488384, 95397069, 6107652096, 390925536461, 25019807039488, 1601276813298893, 102481862653378560, 6558841555443829965, 419765897078413197312, 26865018013498439683277, 1719361162471579523743744, 110039114551903957516340429, 7042503333781419160403705856, 450720213401363880301204131021, 28846093658316937203705496731648, 1846149994142358362867456952552653 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 2 x (384 x - 16 x - 5) F[[5, 2, 1], [3, 2, 2, 1]](x) = -------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[3, 2, 2, 1]](x) = x^2*(384*x^2-16*x-5)/(1+2*x)/(-1+4*x)/(-1+16*x)/ (-1+64*x) The first 20 term , starting with k=1 are 0, 5, 426, 28668, 1856840, 119188080, 7633633056, 488642007488, 31274520190080, 2001592198874880, 128102267232743936, 8198550966961118208, 524707355710517790720, 33581272266673070305280, 2149201449086274735906816, 137548893125828752195043328, 8803129166201954835298549760, 563400266735307744533207777280, 36057617072633817811139112861696, 2307687492673750294488437061582848 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [3, 2, 1, 1, 1]](x) = 2 3 2 x (960 x - 184 x - 43 x - 4) - ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[3, 2, 1, 1, 1]](x) = -x^2*(960*x^3-184*x^2-43*x-4)/(-1+x)/(1+x)/(1 +2*x)/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 4, 371, 25906, 1692711, 108892170, 6978043767, 446737956202, 28593519801527, 1830021917512426, 117121989126293943, 7495816686579247338, 479732418061048208823, 30702877157826841919722, 1964984176531634524073399, 125758987912916077948641514, 8048575236264892491827932599, 515108815278365335561189761258, 32966964180333976933400808222135, 2109885707581672051057956115357930 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [3, 1, 1, 1, 1, 1]](x) = 2 3 2 x (864 x - 268 x - 28 x - 1) - --------------------------------------------------- (-1 + x) (1 + x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[3, 1, 1, 1, 1, 1]](x) = -x^2*(864*x^3-268*x^2-28*x-1)/(-1+x)/(1+x) /(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 112, 8333, 552704, 35686605, 2288971776, 146574707917, 9382069731328, 600473078385869, 38430606869135360, 2459564117276282061, 157412187948158287872, 10074381379761951526093, 644760429922042840612864, 41264667860887192107535565, 2640938748630803512700174336, 169020080000915796349437136077, 10817285121475320911156150796288, 692306247797087897431493070081229 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [2, 2, 2, 2]](x) = 2 4 3 2 x (640 x + 1400 x - 341 x + 3 x - 1) - ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[2, 2, 2, 2]](x) = -x^2*(640*x^4+1400*x^3-341*x^2+3*x-1)/(-1+x)/(1+ x)/(1+2*x)/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 79, 5644, 369909, 23814320, 1526353773, 97722436288, 6254808594253, 400316912675584, 25620429012956877, 1639709802454756352, 104941464887104074957, 6716254353254622343168, 429840288215975079668941, 27509778599545272558862336, 1760625832830463320977263821, 112680053340502706569322627072, 7211523414421822084830719298765, 461537498533070995259333760778240 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 2 x (256 x - 72 x - 1) F[[5, 2, 1], [2, 2, 2, 1, 1]](x) = -------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[2, 2, 2, 1, 1]](x) = x^2*(256*x^2-72*x-1)/(1+2*x)/(-1+4*x)/(-1+16* x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 154, 11196, 738376, 47605360, 3052334880, 195438907840, 12509521747072, 800632298260224, 51240833592357376, 3279419213972184064, 209882923519209228288, 13432508606429255200768, 859680574830670300782592, 55019557173470067273351168, 3521251665250998996015284224, 225360106674446570801898913792, 14423046828738702692266626121728, 923074997064462926880323032317952 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [2, 2, 1, 1, 1, 1]](x) = 3 2 x (1792 x - 680 x + 103) ----------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[2, 2, 1, 1, 1, 1]](x) = x^3*(1792*x^2-680*x+103)/(-1+x)/(1+2*x)/(-\ 1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 0, 103, 7869, 525345, 33970585, 2179706793, 139590698889, 8935236331465, 571878031484745, 36600560518031433, 2342441737212520009, 149916365006579558985, 9594648861620911596105, 614057551162936131187273, 39299683658735079703302729, 2515179760307959788669112905, 160971504758092061520290222665, 10302176306092014098197857866313, 659339283615074856859738610111049 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 3 32 x F[[5, 2, 1], [2, 1, 1, 1, 1, 1, 1]](x) = - ---------------------------------- (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -32*x^3/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 0, 32, 2688, 182784, 11872256, 762617856, 48852271104, 3127261134848, 200156165701632, 12810177856143360, 819854314821386240, 52470723061053652992, 3358127026507326947328, 214920141706067751993344, 13754889261341919512887296, 880312915800340191579734016, 56340026660413089779541868544, 3605761707053498826323140804608, 230768749264016902172150146793472 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[5, 2, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (960 x - 184 x - 43 x - 4) - ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 16 x) (-1 + 64 x) and in Maple notation F[[5, 2, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(960*x^3-184*x^2-43*x-4)/(-1+x) /(1+x)/(1+2*x)/(-1+4*x)/(-1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 0, 4, 371, 25906, 1692711, 108892170, 6978043767, 446737956202, 28593519801527, 1830021917512426, 117121989126293943, 7495816686579247338, 479732418061048208823, 30702877157826841919722, 1964984176531634524073399, 125758987912916077948641514, 8048575236264892491827932599, 515108815278365335561189761258, 32966964180333976933400808222135 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 Regarding Lambda=, [5, 1, 1, 1] 8 7 6 5 4 F[[5, 1, 1, 1], [8]](x) = (8025 x - 3375 x - 16336 x + 8621 x + 2148 x 3 2 - 1259 x - 36 x + 37 x - 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[5, 1, 1, 1],[8]](x) = (8025*x^8-3375*x^7-16336*x^6+8621*x^5+2148*x^4-1259*x^ 3-36*x^2+37*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35 *x) The first 20 term , starting with k=1 are 0, 1, 1, 45, 1297, 45741, 1595571, 55853575, 1954749427, 68416438611, 2394572269441, 83810034568305, 2933351133334557, 102667289794349281, 3593355140892554311, 125767429934422739235, 4401860047657091410687, 154065101668077712349751, 5392278558381527653226181, 188729749543355455046654365 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [7, 1]](x) = 2 3 2 x (265 x - 27 x - 27 x + 1) - ------------------------------------------------------------- (1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[7, 1]](x) = -x^2*(265*x^3-27*x^2-27*x+1)/(1+x)/(-1+5*x)/(-1+3*x )/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 7, 280, 9112, 319621, 11169907, 390962440, 13683268312, 478914766561, 16762006435207, 586670234472400, 20533457946968512, 718671028373543701, 25153485986587549507, 880372009536293154160, 30813020333608122309712, 1078455711676427393867041, 37745949908670905548432807, 1321108246803485271030751720 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [6, 2]](x) = 2 5 4 3 2 x (1125 x + 50 x - 710 x + 32 x + 49 x - 2) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[6, 2]](x) = -x^2*(1125*x^5+50*x^4-710*x^3+32*x^2+49*x-2)/(1+x)/ (-1+x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 21, 773, 26050, 912512, 31914311, 1117018163, 39095058660, 1368327469022, 47891447107801, 1676200659026753, 58667022709300070, 2053345795080546332, 71867102818912780491, 2515348598668311722543, 88037200953168337854280, 3081302033361050829808442, 107845571167631215507268381, 3774594990867096516930081533 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [6, 1, 1]](x) = 2 4 3 2 x (1305 x - 94 x - 114 x - 10 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[6, 1, 1]](x) = x^2*(1305*x^4-94*x^3-114*x^2-10*x+1)/(1+x)/(-1+x )/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 25, 796, 27416, 957841, 33511485, 1172862496, 41049846856, 1436743685641, 50286020333945, 1760010688125196, 61600373866436496, 2156013084738862441, 75460457960399328205, 2641116028599340028896, 92439061000840269572336, 3235367135029043738808241, 113237849726013114082238265, 3963324740410449852409853596 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[5, 1, 1, 1], [5, 3]](x) = - x 6 5 4 3 2 (2625 x - 930 x - 1876 x + 672 x + 97 x - 46 x + 2)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[5, 1, 1, 1],[5, 3]](x) = -x^2*(2625*x^6-930*x^5-1876*x^4+672*x^3+97*x^2-46*x +2)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 28, 1063, 36433, 1277062, 44679138, 1563814373, 54733060123, 1915658183452, 67048025405548, 2346680915838283, 82133831779430613, 2874684112943064242, 100613943946138571158, 3521488038131396427793, 123252081334427193745903, 4313822846705365185019432, 150983799634683489763207968, 5284432987213932474490622903 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 2 2 x (25 x + 6 x - 1) F[[5, 1, 1, 1], [5, 2, 1]](x) = ----------------------------------------- (-1 + 2 x) (1 + x) (-1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[5, 2, 1]](x) = 2*x^2*(25*x^2+6*x-1)/(-1+2*x)/(1+x)/(-1+5*x)/(-1 +35*x) The first 20 term , starting with k=1 are 0, 2, 70, 2394, 83434, 2918222, 102127590, 3574414034, 125104231714, 4378646809782, 153252631835710, 5363842081705274, 187734472696939194, 6570706543579099742, 229974729021199540630, 8049115515721638990114, 281719043050155639633874, 9860166506754938761364102, 345105827736420313517194350, 12078703970774698257446172554 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [5, 1, 1, 1]](x) = - x 6 5 4 3 2 (7725 x - 4950 x - 1569 x + 962 x + 43 x - 36 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[5, 1, 1, 1],[5, 1, 1, 1]](x) = -x*(7725*x^6-4950*x^5-1569*x^4+962*x^3+43*x^2 -36*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 1, 45, 1297, 45741, 1595571, 55853575, 1954749427, 68416438611, 2394572269441, 83810034568305, 2933351133334557, 102667289794349281, 3593355140892554311, 125767429934422739235, 4401860047657091410687, 154065101668077712349751, 5392278558381527653226181, 188729749543355455046654365, 6605541234017411122637841817 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [4, 4]](x) = 2 3 2 x (105 x - 19 x - 19 x + 1) - ------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[4, 4]](x) = -x^2*(105*x^3-19*x^2-19*x+1)/(1+x)/(-1+x)/(-1+3*x)/ (1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 14, 529, 18208, 638461, 22339254, 781905349, 27366521168, 957829044841, 33524012470894, 1173340456737769, 41066915883830928, 1437342056441911621, 50306971972921383734, 1760744019064956913789, 61626040667209892171488, 2156911423352664052870801, 75491899817341652199573774, 2642216493606965773689921409 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[5, 1, 1, 1], [4, 3, 1]](x) = x 6 5 4 3 2 (2625 x + 225 x - 840 x + 142 x + 25 x + x - 2)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[5, 1, 1, 1],[4, 3, 1]](x) = x^2*(2625*x^6+225*x^5-840*x^4+142*x^3+25*x^2+x-2 )/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 73, 2606, 91135, 3191472, 111698853, 3909507086, 136832672455, 4789144743392, 167620064038933, 5866702271765766, 205334579461440975, 7186710281912362112, 251534859865665661813, 8803720095317362621646, 308130203336075943746695, 10784557116763134787339632, 377459499086708923199087493, 13211082468034824232169642726 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [4, 2, 2]](x) = 2 2 2 x (37 x - 10 x + 1) - --------------------------------------------------- (1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[4, 2, 2]](x) = -2*x^2*(37*x^2-10*x+1)/(1+x)/(-1+5*x)/(-1+3*x)/( 1+3*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 58, 2084, 72916, 2553062, 89359438, 3127601864, 109466148136, 3831315691082, 134096051496418, 4693361814744044, 164267663575880956, 5749368225462470702, 201227887892701591798, 7042976076252198260624, 246504162668864989239376, 8627645693410465476619922, 301967599269367244484317578, 10568865974427858326388103604 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 2 x (75 x - 35 x - 2) F[[5, 1, 1, 1], [4, 2, 1, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[4, 2, 1, 1]](x) = -x^2*(75*x^2-35*x-2)/(1+x)/(-1+3*x)/(1+5*x)/( -1+35*x) The first 20 term , starting with k=1 are 0, 2, 99, 3329, 117330, 4102772, 143616909, 5026495259, 175927825620, 6157471465142, 215511513516519, 7542902912131589, 264001602230047110, 9240056076526567112, 323401962686061634929, 11319068693974017424319, 396167404289281366237800, 13865859150123894208576682, 485305070254341065865476139, 16985677458901913464014885449 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [4, 1, 1, 1, 1]](x) = 2 4 3 2 x (1125 x + 130 x - 174 x + 6 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[4, 1, 1, 1, 1]](x) = x^2*(1125*x^4+130*x^3-174*x^2+6*x+1)/(1+x) /(-1+x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 41, 1296, 45700, 1595501, 55852881, 1954747096, 68416423760, 2394572205201, 83810034217921, 2933351131671896, 102667289785771620, 3593355140850465901, 125767429934209911161, 4401860047656034455696, 154065101668072406073280, 5392278558381501186457601, 188729749543355322519188601, 6605541234017410460581818496 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [3, 3, 2]](x) = 2 3 2 x (355 x - 79 x - 3 x - 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[3, 3, 2]](x) = -x^2*(355*x^3-79*x^2-3*x-1)/(1+x)/(-1+x)/(-1+5*x )/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 41, 1558, 54622, 1914751, 67017951, 2345700988, 82099570412, 2873486764621, 100572037605061, 3520021361024818, 123200747656479402, 4312026169096554091, 150920915918890410971, 5282232057189146005048, 184878122001632847357592, 6470734270057849083251161, 226475699452025035998939681, 7926649480820893744573153678 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [3, 3, 1, 1]](x) = 2 4 3 2 x (105 x + 234 x - 142 x + 18 x + 1) -------------------------------------------------------------- (-1 + 2 x) (1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[3, 3, 1, 1]](x) = x^2*(105*x^4+234*x^3-142*x^2+18*x+1)/(-1+2*x) /(1+x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 59, 2073, 72917, 2552911, 89359179, 3127599133, 109466138897, 3831315631851, 134096051239799, 4693361813343193, 164267663569231677, 5749368225428162791, 201227887892533243619, 7042976076251346959253, 246504162668860761441257, 8627645693410444251557731, 301967599269367138617330639, 10568865974427857796278415313 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [3, 2, 2, 1]](x) = 2 4 3 2 2 x (625 x - 40 x - 40 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[3, 2, 2, 1]](x) = -2*x^2*(625*x^4-40*x^3-40*x^2-1)/(1+x)/(-1+x) /(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 70, 2600, 91080, 3191282, 111697670, 3909501920, 136832644480, 4789144610162, 167620063353270, 5866702268396840, 205334579444419880, 7186710281827789442, 251534859865241206870, 8803720095315245135360, 308130203336065341977280, 10784557116763081821561122, 377459499086708658241098470, 13211082468034822907767205480 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 2 x F[[5, 1, 1, 1], [3, 2, 1, 1, 1]](x) = - ------------------- (1 + x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[3, 2, 1, 1, 1]](x) = -2*x^2/(1+x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 68, 2382, 83368, 2917882, 102125868, 3574405382, 125104188368, 4378646592882, 153252630750868, 5363842076280382, 187734472669813368, 6570706543443467882, 229974729020521375868, 8049115515718248155382, 281719043050138685438368, 9860166506754853990342882, 345105827736419889662000868, 12078703970774696138170030382 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [3, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (1575 x - 525 x + 62 x - 12 x - 13 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[3, 1, 1, 1, 1, 1]](x) = x^2*(1575*x^5-525*x^4+62*x^3-12*x^2-13* x+1)/(1+x)/(-1+x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 22, 793, 27362, 957691, 33510372, 1172858023, 41049821212, 1436743567261, 50286019712522, 1760010685106653, 61600373851078062, 2156013084662867431, 75460457960016961672, 2641116028597435370683, 92439061000830724757912, 3235367135028996079306201, 113237849726012875591017822, 3963324740410448660534882113 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [2, 2, 2, 2]](x) = 2 4 3 2 x (755 x - 114 x - 122 x + 26 x - 1) ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[2, 2, 2, 2]](x) = x^2*(755*x^4-114*x^3-122*x^2+26*x-1)/(1+x)/(-\ 1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 14, 527, 18194, 638381, 22338834, 781903207, 27366510374, 957828990701, 33524012199854, 1173340455381887, 41066915877050154, 1437342056408005021, 50306971972751845274, 1760744019064109210567, 61626040667205653633534, 2156911423352642860137341, 75491899817341546235819094, 2642216493606965243870973247 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [2, 2, 2, 1, 1]](x) = 2 4 3 2 x (535 x + 72 x - 70 x + 8 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[2, 2, 2, 1, 1]](x) = -x^2*(535*x^4+72*x^3-70*x^2+8*x-1)/(1+x)/( -1+x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 27, 1050, 36398, 1276811, 44678137, 1563808660, 54733033788, 1915658045301, 67048024734647, 2346680912425070, 82133831762542378, 2874684112858092991, 100613943945715311957, 3521488038129275354280, 123252081334416602738168, 4313822846705312186955881, 150983799634683224902074067, 5284432987213931149797620290 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[5, 1, 1, 1], [2, 2, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (375 x - 975 x - 15 x + 110 x - 55 x + 17 x - 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[5, 1, 1, 1],[2, 2, 1, 1, 1, 1]](x) = x^2*(375*x^6-975*x^5-15*x^4+110*x^3-55* x^2+17*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 20, 760, 26015, 912261, 31913310, 1117012450, 39095032325, 1368327330871, 47891446436900, 1676200655613540, 58667022692411835, 2053345794995575081, 71867102818489521290, 2515348598666190649030, 88037200953157746846545, 3081302033360997831744891, 107845571167630950646134480, 3774594990867095192237078920 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (2625 x + 105 x - 484 x - 22 x + 29 x - 1) - ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -x^2*(2625*x^5+105*x^4-484*x^3-22*x^ 2+29*x-1)/(1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 7, 275, 9097, 319501, 11169417, 390959605, 13683255187, 478914697571, 16762006099927, 586670232766135, 20533457938525077, 718671028331059441, 25153485986375922637, 880372009535232622865, 30813020333602826816767, 1078455711676400894857111, 37745949908670773117909547, 1321108246803484608684337795 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[5, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (1125 x + 130 x - 174 x + 6 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[5, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(1125*x^4+130*x^3-174*x^2+6*x +1)/(1+x)/(-1+x)/(-1+5*x)/(-1+3*x)/(1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 0, 1, 41, 1296, 45700, 1595501, 55852881, 1954747096, 68416423760, 2394572205201, 83810034217921, 2933351131671896, 102667289785771620, 3593355140850465901, 125767429934209911161, 4401860047656034455696, 154065101668072406073280, 5392278558381501186457601, 188729749543355322519188601 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 Regarding Lambda=, [4, 4] F[[4, 4], [8]](x) = 7 6 5 4 3 2 682 x - 193 x - 1395 x + 674 x + 239 x - 160 x + 24 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[8]](x) = (682*x^7-193*x^6-1395*x^5+674*x^4+239*x^3-160*x^2+24*x-1)/(1 +x)/(-1+2*x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 1, 7, 36, 320, 3362, 41056, 538882, 7332304, 101383458, 1411779920, 19719445538, 275799669328, 3859560890914, 54024049478224, 756277889630754, 10587537696717392, 148223411443933730, 2075115063311333968 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [7, 1]](x) = 4 3 2 x (70 x + 11 x - 20 x + 4) - ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[7, 1]](x) = -x^4*(70*x^3+11*x^2-20*x+4)/(1+x)/(-1+2*x)/(-1+x)/(1+2*x) /(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 4, 76, 1199, 17626, 252063, 3561642, 50063983, 702119242, 9837082607, 137763924298, 1928964710127, 27007129210698, 378109567196911, 5293592564760394, 74110647948898031, 1037551184730955594, 14525729271683739375 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [6, 2]](x) = 2 5 4 3 2 x (164 x - 220 x - 89 x + 99 x - 20 x + 1) - ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[6, 2]](x) = -x^2*(164*x^5-220*x^4-89*x^3+99*x^2-20*x+1)/(1+x)/(-1+2*x )/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 4, 36, 355, 4256, 55319, 750016, 10355335, 144116736, 2012521159, 28144772096, 393844239815, 5512726306816, 77171619283399, 1080363413323776, 15124852395684295, 211746521791594496, 2964442836980036039, 41502148918653485056 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [6, 1, 1]](x) = 4 2 3 x (14 x - 17 x + 4) -------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[6, 1, 1]](x) = 3*x^4*(14*x^2-17*x+4)/(1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/ (-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 12, 225, 3585, 52815, 755937, 10683855, 150187665, 2106340335, 29511178257, 413291493615, 5786893013265, 81021383159535, 1134328683700497, 15880777622703855, 222331943560384785, 3112653553047564015, 43577187810470007057 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [5, 3]](x) = 4 3 2 x (196 x - 20 x - 61 x + 15) - ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[5, 3]](x) = -x^4*(196*x^3-20*x^2-61*x+15)/(1+x)/(-1+2*x)/(-1+x)/(1+2* x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 15, 299, 4771, 70399, 1007811, 14244783, 200248707, 2808447983, 39348214147, 551055231727, 7715856977283, 108028509388527, 1512438238965123, 21174370139746031, 296442591318388099, 4150204737014984431, 58102917079099518339 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [5, 2, 1]](x) = 3 3 2 x (32 x + 4 x - 28 x + 5) ------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[5, 2, 1]](x) = x^3*(32*x^3+4*x^2-28*x+5)/(1+x)/(-1+2*x)/(-1+x)/(1+2*x )/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 5, 62, 865, 11990, 167505, 2343222, 32798225, 459145910, 6427927825, 89990523062, 1259865465105, 17638109053110, 246933496942865, 3457068837888182, 48398963253317905, 677585483637547190, 9486196763290571025, 132806754655525888182 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [5, 1, 1, 1]](x) = 2 5 4 3 2 x (294 x - 165 x - 154 x + 109 x - 20 x + 1) - ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[5, 1, 1, 1]](x) = -x^2*(294*x^5-165*x^4-154*x^3+109*x^2-20*x+1)/(1+x) /(-1+2*x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 4, 46, 530, 6921, 93744, 1294441, 18014560, 251565241, 3518096384, 49230530361, 689090787840, 9646452411961, 135045426663424, 1890606549466681, 26468315223941120, 370555354622529081, 5187768614831652864, 72628722513115385401 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [4, 4]](x) = 5 4 3 2 x (662 x - 463 x - 189 x + 142 x - 23 x + 1) - ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[4, 4]](x) = -x*(662*x^5-463*x^4-189*x^3+142*x^2-23*x+1)/(1+x)/(-1+2*x )/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 1, 7, 36, 320, 3362, 41056, 538882, 7332304, 101383458, 1411779920, 19719445538, 275799669328, 3859560890914, 54024049478224, 756277889630754, 10587537696717392, 148223411443933730, 2075115063311333968, 29051534708753834530 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [4, 3, 1]](x) = 3 4 3 2 x (196 x - 90 x - 72 x + 35 x - 4) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[4, 3, 1]](x) = x^3*(196*x^4-90*x^3-72*x^2+35*x-4)/(1+x)/(-1+2*x)/(-1+ x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 4, 61, 900, 12855, 181664, 2553831, 35819200, 501870535, 7028634624, 98415710151, 1377909478400, 19291272233415, 270081057808384, 3781154325795271, 52936277809152000, 741108593412567495, 10375524534668754944, 145257368856263029191 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [4, 2, 2]](x) = 2 5 4 3 2 x (56 x + 64 x - 22 x - 48 x + 16 x - 1) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[4, 2, 2]](x) = x^2*(56*x^5+64*x^4-22*x^3-48*x^2+16*x-1)/(1+x)/(-1+2*x )/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 8, 81, 910, 11441, 152278, 2084881, 28906550, 403005201, 5631965878, 78786941201, 1102653913270, 15434976145681, 216076597887158, 3024993976455441, 42349445366079670, 592889413538091281, 8300434860964929718, 116205986485917258001 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 3 3 x F[[4, 4], [4, 2, 1, 1]](x) = - --------------------------------- (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[4, 4],[4, 2, 1, 1]](x) = -3*x^3/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 66, 1080, 16080, 230928, 3267936, 45960960, 644712960, 9033539328, 126514899456, 1771480688640, 24802362224640, 347242866659328, 4861458906341376, 68060777327493120, 952852998417285120, 13339954672836476928, 186759441589677981696 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [4, 1, 1, 1, 1]](x) = 3 4 3 2 x (126 x + 65 x - 177 x + 55 x - 4) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[4, 1, 1, 1, 1]](x) = x^3*(126*x^4+65*x^3-177*x^2+55*x-4)/(1+x)/(-1+2* x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 4, 41, 525, 6880, 93639, 1293936, 18012775, 251557760, 3518067399, 49230413056, 689090322375, 9646450544640, 135045419209159, 1890606519627776, 26468315104645575, 370555354145259520, 5187768612922814919, 72628722505479684096 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 4 21 x F[[4, 4], [3, 3, 2]](x) = - ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[4, 4],[3, 3, 2]](x) = -21*x^4/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 21, 441, 7119, 105441, 1511055, 21364497, 300362223, 4212628497, 59022146799, 826582149393, 11573782671087, 162042752901393, 2268657313713903, 31761555030675729, 444663886261776111, 6225307102659219729, 87154375607196118767 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [3, 3, 1, 1]](x) = 2 5 4 3 2 x (280 x - 312 x + 66 x + 46 x - 16 x + 1) - ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[3, 3, 1, 1]](x) = -x^2*(280*x^5-312*x^4+66*x^3+46*x^2-16*x+1)/(1+x)/( -1+2*x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 8, 79, 906, 11415, 152194, 2084503, 28905122, 402999319, 5631942690, 78786847767, 1102653540898, 15434974653463, 216076591923746, 3024993952590871, 42349445270643234, 592889413156301847, 8300434859437859362, 116205986479808801815 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [3, 2, 2, 1]](x) = 3 4 3 2 x (28 x - 134 x + 30 x + 14 x - 3) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[3, 2, 2, 1]](x) = x^3*(28*x^4-134*x^3+30*x^2+14*x-3)/(1+x)/(-1+2*x)/( -1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 58, 885, 12800, 181433, 2552928, 35815545, 501856000, 7028576313, 98415477248, 1377908546105, 19291268505600, 270081042894393, 3781154266144768, 52936277570539065, 741108592458137600, 10375524530850991673, 145257368840992063488 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [3, 2, 1, 1, 1]](x) = 3 2 x (2 x - 3) (16 x - 2 x - 1) ------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[3, 2, 1, 1, 1]](x) = x^3*(2*x-3)*(16*x^2-2*x-1)/(1+x)/(-1+2*x)/(-1+x) /(1+2*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 58, 839, 11906, 167127, 2341794, 32792343, 459122722, 6427834391, 89990150690, 1259863972887, 17638103089698, 246933473078295, 3457068742451746, 48398962871528471, 677585482110476834, 9486196757182114839, 132806754631092412962 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [3, 1, 1, 1, 1, 1]](x) = 4 2 3 x (70 x - 3 x - 3) - -------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[3, 1, 1, 1, 1, 1]](x) = -3*x^4*(70*x^2-3*x-3)/(1+x)/(-1+2*x)/(1+2*x)/ (-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 9, 216, 3534, 52626, 755118, 10680642, 150174558, 2106288162, 29510968542, 413290655778, 5786889657822, 81021369741858, 1134328630013406, 15880777407971874, 222331942701391326, 3112653549611655714, 43577187796726111710 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [2, 2, 2, 2]](x) = 2 4 3 2 x (94 x + 115 x - 98 x + 20 x - 1) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[2, 2, 2, 2]](x) = x^2*(94*x^4+115*x^3-98*x^2+20*x-1)/(1+x)/(-1+2*x)/( -1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 4, 35, 305, 3341, 40929, 538525, 7330705, 101377661, 1411756049, 19719352445, 275799294225, 3859559400061, 54024043503889, 756277865771645, 10587537601237265, 148223411062166141, 2075115061784088849, 29051534702645465725 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [2, 2, 2, 1, 1]](x) = 4 3 2 x (28 x - 148 x - 23 x + 13) - ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[2, 2, 2, 1, 1]](x) = -x^4*(28*x^3-148*x^2-23*x+13)/(1+x)/(-1+2*x)/(-1 +x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 13, 289, 4721, 70189, 1006929, 14241213, 200234257, 2808390013, 39347981585, 551054300797, 7715853250833, 108028494479997, 1512438179320081, 21174369901154941, 296442590363980049, 4150204733197308541, 58102917063828640017 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [2, 2, 1, 1, 1, 1]](x) = 3 4 3 2 x (4 x + 12 x - 121 x + 43 x - 3) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[2, 2, 1, 1, 1, 1]](x) = x^3*(4*x^4+12*x^3-121*x^2+43*x-3)/(1+x)/(-1+2 *x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 29, 340, 4185, 55088, 749049, 10351680, 144101945, 2012462848, 28144538169, 393843307520, 5512722574905, 77171604369408, 1080363353656889, 15124852157071360, 211746520837099065, 2964442833162272768, 41502148903382257209 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [2, 1, 1, 1, 1, 1, 1]](x) = 4 3 2 x (14 x + 53 x + x - 3) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[2, 1, 1, 1, 1, 1, 1]](x) = x^4*(14*x^3+53*x^2+x-3)/(1+x)/(-1+2*x)/(-1 +x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 71, 1174, 17521, 251622, 3559857, 50056758, 702090257, 9836966326, 137763458833, 1928962846902, 27007121756433, 378109537374390, 5293592445464849, 74110647471694006, 1037551182822117649, 14525729264048300214 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[4, 4], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (94 x + 115 x - 98 x + 20 x - 1) ----------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (-1 + 14 x) and in Maple notation F[[4, 4],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(94*x^4+115*x^3-98*x^2+20*x-1)/(1+x )/(-1+2*x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 1, 4, 35, 305, 3341, 40929, 538525, 7330705, 101377661, 1411756049, 19719352445, 275799294225, 3859559400061, 54024043503889, 756277865771645, 10587537601237265, 148223411062166141, 2075115061784088849 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 Regarding Lambda=, [4, 3, 1] 8 7 6 5 4 F[[4, 3, 1], [8]](x) = (31450 x + 4825 x - 62973 x - 20204 x + 14039 x 3 2 + 4992 x - 4 x - 71 x + 1)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[8]](x) = (31450*x^8+4825*x^7-62973*x^6-20204*x^5+14039*x^4+4992*x^ 3-4*x^2-71*x+1)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70* x) The first 20 term , starting with k=1 are 0, 1, 8, 608, 41733, 2918666, 204258288, 14297692753, 1000834193213, 70058352468656, 4904084253292818, 343285893577242023, 24030012508676465343, 1682100875190991973446, 117747061259201292931748, 8242294288102431024153293, 576960600166753469923983873, 40387242011668576400201111036, 2827106940816758680499710317078, 197897485857172690972498585052563 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [7, 1]](x) = 2 5 4 3 2 x (5250 x + 175 x - 2370 x - 32 x + 84 x - 2) - ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[7, 1]](x) = -x^2*(5250*x^5+175*x^4-2370*x^3-32*x^2+84*x-2)/(-1+x)/ (1+2*x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 60, 4218, 292070, 20428947, 1429795500, 100083704303, 7005837990960, 490408453266267, 34328589634743140, 2403001253649070263, 168210087546859339800, 11774706126197992871987, 824229428813020460695980, 57696060016703126829638223, 4038724201167135407652695840, 282710694081678645877866340107, 19789748585717296874779482460020, 1385282401000208697917760664526183 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [6, 2]](x) = - x 6 5 4 3 2 (4500 x + 4700 x - 4495 x - 2086 x + 380 x + 110 x - 4)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[6, 2]](x) = -x^2*(4500*x^6+4700*x^5-4495*x^4-2086*x^3+380*x^2+110* x-4)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 174, 11994, 834310, 58364889, 4085102739, 285953131804, 20016677050740, 1401166979324199, 98081684374234429, 6865717864587073914, 480600250104148314270, 33642017503125090364309, 2354941225177082887108119, 164845885761979168875424824, 11539212003334374989045703400, 807744840233364583387746909219, 56542138816335104166396030505809, 3957949717143453125001347454208534 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [6, 1, 1]](x) = 2 5 4 3 2 x (1750 x - 2105 x - 1231 x + 384 x + 99 x - 4) ------------------------------------------------------------------------ (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[6, 1, 1]](x) = x^2*(1750*x^5-2105*x^4-1231*x^3+384*x^2+99*x-4)/(1+ 2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 181, 12586, 875906, 61282159, 4289347146, 300250685641, 21017509855096, 1471225317903859, 102985768488638486, 7209003756775426621, 504630262598935891236, 35324118378177193447159, 2472688286434895291153026, 173088180050067711010682401, 12116172603500989570080806576, 848132082245031770899059104059, 59369245757151848958006851966766, 4155847203000625677084957150262981 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [5, 3]](x) = - x 6 5 4 3 2 (3500 x - 9900 x - 5385 x + 1690 x + 746 x + 38 x - 4)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[5, 3]](x) = -x^2*(3500*x^6-9900*x^5-5385*x^4+1690*x^3+746*x^2+38*x -4)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 246, 16740, 1167860, 81707599, 5719118181, 400334096950, 28023345138930, 1961633743060869, 137314357847173991, 9612005007639217660, 672840350118053562900, 47098824504097233584539, 3296917715245138826487801, 230784240066743055889934370, 16154896804667847220460861270, 1130842776326707638898053622609, 79158994342869118055508300277611, 5541129604000834097222464225295080 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 2 x (500 x + 130 x - 9) F[[4, 3, 1], [5, 2, 1]](x) = ------------------------------------------ (-1 + x) (1 + 5 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[5, 2, 1]](x) = x^2*(500*x^2+130*x-9)/(-1+x)/(1+5*x)/(-1+10*x)/(-1+ 70*x) The first 20 term , starting with k=1 are 0, 9, 554, 38229, 2668854, 186755729, 13072221354, 915048893229, 64053355533854, 4483734222330729, 313861388888346354, 21970297155558268229, 1537920800222208658854, 107654456008888956705729, 7535811920555555216471354, 527506834438222223917643229, 36925478410668888880411783854, 2584783488746755555597941080729, 180934844212272222222010294596354, 12665439094859048888889948527018229 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [5, 1, 1, 1]](x) = - x 6 5 4 3 2 (8750 x + 1125 x - 940 x - 38 x + 367 x + 56 x - 5)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[5, 1, 1, 1]](x) = -x^2*(8750*x^6+1125*x^5-940*x^4-38*x^3+367*x^2+ 56*x-5)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 5, 299, 20887, 1459235, 102129595, 7148844234, 500417106782, 35029176185435, 2452042126882585, 171642946787472644, 12015006254343843052, 841050437595468458085, 58873530629600782010375, 4121147144051214842738654, 288480300083376738275121322, 20193621005834288183669109935, 1413553470408379340331475490965, 98948742928586345485843338282264, 6926412005001042100695780445267592 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [4, 4]](x) = - x 6 5 4 3 2 (7700 x + 250 x - 5358 x + 112 x + 384 x + 19 x - 2)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[4, 4]](x) = -x^2*(7700*x^6+250*x^5-5358*x^4+112*x^3+384*x^2+19*x-2 )/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 123, 8359, 583882, 40853002, 2859552633, 200166976739, 14011671885852, 980816864534452, 68657178854389543, 4806002503123972669, 336420175052088121722, 23549412251979144121502, 1648458857621875107138453, 115392120033364582820005399, 8077448402333854169143847992, 565421388163353124987971986152, 39579497171434552083392041827363, 2770564802000416979166378850734929 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [4, 3, 1]](x) = - x 6 5 4 3 2 (24550 x + 13975 x - 7027 x - 3580 x - 35 x + 63 x - 1)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[4, 3, 1]](x) = -x*(24550*x^6+13975*x^5-7027*x^4-3580*x^3-35*x^2+63 *x-1)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 1, 8, 608, 41733, 2918666, 204258288, 14297692753, 1000834193213, 70058352468656, 4904084253292818, 343285893577242023, 24030012508676465343, 1682100875190991973446, 117747061259201292931748, 8242294288102431024153293, 576960600166753469923983873, 40387242011668576400201111036, 2827106940816758680499710317078, 197897485857172690972498585052563, 13852824010002084201387518528160803 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [4, 2, 2]](x) = 2 2 x (590 x + 38 x - 7) ---------------------------------------------------- (-1 + x) (1 + 5 x) (1 + 2 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[4, 2, 2]](x) = x^2*(590*x^2+38*x-7)/(-1+x)/(1+5*x)/(1+2*x)/(-1+10* x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 7, 480, 33369, 2334616, 163403997, 11438125860, 800667079009, 56046679188336, 3923267374891557, 274628714583875740, 19224010004163954249, 1345680700125013562856, 94197649007083265517517, 6593835430479167005748820, 461568480133374998304583089, 32309793609334583341810430176, 2261685552653404166624281157877, 158317988685738125000211927593100, 11082259208001667083332273695269529 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 x (5 x - 11) F[[4, 3, 1], [4, 2, 1, 1]](x) = - -------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[4, 2, 1, 1]](x) = -x^2*(5*x-11)/(1+2*x)/(-1+2*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 11, 765, 53594, 3751560, 262609376, 18382656240, 1286785937504, 90075015624960, 6305251093750016, 441367576562499840, 30895730359375000064, 2162701125156249999360, 151389078760937500000256, 10597235513265624999997440, 741806485928593750000001024, 51926454015001562499999989760, 3634851781050109375000000004096, 254439624673507656249999999959040, 17810773727145535937500000000016384 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [4, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (1750 x - 2325 x - 2125 x - 408 x - 10 x + 9 x + 4)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[4, 1, 1, 1, 1]](x) = x^2*(1750*x^6-2325*x^5-2125*x^4-408*x^3-10*x^ 2+9*x+4)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 293, 20813, 1458560, 102122569, 7148775123, 500416410988, 35029169246420, 2452042057416359, 171642946093115453, 12015006247399049338, 841050437526025411230, 58873530628906331974549, 4121147144044270420661783, 288480300083307293741202488, 20193621005833593739582571240, 1413553470408372395885599407139, 98948742928586276041404620428113, 6926412005001041406251313094396438 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [3, 3, 2]](x) = 2 3 2 x (175 x + 70 x - 39 x - 5) - -------------------------------------------------- (1 + 2 x) (1 + x) (1 + 5 x) (-1 + 2 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[3, 3, 2]](x) = -x^2*(175*x^3+70*x^2-39*x-5)/(1+2*x)/(1+x)/(1+5*x)/ (-1+2*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 5, 359, 25001, 1750780, 122550785, 8578574214, 600500097671, 42035007324200, 2942450510253965, 205971535729980394, 14418007501037597891, 1009260525072937011420, 70648236755104064942345, 4945376572857292175291774, 346176360100010414123538911, 24232345207000729179382319440, 1696264164490051041603088393925, 118738491514303572916984558086354, 8311694406001250104165077209532731 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [3, 3, 1, 1]](x) = 2 3 2 x (1400 x + 500 x - 31 x - 6) ---------------------------------------------------- (-1 + x) (1 + 5 x) (1 + 2 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[3, 3, 1, 1]](x) = x^2*(1400*x^3+500*x^2-31*x-6)/(-1+x)/(1+5*x)/(1+ 2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 6, 475, 33312, 2334063, 163398436, 11438070315, 800666523432, 56046673632823, 3923267319335916, 274628714028320355, 19224009998608398352, 1345680700069458007983, 94197649006527709960596, 6593835430473611450195995, 461568480133319442749022072, 32309793609334027786254885543, 2261685552653398611068725580476, 158317988685738069444656372081235, 11082259208001666527776718139626592 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [3, 2, 2, 1]](x) = x 5 4 3 2 (2750 x + 545 x - 46 x - 172 x + 20 x + 8)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[3, 2, 2, 1]](x) = x^2*(2750*x^5+545*x^4-46*x^3-172*x^2+20*x+8)/(-1 +x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 8, 588, 41616, 2917190, 204244743, 14297552493, 1000832809796, 70058338557900, 4904084114491353, 343285892188003523, 24030012494788974726, 1682100875052097491810, 117747061257812426413163, 8242294288088542045784553, 576960600166614581393011656, 40387242011667187509880560920, 2827106940816744791616548062173, 197897485857172552083586789649583, 13852824010002082812498721265284586 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 2 x (200 x + 7) F[[4, 3, 1], [3, 2, 1, 1, 1]](x) = - ------------------------------------------ (-1 + x) (1 + 5 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[3, 2, 1, 1, 1]](x) = -x^2*(200*x^2+7)/(-1+x)/(1+5*x)/(-1+10*x)/(-1 +70*x) The first 20 term , starting with k=1 are 0, 7, 532, 38007, 2666632, 186733507, 13071999132, 915046671007, 64053333311632, 4483734000108507, 313861386666124132, 21970297133336046007, 1537920799999986436632, 107654456006666734483507, 7535811920533332994249132, 527506834438000001695421007, 36925478410666666658189561632, 2584783488746733333375718858507, 180934844212271999999788072374132, 12665439094859046666667726304796007 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [3, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (2450 x - 225 x - 558 x - 256 x - 32 x - 2) ------------------------------------------------------------------------ (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[3, 1, 1, 1, 1, 1]](x) = x^2*(2450*x^5-225*x^4-558*x^3-256*x^2-32*x -2)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 172, 12446, 874718, 61269407, 4289223162, 300249431561, 21017497371448, 1471225192838387, 102985767238900502, 7209003744274378301, 504630262473940085028, 35324118376927176670967, 2472688286422395358259842, 173088180049942710742251041, 12116172603499739571154540208, 848132082245019270894764153147, 59369245757151723958024031803182, 4155847203000624427084888430851781 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [2, 2, 2, 2]](x) = - x 6 5 4 3 2 (3500 x + 10670 x - 2216 x - 2525 x - 139 x + 27 x - 2)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[2, 2, 2, 2]](x) = -x^2*(3500*x^6+10670*x^5-2216*x^4-2525*x^3-139*x ^2+27*x-2)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 115, 8314, 583284, 40847617, 2859496395, 200166423914, 14011666319374, 980816809022587, 68657178298659225, 4806002497569116164, 336420174996529769964, 23549412251423599750757, 1648458857616319506843655, 115392120033309027443406814, 8077448402333298612872464554, 565421388163347569435279742127, 39579497171434496527825033025685, 2770564802000416423610869108163864 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [2, 2, 2, 1, 1]](x) = x 6 5 4 3 2 (3500 x - 700 x - 1045 x + 1064 x + 266 x + 17 x + 3)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[2, 2, 2, 1, 1]](x) = x^2*(3500*x^6-700*x^5-1045*x^4+1064*x^3+266*x ^2+17*x+3)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 3, 230, 16611, 1166430, 81693878, 5718978615, 400332710781, 28023331239140, 1961633604215628, 137314356458110425, 9612004993751027651, 672840349979161878150, 47098824502708355879778, 3296917715231249892861035, 230784240066604167179999721, 16154896804666458330856149960, 1130842776326693750012028034328, 79158994342868979166607958164445, 5541129604000832708333621149346991 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[4, 3, 1], [2, 2, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (11500 x + 2500 x - 3065 x - 1336 x - 261 x - 21 x - 2)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[2, 2, 1, 1, 1, 1]](x) = -x^2*(11500*x^6+2500*x^5-3065*x^4-1336*x^3 -261*x^2-21*x-2)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70 *x) The first 20 term , starting with k=1 are 0, 2, 163, 11844, 832965, 58350827, 4084964538, 285951740174, 20016663172795, 1401166840391577, 98081682985520388, 6865717850697485804, 480600249965262221925, 33642017501736190289927, 2354941225163194042959838, 164845885761840279807576234, 11539212003332986100872647855, 807744840233350694495994697877, 56542138816334965277518594884888, 3957949717143451736112412752291464 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[4, 3, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (12250 x - 775 x - 2245 x + 28 x + 57) ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x) and in Maple notation F[[4, 3, 1],[2, 1, 1, 1, 1, 1, 1]](x) = x^3*(12250*x^4-775*x^3-2245*x^2+28*x+57 )/(-1+x)/(1+2*x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 0, 57, 4132, 291440, 20421745, 1429727082, 100083005757, 7005831062910, 490408383756265, 34328588940560882, 2403001246703577157, 168210087477419089830, 11774706125503531649985, 824229428806076083361082, 57696060016633682116756957, 4038724201166440964281995950, 282710694081671701429126922905, 19789748585717227430352217895682, 1385282401000208003473247500583157 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 3 F[[4, 3, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (2750 x + 545 x - 46 x - 172 x + 20 x + 8)/((-1 + x) (1 + 2 x) (1 + x) (1 + 4 x) (1 + 5 x) (-1 + 2 x) (-1 + 10 x) (-1 + 70 x)) and in Maple notation F[[4, 3, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(2750*x^5+545*x^4-46*x^3-172*x^2 +20*x+8)/(-1+x)/(1+2*x)/(1+x)/(1+4*x)/(1+5*x)/(-1+2*x)/(-1+10*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 0, 8, 588, 41616, 2917190, 204244743, 14297552493, 1000832809796, 70058338557900, 4904084114491353, 343285892188003523, 24030012494788974726, 1682100875052097491810, 117747061257812426413163, 8242294288088542045784553, 576960600166614581393011656, 40387242011667187509880560920, 2827106940816744791616548062173, 197897485857172552083586789649583 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 Regarding Lambda=, [4, 2, 2] F[[4, 2, 2], [8]](x) = 7 6 5 4 3 2 7648 x + 2896 x - 14762 x - 5234 x + 1894 x + 304 x - 62 x + 1 --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[8]](x) = (7648*x^7+2896*x^6-14762*x^5-5234*x^4+1894*x^3+304*x^2-62 *x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 6, 259, 13752, 765647, 42840288, 2398787455, 134329951104, 7522460377663, 421257646149120, 23590427108887871, 1321063909495289856, 73979578862989179199, 4142856415777438359552, 231999959279138060249407, 12991997719596543804407808, 727551872297124971007502655, 40742904848636746525448208384, 2281602671523639790913585670463 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [7, 1]](x) = 2 5 4 3 2 x (1120 x + 1600 x - 618 x - 209 x + 33 x - 1) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[7, 1]](x) = -x^2*(1120*x^5+1600*x^4-618*x^3-209*x^2+33*x-1)/(1+x)/ (-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 29, 1704, 95529, 5353760, 299837433, 16791157888, 940306847033, 52657200198144, 2948803343854137, 165132988329347072, 9247447355010055737, 517857051949278896128, 28999994909709001117241, 1623999714948102049529856, 90943984037128893180317241, 5092863106079499492120461312, 285200333940454223263114956345, 15971218700665454517117677731840 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [6, 2]](x) = 2 5 4 3 2 x (1152 x - 1312 x - 1780 x - 75 x + 93 x - 3) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[6, 2]](x) = -x^2*(1152*x^5-1312*x^4-1780*x^3-75*x^2+93*x-3)/(1+x)/ (-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 3, 93, 4932, 273535, 15301008, 856715943, 47975034432, 2686593389895, 150449162565888, 8425152564428743, 471808539310482432, 26421278166989056455, 1479591577076468256768, 82857128314082587660743, 4639999185571032066834432, 259839954391837048471286215, 14551037445941748803986587648, 814858096972728925667466047943, 45632053430472747779616080658432 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [6, 1, 1]](x) = 2 4 3 2 x (112 x - 1368 x + 79 x + 21 x + 1) ----------------------------------------------------------- (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[6, 1, 1]](x) = x^2*(112*x^4-1368*x^3+79*x^2+21*x+1)/(1+x)/(-1+x)/( 1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 85, 5088, 286541, 16060928, 899511501, 50373468160, 2820920528077, 157971600506880, 8846410031353037, 495398964986642432, 27742342065026813133, 1553571155847814316032, 86999984729126949670093, 4871999144844305790664704, 272831952111386678681980109, 15278589318238498470634717184, 855601001821362669775601061069, 47913656101996363551261407051776 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [5, 3]](x) = 2 5 4 3 2 x (896 x + 448 x + 500 x + 85 x - 6 x + 2) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[5, 3]](x) = x^2*(896*x^5+448*x^4+500*x^3+85*x^2-6*x+2)/(1+x)/(-1+x )/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 118, 6795, 382136, 21414743, 1199349984, 67164626951, 3761227391872, 210628800719559, 11795213375475200, 660531953316222407, 36989789420041156608, 2071428207797096939975, 115999979638836019388416, 6495998859792407899845063, 363775936148515572959903744, 20371452424317997963709608391, 1140801335761816893056277676032, 63884874802661818068394355749319 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [5, 2, 1]](x) = 2 4 3 2 x (256 x + 432 x + 96 x - 9 x - 5) ----------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[5, 2, 1]](x) = x^2*(256*x^4+432*x^3+96*x^2-9*x-5)/(1+x)/(-1+x)/(1+ 2*x)/(1+4*x)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 279, 15615, 874179, 48954023, 2741420403, 153519543623, 8597094362163, 481437284302791, 26960487919654707, 1509787323501029831, 84548090116036805427, 4734693046498067042759, 265142810603891420394291, 14847997393817919637418439, 831487854053803494350795571, 46563319827012995685171229127, 2607545910312727758284072104755, 146022570977512754463932469768647 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [5, 1, 1, 1]](x) = 2 5 4 3 2 x (1120 x + 464 x + 238 x + 74 x + 32 x - 3) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[5, 1, 1, 1]](x) = -x^2*(1120*x^5+464*x^4+238*x^3+74*x^2+32*x-3)/(1 +x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 3, 154, 8565, 478280, 26773673, 1499229984, 83956129785, 4701537016960, 263286023214393, 14744016897995264, 825664943076059705, 46237236786499799040, 2589285259837983161913, 144999974549277953695744, 8119998574746373713120825, 454719920185691377443307520, 25464315530397872751026802233, 1426001669702274118700055527424, 79856093503327296604633691033145 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [4, 4]](x) = 2 5 4 3 2 x (2240 x + 2544 x + 852 x - 193 x - 55 x + 2) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[4, 4]](x) = x^2*(2240*x^5+2544*x^4+852*x^3-193*x^2-55*x+2)/(1+x)/( -1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 69, 3479, 191743, 10712771, 599718567, 33582662003, 1880616490311, 105314422713395, 5897606866665415, 330265978089510707, 18494894721473358279, 1035714103990170338099, 57999989820151009989063, 3247999429902067946320691, 181887968074304698856763847, 10185726212159374280773874483, 570400667880911448925981536711, 31942437401330933053378392437555 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [4, 3, 1]](x) = 2 5 4 3 2 x (3136 x + 912 x - 364 x + 163 x + 8 x - 5) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[4, 3, 1]](x) = -x^2*(3136*x^5+912*x^4-364*x^3+163*x^2+8*x-5)/(1+x) /(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 302, 17046, 955960, 53541890, 2998417632, 167911910066, 9403071259520, 526572024059250, 29488033617382912, 1651329884720463986, 92474473561551943680, 5178570519584340356210, 289999949097822989115392, 16239997149486883364232306, 909439840371335843822141440, 50928631060795370202084678770, 2852003339404545235023265923072, 159712187006654569190069369511026 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [4, 2, 2]](x) = 5 4 3 2 x (6688 x + 2224 x - 1462 x - 190 x + 56 x - 1) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[4, 2, 2]](x) = -x*(6688*x^5+2224*x^4-1462*x^3-190*x^2+56*x-1)/(1+x )/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 6, 259, 13752, 765647, 42840288, 2398787455, 134329951104, 7522460377663, 421257646149120, 23590427108887871, 1321063909495289856, 73979578862989179199, 4142856415777438359552, 231999959279138060249407, 12991997719596543804407808, 727551872297124971007502655, 40742904848636746525448208384, 2281602671523639790913585670463, 127769749605323684175154194677760 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 6 x F[[4, 2, 2], [4, 2, 1, 1]](x) = --------------------- (8 x - 1) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[4, 2, 1, 1]](x) = 6*x^2/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 6, 384, 21888, 1228800, 68837376, 3855089664, 215886594048, 12089661849600, 677021164240896, 37913186002796544, 2123138422599057408, 118895751717086822400, 6658162096569178914816, 372857077411172554113024, 20879996335052051309395968, 1169279794763125979558707200, 65479668506736743705147867136, 3666861436377271158287162671104, 205344240437127292950472166473728 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [4, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (1120 x - 1328 x - 1746 x - 58 x + 91 x - 4) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[4, 1, 1, 1, 1]](x) = -x^2*(1120*x^5-1328*x^4-1746*x^3-58*x^2+91*x-\ 4)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 4, 157, 8580, 478335, 26773904, 1499230887, 83956133440, 4701537031495, 263286023272704, 14744016898228167, 825664943076992000, 46237236786503526855, 2589285259837998075904, 144999974549278013346247, 8119998574746373951733760, 454719920185691378397737415, 25464315530397872754844565504, 1426001669702274118715326493127, 79856093503327296604694775070720 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 2 x (28 x + 1) F[[4, 2, 2], [3, 3, 2]](x) = ------------------------------- (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[3, 3, 2]](x) = 2*x^2*(28*x+1)/(1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 176, 10176, 573184, 32121856, 1799024640, 100746936320, 5641841082368, 315943201013760, 17692820063125504, 990797929973284864, 55484684130060337152, 3107142311695628632064, 173999969458254006714368, 9743998289688611581329408, 545663904222773359081947136, 30557178636476996941269434368, 1711202003642725339578689912832, 95827312203992727102522814103552 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [3, 3, 1, 1]](x) = 2 4 3 2 x (480 x + 336 x - 378 x - 58 x + 5) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[3, 3, 1, 1]](x) = x^2*(480*x^4+336*x^3-378*x^2-58*x+5)/(1+x)/(-1+x )/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 252, 13731, 765552, 42839931, 2398785984, 134329945307, 7522460354304, 421257646056027, 23590427108514816, 1321063909493799003, 73979578862983213056, 4142856415777414500443, 231999959279137964802048, 12991997719596543422640219, 727551872297124969480388608, 40742904848636746519339839579, 2281602671523639790889152020480, 127769749605323684175056460427355 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [3, 2, 2, 1]](x) = 2 5 4 3 2 x (448 x + 3056 x + 628 x - 281 x - 6 x + 5) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[3, 2, 2, 1]](x) = x^2*(448*x^5+3056*x^4+628*x^3-281*x^2-6*x+5)/(1+ x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 304, 17052, 955990, 53542000, 2998418094, 167911911872, 9403071266830, 526572024088320, 29488033617499534, 1651329884720929792, 92474473561553808270, 5178570519584347811840, 289999949097823018943374, 16239997149486883483533312, 909439840371335844299367310, 50928631060795370203993538560, 2852003339404545235030901449614, 159712187006654569190099911442432 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [3, 2, 1, 1, 1]](x) = 2 4 3 2 x (256 x - 16 x - 8 x - 7 x - 5) ----------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[3, 2, 1, 1, 1]](x) = x^2*(256*x^4-16*x^3-8*x^2-7*x-5)/(1+x)/(-1+x) /(1+2*x)/(1+4*x)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 277, 15611, 874153, 48953939, 2741420025, 153519542195, 8597094356281, 481437284279603, 26960487919561273, 1509787323500657459, 84548090116035313209, 4734693046498061079347, 265142810603891396529721, 14847997393817919541982003, 831487854053803493969006137, 46563319827012995683644158771, 2607545910312727758277963648569, 146022570977512754463908036293427 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [3, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (1680 x - 232 x - 331 x + 40 x - 2) - ----------------------------------------------------------- (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[3, 1, 1, 1, 1, 1]](x) = -x^2*(1680*x^4-232*x^3-331*x^2+40*x-2)/(1+ x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 88, 5101, 286592, 16061133, 899512320, 50373471437, 2820920541184, 157971600559309, 8846410031562752, 495398964987481293, 27742342065030168576, 1553571155847827737805, 86999984729127003357184, 4871999144844306005413069, 272831952111386679540973568, 15278589318238498474070691021, 855601001821362669789344956416, 47913656101996363551316382633165 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [2, 2, 2, 2]](x) = 2 5 4 3 2 x (1344 x - 2160 x - 972 x + 195 x + 55 x - 2) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[2, 2, 2, 2]](x) = -x^2*(1344*x^5-2160*x^4-972*x^3+195*x^2+55*x-2)/ (1+x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 69, 3477, 191739, 10712745, 599718483, 33582661625, 1880616488883, 105314422707513, 5897606866642227, 330265978089417273, 18494894721472985907, 1035714103990168845881, 57999989820151004025651, 3247999429902067922456121, 181887968074304698761327411, 10185726212159374280392085049, 570400667880911448924454466355, 31942437401330933053372283981369 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [2, 2, 2, 1, 1]](x) = 2 5 4 3 2 x (896 x - 1344 x - 1484 x - 47 x + 53 x + 1) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[2, 2, 2, 1, 1]](x) = x^2*(896*x^5-1344*x^4-1484*x^3-47*x^2+53*x+1) /(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 115, 6780, 382081, 21414512, 1199349081, 67164623296, 3761227377337, 210628800661248, 11795213375242297, 660531953315290112, 36989789420037428793, 2071428207797082025984, 115999979638835959737913, 6495998859792407661232128, 363775936148515572005473849, 20371452424317997959891845120, 1140801335761816893041006710329, 63884874802661818068333271711744 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [2, 2, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (1152 x + 480 x + 204 x + 57 x + 34 x - 2) - --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[2, 2, 1, 1, 1, 1]](x) = -x^2*(1152*x^5+480*x^4+204*x^3+57*x^2+34*x -2)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 90, 4917, 273480, 15300777, 856715040, 47975030777, 2686593375360, 150449162507577, 8425152564195840, 471808539309550137, 26421278166985328640, 1479591577076453342777, 82857128314082528010240, 4639999185571031828221497, 259839954391837047516856320, 14551037445941748800168824377, 814858096972728925652195082240, 45632053430472747779554996620857 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [2, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (2464 x + 576 x - 1102 x - 41 x + 28) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[2, 1, 1, 1, 1, 1, 1]](x) = x^3*(2464*x^4+576*x^3-1102*x^2-41*x+28) /(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 28, 1695, 95504, 5353639, 299836992, 16791156039, 940306839808, 52657200168903, 2948803343737856, 165132988328880583, 9247447355008192512, 517857051949271437767, 28999994909708971294720, 1623999714948101930217927, 90943984037128892703113216, 5092863106079499490211557831, 285200333940454223255479517184, 15971218700665454517087135625671 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[4, 2, 2], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (480 x + 336 x - 378 x - 58 x + 5) --------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 2, 2],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(480*x^4+336*x^3-378*x^2-58*x+5) /(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(8*x-1)/(-1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 5, 252, 13731, 765552, 42839931, 2398785984, 134329945307, 7522460354304, 421257646056027, 23590427108514816, 1321063909493799003, 73979578862983213056, 4142856415777414500443, 231999959279137964802048, 12991997719596543422640219, 727551872297124969480388608, 40742904848636746519339839579, 2281602671523639790889152020480 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 Regarding Lambda=, [4, 2, 1, 1] 4 3 2 816 x + 640 x - 457 x - 85 x + 1 F[[4, 2, 1, 1], [8]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[8]](x) = (816*x^4+640*x^3-457*x^2-85*x+1)/(1+x)/(-1+2*x)/(1+6*x )/(-1+90*x) The first 20 term , starting with k=1 are 0, 1, 17, 1639, 146393, 13180951, 1186250057, 106762714039, 9608643001673, 864777877704631, 77830008948058697, 7004700805597363639, 630423072502130106953, 56738076525201505078711, 5106426887268076683825737, 459578419854127254182784439, 41362057786871450760617628233, 3722585200818430581150580043191, 335032668073658752227382234223177, 30152940126629287700921420880768439 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 2 x (24 x - 1) F[[4, 2, 1, 1], [7, 1]](x) = - -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[7, 1]](x) = -2*x^2*(24*x-1)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 2, 124, 11408, 1025056, 92264672, 8303761984, 747338928128, 67260501431296, 6053445141412352, 544810062651526144, 49032905639090843648, 4412961507515454939136, 397166535676407270367232, 35744988210876556377800704, 3217048938978890661733203968, 289534404508100156029600792576, 26058096405729014063822395277312, 2345228676515611265617065628401664, 211070580886405013906297606229721088 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 5 x (12 x - 1) F[[4, 2, 1, 1], [6, 2]](x) = ----------------------------- (1 + x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[6, 2]](x) = 5*x^2*(12*x-1)/(1+x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 5, 355, 32585, 2928775, 263613065, 23725035895, 2135254070345, 192172861292215, 17295557546532425, 1556600179006519735, 140094016111675167305, 12608461450044234710455, 1134761530504020306022985, 102128537745361592449576375, 9191568397082544731016827465, 827241155737429017328184749495, 74451704016368611610316605788745, 6700653361473175044623814650981815, 603058802532585754017971397808394825 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 4 x (9 x + 1) F[[4, 2, 1, 1], [6, 1, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[6, 1, 1]](x) = 4*x^2*(9*x+1)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 4, 380, 34168, 3075488, 276792064, 24911297600, 2242016714368, 201781504713728, 18160335421717504, 1634430187969694720, 147098716917181831168, 13238884522546909011968, 1191499607029218545926144, 107234964632629688724439040, 9651146816936671867653357568, 868603213524300468794079838208, 78174289217187042187235520937984, 7035686029546833796876586874306560, 633211742659215041718740478754029568 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 6 x F[[4, 2, 1, 1], [5, 3]](x) = - --------------------- (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[5, 3]](x) = -6*x^2/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 6, 504, 45576, 4100544, 369056736, 33215059584, 2989355642496, 269042006145024, 24213780563129856, 2179240250621220864, 196131622556272674816, 17651846030062363951104, 1588666142705625816293376, 142979952843506245102239744, 12868195755915562529386561536, 1158137618032400624823680630784, 104232385622916056251057916215296, 9380914706062445062493652502708224, 844282323545620055625038084983750656 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 13 x F[[4, 2, 1, 1], [5, 2, 1]](x) = - ------------------- (1 + x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[5, 2, 1]](x) = -13*x^2/(1+x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 13, 1157, 104143, 9372857, 843557143, 75920142857, 6832812857143, 614953157142857, 55345784142857143, 4981120572857142857, 448300851557142857143, 40347076640142857142857, 3631236897612857142857143, 326811320785157142857142857, 29413018870664142857142857143, 2647171698359772857142857142857, 238245452852379557142857142857143, 21442090756714160142857142857142857, 1929788168104274412857142857142857143 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 6 x (20 x + 1) F[[4, 2, 1, 1], [5, 1, 1, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[5, 1, 1, 1]](x) = 6*x^2*(20*x+1)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 6, 636, 56928, 5125920, 461319456, 41518833216, 3736694500608, 336302507996160, 30267225702022656, 2724050313287863296, 245164528195272818688, 22064807537578363084800, 1985832678382029821485056, 178724941054382821071077376, 16085244694894453073573511168, 1447672022540500781558558883840, 130290482028645070310648646598656, 11726143382578056328136108120211456, 1055352904432025069531183351278338048 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 4 x (24 x - 1) F[[4, 2, 1, 1], [4, 4]](x) = - -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[4, 4]](x) = -4*x^2*(24*x-1)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 4, 248, 22816, 2050112, 184529344, 16607523968, 1494677856256, 134521002862592, 12106890282824704, 1089620125303052288, 98065811278181687296, 8825923015030909878272, 794333071352814540734464, 71489976421753112755601408, 6434097877957781323466407936, 579068809016200312059201585152, 52116192811458028127644790554624, 4690457353031222531234131256803328, 422141161772810027812595212459442176 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 2 x (30 x + 7) F[[4, 2, 1, 1], [4, 3, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[4, 3, 1]](x) = 2*x^2*(30*x+7)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 14, 1264, 113912, 10251520, 922640864, 83037654784, 7473389071232, 672605015572480, 60534451406564864, 5448100626560610304, 490329056390636337152, 44129615075156181975040, 3971665356764062908145664, 357449882108765622551117824, 32170489389788906264693276672, 2895344045081001562411840307200, 260580964057290140625528958091264, 23452286765156112656246826251321344, 2110705808864050139062519042491809792 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 12 x F[[4, 2, 1, 1], [4, 2, 2]](x) = - --------------------- (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[4, 2, 2]](x) = -12*x^2/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 12, 1008, 91152, 8201088, 738113472, 66430119168, 5978711284992, 538084012290048, 48427561126259712, 4358480501242441728, 392263245112545349632, 35303692060124727902208, 3177332285411251632586752, 285959905687012490204479488, 25736391511831125058773123072, 2316275236064801249647361261568, 208464771245832112502115832430592, 18761829412124890124987305005416448, 1688564647091240111250076169967501312 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 x (264 x + 68 x - 1) F[[4, 2, 1, 1], [4, 2, 1, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[4, 2, 1, 1]](x) = -x*(264*x^2+68*x-1)/(1+x)/(-1+2*x)/(1+6*x)/(-\ 1+90*x) The first 20 term , starting with k=1 are 1, 17, 1639, 146393, 13180951, 1186250057, 106762714039, 9608643001673, 864777877704631, 77830008948058697, 7004700805597363639, 630423072502130106953, 56738076525201505078711, 5106426887268076683825737, 459578419854127254182784439, 41362057786871450760617628233, 3722585200818430581150580043191, 335032668073658752227382234223177, 30152940126629287700921420880768439, 2713764611396635893080185760430461513 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 6 x (20 x + 1) F[[4, 2, 1, 1], [4, 1, 1, 1, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[4, 1, 1, 1, 1]](x) = 6*x^2*(20*x+1)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 6, 636, 56928, 5125920, 461319456, 41518833216, 3736694500608, 336302507996160, 30267225702022656, 2724050313287863296, 245164528195272818688, 22064807537578363084800, 1985832678382029821485056, 178724941054382821071077376, 16085244694894453073573511168, 1447672022540500781558558883840, 130290482028645070310648646598656, 11726143382578056328136108120211456, 1055352904432025069531183351278338048 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 8 x (9 x + 1) F[[4, 2, 1, 1], [3, 3, 2]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[3, 3, 2]](x) = 8*x^2*(9*x+1)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 8, 760, 68336, 6150976, 553584128, 49822595200, 4484033428736, 403563009427456, 36320670843435008, 3268860375939389440, 294197433834363662336, 26477769045093818023936, 2382999214058437091852288, 214469929265259377448878080, 19302293633873343735306715136, 1737206427048600937588159676416, 156348578434374084374471041875968, 14071372059093667593753173748613120, 1266423485318430083437480957508059136 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 12 x F[[4, 2, 1, 1], [3, 3, 1, 1]](x) = - --------------------- (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[3, 3, 1, 1]](x) = -12*x^2/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 12, 1008, 91152, 8201088, 738113472, 66430119168, 5978711284992, 538084012290048, 48427561126259712, 4358480501242441728, 392263245112545349632, 35303692060124727902208, 3177332285411251632586752, 285959905687012490204479488, 25736391511831125058773123072, 2316275236064801249647361261568, 208464771245832112502115832430592, 18761829412124890124987305005416448, 1688564647091240111250076169967501312 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 2 x (30 x + 7) F[[4, 2, 1, 1], [3, 2, 2, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[3, 2, 2, 1]](x) = 2*x^2*(30*x+7)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 14, 1264, 113912, 10251520, 922640864, 83037654784, 7473389071232, 672605015572480, 60534451406564864, 5448100626560610304, 490329056390636337152, 44129615075156181975040, 3971665356764062908145664, 357449882108765622551117824, 32170489389788906264693276672, 2895344045081001562411840307200, 260580964057290140625528958091264, 23452286765156112656246826251321344, 2110705808864050139062519042491809792 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 13 x F[[4, 2, 1, 1], [3, 2, 1, 1, 1]](x) = - ------------------- (1 + x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[3, 2, 1, 1, 1]](x) = -13*x^2/(1+x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 13, 1157, 104143, 9372857, 843557143, 75920142857, 6832812857143, 614953157142857, 55345784142857143, 4981120572857142857, 448300851557142857143, 40347076640142857142857, 3631236897612857142857143, 326811320785157142857142857, 29413018870664142857142857143, 2647171698359772857142857142857, 238245452852379557142857142857143, 21442090756714160142857142857142857, 1929788168104274412857142857142857143 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 4 x (9 x + 1) F[[4, 2, 1, 1], [3, 1, 1, 1, 1, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[3, 1, 1, 1, 1, 1]](x) = 4*x^2*(9*x+1)/(-1+2*x)/(1+6*x)/(-1+90*x ) The first 20 term , starting with k=1 are 0, 4, 380, 34168, 3075488, 276792064, 24911297600, 2242016714368, 201781504713728, 18160335421717504, 1634430187969694720, 147098716917181831168, 13238884522546909011968, 1191499607029218545926144, 107234964632629688724439040, 9651146816936671867653357568, 868603213524300468794079838208, 78174289217187042187235520937984, 7035686029546833796876586874306560, 633211742659215041718740478754029568 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 4 x (24 x - 1) F[[4, 2, 1, 1], [2, 2, 2, 2]](x) = - -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[2, 2, 2, 2]](x) = -4*x^2*(24*x-1)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 4, 248, 22816, 2050112, 184529344, 16607523968, 1494677856256, 134521002862592, 12106890282824704, 1089620125303052288, 98065811278181687296, 8825923015030909878272, 794333071352814540734464, 71489976421753112755601408, 6434097877957781323466407936, 579068809016200312059201585152, 52116192811458028127644790554624, 4690457353031222531234131256803328, 422141161772810027812595212459442176 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 6 x F[[4, 2, 1, 1], [2, 2, 2, 1, 1]](x) = - --------------------- (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[2, 2, 2, 1, 1]](x) = -6*x^2/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 6, 504, 45576, 4100544, 369056736, 33215059584, 2989355642496, 269042006145024, 24213780563129856, 2179240250621220864, 196131622556272674816, 17651846030062363951104, 1588666142705625816293376, 142979952843506245102239744, 12868195755915562529386561536, 1158137618032400624823680630784, 104232385622916056251057916215296, 9380914706062445062493652502708224, 844282323545620055625038084983750656 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 5 x (12 x - 1) F[[4, 2, 1, 1], [2, 2, 1, 1, 1, 1]](x) = ----------------------------- (1 + x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[2, 2, 1, 1, 1, 1]](x) = 5*x^2*(12*x-1)/(1+x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 5, 355, 32585, 2928775, 263613065, 23725035895, 2135254070345, 192172861292215, 17295557546532425, 1556600179006519735, 140094016111675167305, 12608461450044234710455, 1134761530504020306022985, 102128537745361592449576375, 9191568397082544731016827465, 827241155737429017328184749495, 74451704016368611610316605788745, 6700653361473175044623814650981815, 603058802532585754017971397808394825 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 2 2 x (24 x - 1) F[[4, 2, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = - -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -2*x^2*(24*x-1)/(-1+2*x)/(1+6*x)/(-1 +90*x) The first 20 term , starting with k=1 are 0, 2, 124, 11408, 1025056, 92264672, 8303761984, 747338928128, 67260501431296, 6053445141412352, 544810062651526144, 49032905639090843648, 4412961507515454939136, 397166535676407270367232, 35744988210876556377800704, 3217048938978890661733203968, 289534404508100156029600792576, 26058096405729014063822395277312, 2345228676515611265617065628401664, 211070580886405013906297606229721088 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 F[[4, 2, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 2 2 x (264 x + 68 x - 1) - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation F[[4, 2, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^2*(264*x^2+68*x-1)/(1+x)/(-1+2 *x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 0, 1, 17, 1639, 146393, 13180951, 1186250057, 106762714039, 9608643001673, 864777877704631, 77830008948058697, 7004700805597363639, 630423072502130106953, 56738076525201505078711, 5106426887268076683825737, 459578419854127254182784439, 41362057786871450760617628233, 3722585200818430581150580043191, 335032668073658752227382234223177, 30152940126629287700921420880768439 ---------------------------------- Their sum is 4 3 2 624 x + 520 x - 384 x - 84 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 90 x) and in Maple notation (624*x^4+520*x^3-384*x^2-84*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+90*x) The first 20 term , starting with k=1 are 1, 159, 13785, 1243383, 111887433, 10069969719, 906296667465, 81566703709623, 7341003312028233, 660690298213545399, 59462126838433014345, 5351591415463687630263, 481643227391703588508233, 43347890465253492754673079, 3901310141872813329186263625, 351117912768553205739168165303, 31600612149169788479850705359433, 2844055093425280963406610053688759, 255964958408275286705274625392415305, 23036846256744775803482637961931746743 Regarding Lambda=, [4, 1, 1, 1, 1] 8 7 6 5 4 F[[4, 1, 1, 1, 1], [8]](x) = (5350 x - 1720 x - 11432 x + 4101 x + 2939 x 3 2 - 1103 x - 76 x + 38 x - 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[8]](x) = (5350*x^8-1720*x^7-11432*x^6+4101*x^5+2939*x^4-1103 *x^3-76*x^2+38*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1 +35*x) The first 20 term , starting with k=1 are 0, 1, 1, 45, 1296, 45741, 1595501, 55853575, 1954747096, 68416438611, 2394572205201, 83810034568305, 2933351131671896, 102667289794349281, 3593355140850465901, 125767429934422739235, 4401860047656034455696, 154065101668077712349751, 5392278558381501186457601, 188729749543355455046654365 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[4, 1, 1, 1, 1], [7, 1]](x) = x 5 4 3 2 (755 x + 106 x - 122 x - 89 x + 31 x - 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[7, 1]](x) = x^2*(755*x^5+106*x^4-122*x^3-89*x^2+31*x-1)/(-1+ x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 7, 280, 9097, 319621, 11169417, 390962440, 13683255187, 478914766561, 16762006099927, 586670234472400, 20533457938525077, 718671028373543701, 25153485986375922637, 880372009536293154160, 30813020333602826816767, 1078455711676427393867041, 37745949908670773117909547, 1321108246803485271030751720 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [6, 2]](x) = 2 5 4 3 2 x (1500 x - 1525 x + 200 x + 221 x - 58 x + 2) - ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[6, 2]](x) = -x^2*(1500*x^5-1525*x^4+200*x^3+221*x^2-58*x+2)/ (-1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 20, 773, 26015, 912512, 31913310, 1117018163, 39095032325, 1368327469022, 47891446436900, 1676200659026753, 58667022692411835, 2053345795080546332, 71867102818489521290, 2515348598668311722543, 88037200953157746846545, 3081302033361050829808442, 107845571167630950646134480, 3774594990867096516930081533 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [6, 1, 1]](x) = 2 3 2 x (160 x - 39 x + 16 x - 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[6, 1, 1]](x) = -x^2*(160*x^3-39*x^2+16*x-1)/(1+x)/(-1+x)/(-1 +5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 22, 796, 27362, 957841, 33510372, 1172862496, 41049821212, 1436743685641, 50286019712522, 1760010688125196, 61600373851078062, 2156013084738862441, 75460457960016961672, 2641116028599340028896, 92439061000830724757912, 3235367135029043738808241, 113237849726012875591017822, 3963324740410449852409853596 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [5, 3]](x) = 2 5 4 3 2 x (1750 x - 1670 x + 71 x + 238 x - 51 x + 2) - ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[5, 3]](x) = -x^2*(1750*x^5-1670*x^4+71*x^3+238*x^2-51*x+2)/( -1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 27, 1063, 36398, 1277062, 44678137, 1563814373, 54733033788, 1915658183452, 67048024734647, 2346680915838283, 82133831762542378, 2874684112943064242, 100613943945715311957, 3521488038131396427793, 123252081334416602738168, 4313822846705365185019432, 150983799634683224902074067, 5284432987213932474490622903 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [5, 2, 1]](x) = 2 4 3 2 2 x (250 x - 160 x - 21 x - 2 x + 1) -------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[5, 2, 1]](x) = 2*x^2*(250*x^4-160*x^3-21*x^2-2*x+1)/(-1+x)/( 1+2*x)/(-1+2*x)/(-1+5*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 68, 2394, 83368, 2918222, 102125868, 3574414034, 125104188368, 4378646809782, 153252630750868, 5363842081705274, 187734472669813368, 6570706543579099742, 229974729020521375868, 8049115515721638990114, 281719043050138685438368, 9860166506754938761364102, 345105827736419889662000868, 12078703970774698257446172554 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[4, 1, 1, 1, 1], [5, 1, 1, 1]](x) = x 5 4 3 2 (1500 x + 35 x - 629 x + 186 x - 3 x - 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[5, 1, 1, 1]](x) = x^2*(1500*x^5+35*x^4-629*x^3+186*x^2-3*x-1 )/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 41, 1297, 45700, 1595571, 55852881, 1954749427, 68416423760, 2394572269441, 83810034217921, 2933351133334557, 102667289785771620, 3593355140892554311, 125767429934209911161, 4401860047657091410687, 154065101668072406073280, 5392278558381527653226181, 188729749543355322519188601, 6605541234017411122637841817 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[4, 1, 1, 1, 1], [4, 4]](x) = - x 6 5 4 3 2 (2100 x - 2270 x - 297 x + 282 x + 72 x - 24 x + 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[4, 4]](x) = -x^2*(2100*x^6-2270*x^5-297*x^4+282*x^3+72*x^2-\ 24*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 14, 529, 18194, 638461, 22338834, 781905349, 27366510374, 957829044841, 33524012199854, 1173340456737769, 41066915877050154, 1437342056441911621, 50306971972751845274, 1760744019064956913789, 61626040667205653633534, 2156911423352664052870801, 75491899817341546235819094, 2642216493606965773689921409 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[4, 1, 1, 1, 1], [4, 3, 1]](x) = 2 x 6 5 4 3 2 (875 x - 1150 x + 180 x + 209 x - 48 x + 3 x - 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[4, 3, 1]](x) = 2*x^2*(875*x^6-1150*x^5+180*x^4+209*x^3-48*x^ 2+3*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 70, 2606, 91080, 3191472, 111697670, 3909507086, 136832644480, 4789144743392, 167620063353270, 5866702271765766, 205334579444419880, 7186710281912362112, 251534859865241206870, 8803720095317362621646, 308130203336065341977280, 10784557116763134787339632, 377459499086708658241098470, 13211082468034824232169642726 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [4, 2, 2]](x) = 2 4 3 2 x (685 x - 419 x - 11 x + 19 x - 2) ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[4, 2, 2]](x) = x^2*(685*x^4-419*x^3-11*x^2+19*x-2)/(-1+x)/(1 +2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 59, 2084, 72917, 2553062, 89359179, 3127601864, 109466138897, 3831315691082, 134096051239799, 4693361814744044, 164267663569231677, 5749368225462470702, 201227887892533243619, 7042976076252198260624, 246504162668860761441257, 8627645693410465476619922, 301967599269367138617330639, 10568865974427858326388103604 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [4, 2, 1, 1]](x) = 2 2 x (75 x - 35 x - 2) - ---------------------------------------- (1 + x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[4, 2, 1, 1]](x) = -x^2*(75*x^2-35*x-2)/(1+x)/(-1+3*x)/(1+5*x )/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 99, 3329, 117330, 4102772, 143616909, 5026495259, 175927825620, 6157471465142, 215511513516519, 7542902912131589, 264001602230047110, 9240056076526567112, 323401962686061634929, 11319068693974017424319, 396167404289281366237800, 13865859150123894208576682, 485305070254341065865476139, 16985677458901913464014885449 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [4, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (5150 x - 2080 x - 2013 x + 801 x + 82 x - 37 x + 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[4, 1, 1, 1, 1]](x) = -x*(5150*x^6-2080*x^5-2013*x^4+801*x^3+ 82*x^2-37*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x ) The first 20 term , starting with k=1 are 1, 1, 45, 1296, 45741, 1595501, 55853575, 1954747096, 68416438611, 2394572205201, 83810034568305, 2933351131671896, 102667289794349281, 3593355140850465901, 125767429934422739235, 4401860047656034455696, 154065101668077712349751, 5392278558381501186457601, 188729749543355455046654365, 6605541234017410460581818496 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [3, 3, 2]](x) = 2 3 2 x (355 x - 79 x - 3 x - 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[3, 3, 2]](x) = -x^2*(355*x^3-79*x^2-3*x-1)/(1+x)/(-1+x)/(-1+ 5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 41, 1558, 54622, 1914751, 67017951, 2345700988, 82099570412, 2873486764621, 100572037605061, 3520021361024818, 123200747656479402, 4312026169096554091, 150920915918890410971, 5282232057189146005048, 184878122001632847357592, 6470734270057849083251161, 226475699452025035998939681, 7926649480820893744573153678 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [3, 3, 1, 1]](x) = 2 5 4 3 2 x (350 x - 970 x + 293 x + 75 x - 19 x - 1) ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[3, 3, 1, 1]](x) = x^2*(350*x^5-970*x^4+293*x^3+75*x^2-19*x-1 )/(-1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 58, 2073, 72916, 2552911, 89359438, 3127599133, 109466148136, 3831315631851, 134096051496418, 4693361813343193, 164267663575880956, 5749368225428162791, 201227887892701591798, 7042976076251346959253, 246504162668864989239376, 8627645693410444251557731, 301967599269367244484317578, 10568865974427857796278415313 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[4, 1, 1, 1, 1], [3, 2, 2, 1]](x) = x 5 4 3 2 (1075 x - 330 x - 90 x + 24 x + 3 x - 2)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[3, 2, 2, 1]](x) = x^2*(1075*x^5-330*x^4-90*x^3+24*x^2+3*x-2) /(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 73, 2600, 91135, 3191282, 111698853, 3909501920, 136832672455, 4789144610162, 167620064038933, 5866702268396840, 205334579461440975, 7186710281827789442, 251534859865665661813, 8803720095315245135360, 308130203336075943746695, 10784557116763081821561122, 377459499086708923199087493, 13211082468034822907767205480 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [3, 2, 1, 1, 1]](x) = 2 4 3 2 2 x (100 x - 95 x + 63 x + x - 1) - -------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[3, 2, 1, 1, 1]](x) = -2*x^2*(100*x^4-95*x^3+63*x^2+x-1)/(-1+ x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 2, 70, 2382, 83434, 2917882, 102127590, 3574405382, 125104231714, 4378646592882, 153252631835710, 5363842076280382, 187734472696939194, 6570706543443467882, 229974729021199540630, 8049115515718248155382, 281719043050155639633874, 9860166506754853990342882, 345105827736420313517194350, 12078703970774696138170030382 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (525 x + 245 x - 78 x - 13 x + 1) ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1]](x) = x^2*(525*x^4+245*x^3-78*x^2-13*x+1)/ (1+x)/(-1+x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 25, 793, 27416, 957691, 33511485, 1172858023, 41049846856, 1436743567261, 50286020333945, 1760010685106653, 61600373866436496, 2156013084662867431, 75460457960399328205, 2641116028597435370683, 92439061000840269572336, 3235367135028996079306201, 113237849726013114082238265, 3963324740410448660534882113 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[4, 1, 1, 1, 1], [2, 2, 2, 2]](x) = x 5 4 3 2 (40 x + 1059 x - 372 x - 70 x + 24 x - 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[2, 2, 2, 2]](x) = x^2*(40*x^5+1059*x^4-372*x^3-70*x^2+24*x-1 )/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 14, 527, 18208, 638381, 22339254, 781903207, 27366521168, 957828990701, 33524012470894, 1173340455381887, 41066915883830928, 1437342056408005021, 50306971972921383734, 1760744019064109210567, 61626040667209892171488, 2156911423352642860137341, 75491899817341652199573774, 2642216493606965243870973247 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [2, 2, 2, 1, 1]](x) = 2 4 3 2 x (305 x - 299 x + 72 x - 11 x + 1) - ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[2, 2, 2, 1, 1]](x) = -x^2*(305*x^4-299*x^3+72*x^2-11*x+1)/(-\ 1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 28, 1050, 36433, 1276811, 44679138, 1563808660, 54733060123, 1915658045301, 67048025405548, 2346680912425070, 82133831779430613, 2874684112858092991, 100613943946138571158, 3521488038129275354280, 123252081334427193745903, 4313822846705312186955881, 150983799634683489763207968, 5284432987213931149797620290 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 F[[4, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (250 x - 450 x + 170 x - 55 x + 18 x - 1) ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation F[[4, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1]](x) = x^2*(250*x^5-450*x^4+170*x^3-55*x^2+ 18*x-1)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 0, 1, 21, 760, 26050, 912261, 31914311, 1117012450, 39095058660, 1368327330871, 47891447107801, 1676200655613540, 58667022709300070, 2053345794995575081, 71867102818912780491, 2515348598666190649030, 88037200953168337854280, 3081302033360997831744891, 107845571167631215507268381, 3774594990867095192237078920 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 2 F[[4, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (1750 x - 280 x - 1171 x + 327 x + 84 x - 31 x + 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -x^2*(1750*x^6-280*x^5-1171*x^4+ 327*x^3+84*x^2-31*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/ (-1+35*x) The first 20 term , starting with k=1 are 0, 1, 7, 275, 9112, 319501, 11169907, 390959605, 13683268312, 478914697571, 16762006435207, 586670232766135, 20533457946968512, 718671028331059441, 25153485986587549507, 880372009535232622865, 30813020333608122309712, 1078455711676400894857111, 37745949908670905548432807, 1321108246803484608684337795 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 3 F[[4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (1500 x + 35 x - 629 x + 186 x - 3 x - 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x)) and in Maple notation F[[4, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(1500*x^5+35*x^4-629*x^3+ 186*x^2-3*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x ) The first 20 term , starting with k=1 are 0, 0, 1, 41, 1297, 45700, 1595571, 55852881, 1954749427, 68416423760, 2394572269441, 83810034217921, 2933351133334557, 102667289785771620, 3593355140892554311, 125767429934209911161, 4401860047657091410687, 154065101668072406073280, 5392278558381527653226181, 188729749543355322519188601 ---------------------------------- Their sum is 7 6 5 4 3 2 1800 x - 1940 x - 2635 x + 3245 x - 636 x - 144 x + 39 x - 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (1 + 5 x) (-1 + 35 x) and in Maple notation (1800*x^7-1940*x^6-2635*x^5+3245*x^4-636*x^3-144*x^2+39*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+5*x)/(-1+3*x)/(1+5*x)/(-1+35*x) The first 20 term , starting with k=1 are 1, 29, 811, 28551, 995121, 34835009, 1219125151, 42669533411, 1493431186981, 52270095427149, 1829453278034691, 64030864828678271, 2241080267456954041, 78437809363433143289, 2745323327681499513031, 96086316468913503523131, 3363021076411006189808301, 117705737674386742392095429, 4119700818603511823584648171, 144189528651122951971269235991 Regarding Lambda=, [3, 3, 2] 5 4 3 2 760 x + 912 x - 146 x - 291 x - 35 x + 1 F[[3, 3, 2], [8]](x) = -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[8]](x) = (760*x^5+912*x^4-146*x^3-291*x^2-35*x+1)/(1+x)/(-1+2*x)/( 1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 1, 87, 3201, 136399, 5716241, 240154223, 10086041617, 423616350447, 17791871090961, 747258679539439, 31384863978291473, 1318164290462240495, 55362900179169906961, 2325241807646600490735, 97660155920428433412369, 4101726548662366923452143, 172272515043793174460961041, 7235445631839470745292304111 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 3 2 x (56 x + 124 x + 24 x - 1) F[[3, 3, 2], [7, 1]](x) = - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[7, 1]](x) = -x^2*(56*x^3+124*x^2+24*x-1)/(1+x)/(-1+2*x)/(1+2*x)/(1 +6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 11, 553, 22623, 953361, 40021807, 1681030289, 70602585071, 2965312688401, 124543108217583, 5230810693282065, 219694048228974319, 9227150030950043921, 387540301267903065839, 16276692653443920990481, 683621091443492728073967, 28712085840633606298472721, 1205907605306569994218958575, 50648119422876188579090469137 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (22 x - 1) F[[3, 3, 2], [6, 2]](x) = - -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[6, 2]](x) = -2*x^2*(22*x-1)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 2, 32, 1576, 64640, 2723872, 114348032, 4802943616, 201721671680, 8472321966592, 355837452050432, 14945173409376256, 627697280654213120, 26363285802714406912, 1107258003622580191232, 46504836152696917098496, 1953203118409979223080960, 82034530973238875138424832, 3445450300875914269197074432, 144708912636789110225972494336 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (44 x + 2 x + 1) F[[3, 3, 2], [6, 1, 1]](x) = -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[6, 1, 1]](x) = x^2*(44*x^2+2*x+1)/(1+x)/(-1+2*x)/(1+2*x)/(1+6*x)/( -1+42*x) The first 20 term , starting with k=1 are 0, 1, 37, 1631, 68001, 2859279, 120070097, 5043062767, 211807923217, 8895937057007, 373629330699537, 15692432043564783, 659082144904601873, 27681450091544055535, 1162620903811545616657, 48830077960284744445679, 2050863274330760295223569, 86136257521899126229364463, 3617722815919720138652586257, 151944358268628504801296969455 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (28 x - 16 x - 1) F[[3, 3, 2], [5, 3]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[5, 3]](x) = -x^2*(28*x^2-16*x-1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 53, 2151, 90817, 3811471, 160098897, 6724051055, 282410760209, 11861248233711, 498172447986961, 20923242682427119, 878776193460093201, 36908600120534994671, 1550161205091203305745, 65106770613658137685743, 2734484365774676189778193, 114848343362530193528909551, 4823630421226305366865023249, 202592477691504601976426393327 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (44 x + 4 x + 3) F[[3, 3, 2], [5, 2, 1]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[5, 2, 1]](x) = x^2*(44*x^2+4*x+3)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 3, 115, 4953, 207359, 8713265, 365932335, 15369307537, 645510021103, 27111426261777, 1138679870747375, 47824554764884241, 2008631298964193007, 84362514563461820689, 3543225611623602269935, 148815475688442060706065, 6250249978913061957791471, 262510499114357629779120401, 11025440962802966285413183215, 463068520437724908979215733009 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 2 x (28 x - 6 x + 1) F[[3, 3, 2], [5, 1, 1, 1]](x) = ------------------------------------------ (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[5, 1, 1, 1]](x) = 2*x^2*(28*x^2-6*x+1)/(-1+2*x)/(1+2*x)/(1+6*x)/(-\ 1+42*x) The first 20 term , starting with k=1 are 0, 2, 60, 2728, 113280, 4765792, 200114880, 8405116288, 353013135360, 14826562181632, 622715548646400, 26154053421058048, 1098470241416970240, 46135750153117622272, 1937701506349310853120, 81383463267160831787008, 3418105457217816279121920, 143560429203165915659763712, 6029538026532862666089431040, 253240597114380866725484167168 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (28 x - 14 x + 1) F[[3, 3, 2], [4, 4]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[4, 4]](x) = x^2*(28*x^2-14*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 23, 1097, 45279, 1906513, 80044783, 3362053521, 141205212143, 5930625124625, 249086217946863, 10461621377493265, 439388096512368367, 18454300061573566737, 775080602537765236463, 32553385306876087341329, 1367242182887055983898351, 57424171681266789430399249, 2411815210613142527436844783, 101296238845752361924187197713 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (28 x + 1) F[[3, 3, 2], [4, 3, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[4, 3, 1]](x) = 2*x^2*(28*x+1)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 2, 132, 5376, 227040, 9528672, 400247232, 16810127616, 706026900480, 29653120584192, 1245431119967232, 52308106706067456, 2196940483650232320, 92271500301337485312, 3875403012728008261632, 162766926534145344208896, 6836210914436690474434560, 287120858406325483822252032, 12059076053065763417162514432, 506481194228761504941065895936 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (56 x + 10 x - 3) F[[3, 3, 2], [4, 2, 2]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[4, 2, 2]](x) = -x^2*(56*x^2+10*x-3)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42* x) The first 20 term , starting with k=1 are 0, 3, 101, 4335, 181441, 7624111, 320190801, 13448144111, 564821268497, 23722497979119, 996344886904081, 41846485419273967, 1757552386593669393, 73817200243029094127, 3100322410170651988241, 130213541227386803121903, 5468968731548929213075729, 229696686725062926056746735, 9647260842452595499736568081, 405184955383009295356813831919 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 4 x (36 x - 6 x - 1) F[[3, 3, 2], [4, 2, 1, 1]](x) = - ------------------------------------------ (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[4, 2, 1, 1]](x) = -4*x^2*(36*x^2-6*x-1)/(-1+2*x)/(1+2*x)/(1+6*x)/( -1+42*x) The first 20 term , starting with k=1 are 0, 4, 168, 6928, 291840, 12251584, 514601088, 21613036288, 907748782080, 38125441291264, 1601268579575808, 67253280070094848, 2824637764576542720, 118634786102419308544, 4982661016360383971328, 209271762686783488196608, 8789414032847022336245760, 369155389379562243128295424, 15504526353941690381354139648, 651190106865550538997071085568 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 2 x (28 x - 6 x + 1) F[[3, 3, 2], [4, 1, 1, 1, 1]](x) = ------------------------------------------ (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[4, 1, 1, 1, 1]](x) = 2*x^2*(28*x^2-6*x+1)/(-1+2*x)/(1+2*x)/(1+6*x) /(-1+42*x) The first 20 term , starting with k=1 are 0, 2, 60, 2728, 113280, 4765792, 200114880, 8405116288, 353013135360, 14826562181632, 622715548646400, 26154053421058048, 1098470241416970240, 46135750153117622272, 1937701506349310853120, 81383463267160831787008, 3418105457217816279121920, 143560429203165915659763712, 6029538026532862666089431040, 253240597114380866725484167168 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 F[[3, 3, 2], [3, 3, 2]](x) = 3 2 x (248 x + 240 x + 34 x - 1) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[3, 3, 2]](x) = -x*(248*x^3+240*x^2+34*x-1)/(1+x)/(-1+2*x)/(1+2*x)/ (1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 1, 87, 3201, 136399, 5716241, 240154223, 10086041617, 423616350447, 17791871090961, 747258679539439, 31384863978291473, 1318164290462240495, 55362900179169906961, 2325241807646600490735, 97660155920428433412369, 4101726548662366923452143, 172272515043793174460961041, 7235445631839470745292304111, 303888716537256826794672197905 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (56 x + 10 x - 3) F[[3, 3, 2], [3, 3, 1, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[3, 3, 1, 1]](x) = -x^2*(56*x^2+10*x-3)/(1+x)/(-1+2*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 0, 3, 101, 4335, 181441, 7624111, 320190801, 13448144111, 564821268497, 23722497979119, 996344886904081, 41846485419273967, 1757552386593669393, 73817200243029094127, 3100322410170651988241, 130213541227386803121903, 5468968731548929213075729, 229696686725062926056746735, 9647260842452595499736568081, 405184955383009295356813831919 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (28 x + 1) F[[3, 3, 2], [3, 2, 2, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[3, 2, 2, 1]](x) = 2*x^2*(28*x+1)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 2, 132, 5376, 227040, 9528672, 400247232, 16810127616, 706026900480, 29653120584192, 1245431119967232, 52308106706067456, 2196940483650232320, 92271500301337485312, 3875403012728008261632, 162766926534145344208896, 6836210914436690474434560, 287120858406325483822252032, 12059076053065763417162514432, 506481194228761504941065895936 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (44 x + 4 x + 3) F[[3, 3, 2], [3, 2, 1, 1, 1]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[3, 2, 1, 1, 1]](x) = x^2*(44*x^2+4*x+3)/(1+x)/(-1+2*x)/(1+6*x)/(-1 +42*x) The first 20 term , starting with k=1 are 0, 3, 115, 4953, 207359, 8713265, 365932335, 15369307537, 645510021103, 27111426261777, 1138679870747375, 47824554764884241, 2008631298964193007, 84362514563461820689, 3543225611623602269935, 148815475688442060706065, 6250249978913061957791471, 262510499114357629779120401, 11025440962802966285413183215, 463068520437724908979215733009 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 F[[3, 3, 2], [3, 1, 1, 1, 1, 1]](x) = 2 2 x (44 x + 2 x + 1) -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[3, 1, 1, 1, 1, 1]](x) = x^2*(44*x^2+2*x+1)/(1+x)/(-1+2*x)/(1+2*x)/ (1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 37, 1631, 68001, 2859279, 120070097, 5043062767, 211807923217, 8895937057007, 373629330699537, 15692432043564783, 659082144904601873, 27681450091544055535, 1162620903811545616657, 48830077960284744445679, 2050863274330760295223569, 86136257521899126229364463, 3617722815919720138652586257, 151944358268628504801296969455 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (28 x - 14 x + 1) F[[3, 3, 2], [2, 2, 2, 2]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[2, 2, 2, 2]](x) = x^2*(28*x^2-14*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 0, 1, 23, 1097, 45279, 1906513, 80044783, 3362053521, 141205212143, 5930625124625, 249086217946863, 10461621377493265, 439388096512368367, 18454300061573566737, 775080602537765236463, 32553385306876087341329, 1367242182887055983898351, 57424171681266789430399249, 2411815210613142527436844783, 101296238845752361924187197713 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (28 x - 16 x - 1) F[[3, 3, 2], [2, 2, 2, 1, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[2, 2, 2, 1, 1]](x) = -x^2*(28*x^2-16*x-1)/(1+x)/(-1+2*x)/(1+6*x)/( -1+42*x) The first 20 term , starting with k=1 are 0, 1, 53, 2151, 90817, 3811471, 160098897, 6724051055, 282410760209, 11861248233711, 498172447986961, 20923242682427119, 878776193460093201, 36908600120534994671, 1550161205091203305745, 65106770613658137685743, 2734484365774676189778193, 114848343362530193528909551, 4823630421226305366865023249, 202592477691504601976426393327 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 2 2 x (22 x - 1) F[[3, 3, 2], [2, 2, 1, 1, 1, 1]](x) = - -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[2, 2, 1, 1, 1, 1]](x) = -2*x^2*(22*x-1)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 2, 32, 1576, 64640, 2723872, 114348032, 4802943616, 201721671680, 8472321966592, 355837452050432, 14945173409376256, 627697280654213120, 26363285802714406912, 1107258003622580191232, 46504836152696917098496, 1953203118409979223080960, 82034530973238875138424832, 3445450300875914269197074432, 144708912636789110225972494336 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 F[[3, 3, 2], [2, 1, 1, 1, 1, 1, 1]](x) = 2 3 2 x (56 x + 124 x + 24 x - 1) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[2, 1, 1, 1, 1, 1, 1]](x) = -x^2*(56*x^3+124*x^2+24*x-1)/(1+x)/(-1+ 2*x)/(1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 11, 553, 22623, 953361, 40021807, 1681030289, 70602585071, 2965312688401, 124543108217583, 5230810693282065, 219694048228974319, 9227150030950043921, 387540301267903065839, 16276692653443920990481, 683621091443492728073967, 28712085840633606298472721, 1205907605306569994218958575, 50648119422876188579090469137 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 F[[3, 3, 2], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 2 3 2 x (248 x + 240 x + 34 x - 1) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 2],[1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^2*(248*x^3+240*x^2+34*x-1)/(1+x)/ (-1+2*x)/(1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 1, 87, 3201, 136399, 5716241, 240154223, 10086041617, 423616350447, 17791871090961, 747258679539439, 31384863978291473, 1318164290462240495, 55362900179169906961, 2325241807646600490735, 97660155920428433412369, 4101726548662366923452143, 172272515043793174460961041, 7235445631839470745292304111 ---------------------------------- Their sum is 4 3 2 184 x + 48 x - 216 x - 36 x + 1 ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (184*x^4+48*x^3-216*x^2-36*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 39, 1385, 59103, 2475601, 104013295, 4368325521, 183471059951, 7705776171281, 323642649237231, 13592990967630097, 570905622442319599, 23978036131765997841, 1007077517599039876847, 42297255738770465853713, 1776484741030694817361647, 74612359123275170815480081, 3133719083177641243323068143, 131616201493460427805112733969, 5527880462725340994301434195695 Regarding Lambda=, [3, 3, 1, 1] F[[3, 3, 1, 1], [8]](x) = 6 5 4 3 2 1912 x - 710 x - 3636 x + 1413 x + 312 x - 62 x + 1 ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[8]](x) = (1912*x^6-710*x^5-3636*x^4+1413*x^3+312*x^2-62*x+1)/(1 +x)/(-1+x)/(-1+2*x)/(1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 5, 259, 13731, 765647, 42839931, 2398787455, 134329945307, 7522460377663, 421257646056027, 23590427108887871, 1321063909493799003, 73979578862989179199, 4142856415777414500443, 231999959279138060249407, 12991997719596543422640219, 727551872297124971007502655, 40742904848636746519339839579, 2281602671523639790913585670463 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 x (-1 + 20 x) (-1 + 14 x) F[[3, 3, 1, 1], [7, 1]](x) = - ------------------------------------------ (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[7, 1]](x) = -x^2*(-1+20*x)*(-1+14*x)/(-1+2*x)/(1+4*x)/(8*x-1)/( -1+56*x) The first 20 term , starting with k=1 are 0, 1, 28, 1704, 95504, 5353760, 299836992, 16791157888, 940306839808, 52657200198144, 2948803343737856, 165132988329347072, 9247447355008192512, 517857051949278896128, 28999994909708971294720, 1623999714948102049529856, 90943984037128892703113216, 5092863106079499492120461312, 285200333940454223255479517184, 15971218700665454517117677731840 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 2 3 x (96 x - 32 x + 1) F[[3, 3, 1, 1], [6, 2]](x) = - ------------------------------------------ (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[6, 2]](x) = -3*x^2*(96*x^2-32*x+1)/(-1+2*x)/(1+4*x)/(8*x-1)/(-1 +56*x) The first 20 term , starting with k=1 are 0, 3, 90, 4932, 273480, 15301008, 856715040, 47975034432, 2686593375360, 150449162565888, 8425152564195840, 471808539310482432, 26421278166985328640, 1479591577076468256768, 82857128314082528010240, 4639999185571032066834432, 259839954391837047516856320, 14551037445941748803986587648, 814858096972728925652195082240, 45632053430472747779616080658432 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 x (28 x + 1) F[[3, 3, 1, 1], [6, 1, 1]](x) = ------------------------------- (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[6, 1, 1]](x) = x^2*(28*x+1)/(1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 88, 5088, 286592, 16060928, 899512320, 50373468160, 2820920541184, 157971600506880, 8846410031562752, 495398964986642432, 27742342065030168576, 1553571155847814316032, 86999984729127003357184, 4871999144844305790664704, 272831952111386679540973568, 15278589318238498470634717184, 855601001821362669789344956416, 47913656101996363551261407051776 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 3 2 x (224 x - 280 x + 7 x - 2) F[[3, 3, 1, 1], [5, 3]](x) = -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[5, 3]](x) = x^2*(224*x^3-280*x^2+7*x-2)/(1+x)/(-1+2*x)/(1+4*x)/ (8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 115, 6795, 382081, 21414743, 1199349081, 67164626951, 3761227377337, 210628800719559, 11795213375242297, 660531953316222407, 36989789420037428793, 2071428207797096939975, 115999979638835959737913, 6495998859792407899845063, 363775936148515572005473849, 20371452424317997963709608391, 1140801335761816893041006710329, 63884874802661818068394355749319 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 2 x (64 x + 12 x + 5) F[[3, 3, 1, 1], [5, 2, 1]](x) = ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[5, 2, 1]](x) = x^2*(64*x^2+12*x+5)/(1+x)/(-1+2*x)/(1+4*x)/(-1+ 56*x) The first 20 term , starting with k=1 are 0, 5, 277, 15615, 874153, 48954023, 2741420025, 153519543623, 8597094356281, 481437284302791, 26960487919561273, 1509787323501029831, 84548090116035313209, 4734693046498067042759, 265142810603891396529721, 14847997393817919637418439, 831487854053803493969006137, 46563319827012995685171229127, 2607545910312727758277963648569, 146022570977512754463932469768647 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [5, 1, 1, 1]](x) = 2 3 2 x (280 x - 262 x - 26 x + 3) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[5, 1, 1, 1]](x) = -x^2*(280*x^3-262*x^2-26*x+3)/(1+x)/(-1+2*x)/ (1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 3, 157, 8565, 478335, 26773673, 1499230887, 83956129785, 4701537031495, 263286023214393, 14744016898228167, 825664943076059705, 46237236786503526855, 2589285259837983161913, 144999974549278013346247, 8119998574746373713120825, 454719920185691378397737415, 25464315530397872751026802233, 1426001669702274118715326493127, 79856093503327296604633691033145 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 2 x (140 x - 65 x + 2) F[[3, 3, 1, 1], [4, 4]](x) = ----------------------------------------- (-1 + x) (-1 + 2 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[4, 4]](x) = x^2*(140*x^2-65*x+2)/(-1+x)/(-1+2*x)/(8*x-1)/(-1+56 *x) The first 20 term , starting with k=1 are 0, 2, 69, 3479, 191739, 10712771, 599718483, 33582662003, 1880616488883, 105314422713395, 5897606866642227, 330265978089510707, 18494894721472985907, 1035714103990170338099, 57999989820151004025651, 3247999429902067946320691, 181887968074304698761327411, 10185726212159374280773874483, 570400667880911448924454466355, 31942437401330933053378392437555 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [4, 3, 1]](x) = 2 3 2 x (784 x - 248 x - x + 5) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[4, 3, 1]](x) = -x^2*(784*x^3-248*x^2-x+5)/(1+x)/(-1+2*x)/(1+4*x )/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 304, 17046, 955990, 53541890, 2998418094, 167911910066, 9403071266830, 526572024059250, 29488033617499534, 1651329884720463986, 92474473561553808270, 5178570519584340356210, 289999949097823018943374, 16239997149486883364232306, 909439840371335844299367310, 50928631060795370202084678770, 2852003339404545235030901449614, 159712187006654569190069369511026 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 6 (-1 + 20 x) x F[[3, 3, 1, 1], [4, 2, 2]](x) = ------------------------------------------ (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[4, 2, 2]](x) = 6*(-1+20*x)*x^2/(-1+2*x)/(1+4*x)/(8*x-1)/(-1+56* x) The first 20 term , starting with k=1 are 0, 6, 252, 13752, 765552, 42840288, 2398785984, 134329951104, 7522460354304, 421257646149120, 23590427108514816, 1321063909495289856, 73979578862983213056, 4142856415777438359552, 231999959279137964802048, 12991997719596543804407808, 727551872297124969480388608, 40742904848636746525448208384, 2281602671523639790889152020480, 127769749605323684175154194677760 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 6 x F[[3, 3, 1, 1], [4, 2, 1, 1]](x) = --------------------- (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[4, 2, 1, 1]](x) = 6*x^2/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 6, 384, 21888, 1228800, 68837376, 3855089664, 215886594048, 12089661849600, 677021164240896, 37913186002796544, 2123138422599057408, 118895751717086822400, 6658162096569178914816, 372857077411172554113024, 20879996335052051309395968, 1169279794763125979558707200, 65479668506736743705147867136, 3666861436377271158287162671104, 205344240437127292950472166473728 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [4, 1, 1, 1, 1]](x) = 2 2 x (-1 + 20 x) (7 x - 2) - ------------------------------------------ (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[4, 1, 1, 1, 1]](x) = -2*x^2*(-1+20*x)*(7*x-2)/(-1+2*x)/(1+4*x)/ (8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 4, 154, 8580, 478280, 26773904, 1499229984, 83956133440, 4701537016960, 263286023272704, 14744016897995264, 825664943076992000, 46237236786499799040, 2589285259837998075904, 144999974549277953695744, 8119998574746373951733760, 454719920185691377443307520, 25464315530397872754844565504, 1426001669702274118700055527424, 79856093503327296604694775070720 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 2 x (28 x + 1) F[[3, 3, 1, 1], [3, 3, 2]](x) = ------------------------------- (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[3, 3, 2]](x) = 2*x^2*(28*x+1)/(1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 176, 10176, 573184, 32121856, 1799024640, 100746936320, 5641841082368, 315943201013760, 17692820063125504, 990797929973284864, 55484684130060337152, 3107142311695628632064, 173999969458254006714368, 9743998289688611581329408, 545663904222773359081947136, 30557178636476996941269434368, 1711202003642725339578689912832, 95827312203992727102522814103552 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [3, 3, 1, 1]](x) = 4 3 2 x (1672 x - 698 x - 260 x + 57 x - 1) - ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[3, 3, 1, 1]](x) = -x*(1672*x^4-698*x^3-260*x^2+57*x-1)/(1+x)/(-\ 1+x)/(-1+2*x)/(1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 1, 5, 259, 13731, 765647, 42839931, 2398787455, 134329945307, 7522460377663, 421257646056027, 23590427108887871, 1321063909493799003, 73979578862989179199, 4142856415777414500443, 231999959279138060249407, 12991997719596543422640219, 727551872297124971007502655, 40742904848636746519339839579, 2281602671523639790913585670463, 127769749605323684175056460427355 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 x (28 x - 5) F[[3, 3, 1, 1], [3, 2, 2, 1]](x) = -------------------------------- (-1 + 2 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[3, 2, 2, 1]](x) = x^2*(28*x-5)/(-1+2*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 302, 17052, 955960, 53542000, 2998417632, 167911911872, 9403071259520, 526572024088320, 29488033617382912, 1651329884720929792, 92474473561551943680, 5178570519584347811840, 289999949097822989115392, 16239997149486883483533312, 909439840371335843822141440, 50928631060795370203993538560, 2852003339404545235023265923072, 159712187006654569190099911442432 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 2 x (16 x - 5) F[[3, 3, 1, 1], [3, 2, 1, 1, 1]](x) = ------------------------------- (-1 + x) (-1 + 2 x) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[3, 2, 1, 1, 1]](x) = x^2*(16*x-5)/(-1+x)/(-1+2*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 5, 279, 15611, 874179, 48953939, 2741420403, 153519542195, 8597094362163, 481437284279603, 26960487919654707, 1509787323500657459, 84548090116036805427, 4734693046498061079347, 265142810603891420394291, 14847997393817919541982003, 831487854053803494350795571, 46563319827012995683644158771, 2607545910312727758284072104755, 146022570977512754463908036293427 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [3, 1, 1, 1, 1, 1]](x) = 2 2 x (420 x - 37 x + 2) - ---------------------------------------- (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[3, 1, 1, 1, 1, 1]](x) = -x^2*(420*x^2-37*x+2)/(-1+x)/(1+4*x)/(8 *x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 85, 5101, 286541, 16061133, 899511501, 50373471437, 2820920528077, 157971600559309, 8846410031353037, 495398964987481293, 27742342065026813133, 1553571155847827737805, 86999984729126949670093, 4871999144844306005413069, 272831952111386678681980109, 15278589318238498474070691021, 855601001821362669775601061069, 47913656101996363551316382633165 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [2, 2, 2, 2]](x) = 2 3 2 x (336 x - 232 x - 53 x + 2) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[2, 2, 2, 2]](x) = -x^2*(336*x^3-232*x^2-53*x+2)/(1+x)/(-1+2*x)/ (1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 69, 3477, 191743, 10712745, 599718567, 33582661625, 1880616490311, 105314422707513, 5897606866665415, 330265978089417273, 18494894721473358279, 1035714103990168845881, 57999989820151009989063, 3247999429902067922456121, 181887968074304698856763847, 10185726212159374280392085049, 570400667880911448925981536711, 31942437401330933053372283981369 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [2, 2, 2, 1, 1]](x) = 2 2 x (224 x - 56 x - 1) ------------------------------------------ (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[2, 2, 2, 1, 1]](x) = x^2*(224*x^2-56*x-1)/(-1+2*x)/(1+4*x)/(8*x -1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 118, 6780, 382136, 21414512, 1199349984, 67164623296, 3761227391872, 210628800661248, 11795213375475200, 660531953315290112, 36989789420041156608, 2071428207797082025984, 115999979638836019388416, 6495998859792407661232128, 363775936148515572959903744, 20371452424317997959891845120, 1140801335761816893056277676032, 63884874802661818068333271711744 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [2, 2, 1, 1, 1, 1]](x) = 2 3 2 x (288 x - 256 x - 29 x + 2) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[2, 2, 1, 1, 1, 1]](x) = -x^2*(288*x^3-256*x^2-29*x+2)/(1+x)/(-1 +2*x)/(1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 93, 4917, 273535, 15300777, 856715943, 47975030777, 2686593389895, 150449162507577, 8425152564428743, 471808539309550137, 26421278166989056455, 1479591577076453342777, 82857128314082587660743, 4639999185571031828221497, 259839954391837048471286215, 14551037445941748800168824377, 814858096972728925667466047943, 45632053430472747779554996620857 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 3 2 x (616 x + 74 x - 29) -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = x^3*(616*x^2+74*x-29)/(1+x)/(-1+2*x) /(1+4*x)/(8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 29, 1695, 95529, 5353639, 299837433, 16791156039, 940306847033, 52657200168903, 2948803343854137, 165132988328880583, 9247447355010055737, 517857051949271437767, 28999994909709001117241, 1623999714948101930217927, 90943984037128893180317241, 5092863106079499490211557831, 285200333940454223263114956345, 15971218700665454517087135625671 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 F[[3, 3, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 6 x (-1 + 20 x) ------------------------------------------ (-1 + 2 x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation F[[3, 3, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = 6*x^3*(-1+20*x)/(-1+2*x)/(1+4*x)/ (8*x-1)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 6, 252, 13752, 765552, 42840288, 2398785984, 134329951104, 7522460354304, 421257646149120, 23590427108514816, 1321063909495289856, 73979578862983213056, 4142856415777438359552, 231999959279137964802048, 12991997719596543804407808, 727551872297124969480388608, 40742904848636746525448208384, 2281602671523639790889152020480 ---------------------------------- Their sum is 5 4 3 2 816 x + 152 x - 1602 x - 194 x + 59 x - 1 ------------------------------------------------ (1 + x) (-1 + x) (1 + 4 x) (8 x - 1) (-1 + 56 x) and in Maple notation (816*x^5+152*x^4-1602*x^3-194*x^2+59*x-1)/(1+x)/(-1+x)/(1+4*x)/(8*x-1)/(-1+56*x ) The first 20 term , starting with k=1 are 1, 63, 3339, 186495, 10436443, 584396095, 32725770331, 1832640063807, 102627818123355, 5747157614717247, 321840824809009243, 18023086176437791039, 1009292825777364381787, 56520398242708062464319, 3165142301585053265014875, 177247968888710210935580991, 9925886257767349581315944539, 555849630434968198928418466111, 31127579304358192118095885480027, 1743144441044058542441778610694463 Regarding Lambda=, [3, 2, 2, 1] 8 7 6 5 4 F[[3, 2, 2, 1], [8]](x) = (31450 x + 745 x - 62814 x - 2477 x + 17255 x 3 2 + 744 x - 784 x - 59 x + 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[8]](x) = (31450*x^8+745*x^7-62814*x^6-2477*x^5+17255*x^4+744*x^ 3-784*x^2-59*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+ 70*x) The first 20 term , starting with k=1 are 0, 1, 8, 608, 41616, 2918666, 204244743, 14297692753, 1000832809796, 70058352468656, 4904084114491353, 343285893577242023, 24030012494788974726, 1682100875190991973446, 117747061257812426413163, 8242294288102431024153293, 576960600166614581393011656, 40387242011668576400201111036, 2827106940816744791616548062173, 197897485857172690972498585052563 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [7, 1]](x) = - x 6 5 4 3 2 (5250 x - 7175 x - 3980 x + 577 x + 715 x + 61 x - 2)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[7, 1]](x) = -x^2*(5250*x^6-7175*x^5-3980*x^4+577*x^3+715*x^2+61 *x-2)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 57, 4218, 291440, 20428947, 1429727082, 100083704303, 7005831062910, 490408453266267, 34328588940560882, 2403001253649070263, 168210087477419089830, 11774706126197992871987, 824229428806076083361082, 57696060016703126829638223, 4038724201166440964281995950, 282710694081678645877866340107, 19789748585717227430352217895682, 1385282401000208697917760664526183 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [6, 2]](x) = - x 6 5 4 3 2 (4500 x - 1200 x - 5865 x - 544 x + 763 x + 73 x - 4)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[6, 2]](x) = -x^2*(4500*x^6-1200*x^5-5865*x^4-544*x^3+763*x^2+73 *x-4)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 163, 11994, 832965, 58364889, 4084964538, 285953131804, 20016663172795, 1401166979324199, 98081682985520388, 6865717864587073914, 480600249965262221925, 33642017503125090364309, 2354941225163194042959838, 164845885761979168875424824, 11539212003332986100872647855, 807744840233364583387746909219, 56542138816334965277518594884888, 3957949717143453125001347454208534 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [6, 1, 1]](x) = 2 5 4 3 2 x (1750 x - 1685 x + 458 x + 762 x + 60 x - 4) ------------------------------------------------------------------------ (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[6, 1, 1]](x) = x^2*(1750*x^5-1685*x^4+458*x^3+762*x^2+60*x-4)/( 1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 172, 12586, 874718, 61282159, 4289223162, 300250685641, 21017497371448, 1471225317903859, 102985767238900502, 7209003756775426621, 504630262473940085028, 35324118378177193447159, 2472688286422395358259842, 173088180050067711010682401, 12116172603499739571154540208, 848132082245031770899059104059, 59369245757151723958024031803182, 4155847203000625677084957150262981 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [5, 3]](x) = - x 6 5 4 3 2 (3500 x + 2000 x - 1795 x - 1400 x - 30 x + 6 x - 4)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[5, 3]](x) = -x^2*(3500*x^6+2000*x^5-1795*x^4-1400*x^3-30*x^2+6* x-4)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 230, 16740, 1166430, 81707599, 5718978615, 400334096950, 28023331239140, 1961633743060869, 137314356458110425, 9612005007639217660, 672840349979161878150, 47098824504097233584539, 3296917715231249892861035, 230784240066743055889934370, 16154896804666458330856149960, 1130842776326707638898053622609, 79158994342868979166607958164445, 5541129604000834097222464225295080 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [5, 2, 1]](x) = 2 3 2 x (500 x - 40 x + 37 x + 9) ------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[5, 2, 1]](x) = x^2*(500*x^3-40*x^2+37*x+9)/(1+x)/(-1+x)/(1+5*x) /(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 9, 532, 38229, 2666632, 186755729, 13071999132, 915048893229, 64053333311632, 4483734222330729, 313861386666124132, 21970297155558268229, 1537920799999986436632, 107654456008888956705729, 7535811920533332994249132, 527506834438222223917643229, 36925478410666666658189561632, 2584783488746755555597941080729, 180934844212271999999788072374132, 12665439094859048888889948527018229 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [5, 1, 1, 1]](x) = - x 6 5 4 3 2 (8750 x + 3225 x - 715 x - 197 x + 325 x + 2 x - 5)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[5, 1, 1, 1]](x) = -x^2*(8750*x^6+3225*x^5-715*x^4-197*x^3+325*x ^2+2*x-5)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 5, 293, 20887, 1458560, 102129595, 7148775123, 500417106782, 35029169246420, 2452042126882585, 171642946093115453, 12015006254343843052, 841050437526025411230, 58873530629600782010375, 4121147144044270420661783, 288480300083376738275121322, 20193621005833593739582571240, 1413553470408379340331475490965, 98948742928586276041404620428113, 6926412005001042100695780445267592 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [4, 4]](x) = - x 6 5 4 3 2 (7700 x + 9770 x - 4664 x - 1418 x - 4 x + 3 x - 2)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[4, 4]](x) = -x^2*(7700*x^6+9770*x^5-4664*x^4-1418*x^3-4*x^2+3*x -2)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 115, 8359, 583284, 40853002, 2859496395, 200166976739, 14011666319374, 980816864534452, 68657178298659225, 4806002503123972669, 336420174996529769964, 23549412251979144121502, 1648458857616319506843655, 115392120033364582820005399, 8077448402333298612872464554, 565421388163353124987971986152, 39579497171434496527825033025685, 2770564802000416979166378850734929 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [4, 3, 1]](x) = x 5 4 3 2 (2750 x - 5635 x - 277 x + 761 x + 116 x + 8)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[4, 3, 1]](x) = x^2*(2750*x^5-5635*x^4-277*x^3+761*x^2+116*x+8)/ (-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 8, 588, 41733, 2917190, 204258288, 14297552493, 1000834193213, 70058338557900, 4904084253292818, 343285892188003523, 24030012508676465343, 1682100875052097491810, 117747061259201292931748, 8242294288088542045784553, 576960600166753469923983873, 40387242011667187509880560920, 2827106940816758680499710317078, 197897485857172552083586789649583, 13852824010002084201387518528160803 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [4, 2, 2]](x) = 2 4 3 2 x (2160 x + 1182 x - 209 x - 90 x - 7) ---------------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[4, 2, 2]](x) = x^2*(2160*x^4+1182*x^3-209*x^2-90*x-7)/(-1+x)/(1 +x)/(1+2*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 7, 475, 33369, 2334063, 163403997, 11438070315, 800667079009, 56046673632823, 3923267374891557, 274628714028320355, 19224010004163954249, 1345680700069458007983, 94197649007083265517517, 6593835430473611450195995, 461568480133374998304583089, 32309793609334027786254885543, 2261685552653404166624281157877, 158317988685738069444656372081235, 11082259208001667083332273695269529 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 x (5 x - 11) F[[3, 2, 2, 1], [4, 2, 1, 1]](x) = - -------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[4, 2, 1, 1]](x) = -x^2*(5*x-11)/(1+2*x)/(-1+2*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 11, 765, 53594, 3751560, 262609376, 18382656240, 1286785937504, 90075015624960, 6305251093750016, 441367576562499840, 30895730359375000064, 2162701125156249999360, 151389078760937500000256, 10597235513265624999997440, 741806485928593750000001024, 51926454015001562499999989760, 3634851781050109375000000004096, 254439624673507656249999999959040, 17810773727145535937500000000016384 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [4, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (1750 x + 1875 x - 1180 x - 267 x + 32 x + 63 x + 4)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[4, 1, 1, 1, 1]](x) = x^2*(1750*x^6+1875*x^5-1180*x^4-267*x^3+32 *x^2+63*x+4)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 4, 299, 20813, 1459235, 102122569, 7148844234, 500416410988, 35029176185435, 2452042057416359, 171642946787472644, 12015006247399049338, 841050437595468458085, 58873530628906331974549, 4121147144051214842738654, 288480300083307293741202488, 20193621005834288183669109935, 1413553470408372395885599407139, 98948742928586345485843338282264, 6926412005001041406251313094396438 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [3, 3, 2]](x) = 2 3 2 x (175 x + 70 x - 39 x - 5) - -------------------------------------------------- (1 + 2 x) (1 + x) (1 + 5 x) (-1 + 2 x) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[3, 3, 2]](x) = -x^2*(175*x^3+70*x^2-39*x-5)/(1+2*x)/(1+x)/(1+5* x)/(-1+2*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 5, 359, 25001, 1750780, 122550785, 8578574214, 600500097671, 42035007324200, 2942450510253965, 205971535729980394, 14418007501037597891, 1009260525072937011420, 70648236755104064942345, 4945376572857292175291774, 346176360100010414123538911, 24232345207000729179382319440, 1696264164490051041603088393925, 118738491514303572916984558086354, 8311694406001250104165077209532731 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [3, 3, 1, 1]](x) = 2 5 4 3 2 2 x (1400 x + 990 x - 353 x - 441 x - 75 x - 3) ---------------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[3, 3, 1, 1]](x) = 2*x^2*(1400*x^5+990*x^4-353*x^3-441*x^2-75*x-\ 3)/(-1+x)/(1+x)/(1+2*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 6, 480, 33312, 2334616, 163398436, 11438125860, 800666523432, 56046679188336, 3923267319335916, 274628714583875740, 19224009998608398352, 1345680700125013562856, 94197649006527709960596, 6593835430479167005748820, 461568480133319442749022072, 32309793609334583341810430176, 2261685552653398611068725580476, 158317988685738125000211927593100, 11082259208001666527776718139626592 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [3, 2, 2, 1]](x) = - x 6 5 4 3 2 (24550 x + 1255 x - 10306 x - 259 x + 649 x + 51 x - 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[3, 2, 2, 1]](x) = -x*(24550*x^6+1255*x^5-10306*x^4-259*x^3+649* x^2+51*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 8, 608, 41616, 2918666, 204244743, 14297692753, 1000832809796, 70058352468656, 4904084114491353, 343285893577242023, 24030012494788974726, 1682100875190991973446, 117747061257812426413163, 8242294288102431024153293, 576960600166614581393011656, 40387242011668576400201111036, 2827106940816744791616548062173, 197897485857172690972498585052563, 13852824010002082812498721265284586 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [3, 2, 1, 1, 1]](x) = 2 3 2 x (200 x - 530 x - 169 x - 7) - ------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[3, 2, 1, 1, 1]](x) = -x^2*(200*x^3-530*x^2-169*x-7)/(1+x)/(-1+x )/(1+5*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 7, 554, 38007, 2668854, 186733507, 13072221354, 915046671007, 64053355533854, 4483734000108507, 313861388888346354, 21970297133336046007, 1537920800222208658854, 107654456006666734483507, 7535811920555555216471354, 527506834438000001695421007, 36925478410668888880411783854, 2584783488746733333375718858507, 180934844212272222222010294596354, 12665439094859046666667726304796007 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [3, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (2450 x - 1905 x - 1359 x - 262 x - 65 x - 2) ------------------------------------------------------------------------ (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[3, 1, 1, 1, 1, 1]](x) = x^2*(2450*x^5-1905*x^4-1359*x^3-262*x^2 -65*x-2)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 181, 12446, 875906, 61269407, 4289347146, 300249431561, 21017509855096, 1471225192838387, 102985768488638486, 7209003744274378301, 504630262598935891236, 35324118376927176670967, 2472688286434895291153026, 173088180049942710742251041, 12116172603500989570080806576, 848132082245019270894764153147, 59369245757151848958006851966766, 4155847203000624427084888430851781 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 2 F[[3, 2, 2, 1], [2, 2, 2, 2]](x) = - x 6 5 4 3 2 (3500 x - 2210 x - 5682 x + 1609 x + 513 x - 5 x - 2)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[2, 2, 2, 2]](x) = -x^2*(3500*x^6-2210*x^5-5682*x^4+1609*x^3+513 *x^2-5*x-2)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 123, 8314, 583882, 40847617, 2859552633, 200166423914, 14011671885852, 980816809022587, 68657178854389543, 4806002497569116164, 336420175052088121722, 23549412251423599750757, 1648458857621875107138453, 115392120033309027443406814, 8077448402333854169143847992, 565421388163347569435279742127, 39579497171434552083392041827363, 2770564802000416423610869108163864 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [2, 2, 2, 1, 1]](x) = 2 5 4 3 2 x (1750 x + 6475 x + 3130 x + 108 x - 75 x - 3) ---------------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[2, 2, 2, 1, 1]](x) = x^2*(1750*x^5+6475*x^4+3130*x^3+108*x^2-75 *x-3)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 3, 246, 16611, 1167860, 81693878, 5719118181, 400332710781, 28023345138930, 1961633604215628, 137314357847173991, 9612004993751027651, 672840350118053562900, 47098824502708355879778, 3296917715245138826487801, 230784240066604167179999721, 16154896804667847220460861270, 1130842776326693750012028034328, 79158994342869118055508300277611, 5541129604000832708333621149346991 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 F[[3, 2, 2, 1], [2, 2, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (5750 x + 4675 x + 770 x + 128 x + 60 x + 2) - ---------------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[3, 2, 2, 1],[2, 2, 1, 1, 1, 1]](x) = -x^2*(5750*x^5+4675*x^4+770*x^3+128*x^2 +60*x+2)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 174, 11844, 834310, 58350827, 4085102739, 285951740174, 20016677050740, 1401166840391577, 98081684374234429, 6865717850697485804, 480600250104148314270, 33642017501736190289927, 2354941225177082887108119, 164845885761840279807576234, 11539212003334374989045703400, 807744840233350694495994697877, 56542138816335104166396030505809, 3957949717143451736112412752291464 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 3 F[[3, 2, 2, 1], [2, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (12250 x - 3225 x - 6305 x + 1182 x + 592 x + 60)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[2, 1, 1, 1, 1, 1, 1]](x) = x^3*(12250*x^5-3225*x^4-6305*x^3+ 1182*x^2+592*x+60)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+ 70*x) The first 20 term , starting with k=1 are 0, 0, 60, 4132, 292070, 20421745, 1429795500, 100083005757, 7005837990960, 490408383756265, 34328589634743140, 2403001246703577157, 168210087546859339800, 11774706125503531649985, 824229428813020460695980, 57696060016633682116756957, 4038724201167135407652695840, 282710694081671701429126922905, 19789748585717296874779482460020, 1385282401000208003473247500583157 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 3 F[[3, 2, 2, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (2750 x - 5635 x - 277 x + 761 x + 116 x + 8)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (-1 + 2 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[3, 2, 2, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(2750*x^5-5635*x^4-277*x^3+ 761*x^2+116*x+8)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(-1+2*x)/(10*x+1)/(-1+70 *x) The first 20 term , starting with k=1 are 0, 0, 8, 588, 41733, 2917190, 204258288, 14297552493, 1000834193213, 70058338557900, 4904084253292818, 343285892188003523, 24030012508676465343, 1682100875052097491810, 117747061259201292931748, 8242294288088542045784553, 576960600166753469923983873, 40387242011667187509880560920, 2827106940816758680499710317078, 197897485857172552083586789649583 ---------------------------------- Their sum is 6 5 4 3 2 480 x - 49 x - 1150 x - 220 x + 324 x + 64 x - 1 ----------------------------------------------------------- (-1 + 70 x) (-1 + 2 x) (1 + 5 x) (1 + x) (1 + 2 x) (-1 + x) and in Maple notation (480*x^6-49*x^5-1150*x^4-220*x^3+324*x^2+64*x-1)/(-1+70*x)/(-1+2*x)/(1+5*x)/(1+ x)/(1+2*x)/(-1+x) The first 20 term , starting with k=1 are 1, 96, 6490, 455001, 31846370, 2229262951, 156048320740, 10923382875501, 764636799164470, 53524575952094451, 3746720316593695640, 262270422161823189001, 18358929551326300726570, 1285125068592847663035951, 89958754801499303351506540, 6297112836104951399909142501, 440797898527346597167121032670, 30855852896914261805831061537451, 2159909702783998326387511359293440, 151193679194879882847229109870336001 Regarding Lambda=, [3, 2, 1, 1, 1] F[[3, 2, 1, 1, 1], [8]](x) = 7 6 5 4 3 2 9344 x - 1200 x - 17902 x + 3583 x + 4308 x - 836 x - 52 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[8]](x) = (9344*x^7-1200*x^6-17902*x^5+3583*x^4+4308*x^3-836* x^2-52*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 4, 464, 25906, 1716020, 108892170, 6984009028, 446737956202, 28595046901124, 1830021917512426, 117122380063762052, 7495816686579247338, 479732518141039929988, 30702877157826841919722, 1964984202152112404232836, 125758987912916077948641514, 8048575242823734829146913412, 515108815278365335561189761258, 32966964182013040571754459970180 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [7, 1]](x) = 2 3 2 2 x (576 x - 96 x - 36 x + 1) - ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[7, 1]](x) = -2*x^2*(576*x^3-96*x^2-36*x+1)/(-1+2*x)/(1+2*x)/ (-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 2, 32, 3144, 182784, 11988768, 762617856, 48882097280, 3127261134848, 200163801199104, 12810177856143360, 819856269508724736, 52470723061053652992, 3358127526907285544960, 214920141706067751993344, 13754889389444308913651712, 880312915800340191579734016, 56340026693207301466136641536, 3605761707053498826323140804608, 230768749272412220363918405009408 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [6, 2]](x) = 2 4 3 2 x (1536 x - 240 x + 186 x - 109 x + 4) --------------------------------------------------------------- (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[6, 2]](x) = x^2*(1536*x^4-240*x^3+186*x^2-109*x+4)/(-1+x)/(-\ 1+2*x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 4, 103, 8781, 525345, 34203609, 2179706793, 139650351241, 8935236331465, 571893302479689, 36600560518031433, 2342445646587197001, 149916365006579558985, 9594649862420828791369, 614057551162936131187273, 39299683914939858504831561, 2515179760307959788669112905, 160971504823680484893479768649, 10302176306092014098197857866313, 659339283631865493243275126542921 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 4 x (24 x - 1) F[[3, 2, 1, 1, 1], [6, 1, 1]](x) = - --------------------------------- (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[6, 1, 1]](x) = -4*x^2*(24*x-1)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 4, 112, 9152, 552704, 35896320, 2288971776, 146628395008, 9382069731328, 600486822281216, 38430606869135360, 2459567635713490944, 157412187948158287872, 10074382280481877000192, 644760429922042840612864, 41264668091471493028904960, 2640938748630803512700174336, 169020080059945377385307701248, 10817285121475320911156150796288, 692306247812199470176675934765056 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [5, 3]](x) = 2 3 2 x (1536 x - 80 x - 106 x + 5) - ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[5, 3]](x) = -x^2*(1536*x^3-80*x^2-106*x+5)/(-1+2*x)/(1+2*x)/ (-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 5, 154, 12108, 738376, 47838384, 3052334880, 195498560192, 12509521747072, 800647569255168, 51240833592357376, 3279423123346861056, 209882923519209228288, 13432509607229172396032, 859680574830670300782592, 55019557429674846074880000, 3521251665250998996015284224, 225360106740034994175088459776, 14423046828738702692266626121728, 923074997081253563263859548749824 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [5, 2, 1]](x) = 2 3 2 x (1920 x - 640 x + 106 x - 9) - --------------------------------------------------------------- (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[5, 2, 1]](x) = -x^2*(1920*x^3-640*x^2+106*x-9)/(-1+x)/(-1+2* x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 9, 371, 27359, 1692711, 109264983, 6978043767, 446833399863, 28593519801527, 1830046351103927, 117121989126293943, 7495822941578728887, 479732418061048208823, 30702878759106709425591, 1964984176531634524073399, 125758988322843724031061431, 8048575236264892491827932599, 515108815383306812958292929975, 32966964180333976933400808222135, 2109885707608537069271614541229495 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [5, 1, 1, 1]](x) = 2 3 2 x (1664 x - 512 x - 46 x + 5) - ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[5, 1, 1, 1]](x) = -x^2*(1664*x^3-512*x^2-46*x+5)/(-1+2*x)/(1 +2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 5, 214, 14796, 928440, 59710640, 3816816864, 244350831296, 15637260100480, 1000803734956800, 64051133616459264, 4099277438168247296, 262353677855260293120, 16790636633736499343360, 1074600724543137390321664, 68774446691016765587767296, 4401564583100977418007183360, 281700133400448083954630328320, 18028808536316908905575283032064, 1153843746345270465436009695543296 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [4, 4]](x) = 2 4 3 2 x (2816 x - 336 x - 202 x + 23 x - 2) -------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[4, 4]](x) = x^2*(2816*x^4-336*x^3-202*x^2+23*x-2)/(1+x)/(-1+ 2*x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 2, 79, 6007, 369909, 23907523, 1526353773, 97746297203, 6254808594253, 400323021073459, 25620429012956877, 1639711366204626739, 104941464887104074957, 6716254753574589219635, 429840288215975079668941, 27509778702027184079467315, 1760625832830463320977263821, 112680053366738075918598419251, 7211523414421822084830719298765, 461537498539787249812748367246131 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [4, 3, 1]](x) = 2 3 2 x (1280 x - 96 x - 42 x + 9) - ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[4, 3, 1]](x) = -x^2*(1280*x^3-96*x^2-42*x+9)/(-1+2*x)/(1+2*x )/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 9, 426, 29580, 1856840, 119421104, 7633633056, 488701659840, 31274520190080, 2001607469869824, 128102267232743936, 8198554876335795200, 524707355710517790720, 33581273267472987500544, 2149201449086274735906816, 137548893382033530996572160, 8803129166201954835298549760, 563400266800896167906397323264, 36057617072633817811139112861696, 2307687492690540930871973578014720 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [4, 2, 2]](x) = 2 3 2 2 x (512 x - 80 x - 20 x - 3) ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[4, 2, 2]](x) = 2*x^2*(512*x^3-80*x^2-20*x-3)/(-1+2*x)/(1+2*x )/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 6, 352, 23480, 1488384, 95490272, 6107652096, 390949397376, 25019807039488, 1601282921696768, 102481862653378560, 6558843119193700352, 419765897078413197312, 26865018413818406559744, 1719361162471579523743744, 110039114654385869036945408, 7042503333781419160403705856, 450720213427599249650479923200, 28846093658316937203705496731648, 1846149994149074617420871559020544 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 9 x F[[3, 2, 1, 1, 1], [4, 2, 1, 1]](x) = -------------------- (-1 + x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[4, 2, 1, 1]](x) = 9*x^2/(-1+x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 9, 585, 37449, 2396745, 153391689, 9817068105, 628292358729, 40210710958665, 2573485501354569, 164703072086692425, 10540996613548315209, 674623783267092173385, 43175922129093899096649, 2763259016262009542185545, 176848577040768610699874889, 11318308930609191084791992905, 724371771558988229426687545929, 46359793379775246683308002939465, 2967026776305615787731712188125769 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [4, 1, 1, 1, 1]](x) = 2 3 2 x (640 x - 752 x + 86 x + 3) - ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[4, 1, 1, 1, 1]](x) = -x^2*(640*x^3-752*x^2+86*x+3)/(-1+2*x)/ (1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 3, 242, 14340, 935720, 59594128, 3818680992, 244321005120, 15637737319040, 1000796099459328, 64051255784417792, 4099275483480908800, 262353709130257704960, 16790636133336540745728, 1074600732549536727867392, 68774446562914376187002880, 4401564585150615648419348480, 281700133367653872268035555328, 18028808536841616292560799137792, 1153843746336875147244241437327360 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 2 4 x F[[3, 2, 1, 1, 1], [3, 3, 2]](x) = ---------------------- (-1 + 4 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[3, 3, 2]](x) = 4*x^2/(-1+4*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 4, 272, 17472, 1118464, 71582720, 4581298176, 293203099648, 18764998443008, 1200959900614656, 76861433640386560, 4919131752988934144, 314824432191308562432, 20148763660243815104512, 1289520874255604435124224, 82529335952358684921692160, 5281877500950955839283265536, 338040160060861173731308863488, 21634570243895115118872486739968, 1384612495609287367608114029264896 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [3, 3, 1, 1]](x) = 2 4 3 2 x (3072 x - 416 x - 448 x + 120 x + 5) - -------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[3, 3, 1, 1]](x) = -x^2*(3072*x^4-416*x^3-448*x^2+120*x+5)/(1 +x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 5, 375, 23117, 1494211, 95397069, 6109143411, 390925536461, 25020188814387, 1601276813298893, 102481960387745587, 6558841555443829965, 419765922098411127603, 26865018013498439683277, 1719361168876698993783603, 110039114551903957516340429, 7042503335421129744733451059, 450720213401363880301204131021, 28846093658736703113293909668659, 1846149994142358362867456952552653 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [3, 2, 2, 1]](x) = 2 3 2 x (768 x + 576 x - 222 x - 5) ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[3, 2, 2, 1]](x) = x^2*(768*x^3+576*x^2-222*x-5)/(-1+2*x)/(1+ 2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 5, 482, 28668, 1871400, 119188080, 7637361312, 488642007488, 31275474627200, 2001592198874880, 128102511568660992, 8198550966961118208, 524707418260512614400, 33581272266673070305280, 2149201465099073410998272, 137548893125828752195043328, 8803129170301231296122880000, 563400266735307744533207777280, 36057617073683232585110145073152, 2307687492673750294488437061582848 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [3, 2, 1, 1, 1]](x) = 5 4 3 2 x (7040 x - 2128 x - 2786 x + 581 x + 48 x - 1) - ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[3, 2, 1, 1, 1]](x) = -x*(7040*x^5-2128*x^4-2786*x^3+581*x^2+ 48*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 1, 4, 464, 25906, 1716020, 108892170, 6984009028, 446737956202, 28595046901124, 1830021917512426, 117122380063762052, 7495816686579247338, 479732518141039929988, 30702877157826841919722, 1964984202152112404232836, 125758987912916077948641514, 8048575242823734829146913412, 515108815278365335561189761258, 32966964182013040571754459970180, 2109885707581672051057956115357930 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [3, 1, 1, 1, 1, 1]](x) = 2 2 x (864 x - 112 x - 1) - ----------------------------------------- (1 + x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[3, 1, 1, 1, 1, 1]](x) = -x^2*(864*x^2-112*x-1)/(1+x)/(-1+4*x )/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 163, 8333, 565811, 35686605, 2292327219, 146574707917, 9382928724787, 600473078385869, 38430826771460915, 2459564117276282061, 157412244243153630003, 10074381379761951526093, 644760444333561648198451, 41264667860887192107535565, 2640938752320152327442084659, 169020080000915796349437136077, 10817285122419794207730079839027, 692306247797087897431493070081229 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [2, 2, 2, 2]](x) = 2 3 2 x (1280 x - 496 x + 50 x + 1) - ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[2, 2, 2, 2]](x) = -x^2*(1280*x^3-496*x^2+50*x+1)/(-1+2*x)/(1 +2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 102, 5644, 375736, 23814320, 1527845088, 97722436288, 6255190369152, 400316912675584, 25620526747323904, 1639709802454756352, 104941489907102005248, 6716254353254622343168, 429840294621094549708800, 27509778599545272558862336, 1760625834470173905307009024, 112680053340502706569322627072, 7211523414841587994419132235776, 461537498533070995259333760778240 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [2, 2, 2, 1, 1]](x) = 2 3 2 x (512 x + 560 x - 158 x - 1) ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[2, 2, 2, 1, 1]](x) = x^2*(512*x^3+560*x^2-158*x-1)/(-1+2*x)/ (1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 1, 210, 11196, 752936, 47605360, 3056063136, 195438907840, 12510476184192, 800632298260224, 51241077928274432, 3279419213972184064, 209882986069204051968, 13432508606429255200768, 859680590843468975874048, 55019557173470067273351168, 3521251669350275456839614464, 225360106674446570801898913792, 14423046829788117466237658333184, 923074997064462926880323032317952 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1]](x) = 3 3 2 x (3584 x - 1808 x - 558 x + 159) --------------------------------------------------------------- (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[2, 2, 1, 1, 1, 1]](x) = x^3*(3584*x^3-1808*x^2-558*x+159)/(-\ 1+x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 0, 159, 7869, 539905, 33970585, 2183435049, 139590698889, 8936190768585, 571878031484745, 36600804853948489, 2342441737212520009, 149916427556574382665, 9594648861620911596105, 614057567175734806278729, 39299683658735079703302729, 2515179764407236249493443145, 160971504758092061520290222665, 10302176307141428872168890077769, 659339283615074856859738610111049 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 3 2 4 x (32 x - 108 x + 15) - ------------------------------------------------------ (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -4*x^3*(32*x^2-108*x+15)/(-1+2*x) /(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 0, 60, 2688, 190064, 11872256, 764481984, 48852271104, 3127738353408, 200156165701632, 12810300024101888, 819854314821386240, 52470754336051064832, 3358127026507326947328, 214920149712467089539072, 13754889261341919512887296, 880312917849978421991899136, 56340026660413089779541868544, 3605761707578206213308656910336, 230768749264016902172150146793472 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 F[[3, 2, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (1920 x - 640 x + 106 x - 9) - --------------------------------------------------------------- (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 16 x) (-1 + 64 x) and in Maple notation F[[3, 2, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(1920*x^3-640*x^2+106*x-9 )/(-1+x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(1+16*x)/(-1+64*x) The first 20 term , starting with k=1 are 0, 0, 9, 371, 27359, 1692711, 109264983, 6978043767, 446833399863, 28593519801527, 1830046351103927, 117121989126293943, 7495822941578728887, 479732418061048208823, 30702878759106709425591, 1964984176531634524073399, 125758988322843724031061431, 8048575236264892491827932599, 515108815383306812958292929975, 32966964180333976933400808222135 ---------------------------------- Their sum is 5 4 3 2 176 x + 118 x - 387 x - 133 x + 65 x - 1 ------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 64 x) and in Maple notation (176*x^5+118*x^4-387*x^3-133*x^2+65*x-1)/(-1+x)/(1+x)/(1+2*x)/(-1+4*x)/(-1+64*x ) The first 20 term , starting with k=1 are 1, 80, 4970, 317924, 20345770, 1302125316, 83336001514, 5333504027780, 341344257490154, 21846032478241412, 1398146078602891498, 89481349030566910596, 5726806337956209516778, 366515605629197118388868, 23456998760268614413426922, 1501247920657191317806922372, 96079866922060244321030545642, 6149111483011855636471510743684, 393543134912758760733878899351786, 25186760634416560686967058428607108 Regarding Lambda=, [3, 1, 1, 1, 1, 1] 8 7 6 5 F[[3, 1, 1, 1, 1, 1], [8]](x) = (9720 x + 2655 x - 21492 x - 8260 x 4 3 2 + 3660 x + 1044 x - 162 x - 16 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1],[8]](x) = (9720*x^8+2655*x^7-21492*x^6-8260*x^5+3660*x^4+ 1044*x^3-162*x^2-16*x+1)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1) /(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 0, 16, 80, 2646, 42115, 972791, 19458285, 416278291, 8666652830, 182638389066, 3829431252790, 80470467858236, 1689399957599445, 35481668918957641, 745076325143010395, 15646949564027419281, 328582809641148294760, 6900267119818388995316 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [7, 1]](x) = 2 5 4 3 2 x (567 x - 171 x + 441 x - 71 x - 17 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[7, 1]](x) = x^2*(567*x^5-171*x^4+441*x^3-71*x^2-17*x+1)/( 1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 1, 74, 586, 17282, 299152, 6734855, 136662247, 2908593713, 60707131903, 1278058252736, 26809440302008, 563260848047444, 11826081744903154, 248369085278903117, 5215557297697976269, 109528437637039212875, 2300081538592981595305, 48301852922613828900398 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [6, 2]](x) = 2 5 4 3 2 x (4050 x - 1206 x - 1035 x + 161 x + 32 x - 2) - ------------------------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[6, 2]](x) = -x^2*(4050*x^5-1206*x^4-1035*x^3+161*x^2+32*x -2)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 2, 165, 1720, 46922, 864042, 19083955, 391437440, 8298807192, 173535873082, 3650715541145, 76605732779160, 1609247223371062, 33789409361100122, 709620392619162735, 14901641611350018880, 312937944724865662532, 6571665548347731633162, 138005257815794036684725 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 F[[3, 1, 1, 1, 1, 1], [6, 1, 1]](x) = x 5 4 3 2 (648 x - 693 x - 144 x - 49 x - 13 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1],[6, 1, 1]](x) = x^2*(648*x^5-693*x^4-144*x^3-49*x^2-13*x+1 )/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 3, 162, 1877, 48758, 912730, 19996369, 411433074, 8710237635, 182246103677, 3832961621996, 80438694336091, 1689685917505132, 35479095278012844, 745099487895371043, 15646741099240030928, 328584685824089507249, 6900250234171772800231, 144905508049965664027510 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [5, 3]](x) = 2 4 3 2 x (927 x + 99 x - 13 x - 14 x + 1) ------------------------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[5, 3]](x) = x^2*(927*x^4+99*x^3-13*x^2-14*x+1)/(-1+x)/(1+ 3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 3, 184, 2459, 63375, 1218427, 26569418, 548921943, 11607631849, 243031357151, 5110180907052, 107254869057427, 2252881259605523, 47305737189892275, 993463355366438686, 20862344299292368511, 438112703992571201997, 9200335509896788537399, 193207327109882444949920 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 x (27 x - 2) F[[3, 1, 1, 1, 1, 1], [5, 2, 1]](x) = - -------------------------------- (-1 + 6 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[5, 2, 1]](x) = -x^2*(27*x-2)/(-1+6*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 2, 9, 396, 5913, 142560, 2808837, 60533244, 1256569281, 26515415448, 555651385245, 11679078977172, 245166152544729, 5149334315266416, 108128381962415733, 2270764573223931180, 47685437894121286257, 1001399752016003659464, 21029344743864318404301, 441616689803496620819268 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 F[[3, 1, 1, 1, 1, 1], [5, 1, 1, 1]](x) = x 6 5 4 3 2 (3402 x - 1224 x - 864 x + 10 x - 66 x - 9 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1],[5, 1, 1, 1]](x) = x^2*(3402*x^6-1224*x^5-864*x^4+10*x^3-\ 66*x^2-9*x+1)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 7, 209, 3435, 76936, 1550612, 33005064, 688272970, 14492009321, 303956398317, 6386277821419, 134081961072605, 2815983240839406, 59133248733128122, 1241819572447792274, 26078017410741386340, 547640099305738043191, 11500426429404379340027, 241509095604263774005629 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [4, 4]](x) = 3 3 2 x (756 x + 87 x - 67 x - 1) ------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[4, 4]](x) = x^3*(756*x^3+87*x^2-67*x-1)/(-1+x)/(1+3*x)/(1 +2*x)/(-1+6*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 1, 81, 1235, 31120, 611331, 13247801, 274687645, 5801142540, 121535955761, 2554885251721, 53629145261055, 1126424416622960, 23653009615497991, 496730379109015041, 10431183660491934665, 219056247340733475580, 4600168690500841777221, 96603655096883993452961 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [4, 3, 1]](x) = 3 4 3 2 x (3402 x - 1530 x - 837 x + 206 x + 9) ------------------------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[4, 3, 1]](x) = x^3*(3402*x^4-1530*x^3-837*x^2+206*x+9)/(-\ 1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 9, 359, 6580, 151890, 3089849, 65939839, 1376128560, 28981496630, 607897702489, 12772464920919, 268163378149940, 5631963216303970, 118266477877012929, 2483639027347536599, 52156034116221454720, 1095280194379795049910, 23000852833418914861169, 483018191056187467594879 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [4, 2, 2]](x) = 2 3 2 x (567 x - 9 x - 9 x + 1) - ----------------------------------------------------- (-1 + 6 x) (1 + 3 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[4, 2, 2]](x) = -x^2*(567*x^3-9*x^2-9*x+1)/(-1+6*x)/(1+3*x )/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 9, 279, 5427, 120123, 2485161, 52633071, 1101978999, 23175518715, 486405331533, 10217187472383, 214537763258091, 4505507026956027, 94613754221420625, 1986906074604751215, 41724873618481880703, 876223738574338756059, 18400686019101014296437, 386414519073657439077567 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [4, 2, 1, 1]](x) = 2 2 x (9 x - 3 x - 1) - -------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[4, 2, 1, 1]](x) = -x^2*(9*x^2-3*x-1)/(-1+x)/(1+x)/(1+3*x) /(-1+21*x) The first 20 term , starting with k=1 are 0, 1, 21, 433, 9120, 191431, 4020321, 84425923, 1772946840, 37231876261, 781869423621, 16419257829613, 344804414621160, 7240892706446491, 152058746837169921, 3193233683575187503, 67057907355095080080, 1408216054456948254121, 29572537143596058619221, 621023280015516795155593 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 2 F[[3, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (1134 x - 2772 x - 2862 x + 29 x + 220 x + 2 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1]](x) = -x^2*(1134*x^6-2772*x^5-2862*x^4+29* x^3+220*x^2+2*x-1)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+ 21*x) The first 20 term , starting with k=1 are 0, 1, 14, 167, 3865, 73276, 1584009, 32706402, 690965030, 14467798001, 304174337104, 6384316527337, 134099613051495, 2815824374424426, 59134678533852299, 1241806704253829972, 26078133224513951260, 547639056981897956551, 11500435810319182257594, 241509011176031564726307 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [3, 3, 2]](x) = 3 2 x (120 x + 49 x + 6) - ----------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 6 x) (1 + 3 x) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[3, 3, 2]](x) = -x^3*(120*x^2+49*x+6)/(-1+x)/(1+x)/(1+2*x) /(-1+6*x)/(1+3*x)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 6, 181, 4126, 88556, 1871492, 39370737, 827207352, 17373867412, 364866352378, 7662284036693, 160908509157578, 3379081956900468, 70960740687713064, 1490175671982902449, 31293689816934445204, 657167490387240034124, 13800517323522174663950, 289810863946305168102405 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [3, 3, 1, 1]](x) = 3 3 2 x (1701 x - 63 x - 27 x - 11) - -------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[3, 3, 1, 1]](x) = -x^3*(1701*x^3-63*x^2-27*x-11)/(-1+x)/( 1+3*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 11, 236, 5735, 117110, 2511551, 52393376, 1104129695, 23156142770, 486579655991, 10215618375116, 214551884602055, 4505379933266030, 94614898059847631, 1986895780044559256, 41724966269480561615, 876222904715221487690, 18400693523832682291271, 386414451531071264862596 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [3, 2, 2, 1]](x) = 3 4 3 2 2 x (567 x + 45 x + 18 x - 14 x + 9) ------------------------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[3, 2, 2, 1]](x) = 2*x^3*(567*x^4+45*x^3+18*x^2-14*x+9)/(-\ 1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 18, 278, 7390, 144600, 3156188, 65342788, 1381508580, 28933076450, 608333543158, 12768542354898, 268198681775570, 5631645483673300, 118269337475471928, 2483613290961405608, 52156265743739680360, 1095278109732131019150, 23000871595248278558498, 483018022199723194318918 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1]](x) = 3 25 x (-1 + 9 x) ----------------------------------------- (-1 + x) (-1 + 6 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1]](x) = 25*x^3*(-1+9*x)/(-1+x)/(-1+6*x)/(9*x +1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 25, 250, 7225, 130750, 2915125, 59576650, 1265178625, 26437931350, 556348742125, 11672802765250, 245222638452025, 5148825942100750, 108132957320906725, 2270723394997512250, 47685808498159056625, 1001396416579663726150, 21029374762791377804125, 441616419633153086220850 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (10692 x + 5445 x - 2565 x - 884 x + 147 x + 16 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1]](x) = -x*(10692*x^6+5445*x^5-2565*x^4-\ 884*x^3+147*x^2+16*x-1)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/ (-1+21*x) The first 20 term , starting with k=1 are 1, 0, 16, 80, 2646, 42115, 972791, 19458285, 416278291, 8666652830, 182638389066, 3829431252790, 80470467858236, 1689399957599445, 35481668918957641, 745076325143010395, 15646949564027419281, 328582809641148294760, 6900267119818388995316, 144905356079147425815900 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [2, 2, 2, 2]](x) = 3 2 x (129 x - 8 x + 4) - ------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[2, 2, 2, 2]](x) = -x^3*(129*x^2-8*x+4)/(-1+x)/(1+3*x)/(1+ 2*x)/(-1+6*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 4, 48, 1553, 28198, 637812, 13008926, 276839161, 5781773976, 121710287600, 2553316220884, 53643266671449, 1126297323530834, 23654153454522868, 496720084554203922, 10431276311495996417, 219055413481664634772, 4600176195232558199616, 96603587554298255086040 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1]](x) = 3 2 2 x (21 x + 5 x - 1) (108 x - 39 x + 11) - ------------------------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1]](x) = -x^3*(21*x^2+5*x-1)*(108*x^2-39*x+11 )/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 11, 93, 3259, 55994, 1284675, 25971547, 554301143, 11559204288, 243467190439, 5106258274601, 107290172616627, 2252563526376982, 47308596787753403, 993437618974926855, 20862575926805213311, 438110619344858743676, 9200354271726103807167, 193207158253417735825909 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1]](x) = 3 4 3 2 x (486 x + 990 x - 351 x - 137 x + 12) ------------------------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1]](x) = x^3*(486*x^4+990*x^3-351*x^2-137* x+12)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 12, 67, 2540, 39480, 930472, 18485537, 396818280, 8250374710, 173971721132, 3646792864407, 76641036471220, 1608929489743940, 33792268960156992, 709594656224063677, 14901873238873625360, 312935860077120919170, 6571684310177143758052, 138005088959329036995347 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 F[[3, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (1701 x - 1827 x + 279 x + 104 x - 7) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x) and in Maple notation F[[3, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -x^3*(1701*x^4-1827*x^3+279*x^ 2+104*x-7)/(1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21*x) The first 20 term , starting with k=1 are 0, 0, 7, 22, 1006, 13531, 332458, 6435373, 139353487, 2884375012, 60925063309, 1276096892224, 26827092214468, 563101981034593, 11827511545029460, 248356217079559975, 5215673111465160349, 109527395313150698674, 2300090919507736085311, 48301768494381183773026 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 3 F[[3, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (648 x - 693 x - 144 x - 49 x - 13 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (9 x + 1) (-1 + 21 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(648*x^5-693*x^4-144*x^ 3-49*x^2-13*x+1)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(9*x+1)/(-1+21* x) The first 20 term , starting with k=1 are 0, 0, 1, 3, 162, 1877, 48758, 912730, 19996369, 411433074, 8710237635, 182246103677, 3832961621996, 80438694336091, 1689685917505132, 35479095278012844, 745099487895371043, 15646741099240030928, 328584685824089507249, 6900250234171772800231 ---------------------------------- Their sum is 5 4 3 2 276 x - 400 x - 404 x - 8 x + 22 x - 1 --------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 21 x) and in Maple notation (276*x^5-400*x^4-404*x^3-8*x^2+22*x-1)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(-1+21*x ) The first 20 term , starting with k=1 are 1, 12, 190, 3783, 77939, 1628530, 34147908, 716804681, 15051073297, 316061637468, 6637228835546, 139381412653099, 2927007307127775, 61467139301809526, 1290809840439870304, 27107006139881034237, 569247125881266908573, 11954189625169487787904, 251037982018535666087382, 5271797621729110142191295 Regarding Lambda=, [2, 2, 2, 2] F[[2, 2, 2, 2], [8]](x) = 7 6 5 4 3 682 x + 169 x - 1406 x - 26 x + 401 x - 16 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[8]](x) = (682*x^7+169*x^6-1406*x^5-26*x^4+401*x^3-16*x+1)/(1+x) /(-1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 1, 7, 35, 320, 3341, 41056, 538525, 7332304, 101377661, 1411779920, 19719352445, 275799669328, 3859559400061, 54024049478224, 756277865771645, 10587537696717392, 148223411062166141, 2075115063311333968 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [7, 1]](x) = 4 2 x (70 x + 11 x + 4) - ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[7, 1]](x) = -x^4*(70*x^2+11*x+4)/(1+x)/(-1+2*x)/(1+2*x)/(-1+6*x )/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 4, 71, 1199, 17521, 252063, 3559857, 50063983, 702090257, 9837082607, 137763458833, 1928964710127, 27007121756433, 378109567196911, 5293592445464849, 74110647948898031, 1037551182822117649, 14525729271683739375 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [6, 2]](x) = 2 3 2 x (164 x - 12 x - 13 x + 1) - ----------------------------------------------------- (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[6, 2]](x) = -x^2*(164*x^3-12*x^2-13*x+1)/(-1+2*x)/(1+2*x)/(-1+6 *x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 3, 36, 340, 4256, 55088, 750016, 10351680, 144116736, 2012462848, 28144772096, 393843307520, 5512726306816, 77171604369408, 1080363413323776, 15124852157071360, 211746521791594496, 2964442833162272768, 41502148918653485056 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [6, 1, 1]](x) = 4 3 2 3 x (14 x + 39 x + 8 x + 4) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[6, 1, 1]](x) = 3*x^4*(14*x^3+39*x^2+8*x+4)/(1+x)/(-1+x)/(-1+2*x )/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 12, 216, 3585, 52626, 755937, 10680642, 150187665, 2106288162, 29511178257, 413290655778, 5786893013265, 81021369741858, 1134328683700497, 15880777407971874, 222331943560384785, 3112653549611655714, 43577187810470007057 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [5, 3]](x) = 4 2 x (196 x + 64 x + 15) - ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[5, 3]](x) = -x^4*(196*x^2+64*x+15)/(1+x)/(-1+2*x)/(1+2*x)/(-1+6 *x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 15, 289, 4771, 70189, 1007811, 14241213, 200248707, 2808390013, 39348214147, 551054300797, 7715856977283, 108028494479997, 1512438238965123, 21174369901154941, 296442591318388099, 4150204733197308541, 58102917079099518339 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [5, 2, 1]](x) = 3 2 x (16 x - 26 x - 3) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[5, 2, 1]](x) = x^3*(16*x^2-26*x-3)/(1+x)/(-1+x)/(-1+2*x)/(1+4*x )/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 62, 839, 11990, 167127, 2343222, 32792343, 459145910, 6427834391, 89990523062, 1259863972887, 17638109053110, 246933473078295, 3457068837888182, 48398962871528471, 677585483637547190, 9486196757182114839, 132806754655525888182 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [5, 1, 1, 1]](x) = 2 4 3 2 x (294 x + 171 x - 30 x - 11 x + 1) - ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[5, 1, 1, 1]](x) = -x^2*(294*x^4+171*x^3-30*x^2-11*x+1)/(1+x)/(-\ 1+2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 4, 46, 525, 6921, 93639, 1294441, 18012775, 251565241, 3518067399, 49230530361, 689090322375, 9646452411961, 135045419209159, 1890606549466681, 26468315104645575, 370555354622529081, 5187768612922814919, 72628722513115385401 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [4, 4]](x) = 2 4 3 2 x (94 x + 141 x - 29 x - 12 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[4, 4]](x) = x^2*(94*x^4+141*x^3-29*x^2-12*x+1)/(1+x)/(-1+x)/(-1 +2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 4, 36, 305, 3362, 40929, 538882, 7330705, 101383458, 1411756049, 19719445538, 275799294225, 3859560890914, 54024043503889, 756277889630754, 10587537601237265, 148223411443933730, 2075115061784088849, 29051534708753834530 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [4, 3, 1]](x) = 3 2 x (98 x - 10 x - 3) --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[4, 3, 1]](x) = x^3*(98*x^2-10*x-3)/(1+x)/(-1+2*x)/(-1+6*x)/(1+4 *x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 61, 885, 12855, 181433, 2553831, 35815545, 501870535, 7028576313, 98415710151, 1377908546105, 19291272233415, 270081042894393, 3781154325795271, 52936277570539065, 741108593412567495, 10375524530850991673, 145257368856263029191 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [4, 2, 2]](x) = 2 3 2 x (14 x + 44 x + 8 x - 1) ------------------------------------------------- (1 + x) (-1 + x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[4, 2, 2]](x) = x^2*(14*x^3+44*x^2+8*x-1)/(1+x)/(-1+x)/(-1+6*x)/ (1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 8, 81, 906, 11441, 152194, 2084881, 28905122, 403005201, 5631942690, 78786941201, 1102653540898, 15434976145681, 216076591923746, 3024993976455441, 42349445270643234, 592889413538091281, 8300434859437859362, 116205986485917258001 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 3 3 x F[[2, 2, 2, 2], [4, 2, 1, 1]](x) = - --------------------------------- (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[4, 2, 1, 1]](x) = -3*x^3/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 3, 66, 1080, 16080, 230928, 3267936, 45960960, 644712960, 9033539328, 126514899456, 1771480688640, 24802362224640, 347242866659328, 4861458906341376, 68060777327493120, 952852998417285120, 13339954672836476928, 186759441589677981696 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [4, 1, 1, 1, 1]](x) = 3 2 x (126 x + 23 x - 4) ----------------------------------------------------- (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[4, 1, 1, 1, 1]](x) = x^3*(126*x^2+23*x-4)/(-1+2*x)/(1+2*x)/(-1+ 6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 4, 41, 530, 6880, 93744, 1293936, 18014560, 251557760, 3518096384, 49230413056, 689090787840, 9646450544640, 135045426663424, 1890606519627776, 26468315223941120, 370555354145259520, 5187768614831652864, 72628722505479684096 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 4 21 x F[[2, 2, 2, 2], [3, 3, 2]](x) = - ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[3, 3, 2]](x) = -21*x^4/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 21, 441, 7119, 105441, 1511055, 21364497, 300362223, 4212628497, 59022146799, 826582149393, 11573782671087, 162042752901393, 2268657313713903, 31761555030675729, 444663886261776111, 6225307102659219729, 87154375607196118767 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [3, 3, 1, 1]](x) = 2 4 3 2 x (140 x - 114 x + 30 x + 10 x - 1) - ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[3, 3, 1, 1]](x) = -x^2*(140*x^4-114*x^3+30*x^2+10*x-1)/(1+x)/(-\ 1+x)/(-1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 1, 8, 79, 910, 11415, 152278, 2084503, 28906550, 402999319, 5631965878, 78786847767, 1102653913270, 15434974653463, 216076597887158, 3024993952590871, 42349445366079670, 592889413156301847, 8300434860964929718, 116205986479808801815 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 3 2 x (7 x - 2) F[[2, 2, 2, 2], [3, 2, 2, 1]](x) = ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[3, 2, 2, 1]](x) = 2*x^3*(7*x-2)/(-1+2*x)/(-1+6*x)/(1+4*x)/(-1+ 14*x) The first 20 term , starting with k=1 are 0, 0, 4, 58, 900, 12800, 181664, 2552928, 35819200, 501856000, 7028634624, 98415477248, 1377909478400, 19291268505600, 270081057808384, 3781154266144768, 52936277809152000, 741108592458137600, 10375524534668754944, 145257368840992063488 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 3 x (8 x + 5) F[[2, 2, 2, 2], [3, 2, 1, 1, 1]](x) = -------------------------------------- (1 + x) (-1 + x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[3, 2, 1, 1, 1]](x) = x^3*(8*x+5)/(1+x)/(-1+x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 5, 58, 865, 11906, 167505, 2341794, 32798225, 459122722, 6427927825, 89990150690, 1259865465105, 17638103089698, 246933496942865, 3457068742451746, 48398963253317905, 677585482110476834, 9486196763290571025, 132806754631092412962 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [3, 1, 1, 1, 1, 1]](x) = 4 3 2 3 x (70 x + 25 x - 27 x - 3) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[3, 1, 1, 1, 1, 1]](x) = -3*x^4*(70*x^3+25*x^2-27*x-3)/(1+x)/(-1 +x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 9, 225, 3534, 52815, 755118, 10683855, 150174558, 2106340335, 29510968542, 413291493615, 5786889657822, 81021383159535, 1134328630013406, 15880777622703855, 222331942701391326, 3112653553047564015, 43577187796726111710 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [2, 2, 2, 2]](x) = 5 4 3 2 x (662 x - 153 x - 338 x + 10 x + 15 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[2, 2, 2, 2]](x) = -x*(662*x^5-153*x^4-338*x^3+10*x^2+15*x-1)/(1 +x)/(-1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 1, 7, 35, 320, 3341, 41056, 538525, 7332304, 101377661, 1411779920, 19719352445, 275799669328, 3859559400061, 54024049478224, 756277865771645, 10587537696717392, 148223411062166141, 2075115063311333968, 29051534702645465725 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [2, 2, 2, 1, 1]](x) = 4 2 x (28 x + 104 x + 13) - ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[2, 2, 2, 1, 1]](x) = -x^4*(28*x^2+104*x+13)/(1+x)/(-1+2*x)/(1+2 *x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 13, 299, 4721, 70399, 1006929, 14244783, 200234257, 2808447983, 39347981585, 551055231727, 7715853250833, 108028509388527, 1512438179320081, 21174370139746031, 296442590363980049, 4150204737014984431, 58102917063828640017 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [2, 2, 1, 1, 1, 1]](x) = 3 3 2 x (4 x + 144 x + 31 x - 4) ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[2, 2, 1, 1, 1, 1]](x) = x^3*(4*x^3+144*x^2+31*x-4)/(1+x)/(-1+2* x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 4, 29, 355, 4185, 55319, 749049, 10355335, 144101945, 2012521159, 28144538169, 393844239815, 5512722574905, 77171619283399, 1080363353656889, 15124852395684295, 211746520837099065, 2964442836980036039, 41502148903382257209 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [2, 1, 1, 1, 1, 1, 1]](x) = 4 2 x (14 x - 31 x - 3) ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[2, 1, 1, 1, 1, 1, 1]](x) = x^4*(14*x^2-31*x-3)/(1+x)/(-1+2*x)/( 1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 76, 1174, 17626, 251622, 3561642, 50056758, 702119242, 9836966326, 137763924298, 1928962846902, 27007129210698, 378109537374390, 5293592564760394, 74110647471694006, 1037551184730955594, 14525729264048300214 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 F[[2, 2, 2, 2], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (94 x + 141 x - 29 x - 12 x + 1) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 14 x) and in Maple notation F[[2, 2, 2, 2],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(94*x^4+141*x^3-29*x^2-12*x+1 )/(1+x)/(-1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+14*x) The first 20 term , starting with k=1 are 0, 0, 1, 4, 36, 305, 3362, 40929, 538882, 7330705, 101383458, 1411756049, 19719445538, 275799294225, 3859560890914, 54024043503889, 756277889630754, 10587537601237265, 148223411443933730, 2075115061784088849 ---------------------------------- Their sum is 4 3 2 24 x + 27 x - 88 x + 20 x - 1 ----------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) and in Maple notation (24*x^4+27*x^3-88*x^2+20*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(-1+14*x) The first 20 term , starting with k=1 are 1, 7, 62, 776, 10448, 144160, 2006224, 28016464, 391809200, 5482806608, 76744170416, 1074327675472, 15040043238832, 210557340125776, 2947783170631088, 41268846842411600, 577763150515930544, 8088679875557456464, 113241492867813803440, 1585380747809455405648 Regarding Lambda=, [2, 2, 2, 1, 1] 8 7 6 5 4 F[[2, 2, 2, 1, 1], [8]](x) = (8272 x + 540 x - 19242 x - 1242 x + 5689 x 3 2 + 360 x - 300 x - 18 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[8]](x) = (8272*x^8+540*x^7-19242*x^6-1242*x^5+5689*x^4+360*x ^3-300*x^2-18*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+ 28*x) The first 20 term , starting with k=1 are 0, 1, 0, 26, 360, 12684, 327744, 9440068, 261668736, 7353102692, 205622765568, 5760076150500, 161255739820032, 4515424601055460, 126429249884700672, 3540045385615775972, 99121006907457896448, 2775390832296994191588, 77710916915412631289856, 2175905937520431111878884 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [7, 1]](x) = 2 5 4 3 2 x (840 x + 1162 x - 300 x - 203 x - 15 x + 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[7, 1]](x) = -x^2*(840*x^5+1162*x^4-300*x^3-203*x^2-15*x+1)/( 1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 1, 146, 2657, 87192, 2308089, 65938720, 1833070009, 51457785728, 1439498247737, 40319143463424, 1128804067628601, 31607833307305984, 885006138081791545, 24780303810242551808, 693847187241094155833, 19427734437187207004160, 543976432296777307885113, 15231341423754083080732672 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[2, 2, 2, 1, 1], [6, 2]](x) = - x 6 5 4 3 2 (2560 x - 1232 x - 2956 x - 194 x + 308 x + 31 x - 2)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[6, 2]](x) = -x^2*(2560*x^6-1232*x^5-2956*x^4-194*x^3+308*x^2 +31*x-2)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 5, 384, 7855, 246112, 6623655, 188098304, 5240308295, 146992476672, 4113149581255, 115194576461824, 3225184238049735, 90307797543084032, 2528591942093722055, 70800838267334098944, 1982420832592886559175, 55507809701486849032192, 1554218407752685617050055, 43518118055964040352497664 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [6, 1, 1]](x) = 2 3 2 x (56 x + 46 x + 10 x - 1) - ----------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[6, 1, 1]](x) = -x^2*(56*x^3+46*x^2+10*x-1)/(1+2*x)/(1+4*x)/( -1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 6, 382, 8352, 257224, 6965280, 197396704, 5503365888, 154331646592, 4318911235584, 120953263023616, 3386453866758144, 94823083244062720, 2655022580867309568, 74340869763882016768, 2081541978389233336320, 58283199144892090974208, 1631929338556987137196032, 45694023854595536762306560 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[2, 2, 2, 1, 1], [5, 3]](x) = x 5 4 3 2 (208 x - 1412 x - 290 x + 16 x - 8 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[5, 3]](x) = x^2*(208*x^5-1412*x^4-290*x^3+16*x^2-8*x+1)/(1+x )/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 10, 497, 11288, 341625, 9301152, 263057593, 7339213696, 205761654329, 5758687261184, 161269628708409, 4515285712164864, 126430638773587513, 3540031496726880256, 99121145796346777145, 2775389443408105275392, 77710930804301520145977, 2175905798631542222880768, 60925365000571842065043001 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [5, 2, 1]](x) = 2 3 2 2 x (68 x + 34 x - 4 x + 1) ------------------------------------------------- (-1 + x) (1 + 2 x) (1 + x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[5, 2, 1]](x) = 2*x^2*(68*x^3+34*x^2-4*x+1)/(-1+x)/(1+2*x)/(1 +x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 24, 1086, 26208, 776014, 21306240, 600796814, 16780088832, 470264709006, 13163189630976, 368611531887502, 10320700670631936, 288983840999908238, 8091505325775224832, 226562571343928484750, 6343747775407775416320, 177624979933639933748110, 4973499015919695922987008, 139257976667973708065334158 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[2, 2, 2, 1, 1], [5, 1, 1, 1]](x) = - x 6 5 4 3 2 (3920 x + 748 x - 1118 x - 572 x - 11 x + 4 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[5, 1, 1, 1]](x) = -x^2*(3920*x^6+748*x^5-1118*x^4-572*x^3-11 *x^2+4*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 14, 564, 14560, 421706, 11677344, 328299154, 9179206400, 257149951986, 7198879604224, 201581827028594, 5644159218647040, 158037777625265266, 4425044579163643904, 123901380161965931634, 3469237325092212244480, 97138658297041419410546, 2719882300372741068816384, 76156705729881434888412274 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[2, 2, 2, 1, 1], [4, 4]](x) = - x 6 5 4 3 2 (672 x - 1304 x - 904 x - 104 x + 143 x + 13 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[4, 4]](x) = -x^2*(672*x^6-1304*x^5-904*x^4-104*x^3+143*x^2+ 13*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 5, 248, 5695, 170208, 4657191, 131460736, 3670295879, 102873904640, 2879412987847, 80634120259584, 2257649799129543, 63215249947942912, 1770016442785518023, 49560565953818427392, 1387694791148139180487, 38855464707707747303424, 1087952906260209829310919, 30462682430841499494645760 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[2, 2, 2, 1, 1], [4, 3, 1]](x) = - x 6 5 4 3 2 (1120 x - 56 x + 236 x + 142 x + 41 x + 4 x - 2)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[4, 3, 1]](x) = -x^2*(1120*x^6-56*x^5+236*x^4+142*x^3+41*x^2+ 4*x-2)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 32, 1137, 29200, 843577, 23356032, 656601017, 18358434560, 514299947577, 14397759557632, 403163654755897, 11288318442885120, 316075555261713977, 8850089158416760832, 247802760324110814777, 6938474650185856122880, 194277316594085702110777, 5439764600745505044037632, 152313411459762915589721657 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [4, 2, 2]](x) = 2 4 3 2 2 x (112 x + 292 x + 93 x - x - 1) - ----------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[4, 2, 2]](x) = -2*x^2*(112*x^4+292*x^3+93*x^2-x-1)/(1+x)/(-1 +x)/(1+2*x)/(1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 26, 888, 23602, 672160, 18712050, 525004160, 14689516658, 411412197888, 11518485283954, 322528146307072, 9030682529848434, 252860166436069376, 7080074104475393138, 198242180481582465024, 5550779997925890006130, 155421850497491929268224, 4351811708374172650380402, 121850728890032573019324416 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 x (20 x + 1) F[[2, 2, 2, 1, 1], [4, 2, 1, 1]](x) = -------------------------------- (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[4, 2, 1, 1]](x) = x^2*(20*x+1)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 48, 1360, 38400, 1075456, 30117888, 843304960, 23612620800, 661153447936, 18512297852928, 518344340930560, 14513641567027200, 406381963893538816, 11378694989354631168, 318603459702198108160, 8920896871666915737600, 249785112406677935620096, 6993983147387068096708608, 195831528126837975427317760 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 3 F[[2, 2, 2, 1, 1], [4, 1, 1, 1, 1]](x) = 2 x 5 4 3 2 (280 x - 1574 x - 437 x + 169 x + 67 x + 10)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[4, 1, 1, 1, 1]](x) = 2*x^3*(280*x^5-1574*x^4-437*x^3+169*x^2 +67*x+10)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 20, 494, 15250, 414760, 11746770, 327604704, 9186150770, 257080507520, 7199574048370, 201574882584064, 5644228663090290, 158037083180820480, 4425051523608083570, 123901310717521485824, 3469238019536656669810, 97138651352596974960640, 2719882369817185513184370, 76156705035436990443945984 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 2 x (56 x + 2 x - 1) F[[2, 2, 2, 1, 1], [3, 3, 2]](x) = - ------------------------------------------ (1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[3, 3, 2]](x) = -x^2*(56*x^2+2*x-1)/(1+2*x)/(1+4*x)/(-1+4*x)/ (-1+28*x) The first 20 term , starting with k=1 are 0, 1, 24, 640, 17952, 501952, 14055552, 393543424, 11019231744, 308538293248, 8639072471040, 241894026047488, 6773032733515776, 189644916488126464, 5310057661734617088, 148681614527764037632, 4163085206778466664448, 116566385789784181964800, 3263858802113974274359296, 91388046459191073524678656 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [3, 3, 1, 1]](x) = 2 4 3 2 x (224 x + 184 x + 66 x + 20 x + 1) ----------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[3, 3, 1, 1]](x) = x^2*(224*x^4+184*x^3+66*x^2+20*x+1)/(1+x)/ (-1+x)/(1+2*x)/(1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 32, 831, 24160, 666599, 18767616, 524448583, 14695072256, 411356642247, 11519040839680, 322522590751175, 9030738085404672, 252859610880512455, 7080079660030951424, 198242124926026904007, 5550780553481445572608, 155421844941936373690823, 4351811763929728205979648, 121850728334477017463681479 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 2 F[[2, 2, 2, 1, 1], [3, 2, 2, 1]](x) = - x 6 5 4 3 2 (1120 x - 3640 x - 1428 x + 878 x + 129 x - 28 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[3, 2, 2, 1]](x) = -x^2*(1120*x^6-3640*x^5-1428*x^4+878*x^3+ 129*x^2-28*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28* x) The first 20 term , starting with k=1 are 0, 1, 46, 1000, 30590, 829696, 23494926, 655212160, 18372323470, 514161058816, 14399148446606, 403149765867520, 11288457331774350, 316074166372827136, 8850103047305651086, 247802621435221934080, 6938476039074745017230, 194277302705196813254656, 5439764739634393932948366, 152313410070874026700963840 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [3, 2, 1, 1, 1]](x) = 3 2 2 x (12 x + 64 x + 23) ------------------------------------------------- (-1 + x) (1 + 2 x) (1 + x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[3, 2, 1, 1, 1]](x) = 2*x^3*(12*x^2+64*x+23)/(-1+x)/(1+2*x)/( 1+x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 46, 864, 28430, 753792, 21528462, 598574592, 16802311054, 470042486784, 13165411853198, 368589309665280, 10320922892854158, 288981618777686016, 8091527547997447054, 226562349121706262528, 6343749997629997638542, 177624957711417711525888, 4973499238141918145209230, 139257974445751485843111936 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [3, 1, 1, 1, 1, 1]](x) = 3 2 6 x (84 x + 5 x - 3) - ----------------------------------------------------- (1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[3, 1, 1, 1, 1, 1]](x) = -6*x^3*(84*x^2+5*x-3)/(1+2*x)/(1+4*x )/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 18, 258, 9600, 244728, 7090272, 196146720, 5515865856, 154206646656, 4320161235456, 120940763023872, 3386578866757632, 94821833244063744, 2655035080867307520, 74340744763882020864, 2081543228389233328128, 58283186644892090990592, 1631929463556987137163264, 45694022604595536762372096 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 3 F[[2, 2, 2, 1, 1], [2, 2, 2, 2]](x) = x 5 4 3 2 (2912 x + 280 x - 1528 x - 200 x + 11 x + 10)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[2, 2, 2, 2]](x) = x^3*(2912*x^5+280*x^4-1528*x^3-200*x^2+11* x+10)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 10, 191, 6248, 164647, 4712736, 130905159, 3675851392, 102818348999, 2879968543232, 80628564703687, 2257705354684416, 63214694392385991, 1770021998341070848, 49560510398262866375, 1387695346703694725120, 38855459152152191726023, 1087952961815765384822784, 30462681875285943939002823 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [2, 2, 2, 1, 1]](x) = - x 6 5 4 3 2 (9648 x + 612 x - 4342 x - 270 x + 275 x + 18 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[2, 2, 2, 1, 1]](x) = -x*(9648*x^6+612*x^5-4342*x^4-270*x^3+ 275*x^2+18*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28* x) The first 20 term , starting with k=1 are 1, 0, 26, 360, 12684, 327744, 9440068, 261668736, 7353102692, 205622765568, 5760076150500, 161255739820032, 4515424601055460, 126429249884700672, 3540045385615775972, 99121006907457896448, 2775390832296994191588, 77710916915412631289856, 2175905937520431111878884, 60925363611682953176285184 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 3 F[[2, 2, 2, 1, 1], [2, 2, 1, 1, 1, 1]](x) = - 2 x 5 4 3 2 (1280 x - 168 x + 74 x + 267 x + 41 x - 9)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[2, 2, 1, 1, 1, 1]](x) = -2*x^3*(1280*x^5-168*x^4+74*x^3+267* x^2+41*x-9)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 18, 242, 9240, 232210, 6762528, 186709362, 5254197120, 146853587570, 4114538469888, 115180687572082, 3225323126937600, 90306408654191730, 2528605830982606848, 70800699378445196402, 1982422221481775431680, 55507795812597960088690, 1554218546641574505873408, 43518116667075151463390322 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 F[[2, 2, 2, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 2 x (700 x + 499 x + 264 x + 26 x - 4) ---------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x) and in Maple notation F[[2, 2, 2, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = 2*x^3*(700*x^4+499*x^3+264*x^2+26 *x-4)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 8, 76, 3352, 80246, 2377536, 65244270, 1840014464, 51388341262, 1440192692224, 40312199018894, 1128873512073216, 31607138862861198, 885013082526236672, 24780234365798105998, 693847881685538603008, 19427727492742762554254, 543976501741221752340480, 15231340729309638636266382 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 3 F[[2, 2, 2, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (208 x - 1412 x - 290 x + 16 x - 8 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (10 x + 1) (-1 + 28 x)) and in Maple notation F[[2, 2, 2, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(208*x^5-1412*x^4-290*x^3+ 16*x^2-8*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+4*x)/(10*x+1)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 10, 497, 11288, 341625, 9301152, 263057593, 7339213696, 205761654329, 5758687261184, 161269628708409, 4515285712164864, 126430638773587513, 3540031496726880256, 99121145796346777145, 2775389443408105275392, 77710930804301520145977, 2175905798631542222880768 ---------------------------------- Their sum is 5 4 3 2 240 x - 484 x - 376 x + 55 x + 26 x - 1 --------------------------------------------------- (-1 + 28 x) (-1 + 4 x) (1 + 4 x) (1 + 2 x) (-1 + x) and in Maple notation (240*x^5-484*x^4-376*x^3+55*x^2+26*x-1)/(-1+28*x)/(-1+4*x)/(1+4*x)/(1+2*x)/(-1+ x) The first 20 term , starting with k=1 are 1, 18, 420, 11684, 326180, 9131620, 255670756, 7158757092, 200444968164, 5612458717412, 157148840413412, 4400167525292260, 123204690649446628, 3449731338083875044, 96592477465408911588, 2704589369029839042788, 75728502332820460550372, 2120398065318947126130916, 59371145828930279012448484, 1662392083210047400033794276 Regarding Lambda=, [2, 2, 1, 1, 1, 1] F[[2, 2, 1, 1, 1, 1], [8]](x) = 7 5 4 3 2 3350 x - 8230 x - 785 x + 1420 x - 128 x - 17 x + 1 ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[8]](x) = (3350*x^7-8230*x^5-785*x^4+1420*x^3-128*x^2-17*x +1)/(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 0, 16, 25, 2361, 25140, 705976, 12011415, 260948071, 5010088930, 102282555486, 2024805502005, 40704384131581, 812004061411920, 16260913155825796, 325009922888407795, 6502281757165011891, 130024801642430950110, 2600704365352864600906 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [7, 1]](x) = 2 4 3 2 x (750 x + 380 x - 72 x - 16 x + 1) - -------------------------------------------------------------- (1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[7, 1]](x) = -x^2*(750*x^4+380*x^3-72*x^2-16*x+1)/(1+x)/(1 +2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 0, 73, 256, 14811, 188580, 4796863, 85441776, 1812622171, 35208930460, 714586264503, 14187514437096, 284791737978931, 5685417019731540, 113812501751345743, 2275208342034133216, 45514583376613646091, 910187500215506759820, 18204791667740621364583 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [6, 2]](x) = 2 4 3 2 x (250 x - 1000 x + 91 x + 34 x - 2) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[6, 2]](x) = -x^2*(250*x^4-1000*x^3+91*x^2+34*x-2)/(1+x)/( -1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 2, 0, 167, 965, 39042, 567060, 13400357, 247059075, 5148978032, 100893666170, 2038694391747, 40565495240985, 813392950304222, 16247024266930080, 325148811777310337, 6500892868276095695, 130038690531319893612, 2600565476463975602790, 52013392858501764652127 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [6, 1, 1]](x) = 2 4 3 2 x (750 x - 150 x + 44 x - 18 x + 1) --------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[6, 1, 1]](x) = x^2*(750*x^4-150*x^3+44*x^2-18*x+1)/(-1+x) /(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 0, 155, 1092, 39831, 605430, 13964665, 260448552, 5395993331, 106042469610, 2139587358525, 42604186837212, 853958434359031, 17060417172497790, 341395835865277985, 6826041679337589072, 136539583396732655931, 2730604166983842293970, 54613958334919927183045 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [5, 3]](x) = 2 4 3 2 x (730 x - 142 x + 55 x - 17 x + 1) ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[5, 3]](x) = x^2*(730*x^4-142*x^3+55*x^2-17*x+1)/(1+x)/(-1 +x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 0, 184, 1549, 51471, 819840, 18473974, 348619299, 7180593541, 141527957230, 2851389741564, 56819448534849, 1138472241971611, 22748611207069020, 455180556024145954, 9101527780076019199, 182051388900201081681, 3640819444500289695210, 72818472222498584935144 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [5, 2, 1]](x) = 2 x (25 x - 2) - ----------------------------------------- (1 + x) (-1 + 5 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[5, 2, 1]](x) = -x^2*(25*x-2)/(1+x)/(-1+5*x)/(10*x+1)/(-1+ 20*x) The first 20 term , starting with k=1 are 0, 2, 3, 372, 4003, 112872, 1921503, 41750372, 801609003, 16365187872, 323968796503, 6512701125372, 129920648484003, 2601746099562872, 52001587640671503, 1040365081060500372, 20803968262445359003, 416112698455083937872, 8321920635132562546503, 166441746032805669875372 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [5, 1, 1, 1]](x) = 2 5 4 3 2 x (1750 x + 450 x - 350 x + 19 x + 13 x - 1) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[5, 1, 1, 1]](x) = -x^2*(1750*x^5+450*x^4-350*x^3+19*x^2+ 13*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 4, 178, 2455, 59171, 1077174, 22573678, 440994565, 8923727641, 177431157244, 3559030862828, 71076404619675, 1422569524322311, 28440972626503714, 568923613152094378, 11377430565838760785, 227559027829506988181, 4551076389148787620584, 91022569445748948936328 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [4, 4]](x) = 2 5 4 3 2 x (1100 x + 100 x - 645 x + 57 x + 16 x - 1) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[4, 4]](x) = -x^2*(1100*x^5+100*x^4-645*x^3+57*x^2+16*x-1) /(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 1, 89, 850, 25066, 416871, 9167299, 175001380, 3583333556, 70833306841, 1424999758909, 28416665023110, 569166656703646, 11374999943189611, 227583333021328919, 4550833331661444040, 91024999991193199336, 1820416666620843017181, 36409166666430390293329 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [4, 3, 1]](x) = 2 5 4 3 2 x (500 x - 50 x + 275 x - 108 x + 11 x - 1) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[4, 3, 1]](x) = -x^2*(500*x^5-50*x^4+275*x^3-108*x^2+11*x-\ 1)/(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 6, 339, 4825, 117856, 2151786, 45133789, 881918715, 17847091986, 354860454916, 7118052251839, 142152761173605, 2845138805514916, 56881944026183646, 1137847220125311489, 22754861100604216495, 455118055502931546646, 9102152777514299933976, 182045138887570067784739 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [4, 2, 2]](x) = 2 3 2 x (300 x - 24 x - 8 x + 1) - -------------------------------------------------------------- (1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[4, 2, 2]](x) = -x^2*(300*x^3-24*x^2-8*x+1)/(1+x)/(1+2*x)/ (-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 1, 8, 249, 4152, 91581, 1749468, 35830369, 708317072, 14249913381, 284166211428, 5691664303689, 113749987834392, 2275833271109581, 45508333016617388, 910249998394060209, 18204166658547470112, 364091666625712808181, 7281749999793798975348, 145635833332296601739929 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 3 15 x F[[2, 2, 1, 1, 1, 1], [4, 2, 1, 1]](x) = ------------------------------ (1 + x) (-1 + 4 x) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[4, 2, 1, 1]](x) = 15*x^3/(1+x)/(-1+4*x)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 15, 345, 7095, 142665, 2856375, 57139785, 1142844855, 22857093705, 457142660535, 9142856356425, 182857139711415, 3657142844559945, 73142857092525495, 1462857142655816265, 29257142856337550775, 585142857139635917385, 11702857142844257955255, 234057142857091317535305 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [4, 1, 1, 1, 1]](x) = 3 4 3 2 x (250 x + 550 x - 105 x - 79 x + 11) ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[4, 1, 1, 1, 1]](x) = x^3*(250*x^4+550*x^3-105*x^2-79*x+11 )/(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 11, 108, 3150, 52225, 1146621, 21879228, 447939020, 8854283175, 178125601731, 3552086418298, 71145849064290, 1421875079877525, 28447917070948841, 568854168707648568, 11378125010283207960, 227552083385062538275, 4551145833593232075951, 91021875001304504470038 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [3, 3, 2]](x) = 3 x (25 x - 6) - ------------------------------------------ (-1 + x) (-1 + 5 x) (-1 + 4 x) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[3, 3, 2]](x) = -x^3*(25*x-6)/(-1+x)/(-1+5*x)/(-1+4*x)/(-1 +20*x) The first 20 term , starting with k=1 are 0, 0, 6, 155, 3276, 66385, 1331946, 26659815, 533299416, 10666498445, 213332497686, 4266662510275, 85333312638756, 1706666563543305, 34133332819114626, 682666664101165535, 13653333320528197296, 273066666602730464965, 5461333333014010238766, 109226666665071482849595 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [3, 3, 1, 1]](x) = 3 3 2 2 x (800 x - 150 x - 30 x + 7) --------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[3, 3, 1, 1]](x) = 2*x^3*(800*x^3-150*x^2-30*x+7)/(-1+x)/( 1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 14, 192, 4710, 86020, 1805034, 35274792, 713872670, 14194357740, 284721767154, 5686108747792, 113805543390630, 2275277715552660, 45513888572175674, 910194442838499192, 18204722214103036590, 364086111070157230780, 7281805555349354574594, 145635277776741046096992 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [3, 2, 2, 1]](x) = 3 4 3 2 x (500 x + 350 x - 325 x + 121 x - 19) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[3, 2, 2, 1]](x) = -x^3*(500*x^4+350*x^3-325*x^2+121*x-19) /(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 19, 202, 6210, 103975, 2290659, 43744932, 895807540, 17708203225, 356249343549, 7104163363462, 142291650061470, 2843749916628075, 56895832915068439, 1137708331236430792, 22756249989493089000, 455104166614042690525, 9102291666403188757329, 182043749998681179026922 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [3, 2, 1, 1, 1]](x) = 3 25 x (8 x - 1) - ----------------------------------------- (1 + x) (-1 + 5 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[3, 2, 1, 1, 1]](x) = -25*x^3*(8*x-1)/(1+x)/(-1+5*x)/(10*x +1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 25, 150, 6225, 90650, 2143725, 39528150, 823831225, 16142965650, 326191018725, 6490478903150, 130142870706225, 2599523877340650, 52023809862893725, 1040142858838278150, 20806190484667581225, 416090476232861715650, 8322142857354784768725, 166439523810583447653150 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1]](x) = 3 3 2 x (350 x + 450 x - 185 x + 12) --------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1]](x) = x^3*(350*x^3+450*x^2-185*x+12)/(-\ 1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 12, 31, 2340, 27335, 730422, 12714681, 272948520, 5270993395, 107292469482, 2127087358781, 42729186836700, 852708434360055, 17072917172495742, 341270835865282081, 6827291679337580880, 136527083396732672315, 2730729166983842261202, 54612708334919927248581 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [2, 2, 2, 2]](x) = 3 4 3 2 x (500 x + 1500 x - 39 x - 87 x + 7) ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[2, 2, 2, 2]](x) = x^3*(500*x^4+1500*x^3-39*x^2-87*x+7)/(1 +x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 7, 32, 1408, 19505, 472437, 8611722, 180556978, 3527777915, 71388862567, 1419444203012, 28472220579348, 568611101146725, 11380555498747897, 227527777465767902, 4551388887217010518, 91019444435637621935, 1820472222176398616427, 36408611110874834650392 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [2, 2, 2, 1, 1]](x) = 3 3 2 x (330 x + 458 x - 174 x + 13) ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[2, 2, 2, 1, 1]](x) = x^3*(330*x^3+458*x^2-174*x+13)/(1+x) /(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 13, 47, 2934, 37590, 958713, 17085117, 362508124, 7041704780, 142916845863, 2837500853187, 56958337422714, 1137083353084770, 22762500095953813, 455041667135265257, 9102916668964891704, 182037500011312225560, 3640958333389178518563, 72817083333609696177327 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1]](x) = 5 4 3 2 x (4650 x + 800 x - 1190 x + 113 x + 17 x - 1) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1]](x) = -x*(4650*x^5+800*x^4-1190*x^3+113 *x^2+17*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 1, 0, 16, 25, 2361, 25140, 705976, 12011415, 260948071, 5010088930, 102282555486, 2024805502005, 40704384131581, 812004061411920, 16260913155825796, 325009922888407795, 6502281757165011891, 130024801642430950110, 2600704365352864600906, 52012003969612875544785 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (1250 x + 120 x - 123 x + 7) --------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = x^3*(1250*x^3+120*x^2-123*x+7) /(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 7, 3, 951, 7865, 258027, 4102413, 92386231, 1743177705, 35903374947, 707641819973, 14256958881711, 284097293534145, 5692361464176667, 113743057306899933, 2275902786478580391, 45507638932169196185, 910256944659951215187, 18204097223296176898293 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 F[[2, 2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (250 x - 1000 x + 91 x + 34 x - 2) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 5 x) (-1 + 4 x) (10 x + 1) (-1 + 20 x) and in Maple notation F[[2, 2, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(250*x^4-1000*x^3+91*x ^2+34*x-2)/(1+x)/(-1+x)/(1+2*x)/(-1+5*x)/(-1+4*x)/(10*x+1)/(-1+20*x) The first 20 term , starting with k=1 are 0, 0, 2, 0, 167, 965, 39042, 567060, 13400357, 247059075, 5148978032, 100893666170, 2038694391747, 40565495240985, 813392950304222, 16247024266930080, 325148811777310337, 6500892868276095695, 130038690531319893612, 2600565476463975602790 ---------------------------------- Their sum is 5 4 3 2 130 x + 28 x - 351 x + 182 x - 28 x + 1 -------------------------------------------------- (-1 + 20 x) (-1 + 4 x) (-1 + 5 x) (-1 + x) (1 + x) and in Maple notation (130*x^5+28*x^4-351*x^3+182*x^2-28*x+1)/(-1+20*x)/(-1+4*x)/(-1+5*x)/(-1+x)/(1+x ) The first 20 term , starting with k=1 are 1, 12, 169, 3112, 60999, 1214382, 24261829, 485116452, 9701763859, 194032593322, 3880639015089, 77612718316992, 1552254065503519, 31045079842239462, 620901589651037149, 12418031757633592732, 248360634978062543979, 4967212698697510088802, 99344253969668720872009, 1986885079372115905119672 Regarding Lambda=, [2, 1, 1, 1, 1, 1, 1] 9 8 7 6 F[[2, 1, 1, 1, 1, 1, 1], [8]](x) = (1854 x - 349 x - 5473 x + 1055 x 5 4 3 2 + 2054 x - 390 x - 218 x + 40 x + 6 x - 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[8]](x) = (1854*x^9-349*x^8-5473*x^7+1055*x^6+2054*x^5-\ 390*x^4-218*x^3+40*x^2+6*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-\ 1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 4, 0, 41, 1, 715, 273, 17686, 25872, 567986, 1733446, 21880951, 99462363, 942800317, 5286836919, 43233294236, 270222716674, 2048310985708 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 2 F[[2, 1, 1, 1, 1, 1, 1], [7, 1]](x) = x 6 5 4 3 2 (6 x + 741 x - 138 x - 158 x + 30 x + 6 x - 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[7, 1]](x) = x^2*(6*x^6+741*x^5-138*x^4-158*x^3+30*x^2+ 6*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 11, 0, 162, 21, 3424, 2835, 97493, 217602, 3462537, 13303290, 142148644, 729972243, 6349172750, 37927017765, 296737912815, 1915783519944, 14196341292463 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 2 F[[2, 1, 1, 1, 1, 1, 1], [6, 2]](x) = - x 6 5 4 3 2 (465 x + 192 x - 298 x - 74 x + 36 x + 4 x - 1)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[6, 2]](x) = -x^2*(465*x^6+192*x^5-298*x^4-74*x^3+36*x^ 2+4*x-1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 13, 0, 256, 90, 6763, 10080, 224986, 699930, 8816413, 40514760, 382733716, 2157925770, 17605674463, 110332535040, 835206380446, 5525576849610, 40257187308913 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 2 F[[2, 1, 1, 1, 1, 1, 1], [6, 1, 1]](x) = - x 6 5 4 3 2 (462 x - 362 x - 198 x + 171 x - 12 x - 8 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[6, 1, 1]](x) = -x^2*(462*x^6-362*x^5-198*x^4+171*x^3-\ 12*x^2-8*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 13, 1, 256, 161, 6784, 12684, 227821, 790042, 9034015, 43910867, 396037006, 2299476543, 18335646706, 116676326950, 873133398211, 5822266334864, 42172970828857 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 4 F[[2, 1, 1, 1, 1, 1, 1], [5, 3]](x) = x 5 4 3 2 (630 x - 367 x - 304 x + 47 x + 36 x - 6)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[5, 3]](x) = x^4*(630*x^5-367*x^4-304*x^3+47*x^2+36*x-6 )/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 6, 0, 199, 154, 6888, 15960, 266853, 1052898, 11353870, 59059000, 514118787, 3088567482, 24149990052, 156303867720, 1157524606801, 7784256595906, 56076624029034 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [5, 2, 1]](x) = 4 3 2 4 x (34 x - 2 x - 17 x + 3) -------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (7 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[5, 2, 1]](x) = 4*x^4*(34*x^3-2*x^2-17*x+3)/(1+x)/(-1+2 *x)/(-1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 12, 4, 400, 624, 14196, 45276, 568440, 2642728, 24869020, 140974548, 1147342560, 7208658912, 54493820484, 360968569420, 2627932540360, 17884611005976, 127726245780588 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 4 F[[2, 1, 1, 1, 1, 1, 1], [5, 1, 1, 1]](x) = x 4 3 2 (315 x + 61 x - 134 x - 14 x + 6)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[5, 1, 1, 1]](x) = x^4*(315*x^4+61*x^3-134*x^2-14*x+6)/ (-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 6, 10, 200, 595, 7161, 31185, 292725, 1598740, 13087316, 80740660, 613581150, 4029574185, 29436826971, 199521019435, 1427747323475, 9832422298930, 69610473450126 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 4 F[[2, 1, 1, 1, 1, 1, 1], [4, 4]](x) = - x 5 4 3 2 (294 x + 93 x - 82 x + 24 x - 6 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[4, 4]](x) = -x^4*(294*x^5+93*x^4-82*x^3+24*x^2-6*x+1)/ (-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 1, 0, 65, 84, 2835, 8442, 121675, 544698, 5432009, 30114084, 251652765, 1561151592, 11950688023, 78611513526, 575822844935, 3904240549566, 27967467026877 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 4 F[[2, 1, 1, 1, 1, 1, 1], [4, 3, 1]](x) = - x 4 3 2 (315 x + 37 x - 68 x + 7 x - 3)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[4, 3, 1]](x) = -x^4*(315*x^4+37*x^3-68*x^2+7*x-3)/(-1+ x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 5, 235, 770, 11613, 53865, 541845, 3044690, 25475923, 158883725, 1215978855, 8016200010, 58694702433, 398342342585, 2852631357265, 19653536590730, 139175134003143 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [4, 2, 2]](x) = 4 2 2 x (28 x - 2 x + 1) - ----------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (7 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[4, 2, 2]](x) = -2*x^4*(28*x^2-2*x+1)/(1+x)/(-1+2*x)/(1 +2*x)/(-1+4*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 2, 6, 160, 756, 8358, 47754, 407780, 2564232, 19715674, 130432302, 955947720, 6497136828, 46532979950, 320787784050, 2271518378380, 15775762809744, 111075284793186 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [4, 2, 1, 1]](x) = 4 3 x - ------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (7 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[4, 2, 1, 1]](x) = -3*x^4/(1+x)/(-1+x)/(-1+3*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 30, 240, 1770, 12663, 89460, 628680, 4408140, 30879123, 216220290, 1513741320, 10596787110, 74179303383, 519260504520, 3634839674160, 25443926146680, 178107628309443 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 3 F[[2, 1, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1]](x) = - x 5 4 3 2 (105 x + 54 x - 17 x + 18 x + 3 x - 1)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1]](x) = -x^3*(105*x^5+54*x^4-17*x^3+18*x^ 2+3*x-1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 1, 35, 80, 1176, 4326, 45130, 223735, 1941401, 11381051, 89250525, 571096890, 4241798926, 28376295676, 204821893220, 1401248313545, 9964901249751, 68948127036201 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [3, 3, 2]](x) = 5 x (14 x - 5) ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[3, 3, 2]](x) = x^5*(14*x-5)/(1+x)/(-1+2*x)/(-1+x)/(-1+ 4*x)/(-1+3*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 66, 616, 4998, 37905, 277452, 1991792, 14144196, 99824725, 702059358, 4927632528, 34546505994, 242038474865, 1695122892984, 11869279994824, 83098655256792 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [3, 3, 1, 1]](x) = 5 2 4 x (14 x - 10 x + 5) -------------------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (7 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[3, 3, 1, 1]](x) = 4*x^5*(14*x^2-10*x+5)/(1+x)/(-1+2*x) /(-1+x)/(1+2*x)/(-1+4*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 20, 80, 1176, 6216, 58548, 353640, 2835272, 18359792, 137213076, 922041120, 6666675288, 45685276728, 325026322004, 2250325644920, 15881726564424, 110545465845024 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [3, 2, 2, 1]](x) = 5 2 15 x (7 x + x - 2) --------------------------------------------------------------------- (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[3, 2, 2, 1]](x) = 15*x^5*(7*x^2+x-2)/(1+x)/(1+3*x)/(1+ 2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 30, 75, 1680, 6720, 79380, 411075, 3708210, 22129140, 175705530, 1131605475, 8438861340, 56579009760, 408927969480, 2799681721275, 19918349297070, 137850876848580 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1]](x) = 3 3 2 2 x (12 x + 2 x + 5 x - 1) ----------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (1 + 5 x) (7 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1]](x) = 2*x^3*(12*x^3+2*x^2+5*x-1)/(1+x)/ (-1+2*x)/(1+2*x)/(-1+4*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 2, 0, 70, 60, 2346, 5544, 88622, 351540, 3727570, 19444128, 168100374, 1011710700, 7886823674, 51102985752, 377922764926, 2543161519140, 18308466199458, 125606969638416 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 3 F[[2, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (42 x + 584 x - 397 x + 20 x + 24 x - 3)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1]](x) = x^3*(42*x^5+584*x^4-397*x^3+20 *x^2+24*x-3)/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 3, 0, 55, 15, 1274, 1491, 38328, 102060, 1411465, 5949042, 59269301, 319444125, 2681843076, 16425607653, 126221141374, 825425468610, 6060757555307, 40980660009324 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 5 F[[2, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2]](x) = - x 4 3 2 (126 x + 131 x + 35 x - 49 x + 9)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[2, 2, 2, 2]](x) = -x^5*(126*x^4+131*x^3+35*x^2-49*x+9) /(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 9, 5, 434, 924, 18501, 69825, 808698, 4097588, 36829793, 217942725, 1730097642, 11104767492, 82844692485, 554646210305, 4010155964066, 27437793186636 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 3 F[[2, 1, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1]](x) = - x 6 5 4 3 2 (210 x + 9 x + 93 x + 23 x - 6 x - 6 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1]](x) = -x^3*(210*x^6+9*x^5+93*x^4+23*x^3 -6*x^2-6*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/( 7*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 0, 35, 9, 1155, 1722, 42295, 133623, 1723799, 7984944, 75947235, 429546117, 3511826683, 22032503766, 166894875455, 1104558828291, 8049117729807, 54752221591788 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 3 F[[2, 1, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1]](x) = - x 5 4 3 2 (45 x - 595 x - 163 x + 92 x + 12 x - 3)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1]](x) = -x^3*(45*x^5-595*x^4-163*x^3+ 92*x^2+12*x-3)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 3, 0, 55, 5, 1273, 1050, 38055, 86835, 1385593, 5403200, 57535855, 297762465, 2582380713, 15484600950, 120934304455, 782208316895, 5790534838633, 38932494306300 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 F[[2, 1, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (3186 x - 623 x - 1603 x + 304 x + 200 x - 37 x - 6 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = -x*(3186*x^7-623*x^6-1603*x ^5+304*x^4+200*x^3-37*x^2-6*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x) /(-1+3*x)/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 0, 4, 0, 41, 1, 715, 273, 17686, 25872, 567986, 1733446, 21880951, 99462363, 942800317, 5286836919, 43233294236, 270222716674, 2048310985708, 13533849421092 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 3 F[[2, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (6 x + 741 x - 138 x - 158 x + 30 x + 6 x - 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (1 + 5 x) (7 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(6*x^6+741*x^5-138*x ^4-158*x^3+30*x^2+6*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x )/(1+5*x)/(7*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 0, 11, 0, 162, 21, 3424, 2835, 97493, 217602, 3462537, 13303290, 142148644, 729972243, 6349172750, 37927017765, 296737912815, 1915783519944 ---------------------------------- Their sum is 6 5 4 3 2 38 x - 67 x - 76 x + 151 x - 76 x + 15 x - 1 ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 4 x) (-1 + 3 x) (7 x - 1) and in Maple notation (38*x^6-67*x^5-76*x^4+151*x^3-76*x^2+15*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+4*x)/(-1 +3*x)/(7*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 422, 2610, 17033, 114685, 785688, 5434476, 37788851, 263538015, 1840906874, 12871138102, 90037801869, 630025968465, 4409233223180, 30860856361488, 216010943710487, 1512016564468435 Regarding Lambda=, [1, 1, 1, 1, 1, 1, 1, 1] 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [8]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[8]](x) = -1/(1+x)/(-1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [7, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[7, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [6, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[6, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [6, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[6, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [5, 3]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[5, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [5, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[5, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [5, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[5, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [4, 4]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[4, 4]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [4, 3, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[4, 3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [4, 2, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[4, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [4, 2, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[4, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [3, 3, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[3, 3, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [3, 3, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[3, 3, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [3, 2, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[3, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[2, 2, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 x F[[1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]](x) = - ---------------- (1 + x) (-1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1]](x) = -x/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ------------------------------------------ Doing the symmetric group on, 9, elements There are, 30, partitions of, 9, here there are in the usual order Regarding Lambda=, [9] 1 F[[9], [9]](x) = - ------ -1 + x and in Maple notation F[[9],[9]](x) = -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [8, 1]](x) = 0 and in Maple notation F[[9],[8, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [7, 2]](x) = 0 and in Maple notation F[[9],[7, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [7, 1, 1]](x) = 0 and in Maple notation F[[9],[7, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [6, 3]](x) = 0 and in Maple notation F[[9],[6, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [6, 2, 1]](x) = 0 and in Maple notation F[[9],[6, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [6, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[6, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [5, 4]](x) = 0 and in Maple notation F[[9],[5, 4]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [5, 3, 1]](x) = 0 and in Maple notation F[[9],[5, 3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [5, 2, 2]](x) = 0 and in Maple notation F[[9],[5, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [5, 2, 1, 1]](x) = 0 and in Maple notation F[[9],[5, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [5, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[5, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [4, 4, 1]](x) = 0 and in Maple notation F[[9],[4, 4, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [4, 3, 2]](x) = 0 and in Maple notation F[[9],[4, 3, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [4, 3, 1, 1]](x) = 0 and in Maple notation F[[9],[4, 3, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [4, 2, 2, 1]](x) = 0 and in Maple notation F[[9],[4, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [4, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[4, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [4, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[4, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [3, 3, 3]](x) = 0 and in Maple notation F[[9],[3, 3, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [3, 3, 2, 1]](x) = 0 and in Maple notation F[[9],[3, 3, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [3, 3, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[3, 3, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [3, 2, 2, 2]](x) = 0 and in Maple notation F[[9],[3, 2, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [3, 2, 2, 1, 1]](x) = 0 and in Maple notation F[[9],[3, 2, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [3, 2, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[3, 2, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [3, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[3, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [2, 2, 2, 2, 1]](x) = 0 and in Maple notation F[[9],[2, 2, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [2, 2, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[2, 2, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [2, 2, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[2, 2, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[2, 1, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[9], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[9],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 Regarding Lambda=, [8, 1] 8 7 6 5 4 3 F[[8, 1], [9]](x) = (2119 x - 2489 x - 4642 x + 8831 x - 5659 x + 1819 x 2 - 315 x + 28 x - 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[9]](x) = (2119*x^8-2489*x^7-4642*x^6+8831*x^5-5659*x^4+1819*x^3-315*x ^2+28*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 1, 1, 4, 11, 41, 162, 715, 3425, 17721, 98208, 579151, 3610399, 23632701, 161309214, 1140267587, 8293469533, 61704992481, 467302279580, 3587497622423 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [8, 1]](x) = - x 6 5 4 3 2 (3641 x - 6583 x + 4566 x - 1579 x + 290 x - 27 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[8, 1]](x) = -x*(3641*x^6-6583*x^5+4566*x^4-1579*x^3+290*x^2-27*x+1)/( 1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 1, 4, 11, 41, 162, 715, 3425, 17721, 98208, 579151, 3610399, 23632701, 161309214, 1140267587, 8293469533, 61704992481, 467302279580, 3587497622423, 27828087318427 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 2 F[[8, 1], [7, 2]](x) = - x 6 5 4 3 2 (1059 x - 3065 x + 2791 x - 1159 x + 243 x - 25 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[7, 2]](x) = -x^2*(1059*x^6-3065*x^5+2791*x^4-1159*x^3+243*x^2-25*x+1) /(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 256, 1274, 6790, 38517, 231376, 1463440, 9686722, 66674699, 474213676, 3464274726, 25854614734, 196221098401, 1508613935656, 11714025673532, 91640527613626 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 2 F[[8, 1], [7, 1, 1]](x) = - x 6 5 4 3 2 (1060 x - 3065 x + 2791 x - 1159 x + 243 x - 25 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[7, 1, 1]](x) = -x^2*(1060*x^6-3065*x^5+2791*x^4-1159*x^3+243*x^2-25*x +1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 1, 3, 13, 55, 256, 1274, 6791, 38545, 231846, 1469600, 9756429, 67391415, 481111996, 3527618006, 26416640627, 201082719165, 1549876414706, 12059261742892, 94497967182585 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 5 4 3 2 F[[8, 1], [6, 3]](x) = x (704 x - 1182 x + 688 x - 181 x + 22 x - 1)/( (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[6, 3]](x) = x^3*(704*x^5-1182*x^4+688*x^3-181*x^2+22*x-1)/(1+x)/(-1+x )/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 200, 1175, 7154, 45283, 297660, 2026079, 14225222, 102578411, 756413840, 5680601863, 43289104410, 333703837619, 2595532752740, 20327171974127, 160031189543918 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 5 4 3 2 F[[8, 1], [6, 2, 1]](x) = x (1415 x - 2366 x + 1376 x - 362 x + 44 x - 2)/ ((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[6, 2, 1]](x) = x^3*(1415*x^5-2366*x^4+1376*x^3-362*x^2+44*x-2)/(1+x)/ (-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 2, 12, 70, 400, 2352, 14357, 91310, 604350, 4148452, 29395927, 213936450, 1591226000, 12041852552, 92367269097, 715901287990, 5592700290850, 43952570600652, 346974396383867 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 5 4 3 2 F[[8, 1], [6, 1, 1, 1]](x) = x (704 x - 1183 x + 688 x - 181 x + 22 x - 1) /((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[6, 1, 1, 1]](x) = x^3*(704*x^5-1183*x^4+688*x^3-181*x^2+22*x-1)/(1+x) /(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 6, 35, 200, 1176, 7182, 45753, 303820, 2095786, 14941938, 109476731, 819757120, 6242627756, 48150725174, 374966316669, 2940768822100, 23184611543086, 183488236683290 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [5, 4]](x) = 4 3 2 x (50 x - 48 x + 13 x - 1) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) (8 x - 1) and in Maple notation F[[8, 1],[5, 4]](x) = -x^4*(50*x^3-48*x^2+13*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x )/(-1+4*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 1, 10, 79, 574, 4067, 28770, 205423, 1487398, 10936123, 81611530, 617260007, 4722903822, 36486904819, 284097002290, 2225907854431, 17525746019446, 138515657408555 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [5, 3, 1]](x) = 4 2 3 x (27 x - 11 x + 1) - ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) (8 x - 1) and in Maple notation F[[8, 1],[5, 3, 1]](x) = -3*x^4*(27*x^2-11*x+1)/(1+x)/(-1+x)/(-1+3*x)/(-1+4*x)/ (-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 30, 240, 1779, 12915, 93780, 687270, 5099853, 38334417, 291612750, 2241488340, 17379471447, 135705944559, 1065609526440, 8404417140450, 66511675276161, 527747849868141 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [5, 2, 2]](x) = 4 2 2 x (13 x - 9 x + 1) - ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) (8 x - 1) and in Maple notation F[[8, 1],[5, 2, 2]](x) = -2*x^4*(13*x^2-9*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/( -1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 2, 20, 160, 1190, 8702, 63840, 473420, 3555530, 27029002, 207667460, 1609732280, 12567552270, 98676594902, 778223851880, 6158639456740, 48866153739410, 388511297090402 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 4 4 3 2 F[[8, 1], [5, 2, 1, 1]](x) = - 3 x (263 x - 297 x + 114 x - 18 x + 1)/( (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[5, 2, 1, 1]](x) = -3*x^4*(263*x^4-297*x^3+114*x^2-18*x+1)/(1+x)/(-1+x )/(-1+2*x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 30, 240, 1785, 13062, 96012, 714360, 5388735, 41170866, 317951634, 2476683300, 19421417925, 153073125390, 1211090364696, 9609321496560, 76406355233355, 608483901756234 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [5, 1, 1, 1, 1]](x) = 4 2 x (15 x - 9 x + 1) - ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation F[[8, 1],[5, 1, 1, 1, 1]](x) = -x^4*(15*x^2-9*x+1)/(1+x)/(-1+2*x)/(-1+x)/(-1+5* x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 1, 10, 80, 595, 4361, 32200, 241410, 1839365, 14211571, 111000890, 873849340, 6917208935, 54958556381, 437728133580, 3491944518870, 27885437563305, 222830044234791 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 3 2 F[[8, 1], [4, 4, 1]](x) = x (356 x - 255 x + 57 x - 4)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[4, 4, 1]](x) = x^5*(356*x^3-255*x^2+57*x-4)/(1+x)/(-1+x)/(-1+2*x)/(-1 +3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 4, 55, 539, 4634, 37590, 297045, 2322793, 18115768, 141457316, 1107741635, 8703927687, 68615048502, 542497488082, 4299888271825, 34151064138821, 271683923044436 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 3 2 F[[8, 1], [4, 3, 2]](x) = x (320 x - 261 x + 65 x - 5)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[4, 3, 2]](x) = x^5*(320*x^3-261*x^2+65*x-5)/(1+x)/(-1+x)/(-1+2*x)/(-1 +3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 75, 791, 7238, 61845, 509985, 4127387, 33082236, 263928665, 2101554455, 16725954063, 133153942194, 1060641723965, 8454284521485, 67432748357219, 538171161518312 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 3 2 F[[8, 1], [4, 3, 1, 1]](x) = 3 x (128 x - 105 x + 26 x - 2)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[4, 3, 1, 1]](x) = 3*x^5*(128*x^3-105*x^2+26*x-2)/(1+x)/(-1+x)/(-1+2*x )/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 6, 90, 951, 8736, 75060, 623070, 5078337, 40988772, 329131374, 2635883250, 21082791483, 168535648008, 1347042531048, 10766566350630, 86062689252789, 688028078672844 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 3 2 F[[8, 1], [4, 2, 2, 1]](x) = x (384 x - 275 x + 65 x - 5)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[4, 2, 2, 1]](x) = x^5*(384*x^3-275*x^2+65*x-5)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 75, 805, 7566, 66633, 566145, 4709045, 38655012, 314635321, 2546867895, 20540346645, 165246975738, 1327172701769, 10646790833325, 85341793883605, 683693689056144 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 3 2 F[[8, 1], [4, 2, 1, 1, 1]](x) = x (291 x - 220 x + 52 x - 4)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[4, 2, 1, 1, 1]](x) = x^5*(291*x^3-220*x^2+52*x-4)/(1+x)/(-1+x)/(-1+2* x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 4, 60, 644, 6069, 53760, 460530, 3867028, 32053263, 263319056, 2149247100, 17458438452, 141302400057, 1140496417792, 9185884827270, 73866259430516, 593245472262051 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 3 2 F[[8, 1], [4, 1, 1, 1, 1, 1]](x) = x (64 x - 55 x + 13 x - 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[4, 1, 1, 1, 1, 1]](x) = x^5*(64*x^3-55*x^2+13*x-1)/(1+x)/(-1+x)/(-1+2 *x)/(-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 1, 15, 161, 1526, 13685, 119245, 1020657, 8623252, 72101029, 597672075, 4918863313, 40243326578, 327663286133, 2657517898505, 21487380464529, 173313964293904 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [3, 3, 3]](x) = 6 x (19 x - 5) - ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation F[[8, 1],[3, 3, 3]](x) = -x^6*(19*x-5)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/ (-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 5, 91, 1092, 10962, 100275, 870177, 7319114, 60404344, 492717225, 3989968983, 32163929616, 258543640446, 2074553057855, 16627608724909, 133177428145398 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 6 2 F[[8, 1], [3, 3, 2, 1]](x) = - x (320 x - 147 x + 16)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[3, 3, 2, 1]](x) = -x^6*(320*x^2-147*x+16)/(1+x)/(-1+x)/(-1+2*x)/(-1+3 *x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 16, 301, 3724, 38430, 360192, 3191727, 27322108, 228788560, 1888414528, 15437975553, 125389698252, 1013911333890, 8173324203424, 65744679788179, 528033539341756 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [3, 3, 1, 1, 1]](x) = 6 10 x - ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 6 x) (8 x - 1) and in Maple notation F[[8, 1],[3, 3, 1, 1, 1]](x) = -10*x^6/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+6*x)/ (8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 10, 190, 2380, 24900, 236670, 2125530, 18420160, 155927200, 1299027730, 10702533270, 87484100340, 711061731900, 5755645257190, 46448124059410, 374002925474920 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 6 2 F[[8, 1], [3, 2, 2, 2]](x) = - x (44 x - 35 x + 5)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[3, 2, 2, 2]](x) = -x^6*(44*x^2-35*x+5)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x) /(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 5, 105, 1414, 15582, 153615, 1414875, 12473648, 106810704, 896640745, 7424301885, 60884807802, 495904893666, 4019484495395, 32465237835135, 261555318269476 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [3, 2, 2, 1, 1]](x) = 6 9 x - ----------------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) (8 x - 1) and in Maple notation F[[8, 1],[3, 2, 2, 1, 1]](x) = -9*x^6/(1+x)/(-1+x)/(-1+3*x)/(-1+4*x)/(-1+6*x)/( 8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 9, 189, 2556, 28350, 281583, 2613303, 23205402, 199999800, 1688455197, 14047850817, 115662533688, 945134312850, 7680706865451, 62165977652331, 501659920229814 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 6 2 F[[8, 1], [3, 2, 1, 1, 1, 1]](x) = - 5 x (13 x - 7 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[3, 2, 1, 1, 1, 1]](x) = -5*x^6*(13*x^2-7*x+1)/(1+x)/(-1+x)/(-1+2*x)/( -1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 5, 105, 1435, 16170, 163485, 1544235, 13937495, 121861740, 1041505465, 8754510765, 72687351555, 597998929710, 4885996555445, 39715195291695, 321561549217615 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 6 2 F[[8, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x (20 x - 7 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^6*(20*x^2-7*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 1, 21, 294, 3430, 35987, 351967, 3275448, 29389360, 256589333, 2194305113, 18471651562, 153631131290, 1266036664439, 10359691543859, 84314386826236 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[8, 1], [2, 2, 2, 2, 1]](x) = 7 14 x ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 6 x) (8 x - 1) and in Maple notation F[[8, 1],[2, 2, 2, 2, 1]](x) = 14*x^7/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1+4*x)/( -1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 14, 322, 4620, 53340, 544698, 5154534, 46406360, 403923520, 3434332902, 28720878186, 237361253220, 1944931437540, 15837629110226, 128377890124078 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 7 F[[8, 1], [2, 2, 2, 1, 1, 1]](x) = 2 x (32 x - 7)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[2, 2, 2, 1, 1, 1]](x) = 2*x^7*(32*x-7)/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x) /(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 14, 328, 4788, 56160, 581658, 5572776, 50706656, 445313440, 3814392582, 32093033544, 266530977804, 2192506311840, 17909045526386, 145522527537832 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 7 F[[8, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 3 x (7 x - 2)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[2, 2, 1, 1, 1, 1, 1]](x) = 3*x^7*(7*x-2)/(1+x)/(-1+x)/(-1+2*x)/(-1+3* x)/(-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 6, 147, 2232, 27090, 288882, 2836449, 26338884, 235194960, 2041946478, 17367180831, 145480838256, 1204904356110, 9894679957194, 80736051888093 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 7 F[[8, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x /((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^7/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/(-1 +4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 1, 28, 470, 6160, 69707, 716716, 6898320, 63343280, 562025893, 4861620764, 41262479050, 345236069360, 2857439568959, 23457047139372 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 8 F[[8, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x /((1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (-1 + 6 x) (8 x - 1)) and in Maple notation F[[8, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^8/(1+x)/(-1+x)/(-1+2*x)/(-1+3*x)/ (-1+4*x)/(-1+5*x)/(-1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 1, 28, 470, 6160, 69707, 716716, 6898320, 63343280, 562025893, 4861620764, 41262479050, 345236069360, 2857439568959 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 4 x) (-1 + 5 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+x)/(-1+2*x)/ (-1+3*x)/(-1+4*x)/(-1+5*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 Regarding Lambda=, [7, 2] 9 8 7 6 5 F[[7, 2], [9]](x) = (322245 x - 351756 x - 623293 x + 1037342 x - 569604 x 4 3 2 + 153734 x - 22352 x + 1751 x - 68 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[9]](x) = (322245*x^9-351756*x^8-623293*x^7+1037342*x^6-569604*x^5+ 153734*x^4-22352*x^3+1751*x^2-68*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/( -1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 2, 16, 172, 2616, 49059, 1059443, 25053803, 626518201, 16194084676, 426637930170, 11362110801054, 304436835281606, 8184836017605353, 220467454152300397, 5944785587153389525, 160391773988776248831, 4328817217712741566290, 116851662456365429597324 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[7, 2], [8, 1]](x) = - x 6 5 4 3 2 (37800 x - 92691 x + 51177 x - 11602 x + 1230 x - 59 x + 1)/((1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x) ) and in Maple notation F[[7, 2],[8, 1]](x) = -x^2*(37800*x^6-92691*x^5+51177*x^4-11602*x^3+1230*x^2-59 *x+1)/(1+x)/(-1+x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 7, 74, 1006, 17187, 347509, 7878988, 192077404, 4891621253, 127784200291, 3386908075662, 90506792123722, 2429667784241959, 65391498434707153, 1762433664205012496, 47538711724413144760, 1282840744112659024905, 34626137330126286831295, 934747305115940967527890 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 7 6 5 4 F[[7, 2], [7, 2]](x) = - x (443205 x - 712989 x + 430099 x - 126452 x 3 2 + 19702 x - 1630 x + 66 x - 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[7, 2]](x) = -x*(443205*x^7-712989*x^6+430099*x^5-126452*x^4+19702*x^3 -1630*x^2+66*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+ 15*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 2, 16, 172, 2616, 49059, 1059443, 25053803, 626518201, 16194084676, 426637930170, 11362110801054, 304436835281606, 8184836017605353, 220467454152300397, 5944785587153389525, 160391773988776248831, 4328817217712741566290, 116851662456365429597324, 3154598919444838159868816 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[7, 2], [7, 1, 1]](x) = x 5 4 3 2 (37800 x - 30816 x + 8561 x - 1039 x + 55 x - 1)/((-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[7, 1, 1]](x) = x^2*(37800*x^5-30816*x^4+8561*x^3-1039*x^2+55*x-1)/(-1 +x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 12, 158, 2540, 48847, 1070944, 25580052, 643774248, 16705005293, 441107537156, 11762947686106, 315412392407476, 8483477779263579, 228565386678592248, 6163949647204820720, 166317009772231259024, 4488915752400543425305, 121176081138844460911420, 3271384604252662465182294 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[7, 2], [6, 3]](x) = x 6 5 4 3 2 (11340 x - 5535 x - 7489 x + 4478 x - 762 x + 49 x - 1)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[6, 3]](x) = x^2*(11340*x^6-5535*x^5-7489*x^4+4478*x^3-762*x^2+49*x-1) /(-1+x)/(1+x)/(-1+2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 14, 209, 3710, 76309, 1742822, 42611193, 1086366110, 28390464557, 752591335430, 20112120825457, 539921635237790, 14531402931624645, 391651552608225638, 10564154789591439401, 285075690453987398750, 7694696909499739305373, 207721620878453196602246, 5607955875747250102267425 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[7, 2], [6, 2, 1]](x) = - x 6 5 4 3 2 (28350 x - 89460 x + 60949 x - 16273 x + 1979 x - 107 x + 2)/((1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x) ) and in Maple notation F[[7, 2],[6, 2, 1]](x) = -x^2*(28350*x^6-89460*x^5+60949*x^4-16273*x^3+1979*x^2 -107*x+2)/(1+x)/(-1+x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 2, 25, 393, 7315, 156107, 3658270, 90953368, 2342885290, 61603024512, 1638787072615, 43882704735443, 1179390129191365, 31762113921971617, 856357772331187060, 23103388036258482618, 623517558173250973540, 16830866706400603509422, 454371803312151593585605, 12267114836954850681650893 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[7, 2], [6, 1, 1, 1]](x) = - x ( 7 6 5 4 3 2 95445 x - 219321 x + 169319 x - 61959 x + 11819 x - 1183 x + 57 x - 1 )/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[6, 1, 1, 1]](x) = -x^2*(95445*x^7-219321*x^6+169319*x^5-61959*x^4+ 11819*x^3-1183*x^2+57*x-1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7* x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 11, 181, 3525, 77951, 1873123, 47342793, 1232004281, 32590886971, 870045244695, 23344338265205, 628109913588157, 16926305928587511, 456521669665859627, 12318768366155687617, 332497105294867926753, 8975778231376875343571, 242321367092168645057119, 6542307315636881765981229 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[7, 2], [5, 4]](x) = 2 4 3 2 x (7875 x - 3931 x + 690 x - 47 x + 1) - --------------------------------------------------------------- (1 + x) (-1 + 5 x) (-1 + 3 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x) and in Maple notation F[[7, 2],[5, 4]](x) = -x^2*(7875*x^4-3931*x^3+690*x^2-47*x+1)/(1+x)/(-1+5*x)/(-\ 1+3*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 9, 145, 2804, 61295, 1452577, 36281211, 936222576, 24632581909, 655435296065, 17552352012317, 471749379456628, 12704784771997683, 342542556413357673, 9241350203790259063, 249406977903558652160, 6732346272368089611017, 181748717619751786306201, 4906845901341923173991049 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[7, 2], [5, 3, 1]](x) = 2 4 3 2 x (279 x - 888 x + 332 x - 36 x + 1) ------------------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x) and in Maple notation F[[7, 2],[5, 3, 1]](x) = x^2*(279*x^4-888*x^3+332*x^2-36*x+1)/(1+x)/(-1+x)/(-1+ 5*x)/(-1+3*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 21, 424, 9180, 214165, 5284149, 135268840, 3541809792, 93976792129, 2512627197597, 67469996840896, 1816127694543084, 48952027617755653, 1320453537180690765, 35633478037814854192, 961822399693617716256, 25964982214630048295137, 700991181369078702672453, 18925811822194057389750328 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 3 2 x (45 x - 179 x + 31 x - 1) F[[7, 2], [5, 2, 2]](x) = - --------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) (-1 + 27 x) and in Maple notation F[[7, 2],[5, 2, 2]](x) = -x^2*(45*x^3-179*x^2+31*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1 +15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 14, 279, 6260, 150741, 3798194, 98481339, 2597969480, 69229841481, 1855475899574, 49891902572799, 1343992170501500, 36241450624393821, 977824097187376154, 26390824579038130659, 712395872958857634320, 19232342709762008191761, 519238065261668293555934, 14018899943536458885890919 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [5, 2, 1, 1]](x) = x 6 5 4 3 2 (76545 x - 256581 x + 204210 x - 65225 x + 9646 x - 642 x + 15)/( (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[5, 2, 1, 1]](x) = x^3*(76545*x^6-256581*x^5+204210*x^4-65225*x^3+9646 *x^2-642*x+15)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15* x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 15, 378, 9100, 226365, 5814578, 152551791, 4052910429, 108447627960, 2913472558381, 78445612756104, 2114769865291538, 57049962996306735, 1539617617143478524, 41558713960384269117, 1121920935353916524827, 30289400903910103506690, 817776866224478095427807, 22079421079903390758975630 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[7, 2], [5, 1, 1, 1, 1]](x) = 3 3 2 x (693 x - 348 x + 75 x - 4) ------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (7 x - 1) (-1 + 27 x) and in Maple notation F[[7, 2],[5, 1, 1, 1, 1]](x) = x^3*(693*x^3-348*x^2+75*x-4)/(1+x)/(-1+x)/(-1+9* x)/(-1+3*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 4, 109, 2870, 75980, 2031414, 54620349, 1472360740, 39729854560, 1072474334624, 28954603823189, 781753650833610, 21107156934967740, 569891472139437634, 15387053579240408629, 415450299144653021480, 11217156735605131927520, 302863219692704974908444, 8177306821506827244216669 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [4, 4, 1]](x) = x 6 5 4 3 2 (65205 x - 144189 x + 99451 x - 30456 x + 4457 x - 299 x + 7)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[4, 4, 1]](x) = x^3*(65205*x^6-144189*x^5+99451*x^4-30456*x^3+4457*x^2 -299*x+7)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-\ 1+27*x) The first 20 term , starting with k=1 are 0, 0, 7, 177, 4243, 104320, 2647855, 68836355, 1817645767, 48452257170, 1298753584843, 34923601883533, 940787847024571, 25368954639965300, 684476315511088471, 18473572194610685511, 498677065705449993055, 13462639486641228104710, 363466641978058876859539, 9813229927035503888967089 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [4, 3, 2]](x) = x 5 4 3 2 (14175 x - 35001 x + 18434 x - 3742 x + 319 x - 9)/((1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[4, 3, 2]](x) = x^3*(14175*x^5-35001*x^4+18434*x^3-3742*x^2+319*x-9)/( 1+x)/(-1+x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 9, 275, 7330, 191846, 5049611, 134035769, 3581224068, 96097753292, 2585442607693, 69666554646623, 1878868731593126, 50697329051747378, 1368344139830331855, 36938018890271279237, 997217265450823671304, 26923226130485040951704, 726902492618589887057297, 19625997995641971876996011 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [4, 3, 1, 1]](x) = x 6 5 4 3 2 (76545 x - 87426 x + 52551 x - 19180 x + 3599 x - 306 x + 9)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[4, 3, 1, 1]](x) = x^3*(76545*x^6-87426*x^5+52551*x^4-19180*x^3+3599*x ^2-306*x+9)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/ (-1+27*x) The first 20 term , starting with k=1 are 0, 0, 9, 306, 8657, 234570, 6304141, 169431858, 4560130833, 122881704210, 3313973890733, 89418038297610, 2413382654624929, 65147629218540570, 1758779241395010285, 47483927547215277762, 1282019268346149606545, 34613817757622238475650, 934562534478726705081997, 25233030187634328797591514 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [4, 2, 2, 1]](x) = - x 6 5 4 3 2 (76545 x - 88101 x + 26856 x - 214 x - 1091 x + 155 x - 6)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[4, 2, 2, 1]](x) = -x^3*(76545*x^6-88101*x^5+26856*x^4-214*x^3-1091*x^ 2+155*x-6)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/( -1+27*x) The first 20 term , starting with k=1 are 0, 0, 6, 253, 7795, 221240, 6101948, 166387035, 4514396661, 122195375950, 3303677368750, 89263582418417, 2411065776012647, 65112875836702980, 1758257939652585312, 47476108015999814599, 1281901975352507867353, 34612058362590520382330, 934536143552615344514834, 25232634323739480055826781 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [4, 2, 1, 1, 1]](x) = - x 6 5 4 3 2 (76545 x - 106056 x + 53820 x - 10343 x + 566 x + 31 x - 3)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[4, 2, 1, 1, 1]](x) = -x^3*(76545*x^6-106056*x^5+53820*x^4-10343*x^3+ 566*x^2+31*x-3)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15 *x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 3, 173, 5948, 178915, 5102566, 141872136, 3892723677, 106044883460, 2877431723969, 77905002052999, 2106660714253186, 56928325779601185, 1537793059141229712, 41531345591604166562, 1121510409828519006875, 30283243021060768456090, 817684497981896531112595, 22078035556265460570017625 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 4 F[[7, 2], [4, 1, 1, 1, 1, 1]](x) = - x 5 4 3 2 (27405 x - 60039 x + 33454 x - 7354 x + 741 x - 31)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[4, 1, 1, 1, 1, 1]](x) = -x^4*(27405*x^5-60039*x^4+33454*x^3-7354*x^2+ 741*x-31)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-\ 1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 31, 1367, 46060, 1397242, 40217233, 1125184893, 30988918390, 846017333744, 22983927709435, 622703795853139, 16845214365747040, 455305297239708006, 12300522784847284837, 332223421600666025705, 8971672976090981880010, 242259788263515961411228, 6541383633210270231970639 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[7, 2], [3, 3, 3]](x) = 3 4 3 2 x (378 x + 96 x - 59 x + 1) ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation F[[7, 2],[3, 3, 3]](x) = x^3*(378*x^4+96*x^3-59*x^2+1)/(-1+x)/(1+x)/(-1+2*x)/(-\ 1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 1, 48, 1530, 43602, 1199093, 32580954, 881599930, 23820894264, 643326734985, 17371308571980, 469038876392450, 12664172719429686, 341933779006695877, 9232222129927656726, 249270088774366767090, 6730293221074097821068, 181717924405881984284969, 4906384026026445928260792 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [3, 3, 2, 1]](x) = - 2 x 5 4 3 2 (14175 x - 15903 x + 5194 x - 562 x + 7 x + 1)/((1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[3, 3, 2, 1]](x) = -2*x^3*(14175*x^5-15903*x^4+5194*x^3-562*x^2+7*x+1) /(1+x)/(-1+x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 2, 146, 5276, 160476, 4576334, 126923230, 3474469784, 94496110232, 2561416324346, 69306152228874, 1873462654548212, 50616237692357428, 1367127768421432838, 36919773314049139478, 996943581782053085360, 26919120875326304063664, 726840913790572986289010, 19625074313218539257373442 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[7, 2], [3, 3, 1, 1, 1]](x) = 4 4 x (45 x - 19) - --------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 15 x) (-1 + 27 x) and in Maple notation F[[7, 2],[3, 3, 1, 1, 1]](x) = -4*x^4*(45*x-19)/(-1+x)/(1+x)/(-1+3*x)/(-1+15*x) /(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 76, 3240, 105520, 3120120, 88310956, 2445415920, 66941544640, 1821151466640, 49377036137836, 1336269174154200, 36125605679715760, 976086423018799560, 26364759466514264716, 712004896271013994080, 19226478059444396634880, 519150095506904249342880, 14017580397214998610115596 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[7, 2], [3, 2, 2, 2]](x) = x 6 5 4 3 2 (133245 x - 150381 x + 52985 x - 7076 x + 361 x - 15 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[3, 2, 2, 2]](x) = x^3*(133245*x^6-150381*x^5+52985*x^4-7076*x^3+361*x ^2-15*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/( -1+27*x) The first 20 term , starting with k=1 are 0, 0, 1, 53, 2215, 73080, 2175229, 61727071, 1710907759, 46850695490, 1274727708397, 34563201500289, 935381780152183, 25287863331437980, 683259944356502605, 18455326619660111507, 498403382043037235887, 13458534231514280360550, 363405063150200921810653, 9812306244612865997941525 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[7, 2], [3, 2, 2, 1, 1]](x) = 4 2 6 x (141 x - 102 x + 13) ------------------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x) and in Maple notation F[[7, 2],[3, 2, 2, 1, 1]](x) = 6*x^4*(141*x^2-102*x+13)/(1+x)/(-1+x)/(-1+5*x)/( -1+3*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 78, 3834, 133194, 4065678, 116972244, 3267263196, 89858104908, 2450844447876, 66543243388050, 1802226331715238, 48743506870162182, 1317325724440909794, 35586560839088745096, 961118641674579107160, 25954425844153734295416, 700832835810980318360232, 18923436638817813253484982 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 4 F[[7, 2], [3, 2, 1, 1, 1, 1]](x) = x 4 3 2 (56700 x - 28845 x + 4147 x + 61 x - 31)/((1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[3, 2, 1, 1, 1, 1]](x) = x^4*(56700*x^4-28845*x^3+4147*x^2+61*x-31)/(1 +x)/(-1+x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 31, 1985, 76705, 2470195, 73147606, 2075877510, 57598306510, 1578718312490, 42981683432281, 1165874860285135, 31559385142027015, 853316841901590885, 23057774086166390056, 622833348953640792860, 16820603568265342710220, 454217856240917248769380, 12264805630890308668116931 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 4 3 2 F[[7, 2], [3, 1, 1, 1, 1, 1, 1]](x) = - x (4725 x - 1971 x + 155 x + 3)/( (-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x) ) and in Maple notation F[[7, 2],[3, 1, 1, 1, 1, 1, 1]](x) = -x^4*(4725*x^3-1971*x^2+155*x+3)/(-1+x)/(-\ 1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 356, 16826, 594412, 18451237, 536938584, 15102955332, 417079219304, 11402535095831, 310006264499932, 8402386165560478, 227349013998127476, 6145704064624852185, 166043326071671529200, 4484810497082860817864, 121114502310032831546128, 3270460921825256202574699 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[7, 2], [2, 2, 2, 2, 1]](x) = 4 2 x (1575 x - 226 x - 9) --------------------------------------------------------------- (1 + x) (-1 + 5 x) (-1 + 3 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x) and in Maple notation F[[7, 2],[2, 2, 2, 2, 1]](x) = x^4*(1575*x^2-226*x-9)/(1+x)/(-1+5*x)/(-1+3*x)/( 7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 9, 730, 29864, 979118, 29168125, 829466652, 23030933928, 631408997956, 17191949550281, 466343302278854, 12623693412209152, 341326185003262914, 9223104627564532077, 249133294234777304536, 6728241017209320437936, 181687138791734788682792, 4905922218918490263803113 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 4 F[[7, 2], [2, 2, 2, 1, 1, 1]](x) = - 2 x 4 3 2 (2835 x + 2844 x - 1886 x + 156 x + 3)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[2, 2, 2, 1, 1, 1]](x) = -2*x^4*(2835*x^4+2844*x^3-1886*x^2+156*x+3)/( -1+x)/(1+x)/(-1+2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 6, 690, 31088, 1064748, 32440810, 933812550, 26102167716, 718266902496, 19597254390494, 532198638890490, 14415557986946584, 389913878439649044, 10538089677067573458, 284684713766143758510, 7688832259182127748492, 207633651123689152389192, 5606636329425789826492102 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 5 3 2 F[[7, 2], [2, 2, 1, 1, 1, 1, 1]](x) = - x (945 x - 9351 x + 3263 x - 265)/( (1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[2, 2, 1, 1, 1, 1, 1]](x) = -x^5*(945*x^3-9351*x^2+3263*x-265)/(1+x)/( -1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 265, 14757, 549077, 17417331, 512060700, 14477653674, 400893573454, 10975955844282, 298644562456955, 8097952181541651, 219164197888889331, 5925236749577988453, 160098541456988185030, 4324418729895027753888, 116785685139894935283908, 3153609259701755840979444 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 5 3 2 F[[7, 2], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - 8 x (4725 x - 1593 x + 87 x + 5)/ ((1 + x) (-1 + x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -8*x^5*(4725*x^3-1593*x^2+87*x+5)/(1+x) /(-1+x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 40, 3336, 142712, 4821144, 146278128, 4204967472, 117486050704, 3232444058448, 88189872821336, 2394914198953848, 64870195675029576, 1754614127903286312, 47421418705340090464, 1281081348953784356064, 34599746403379143386528, 934351441217913311375136 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 6 F[[7, 2], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x 3 2 (945 x - 9351 x + 3263 x - 265)/((1 + x) (-1 + x) (-1 + 2 x) (-1 + 5 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 15 x) (-1 + 27 x)) and in Maple notation F[[7, 2],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^6*(945*x^3-9351*x^2+3263*x-265)/( 1+x)/(-1+x)/(-1+2*x)/(-1+5*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 265, 14757, 549077, 17417331, 512060700, 14477653674, 400893573454, 10975955844282, 298644562456955, 8097952181541651, 219164197888889331, 5925236749577988453, 160098541456988185030, 4324418729895027753888, 116785685139894935283908 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 9 x) (7 x - 1) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(-1+x)/(1+x)/(-1 +2*x)/(-1+3*x)/(-1+9*x)/(7*x-1)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 Regarding Lambda=, [7, 1, 1] 11 10 9 8 F[[7, 1, 1], [9]](x) = (2122176 x - 1720944 x - 4361656 x + 3362164 x 7 6 5 4 3 2 + 589634 x - 913798 x + 169340 x + 23954 x - 10943 x + 1278 x - 61 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[9]](x) = (2122176*x^11-1720944*x^10-4361656*x^9+3362164*x^8+589634 *x^7-913798*x^6+169340*x^5+23954*x^4-10943*x^3+1278*x^2-61*x+1)/(-1+x)/(1+x)/(-\ 1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 1, 16, 162, 2661, 52839, 1237636, 31684780, 849694771, 23303478297, 646023245946, 18001474302318, 502854970483321, 14063668875584875, 393558306893213296, 11016524194709975376, 308419494251036602911, 8635144613735819566173, 241775665401191212905286 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 8 7 6 5 F[[7, 1, 1], [8, 1]](x) = x (47040 x + 138688 x - 93968 x - 36016 x 4 3 2 + 37824 x - 10076 x + 1156 x - 58 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[8, 1]](x) = x^2*(47040*x^8+138688*x^7-93968*x^6-36016*x^5+37824*x^ 4-10076*x^3+1156*x^2-58*x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-\ 1+10*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 7, 74, 1006, 17847, 385813, 9453340, 247909732, 6725744393, 185482492459, 5155529703006, 143840314656298, 4020495451765339, 112477091529267745, 3148020409039595072, 88126004657633623504, 2467269866085977444685, 69079957241329173364471, 1934188583445814780089538 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 8 7 6 5 F[[7, 1, 1], [7, 2]](x) = - x (912576 x - 64944 x - 512920 x + 185100 x 4 3 2 + 7918 x - 12048 x + 1879 x - 108 x + 2)/((1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[7, 2]](x) = -x^2*(912576*x^8-64944*x^7-512920*x^6+185100*x^5+7918* x^4-12048*x^3+1879*x^2-108*x+2)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1+10 *x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 12, 165, 2580, 51793, 1205406, 30741671, 822104904, 22511201829, 623523182370, 17366700269107, 485010978150348, 13563018675076145, 379525504685350854, 10623398021522926623, 297409019926423066512, 8326809817671860278141, 233141706562045841452458, 6527842527265180650460619 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 9 8 7 F[[7, 1, 1], [7, 1, 1]](x) = - x (2393664 x - 1472400 x - 786888 x 6 5 4 3 2 + 746028 x - 125746 x - 25050 x + 10420 x - 1232 x + 60 x - 1)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[7, 1, 1]](x) = -x*(2393664*x^9-1472400*x^8-786888*x^7+746028*x^6-\ 125746*x^5-25050*x^4+10420*x^3-1232*x^2+60*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/( -1+6*x)/(-1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 1, 16, 162, 2661, 52839, 1237636, 31684780, 849694771, 23303478297, 646023245946, 18001474302318, 502854970483321, 14063668875584875, 393558306893213296, 11016524194709975376, 308419494251036602911, 8635144613735819566173, 241775665401191212905286, 6769601591822013961625554 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 F[[7, 1, 1], [6, 3]](x) = - x 7 6 5 4 3 2 (14336 x - 12080 x - 7312 x + 10816 x - 4328 x + 721 x - 48 x + 1)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[6, 3]](x) = -x^2*(14336*x^7-12080*x^6-7312*x^5+10816*x^4-4328*x^3+ 721*x^2-48*x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(-1+10*x)/(-1+4*x)/(-1+ 14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 11, 187, 3625, 81485, 2021807, 53180191, 1443040421, 39781170049, 1105337723083, 30831967561715, 861670257688097, 24104219569926133, 674604481049945639, 18884554133202020359, 528706517024987369053, 14802930494842671751337, 414470146109630930679875, 11604997582626307431685723 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 F[[7, 1, 1], [6, 2, 1]](x) = x 7 6 5 4 3 2 (39200 x + 105224 x - 97920 x + 10570 x + 5600 x - 1342 x + 95 x - 2) /((-1 + x) (1 + x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[6, 2, 1]](x) = x^2*(39200*x^7+105224*x^6-97920*x^5+10570*x^4+5600* x^3-1342*x^2+95*x-2)/(-1+x)/(1+x)/(-1+6*x)/(-1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/ (-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 25, 396, 7625, 173042, 4342625, 115154426, 3139685665, 86779521882, 2414501547425, 67396606150106, 1884223515788705, 52718422089180122, 1475563133484203425, 41308080291783732186, 1156519124790846307745, 32381040772020167756762, 906648265537863206878625, 25385859670144731160136666 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 8 7 6 5 F[[7, 1, 1], [6, 1, 1, 1]](x) = x (122304 x + 52416 x - 153080 x + 51196 x 4 3 2 + 5562 x - 5123 x + 836 x - 51 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[6, 1, 1, 1]](x) = x^2*(122304*x^8+52416*x^7-153080*x^6+51196*x^5+ 5562*x^4-5123*x^3+836*x^2-51*x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2* x)/(-1+10*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 14, 209, 3975, 90292, 2281860, 60875515, 1666349321, 46162546058, 1285999234806, 35920004231761, 1004565672686667, 28111486437661264, 786896372447559752, 22029981740577394247, 616796222460037844013, 17269692171815348318710, 483542988703385345488698, 13539086561006121404543773 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [5, 4]](x) = x 7 6 5 4 3 2 (65856 x - 84784 x + 17568 x + 13860 x - 5108 x + 128 x + 89 x - 4)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[5, 4]](x) = x^3*(65856*x^7-84784*x^6+17568*x^5+13860*x^4-5108*x^3+ 128*x^2+89*x-4)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(1+4*x)/(-1+4*x )/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 4, 115, 2669, 65874, 1705718, 45763389, 1252970023, 34683282448, 965518301792, 26955837599163, 753661465239137, 21087090080460222, 590222469738232426, 16523204303706558937, 462607371926974273211, 12952413529761352853196, 362659278429750479164820, 10154343590234421781242711 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [5, 3, 1]](x) = 3 x 6 5 4 3 2 (6048 x + 6760 x - 8808 x + 604 x + 819 x - 164 x + 6)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[5, 3, 1]](x) = 3*x^3*(6048*x^6+6760*x^5-8808*x^4+604*x^3+819*x^2-\ 164*x+6)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+ 28*x) The first 20 term , starting with k=1 are 0, 0, 18, 390, 9525, 241377, 6396918, 173910492, 4796193735, 133262984559, 3716917868928, 103871319221274, 2905562847403905, 81316077537697821, 2276294592361378698, 63728470309952558136, 1784288271789116649435, 49958547047402544523563, 1398817973412513997352628, 39166604440519840463447478 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 F[[7, 1, 1], [5, 2, 2]](x) = 2 4 3 2 x (912 x - 72 x - 86 x + 27 x - 1) - ----------------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) (-1 + 28 x) and in Maple notation F[[7, 1, 1],[5, 2, 2]](x) = -x^2*(912*x^4-72*x^3-86*x^2+27*x-1)/(-1+x)/(1+x)/(1 +2*x)/(-1+2*x)/(1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 11, 285, 6735, 174381, 4670751, 127864045, 3539155295, 98522606061, 2750597418591, 76904242599405, 2151743950697055, 60226783031682541, 1686041257851880031, 47204833884580816365, 1321674850061571474015, 37006048819881862243821, 1036157509210696034239071, 29012244249456368075154925 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 9 8 7 F[[7, 1, 1], [5, 2, 1, 1]](x) = - x (254016 x - 173264 x + 35464 x 6 5 4 3 2 + 78572 x - 68050 x + 15478 x + 335 x - 435 x + 40 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[5, 2, 1, 1]](x) = -x^2*(254016*x^9-173264*x^8+35464*x^7+78572*x^6-\ 68050*x^5+15478*x^4+335*x^3-435*x^2+40*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6 *x)/(-1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 21, 439, 10510, 270850, 7289345, 200265283, 5556842844, 154913535124, 4328390180719, 121069296258637, 3388211631946778, 94846025083694158, 2655357098694590493, 74345386339141156951, 2081606543729895904312, 58284086392715464490152, 1631941893358166522554667, 45694197955135271414807425 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 F[[7, 1, 1], [5, 1, 1, 1, 1]](x) = 2 4 3 2 x (1680 x - 92 x - 132 x + 30 x - 1) - --------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation F[[7, 1, 1],[5, 1, 1, 1, 1]](x) = -x^2*(1680*x^4-92*x^3-132*x^2+30*x-1)/(-1+x)/ (1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 7, 167, 3755, 97611, 2648667, 73341787, 2045219995, 57183023771, 1600291068827, 44799819742107, 1254311615246235, 35119891943903131, 983348641028824987, 27533678616279113627, 770942167921408024475, 21586372368478976417691, 604418342984060826165147, 16923712770220575958376347 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 7 6 5 4 F[[7, 1, 1], [4, 4, 1]](x) = - x (150528 x + 42784 x - 208888 x + 110976 x 3 2 - 21174 x + 1208 x + 34 x - 3)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[4, 4, 1]](x) = -x^3*(150528*x^7+42784*x^6-208888*x^5+110976*x^4-\ 21174*x^3+1208*x^2+34*x-3)/(-1+x)/(1+x)/(-1+3*x)/(-1+6*x)/(-1+2*x)/(-1+10*x)/(1 +4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 3, 155, 4348, 118700, 3238375, 89206005, 2474507010, 68937290150, 1925135913877, 53830164827615, 1506192824997952, 42158469364808160, 1180226096850056259, 33043355906164561385, 925172117053881108574, 25904231394870023107130, 725310228662094730435921, 20308570696795977865277715 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [4, 3, 2]](x) = - x 7 6 5 4 3 2 (47040 x + 5152 x - 69792 x + 43104 x - 7504 x - 222 x + 129 x - 6)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[4, 3, 2]](x) = -x^3*(47040*x^7+5152*x^6-69792*x^5+43104*x^4-7504*x ^3-222*x^2+129*x-6)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1+10*x)/( -1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 6, 261, 7965, 226258, 6321476, 176250039, 4919102355, 137459932836, 3844514666586, 107580235103737, 3011269598340185, 84301367797471734, 2360234745866745936, 66083673035161748955, 1850301746408596397055, 51807868515244606648552, 1450612143023295990521126, 40617025048853662577587293 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [4, 3, 1, 1]](x) = - x ( 7 6 5 4 3 2 133056 x - 132064 x + 26216 x - 2364 x + 5422 x - 2075 x + 249 x - 9) /((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[4, 3, 1, 1]](x) = -x^3*(133056*x^7-132064*x^6+26216*x^5-2364*x^4+ 5422*x^3-2075*x^2+249*x-9)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1+ 10*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 9, 336, 10082, 286755, 8050207, 225329706, 6304746660, 176437600185, 4938603527825, 138254719295616, 3870735091651558, 108374720677296255, 3034407093300027363, 84962177297909378166, 2378923565247938194376, 66609613235491103510565, 1865065688288591926302421, 52221790219659325892890956 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [4, 2, 2, 1]](x) = x ( 7 6 5 4 3 2 122304 x - 91872 x - 38816 x + 52648 x - 18756 x + 3188 x - 274 x + 9 )/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[4, 2, 2, 1]](x) = x^3*(122304*x^7-91872*x^6-38816*x^5+52648*x^4-\ 18756*x^3+3188*x^2-274*x+9)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1 +10*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 9, 311, 9570, 278566, 7929375, 223600797, 6280318116, 176094257132, 4933788663441, 138187262822323, 3869790410789862, 108361493403810738, 3034221901022683107, 84959584543335243689, 2378887266307752356808, 66609105048071613924184, 1865058573651177744668373, 52221690614654279384941695 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 8 7 6 F[[7, 1, 1], [4, 2, 1, 1, 1]](x) = x (498624 x - 448560 x + 26552 x 5 4 3 2 + 79012 x - 4942 x - 11282 x + 3017 x - 275 x + 9)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[4, 2, 1, 1, 1]](x) = x^3*(498624*x^8-448560*x^7+26552*x^6+79012*x^ 5-4942*x^4-11282*x^3+3017*x^2-275*x+9)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/( -1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 9, 274, 8238, 239285, 6848961, 194109174, 5470713180, 153708055495, 4311515480943, 120833062553984, 3384904432639002, 94799724728737665, 2654708896337330205, 74336311521812329354, 2081479496381329282104, 58282307730399753695195, 1631916992089131903183147, 45693849337389098734677684 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [4, 1, 1, 1, 1, 1]](x) = x ( 7 6 5 4 3 2 28224 x + 12544 x - 47712 x + 33432 x - 10876 x + 1698 x - 114 x + 3) /((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[4, 1, 1, 1, 1, 1]](x) = x^3*(28224*x^7+12544*x^6-47712*x^5+33432*x ^4-10876*x^3+1698*x^2-114*x+3)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/ (-1+10*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 3, 81, 2352, 68600, 1984381, 56748111, 1608789570, 45358053150, 1274744396079, 35762484862181, 1002360691749508, 28080618445965540, 786464231012372097, 22023931823176092891, 616711523992567601766, 17268506395527652831370, 483526387848898936178035, 13538854149124559639441841 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [3, 3, 3]](x) = - x 5 4 3 2 (3472 x - 540 x - 1708 x + 392 x + 3 x + 1)/((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[3, 3, 3]](x) = -x^3*(3472*x^5-540*x^4-1708*x^3+392*x^2+3*x+1)/(1+x )/(-1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 42, 1735, 52956, 1534225, 43448538, 1221585499, 34254242232, 959619228469, 26874336019014, 752531415855583, 21071379606392388, 590003629097769433, 16520151614141048370, 462564745197864071587, 12951817865530691431824, 362650950234890122102717, 10154227106576771752407006 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [3, 3, 2, 1]](x) = x 7 6 5 4 3 2 (47040 x - 5600 x + 20296 x - 34564 x + 13506 x - 1765 x + 46 x + 2)/ ((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[3, 3, 2, 1]](x) = x^3*(47040*x^7-5600*x^6+20296*x^5-34564*x^4+ 13506*x^3-1765*x^2+46*x+2)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1+ 10*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 2, 176, 6601, 206121, 6033328, 172178622, 4861878527, 136657455467, 3833271921094, 107422788293568, 3009065052759493, 84270502417914813, 2359802620104391100, 66077623211797444514, 1850217048505348136299, 51806682742342243050159, 1450595542189121572544146, 40616792637093972760487460 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 F[[7, 1, 1], [3, 3, 1, 1, 1]](x) = 3 4 3 2 x (896 x + 176 x - 192 x + 94 x + 1) ----------------------------------------------------------------------- (-1 + x) (1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (-1 + 14 x) (-1 + 28 x) and in Maple notation F[[7, 1, 1],[3, 3, 1, 1, 1]](x) = x^3*(896*x^4+176*x^3-192*x^2+94*x+1)/(-1+x)/( 1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 132, 4605, 144500, 4252461, 122007732, 3457167085, 97374770100, 2734527715821, 76679266756532, 2148594288899565, 60182687766501300, 1685423924139353581, 47196191212605380532, 1321553852653915416045, 37004354856174677170100, 1036133793718795443381741, 29011912232569759802103732 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 8 7 6 F[[7, 1, 1], [3, 2, 2, 2]](x) = - x (301056 x - 624064 x + 512304 x 5 4 3 2 - 175224 x + 25204 x - 2894 x + 492 x - 18 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[3, 2, 2, 2]](x) = -x^3*(301056*x^8-624064*x^7+512304*x^6-175224*x^ 5+25204*x^4-2894*x^3+492*x^2-18*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1 +2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 79, 3050, 98944, 2952565, 85148553, 2417367184, 68135316538, 1913896191779, 53672736156787, 1503988388256718, 42127604638283892, 1179793975005910993, 33037306106309497981, 925087419291688348652, 25903045622813992448206, 725293627832998310314207, 20308338285066756035047335 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [3, 2, 2, 1, 1]](x) = - 3 x 6 5 4 3 2 (4704 x + 2712 x - 3416 x + 1444 x - 177 x - x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[3, 2, 2, 1, 1]](x) = -3*x^3*(4704*x^6+2712*x^5-3416*x^4+1444*x^3-\ 177*x^2-x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/ (-1+28*x) The first 20 term , starting with k=1 are 0, 0, 3, 150, 5874, 188769, 5650947, 163411164, 4648866708, 131198391663, 3688001473101, 103466417124378, 2899893782682702, 81236708019478557, 2275183403433443415, 63712913570924532792, 1784070476878502182056, 49955497915268610353451, 1398775285542326927113089, 39166006810215349533140406 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [3, 2, 1, 1, 1, 1]](x) = - 2 x 7 6 5 4 3 2 (86240 x - 70168 x + 10268 x - 418 x + 399 x + 10 x - 7 x + 1)/( (-1 + x) (1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[3, 2, 1, 1, 1, 1]](x) = -2*x^3*(86240*x^7-70168*x^6+10268*x^5-418* x^4+399*x^3+10*x^2-7*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1+10*x)/(1+4 *x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 2, 102, 3730, 119780, 3604762, 104870892, 2995996250, 84769549260, 2386372008922, 67002853075532, 1878711335543770, 52641253742526540, 1474482789691755482, 41292955557053597772, 1156307378974809443290, 32378076333416761452620, 906606763414342178196442, 25385278640516996714817612 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[7, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (28224 x - 147856 x + 139192 x - 45300 x + 7118 x - 718 x + 36 x - 1) /((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^3*(28224*x^7-147856*x^6+139192*x^5-\ 45300*x^4+7118*x^3-718*x^2+36*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2 *x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 25, 966, 30766, 937797, 27543411, 792050932, 22498481632, 634765383483, 17843936793397, 500649880705338, 14032800230856498, 393126161539810609, 11010474253799432983, 308334795642510837984, 8633958836601791139364, 241759064541626805651975, 6769369179909984209654169 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 4 F[[7, 1, 1], [2, 2, 2, 2, 1]](x) = - x 6 5 4 3 2 (9408 x + 3248 x + 6528 x - 7004 x + 1420 x - 174 x + 29)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[2, 2, 2, 2, 1]](x) = -x^4*(9408*x^6+3248*x^5+6528*x^4-7004*x^3+ 1420*x^2-174*x+29)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(1+4*x)/(-1+ 4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 0, 29, 1305, 45732, 1417570, 41691951, 1195746195, 33880804994, 954275556300, 26798390788653, 751456919658445, 21056224700901936, 589790343975877590, 16517154480342249035, 462522674023726012455, 12951227756858989232958, 362642677595576061187840, 10154111178474731964055497 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 4 F[[7, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 x 5 4 3 2 (6272 x - 13704 x + 8096 x - 1958 x + 258 x - 17)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[2, 2, 2, 1, 1, 1]](x) = 2*x^4*(6272*x^5-13704*x^4+8096*x^3-1958*x^ 2+258*x-17)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(-1+10*x)/(-1+4*x)/(-1+14*x) /(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 0, 34, 1490, 51604, 1603496, 47323878, 1361052126, 38633334088, 1089268019972, 30606991718842, 858520595889242, 24060124304744892, 673987147337413728, 18875911461226584526, 528585519617331289238, 14801236531135486677616, 414446430617730339735164, 11604665565739699158634530 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 4 F[[7, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (52416 x - 145328 x + 75240 x - 23252 x + 4758 x - 440 x + 21)/( (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -x^4*(52416*x^6-145328*x^5+75240*x^4-\ 23252*x^3+4758*x^2-440*x+21)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-1+10*x) /(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 0, 21, 820, 28401, 885854, 26314407, 760403784, 21649064997, 611463346978, 17197923036723, 482648459704268, 13529945593601073, 379062494605434822, 10616915958768217119, 297318271518042281872, 8325539342775639004029, 233123919930425403714986, 6527593514524054478017995 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 4 F[[7, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (114240 x - 124672 x + 48296 x - 12036 x + 2050 x - 155 x + 6)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^4*(114240*x^6-124672*x^5+48296*x^4 -12036*x^3+2050*x^2-155*x+6)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(-\ 1+10*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 0, 6, 235, 8103, 255650, 7668444, 223145265, 6380385801, 180655534840, 5088000670302, 142895198438135, 4007265566141019, 112291883579090670, 3145427560428463680, 88089705153225804445, 2466761675281155969357, 69072842583603000396740, 1934088978318896324138178 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 4 F[[7, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (28224 x - 147856 x + 139192 x - 45300 x + 7118 x - 718 x + 36 x - 1) /((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 28 x)) and in Maple notation F[[7, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^4*(28224*x^7-147856*x^6+139192 *x^5-45300*x^4+7118*x^3-718*x^2+36*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+6*x)/ (-1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 0, 1, 25, 966, 30766, 937797, 27543411, 792050932, 22498481632, 634765383483, 17843936793397, 500649880705338, 14032800230856498, 393126161539810609, 11010474253799432983, 308334795642510837984, 8633958836601791139364, 241759064541626805651975 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 Regarding Lambda=, [6, 3] 10 9 8 7 F[[6, 3], [9]](x) = (1327232 x - 1202816 x - 1917736 x + 2115468 x 6 5 4 3 2 - 212664 x - 394614 x + 176204 x - 31050 x + 2542 x - 89 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[9]](x) = (1327232*x^10-1202816*x^9-1917736*x^8+2115468*x^7-212664*x^6 -394614*x^5+176204*x^4-31050*x^3+2542*x^2-89*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x) /(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 2, 39, 1062, 40375, 1747170, 80212779, 3778357198, 179933089839, 8608297770138, 412629038421139, 19794813158063814, 949923452860368663, 45591774441368461906, 2188314149488137957819, 105037258719202530062910, 5041752009538675771941247, 242003368279326073952584074, 11616147113846079303686066019 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 7 6 5 4 F[[6, 3], [8, 1]](x) = - x (94208 x - 312512 x + 275504 x - 107320 x 3 2 + 20752 x - 1966 x + 80 x - 1)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[8, 1]](x) = -x^2*(94208*x^7-312512*x^6+275504*x^5-107320*x^4+20752*x^ 3-1966*x^2+80*x-1)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/( -1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 10, 235, 7682, 308959, 13713814, 636548667, 30124690538, 1437428388727, 68825710848158, 3300219319219219, 158342248985981554, 7599062525077964175, 364727693796368359142, 17506383162982249376491, 840295469109293232033530, 40333964063535790974417703, 1936025905980039005577985966, 92929156105684460409421942083 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [7, 2]](x) = - x 6 5 4 3 2 (63360 x - 165056 x + 108120 x - 28268 x + 3238 x - 149 x + 2)/( (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[7, 2]](x) = -x^2*(63360*x^6-165056*x^5+108120*x^4-28268*x^3+3238*x^2-\ 149*x+2)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 25, 673, 24001, 1007409, 45602065, 2134897937, 101403215377, 4845979975441, 232180047911185, 11136106395599121, 534362419890258193, 25645982658737254673, 1230938899656121127185, 59083701839815597822225, 2835990381561889497420049, 136126992180977617557983505, 6534084702014986972060782865, 313635847243343900965123199249 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [7, 1, 1]](x) = - x 7 6 5 4 3 2 (51712 x - 50816 x + 57344 x - 35960 x + 10212 x - 1304 x + 68 x - 1) /((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[7, 1, 1]](x) = -x^2*(51712*x^7-50816*x^6+57344*x^5-35960*x^4+10212*x^ 3-1304*x^2+68*x-1)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/( -1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 22, 653, 24270, 1033889, 47086186, 2209960005, 105079420566, 5023876823177, 240747674881890, 11547922448925677, 554140976788020622, 26595581013901752945, 1276524172363564081434, 61271885956384802133269, 2941025039636788535423238, 141168692177742819205989593, 6776087030039744306368168018, 325251973552105811326934856381 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 8 7 6 5 F[[6, 3], [6, 3]](x) = - x (884608 x - 889216 x - 100952 x + 373940 x 4 3 2 - 156832 x + 28290 x - 2402 x + 87 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[6, 3]](x) = -x*(884608*x^8-889216*x^7-100952*x^6+373940*x^5-156832*x^ 4+28290*x^3-2402*x^2+87*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-\ 1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 1, 2, 39, 1062, 40375, 1747170, 80212779, 3778357198, 179933089839, 8608297770138, 412629038421139, 19794813158063814, 949923452860368663, 45591774441368461906, 2188314149488137957819, 105037258719202530062910, 5041752009538675771941247, 242003368279326073952584074, 11616147113846079303686066019, 557574770193452585808869362486 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 7 6 5 4 F[[6, 3], [6, 2, 1]](x) = x (336000 x - 496704 x + 218664 x - 21300 x 3 2 - 7000 x + 1710 x - 117 x + 2)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[6, 2, 1]](x) = x^2*(336000*x^7-496704*x^6+218664*x^5-21300*x^4-7000*x ^3+1710*x^2-117*x+2)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1) /(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 63, 2118, 85055, 3762822, 174326523, 8242714538, 393157440375, 18821748515742, 902448132081683, 43297597334214258, 2077886434871633295, 99730584311063711462, 4786908757103098424043, 229768434555703907798778, 11028821143058769267243815, 529382140555235754545495982, 25410317260425311385440865603, 1219694718775940300372241348098 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 7 6 5 4 F[[6, 3], [6, 1, 1, 1]](x) = - x (315392 x - 496960 x + 291792 x - 85520 x 3 2 + 14058 x - 1319 x + 64 x - 1)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[6, 1, 1, 1]](x) = -x^2*(315392*x^7-496960*x^6+291792*x^5-85520*x^4+ 14058*x^3-1319*x^2+64*x-1)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/( 8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 26, 1028, 43650, 1975400, 92367114, 4384147328, 209446008546, 10033517743304, 481210796404602, 23090155140783248, 1108168168871431122, 53188886416943990168, 2553002833174245843690, 122542861687017160395488, 5882031874800914481693378, 282337020266371145898060392, 13552166778299906854199877978, 650503801468641640337877747248 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [5, 4]](x) = x 6 5 4 3 2 (32128 x - 35360 x + 8480 x + 1668 x - 721 x + 57 x - 1)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[5, 4]](x) = x^2*(32128*x^6-35360*x^5+8480*x^4+1668*x^3-721*x^2+57*x-1 )/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 27, 862, 34143, 1506118, 69738683, 3297151530, 157263509367, 7528703720350, 360979287649299, 17319039214104178, 831154576203564431, 39892233742535184102, 1914763502986543639275, 91907373823446574419706, 4411528457232843329354535, 211752856222169080269356974, 10164126904170723343738457411, 487877887510380913841627264514 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 2 3 x (392 x - 48 x + 1) F[[6, 3], [5, 3, 1]](x) = - ----------------------------------------- (1 + x) (8 x - 1) (-1 + 20 x) (-1 + 48 x) and in Maple notation F[[6, 3],[5, 3, 1]](x) = -3*x^2*(392*x^2-48*x+1)/(1+x)/(8*x-1)/(-1+20*x)/(-1+48 *x) The first 20 term , starting with k=1 are 0, 3, 81, 2967, 125385, 5690295, 266668617, 12671569335, 605671133769, 29020980866487, 1391982813155913, 66794692884000183, 3205735641252139593, 153867118642663746999, 7385457853748348228169, 354498700171124621929911, 17015872072143613108458057, 816760548742330479252565431, 39204480125227359404495770185, 1881814521722877222618778004919 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [5, 2, 2]](x) = - x 5 4 3 2 (4160 x - 6432 x + 4156 x - 1088 x + 99 x - 2)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[5, 2, 2]](x) = -x^2*(4160*x^5-6432*x^4+4156*x^3-1088*x^2+99*x-2)/(-1+ x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+4*x)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 53, 2078, 90625, 4171562, 196676293, 9369339798, 448306042505, 21490245421922, 1030962890673133, 49474840971832718, 2374564811081718385, 113974559820769075482, 5470687849174525367573, 262591196315932105240838, 12604341014275849502082265, 605007640507462987716444242, 29040352180802667812515623613, 1393936613407416943720692124158 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [5, 2, 1, 1]](x) = x 5 4 3 2 (50112 x - 39968 x + 9236 x - 612 x - 16 x + 1)/((-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[5, 2, 1, 1]](x) = x^2*(50112*x^5-39968*x^4+9236*x^3-612*x^2-16*x+1)/( -1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 69, 3053, 139025, 6500577, 308408401, 14729855953, 705547178769, 33836457785873, 1623553122422033, 77918607085071633, 3739854237873295633, 179508225000968053009, 8616299228715313877265, 413580451527712779653393, 19851823444125397728825617, 952886760732381544939917585, 45738549223429219681604276497, 2195450056890001824332903158033 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 F[[6, 3], [5, 1, 1, 1, 1]](x) = 3 2 x (328 x - 152 x + 17) - ------------------------------------------------------ (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (8 x - 1) (-1 + 48 x) and in Maple notation F[[6, 3],[5, 1, 1, 1, 1]](x) = -x^3*(328*x^2-152*x+17)/(-1+3*x)/(-1+2*x)/(-1+6* x)/(8*x-1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 17, 987, 48845, 2356775, 113225637, 5435650927, 260917911565, 12524113729575, 601157894685557, 28855582453596767, 1385067985982784285, 66483263553699858775, 3191196652394877750277, 153177439329522870101007, 7352517087933830479785005, 352920820221758821728650375, 16940199370651909694799547797, 813129569791351594859438483647 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [4, 4, 1]](x) = x ( 7 6 5 4 3 2 158208 x - 253440 x + 120064 x - 17456 x - 1748 x + 674 x - 52 x + 1) /((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[4, 4, 1]](x) = x^2*(158208*x^7-253440*x^6+120064*x^5-17456*x^4-1748*x ^3+674*x^2-52*x+1)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/( -1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 38, 1463, 63538, 2920987, 137681110, 6558602023, 313814756762, 15043176084083, 721674058185982, 34632388960290463, 1662195370010463106, 79782191892641419819, 3829481494567445423654, 183813837422317378306583, 8823038710002429846574570, 423505348355298868141117795, 20328246526562466251470488526, 975755629385196654269785447183 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 7 6 5 4 F[[6, 3], [4, 3, 2]](x) = - x (110592 x - 326080 x + 268560 x - 103200 x 3 2 + 21040 x - 2282 x + 117 x - 2)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[4, 3, 2]](x) = -x^2*(110592*x^7-326080*x^6+268560*x^5-103200*x^4+ 21040*x^3-2282*x^2+117*x-2)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/ (8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 63, 2690, 122495, 5751990, 273575547, 13081577322, 626917833591, 30072125406878, 1443063635709971, 69259088742658674, 3324276959935669647, 159562108211298582246, 7658917477876546726635, 367626764621235890884346, 17646059215551009590983463, 847010332621633647754264494, 40656485771346553404542012739, 1951511113134832944089128039938 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [4, 3, 1, 1]](x) = x 6 5 4 3 2 (18432 x - 66976 x + 9320 x + 1936 x - 201 x - 18 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[4, 3, 1, 1]](x) = x^2*(18432*x^6-66976*x^5+9320*x^4+1936*x^3-201*x^2-\ 18*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 66, 3228, 153382, 7316840, 350197094, 16788541048, 805426453158, 38651960793144, 1855123641901222, 89042522976920888, 4273972848075087014, 205149331468243972408, 9847140604556011510950, 472662202891302547931448, 22687774816163265761270950, 1089012972722881701090591032, 52272618321634675221417843878, 2509085592057155203382663151928 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[6, 3], [4, 2, 2, 1]](x) = - x 6 5 4 3 2 (36864 x - 27328 x + 11088 x - 2128 x + 284 x - 28 x + 1)/((-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[4, 2, 2, 1]](x) = -x^2*(36864*x^6-27328*x^5+11088*x^4-2128*x^3+284*x^ 2-28*x+1)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 59, 3047, 149615, 7240815, 348673967, 16758067567, 804816940271, 38639770360559, 1854879832552175, 89037646787143407, 4273875324268359407, 205147380992064679663, 9847101595032246726383, 472661422700826536406767, 22687759212353742667575023, 1089012660646691227763404527, 52272612080110865709061566191, 2509085467226679012952285572847 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [4, 2, 1, 1, 1]](x) = x 5 4 3 2 (26496 x - 14848 x + 11528 x - 5212 x + 826 x - 37)/((-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[4, 2, 1, 1, 1]](x) = x^3*(26496*x^5-14848*x^4+11528*x^3-5212*x^2+826* x-37)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 37, 2393, 125713, 6234001, 303075409, 14623190673, 703413850897, 33793791141137, 1622699789176081, 77901540418754833, 3739512904541360401, 179501398334306980113, 8616162695382002913553, 413577720861046202470673, 19851768830792064753406225, 952885668465714879704928529, 45738527378095886353997566225, 2195449619983335157689143070993 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [4, 1, 1, 1, 1, 1]](x) = - 2 x 6 5 4 3 2 (47104 x - 52640 x + 33768 x - 10796 x + 1346 x - 26 x - 3)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[4, 1, 1, 1, 1, 1]](x) = -2*x^3*(47104*x^6-52640*x^5+33768*x^4-10796*x ^3+1346*x^2-26*x-3)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/ (-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 6, 592, 34802, 1797784, 88812234, 4313038912, 208023797218, 10005073342408, 480641907690362, 23078777363703952, 1107940613318671314, 53184335305844061592, 2552911810952068358890, 122541041242572894898912, 5881995465912026308625090, 282336292088593370983557736, 13552152214744351310097539418, 650503510197530529272579474992 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 F[[6, 3], [3, 3, 3]](x) = 2 5 4 3 2 x (2160 x - 5408 x + 3206 x - 667 x + 52 x - 1) - ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (8 x - 1) (-1 + 48 x) and in Maple notation F[[6, 3],[3, 3, 3]](x) = -x^2*(2160*x^5-5408*x^4+3206*x^3-667*x^2+52*x-1)/(-1+x )/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(8*x-1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 14, 624, 29546, 1416028, 67951394, 3261521348, 156551810282, 7514476852380, 360694806386354, 17313350032751092, 831040796098480058, 39889958168435160332, 1914717991727518248194, 91906463600043675952356, 4411510252778969255054474, 211752492133204848820028284, 10164119622392343391500101714, 487877741874820544283031869140 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [3, 3, 2, 1]](x) = x 6 5 4 3 2 (110592 x - 342080 x + 286320 x - 102088 x + 17322 x - 1361 x + 40)/( (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[3, 3, 2, 1]](x) = x^3*(110592*x^6-342080*x^5+286320*x^4-102088*x^3+ 17322*x^2-1361*x+40)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1) /(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 40, 2239, 113592, 5574143, 270019764, 13010465251, 625495607728, 30043680947671, 1442494746762828, 69247710964647083, 3324049404379182024, 159557557100183739679, 7658826455654309591332, 367624944176791386774835, 17646022806662120463485280, 847009604443855869021998567, 40656471207790997845168708476, 1951510821863721832962745730107 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [3, 3, 1, 1, 1]](x) = - x 5 4 3 2 (4160 x - 9184 x + 3628 x - 508 x + 139 x - 21)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[3, 3, 1, 1, 1]](x) = -x^3*(4160*x^5-9184*x^4+3628*x^3-508*x^2+139*x-\ 21)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+4*x)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 21, 1436, 77905, 3917480, 191596501, 9267750676, 446274289305, 21449610472160, 1030150192144381, 49458587003112716, 2374239731714776705, 113968058233460031640, 5470557817428463802661, 262588595681011351015556, 12604289001577436326480105, 605006600253494731839315920, 29040331375723302725515163341, 1393936197305829642102848547196 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [3, 2, 2, 2]](x) = x 6 5 4 3 2 (379392 x - 506880 x + 248128 x - 58800 x + 7412 x - 508 x + 17)/( (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[3, 2, 2, 2]](x) = x^3*(379392*x^6-506880*x^5+248128*x^4-58800*x^3+ 7412*x^2-508*x+17)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/( -1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 17, 1022, 54665, 2743266, 134125789, 6487491822, 312392538209, 15014731654202, 721105169355461, 34621011182745702, 1661967814455840073, 79777640781534036978, 3829390472345238116333, 183812016977872993514462, 8823002301113541196302257, 423504620177521091317777194, 20328231963006910699732710805, 975755338114085543173945330902 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 24 x F[[6, 3], [3, 2, 2, 1, 1]](x) = - --------------------------------- (8 x - 1) (-1 + 20 x) (-1 + 48 x) and in Maple notation F[[6, 3],[3, 2, 2, 1, 1]](x) = -24*x^3/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 24, 1824, 102528, 5233152, 257525760, 12488712192, 602013990912, 28947838009344, 1390519956013056, 66765435741143040, 3205150498394996736, 153855415785520889856, 7385223796605491085312, 354494019028267479072768, 17015778449286470251315200, 816758676285187622109708288, 39204442676084502261638627328, 1881813772740020079761635147776 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [3, 2, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (216960 x - 330432 x + 191928 x - 54004 x + 7102 x - 291 x - 8)/( (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[3, 2, 1, 1, 1, 1]](x) = -x^3*(216960*x^6-330432*x^5+191928*x^4-54004* x^3+7102*x^2-291*x-8)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1 )/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 8, 1011, 62840, 3318415, 165437748, 8064937251, 389601886640, 18750637412055, 901025909888588, 43269152889886891, 2077317545983210440, 99719206533287800095, 4786681201547550325028, 229763883444592826522931, 11028730120836547164326240, 529380320110791310578306535, 25410280851536422498460851068, 1219693990598162522602099213371 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [3, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (169472 x - 202624 x + 70720 x - 5384 x - 1052 x + 122 x + 1)/( (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[3, 1, 1, 1, 1, 1, 1]](x) = x^3*(169472*x^6-202624*x^5+70720*x^4-5384* x^3-1052*x^2+122*x+1)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1 )/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 1, 212, 15397, 856168, 43530865, 2138849804, 103657202013, 4995432393296, 240178786051369, 11536544671380916, 553913421233397589, 26591029902794370104, 1276433150141356774113, 61270065511940417341148, 2940988630747899885150925, 141167963999965042382648992, 6776072466484188754630390297, 325251682280994700231094740100 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [2, 2, 2, 2, 1]](x) = - x 5 4 3 2 (23168 x - 23392 x + 7504 x - 1112 x + 79 x + 4)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[2, 2, 2, 2, 1]](x) = -x^3*(23168*x^5-23392*x^4+7504*x^3-1112*x^2+79*x +4)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 4, 415, 25240, 1328287, 66182900, 3226039523, 155841283504, 7500259261399, 360410398702156, 17307661436093611, 830927020647076808, 39887682631420345631, 1914672480764306503972, 91905553379002070326579, 4411492048343954201856352, 211752128044391301537156583, 10164112340615167784365153148, 487877596239269802715245216827 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[6, 3], [2, 2, 2, 1, 1, 1]](x) = x 5 4 3 2 (128 x - 14080 x + 9816 x - 2268 x + 150 x + 3)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[2, 2, 2, 1, 1, 1]](x) = x^3*(128*x^5-14080*x^4+9816*x^3-2268*x^2+150* x+3)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+ 48*x) The first 20 term , starting with k=1 are 0, 0, 3, 420, 27639, 1493088, 75132923, 3676768076, 177901336383, 8567662820376, 411816339891363, 19778559189343812, 949598373493422887, 45585272854059418064, 2188184117742076376523, 105034658084281775837628, 5041699996840262596273551, 242002328025357818075455752, 11616126308766714216685343603, 557574354091865284191025785524 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 4 F[[6, 3], [2, 2, 1, 1, 1, 1, 1]](x) = 2 x 4 3 2 (30240 x - 13024 x + 2194 x - 752 x + 95)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[2, 2, 1, 1, 1, 1, 1]](x) = 2*x^4*(30240*x^4-13024*x^3+2194*x^2-752*x+ 95)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 0, 190, 14456, 816842, 41792200, 2058706074, 99879400392, 4815503763034, 231570524014280, 11123915919058778, 534118610365050568, 25641106468255470426, 1230841375846574948040, 59081751363625032147802, 2835951372038079615696584, 136126211990501425650115418, 6534069098205463156810350280, 313635535167153424751740429146 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 4 F[[6, 3], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x 5 4 3 2 (157696 x - 143200 x + 58312 x - 10804 x + 467 x + 39)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^4*(157696*x^5-143200*x^4+58312*x^3-\ 10804*x^2+467*x+39)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1 +20*x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 0, 39, 3860, 232703, 12189784, 606071531, 29515163116, 1425237897831, 68581901266208, 3295343128509443, 158244725175526132, 7597112048883757439, 364688684272543924072, 17505602972505999238875, 840279865299769183907708, 40333651987345313829467927, 1936019664456229477950742576, 92929031275208269917960325427 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 4 F[[6, 3], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (128 x - 14080 x + 9816 x - 2268 x + 150 x + 3)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 48 x)) and in Maple notation F[[6, 3],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^4*(128*x^5-14080*x^4+9816*x^3-2268 *x^2+150*x+3)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(-1+4*x)/(8*x-1)/(-1+20 *x)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 0, 3, 420, 27639, 1493088, 75132923, 3676768076, 177901336383, 8567662820376, 411816339891363, 19778559189343812, 949598373493422887, 45585272854059418064, 2188184117742076376523, 105034658084281775837628, 5041699996840262596273551, 242002328025357818075455752, 11616126308766714216685343603 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 2 x) (-1 + 6 x) (1 + 2 x) (-1 + 3 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+2*x)/(-1+6*x)/(1+2*x)/(-1+3*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 Regarding Lambda=, [6, 2, 1] F[[6, 2, 1], [9]](x) = ( 7 6 5 4 3 2 375375 x - 2075 x - 842165 x - 22722 x + 64341 x - 6070 x + 157 x - 1 )/((-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x)) and in Maple notation F[[6, 2, 1],[9]](x) = (375375*x^7-2075*x^6-842165*x^5-22722*x^4+64341*x^3-6070* x^2+157*x-1)/(-1+x)/(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 9, 511, 40739, 3880646, 394221334, 40939299486, 4282881502314, 449153624534671, 47141949967046759, 4949233905858916961, 519646087899927962389, 54562017810082665623196, 5728983122001178492218684, 601542221652438688623244936, 63161898058351997308861540964, 6631998063602041117613189106221, 696359753539924271467371101457109, 73117772611853128474805967729123411 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [8, 1]](x) = 2 4 3 2 x (91875 x + 24400 x - 6165 x + 252 x - 2) - ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[8, 1]](x) = -x^2*(91875*x^4+24400*x^3-6165*x^2+252*x-2)/(1+x)/(-1+ 5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 2, 60, 3699, 314424, 30664332, 3140766900, 327064030029, 34247360579184, 3592680864421302, 377116431143389740, 39593200586405442759, 4157145233687276184744, 436495321101797587135872, 45831836228560114033839780, 4812336767070947252909873889, 505295149251798755353290235104, 53055983276293462968889877016042, 5570877985181134915272076910293020, 584942179384986569610397021018795419 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [7, 2]](x) = 2 3 2 x (57075 x - 11135 x + 474 x - 4) ---------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[7, 2]](x) = x^2*(57075*x^3-11135*x^2+474*x-4)/(-1+x)/(1+x)/(-1+5*x )/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 4, 166, 11499, 1030991, 102492749, 10565953491, 1102658895874, 115543651937866, 12123859071395874, 1272717637552719116, 133625291495397567749, 14030303556225179672241, 1473169552599049596786499, 154682371809335936117172241, 16241633947724471057653427124, 1705371036285400588639974594116, 179063940322117914343494176864624, 18801713086748399790817826986312866, 1974179851461003719691457599401473999 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[6, 2, 1], [7, 1, 1]](x) = x 5 4 3 2 (91875 x + 138775 x + 26335 x - 9923 x + 462 x - 4)/((-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x)) and in Maple notation F[[6, 2, 1],[7, 1, 1]](x) = x^2*(91875*x^5+138775*x^4+26335*x^3-9923*x^2+462*x-\ 4)/(-1+x)/(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 4, 166, 11709, 1061296, 106008594, 10947408766, 1143151393059, 119810895408496, 12572465365065744, 1319840430940718566, 138573854920995932409, 14549926177316018821696, 1527730749070814306544894, 160411326184495986917062366, 16843175163237470280859869759, 1768532899128872204678321540896, 185695937153199142116734722606044, 19498072797150095595232632407160166, 2047297622563018851819426152495045109 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [6, 3]](x) = 2 4 3 2 x (118125 x + 26100 x - 10080 x + 532 x - 5) - ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[6, 3]](x) = -x^2*(118125*x^4+26100*x^3-10080*x^2+532*x-5)/(1+x)/(-\ 1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 5, 248, 19203, 1794544, 180940465, 18740572008, 1958781271823, 205358632009184, 21551700132054045, 2262545238117943768, 237553838234714094043, 24942683646026516355024, 2618965355579875827454025, 274990787391766173977056328, 28874012553237185473972693863, 3031770613790654747204248368064, 318335889797577850037341654852405, 33425267565980735439897961065173688, 3509653064231211752157979541419279283 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [6, 2, 1]](x) = - x 5 4 3 2 (451500 x + 25700 x - 49060 x + 5167 x - 148 x + 1)/((-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x)) and in Maple notation F[[6, 2, 1],[6, 2, 1]](x) = -x*(451500*x^5+25700*x^4-49060*x^3+5167*x^2-148*x+1 )/(-1+x)/(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 9, 511, 40739, 3880646, 394221334, 40939299486, 4282881502314, 449153624534671, 47141949967046759, 4949233905858916961, 519646087899927962389, 54562017810082665623196, 5728983122001178492218684, 601542221652438688623244936, 63161898058351997308861540964, 6631998063602041117613189106221, 696359753539924271467371101457109, 73117772611853128474805967729123411, 7677366071400234762478925867051718039 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [6, 1, 1, 1]](x) = 2 4 3 2 5 x (18375 x + 5915 x - 2178 x + 105 x - 1) - ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[6, 1, 1, 1]](x) = -5*x^2*(18375*x^4+5915*x^3-2178*x^2+105*x-1)/(1+ x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 5, 255, 21105, 2046390, 209416125, 21804743205, 2283164491305, 239512164414540, 25141097011367925, 2639546729787456555, 277143015939558257505, 29099688078859269834690, 3055455748651765730263725, 320822451139279522111237905, 33686343283471495935653319705, 3537065551753171214940808866840, 371391865678746432940027460823525, 38996145292332499552854802837527255, 4094595234557170343687352869353357905 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [5, 4]](x) = 2 3 2 x (1435 x + 563 x - 233 x + 3) -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[5, 4]](x) = x^2*(1435*x^3+563*x^2-233*x+3)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 3, 193, 16152, 1550116, 157656783, 16375244473, 1713145478712, 179661343009936, 18856778384932323, 1979693538316055353, 207858434799562671072, 21824807118626958620356, 2291593248719379884976063, 240616888659759103276861633, 25264759223322553343508377232, 2652799225440542763357426189376, 278543901415965603331694341644003, 29247109044741189811093893305295313, 3070946428560092981309144530063701192 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 2 3 x (875 x - 215 x + 3) F[[6, 2, 1], [5, 3, 1]](x) = -------------------------------------------- (-1 + x) (-1 + 5 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[5, 3, 1]](x) = 3*x^2*(875*x^2-215*x+3)/(-1+x)/(-1+5*x)/(-1+35*x)/( -1+105*x) The first 20 term , starting with k=1 are 0, 9, 669, 59619, 5885244, 604822119, 63046765869, 6603825640869, 692838723687744, 72728362078765869, 7635788381438140869, 801733642761692047119, 84181187684850803375244, 8838995138729889621734619, 928093454680357821750640869, 97449776520417750570774078369, 10232225266908170084402561187744, 1074383608654608578465902805328369, 112810277355757675925032675266265869, 11845079068000388103642310202121734619 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [5, 2, 2]](x) = 2 2 2 x (710 x + 177 x - 3) --------------------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[5, 2, 2]](x) = 2*x^2*(710*x^2+177*x-3)/(-1+x)/(1+x)/(1+3*x)/(-1+35 *x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 6, 468, 43172, 4324720, 446800666, 46658755988, 4890233363112, 513161742662400, 53871036363050126, 5656075686543445108, 593874537481870858252, 62356357099414766114480, 6547401068672191831756386, 687476537273757591035763828, 72185016290955757763120464592, 7579426006252746939780978232960, 795839706006122161789867609463446, 83563168267878257646886375226080148, 8774132637930457125986329052151078132 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [5, 2, 1, 1]](x) = 2 3 2 3 x (13300 x - 4260 x + 245 x - 3) ---------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[5, 2, 1, 1]](x) = 3*x^2*(13300*x^3-4260*x^2+245*x-3)/(-1+x)/(1+x)/ (-1+5*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 9, 705, 66639, 6759855, 701843514, 73421231730, 7699782518514, 808147805231730, 84844011187596639, 8908218670307184855, 935348877053191502889, 98211139238939701716105, 10312152371254170134862264, 1082775395287076843754450480, 113691395376050629653533299764, 11937595774970364954016606012980, 1253447530488914292643966277440389, 131611989795432648710062811772028605, 13819258896813821878131210555632909139 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [5, 1, 1, 1, 1]](x) = 2 4 3 2 x (2625 x - 475 x + 160 x - 129 x + 3) -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[5, 1, 1, 1, 1]](x) = x^2*(2625*x^4-475*x^3+160*x^2-129*x+3)/(-1+x) /(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 3, 237, 23677, 2465440, 258558693, 27143931017, 2850041652587, 299253306039960, 31421581118216383, 3299265777151739197, 346422902996919009297, 36374404760615774751080, 3819312499053743012864273, 401027812388479303518576777, 42107920300607871113619277807, 4421331631561089630273974726800, 464239821313873358627339213718363, 48745181237956086867591241966795757, 5118244029985379884272850056575402117 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[6, 2, 1], [4, 4, 1]](x) = - x 5 4 3 2 (165375 x + 113925 x - 36215 x + 4743 x - 312 x + 4)/((-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x)) and in Maple notation F[[6, 2, 1],[4, 4, 1]](x) = -x^2*(165375*x^5+113925*x^4-36215*x^3+4743*x^2-312* x+4)/(-1+x)/(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 4, 316, 30079, 3025156, 312728734, 32660653816, 3423156232129, 359213113059256, 37709723852250184, 3959252956552895716, 415712175876901100179, 43649449964184228611356, 4583180747989442770437634, 481233576090413941552239616, 50529511403650784854149162229, 5305598204376649174158832081456, 557087794204281407997653237371084, 58494217787514718773991968842845516, 6141892846551319064508004519835938279 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [4, 3, 2]](x) = 2 3 2 x (18900 x - 1665 x - 208 x + 5) ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[4, 3, 2]](x) = x^2*(18900*x^3-1665*x^2-208*x+5)/(1+x)/(-1+5*x)/(1+ 3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 5, 572, 58002, 5976376, 622888615, 65231711412, 6843181304132, 718316706586736, 75415615587943785, 7918371805040964652, 831419658211785200662, 87298735658001837865896, 9166355746570114373012555, 962466950952331804572477092, 101059015764316370667350819592, 10611196162248998837615121477856, 1114175579780915069957245713582925, 116988435273061807489202605374984732, 12283785682533771692746938599415952922 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [4, 3, 1, 1]](x) = 2 3 2 3 x (5250 x + 75 x - 85 x + 2) ------------------------------------------------------- (1 + x) (-1 + 5 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[4, 3, 1, 1]](x) = 3*x^2*(5250*x^3+75*x^2-85*x+2)/(1+x)/(-1+5*x)/(-\ 1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 6, 699, 73026, 7624974, 798722526, 83793561849, 8795707360026, 923456406233724, 96959653088297526, 10180648851054280599, 1068964109722861735026, 112241090768701161702474, 11785309603413539072672526, 1237457335888322192128499349, 129933014231601403631846110026, 13642966283031328247063954671224, 1432511452323201533173529307047526, 150413702235107344390365441640218099, 15793438725627251496049198309580485026 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [4, 2, 2, 1]](x) = 2 5 x (30 x - 1) - ----------------------------------------------- (-1 + 5 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[4, 2, 2, 1]](x) = -5*x^2*(30*x-1)/(-1+5*x)/(-1+15*x)/(-1+35*x)/(-1 +105*x) The first 20 term , starting with k=1 are 0, 5, 650, 71250, 7562500, 796534375, 83716968750, 8793026562500, 923362578125000, 96956369103515625, 10180533911582031250, 1068960086841308593750, 112240949967846679687500, 11785304675383631591796875, 1237457163407275427246093750, 129933008194764766845703125000, 13642966071742045959472656250000, 1432511444928076653107452392578125, 150413701976277973588050842285156250, 15793438716568223517968177795410156250 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [4, 2, 1, 1, 1]](x) = 2 4 3 2 x (55125 x - 11025 x + 2420 x - 114 x + 4) - ---------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[4, 2, 1, 1, 1]](x) = -x^2*(55125*x^4-11025*x^3+2420*x^2-114*x+4)/( -1+x)/(1+x)/(-1+5*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 4, 526, 60384, 6541001, 694184009, 73153151001, 7690399702759, 807819406729126, 84832517240249634, 8907816382151260376, 935334796967740249634, 98210646435948938369751, 10312135123149493570327759, 1082774791603413164758682251, 113691374247122400892496109009, 11937595035457876947399377822876, 1253447504605977212412458658218384, 131611988889529850901960521936416626, 13819258865107223954847632795572280884 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [4, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (73500 x - 5950 x + 1935 x - 22 x + 1) ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[4, 1, 1, 1, 1, 1]](x) = x^2*(73500*x^4-5950*x^3+1935*x^2-22*x+1)/( 1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 134, 16926, 1900444, 204309571, 21626021634, 2276909275376, 239293232052344, 25133434379667621, 2639278537682828734, 277133629215920697826, 29099359543532077316244, 3055444249915314602469671, 320822048683503735690203834, 33686329197519343426175916276, 3537065058744845877185393692144, 371391848423455046118969399435721, 38996144688397301014119678037586934, 4094595213419438394831633038098610726 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [3, 3, 3]](x) = 2 3 2 x (2790 x - 603 x - 2 x - 1) - -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[3, 3, 3]](x) = -x^2*(2790*x^3-603*x^2-2*x-1)/(-1+x)/(1+x)/(-1+5*x) /(1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 124, 13932, 1475066, 155072081, 16285409994, 1710010756672, 179551770065596, 18852945467399241, 1979559418237247264, 207853741077340463612, 21824642845557280163526, 2291587499270062936324201, 240616687430654838529691134, 25264752180328231511912350752, 2652798978936106410807763720856, 278543892788315804665990684870961, 29247108742773528962898234483269604, 3070946417991226083198860784500834092 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [3, 3, 2, 1]](x) = 2 3 2 x (4725 x - 5310 x + 143 x + 2) ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[3, 3, 2, 1]](x) = x^2*(4725*x^3-5310*x^2+143*x+2)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 2, 455, 53844, 5830534, 617782582, 65052992445, 6836926101224, 718097774289644, 75407952956569002, 7918103612937964435, 831410271488155779004, 87298407122674686037554, 9166344247833663448669022, 962466548496556019168695625, 101059001678364218162959679184, 10611195669240673499885137618264, 1114175562525623683136314808770642, 116988434669126608950468116357922015, 12283785661396039743891221947075593764 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [3, 3, 1, 1, 1]](x) = 2 3 2 x (1575 x + 355 x - 161 x - 1) --------------------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[3, 3, 1, 1, 1]](x) = x^2*(1575*x^3+355*x^2-161*x-1)/(-1+x)/(1+x)/( 1+3*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 298, 37217, 4116300, 439505961, 46403441318, 4881297349657, 512848982191480, 53860089746567921, 5655692554966567938, 593861127876680157297, 62355887763233091581060, 6547384641905833223086681, 687475962336935039732324158, 72184996168166968467500076137, 7579425301955139314434264637040, 795839681355705894902732633606241, 83563167405113688305836651071077978, 8774132607733697199049588706726002177 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 4 3 2 F[[6, 2, 1], [3, 2, 2, 2]](x) = - x (90300 x - 41435 x + 595 x + 43 x + 1)/ ((-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) ) and in Maple notation F[[6, 2, 1],[3, 2, 2, 2]](x) = -x^2*(90300*x^4-41435*x^3+595*x^2+43*x+1)/(-1+x) /(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 200, 25926, 2879340, 307622831, 32481935500, 3416901032476, 358994180778440, 37702061220956781, 3958984764450302400, 415702789153273713026, 43649121428857086955540, 4583169249252991896956731, 481233173634638156402771300, 50529497317698632351029587576, 5305597711368323836435206050640, 557087776948990021176754121702681, 58494217183579520235257638771502200, 6141892825413587115652288662224176126 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 x (220 x + 1) F[[6, 2, 1], [3, 2, 2, 1, 1]](x) = -------------------------------------------- (-1 + x) (-1 + 5 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[3, 2, 2, 1, 1]](x) = x^2*(220*x+1)/(-1+x)/(-1+5*x)/(-1+35*x)/(-1+ 105*x) The first 20 term , starting with k=1 are 0, 1, 366, 48916, 5510166, 591692041, 62587201416, 6587740826416, 692275754888916, 72708658169342041, 7635098744600982666, 801709505472354888916, 84180342879723819732666, 8838965570550444278717041, 928092419794077230167388916, 97449740299397929842472076416, 10232223999172476358797550201416, 1074383564283859298069155216217041, 112810275802781451111143648624420166, 11845079013646220235156179964542388916 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 3 3 2 F[[6, 2, 1], [3, 2, 1, 1, 1, 1]](x) = 5 x (18375 x - 6490 x + 533 x - 42)/( (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x)) and in Maple notation F[[6, 2, 1],[3, 2, 1, 1, 1, 1]](x) = 5*x^3*(18375*x^3-6490*x^2+533*x-42)/(-1+x) /(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 210, 30305, 3515845, 381455275, 40492497185, 4267243470630, 448606293669870, 47122793387999450, 4948563425598364660, 519622621090839149455, 54561196471764709758395, 5728954375160050799890125, 601541215512999223206442635, 63161862843471616038346946780, 6631996831081227773240545741420, 696359710401695804414805420847300, 73117771102015132127968553093571110, 7677366018555904890339628275736342605 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 3 F[[6, 2, 1], [3, 1, 1, 1, 1, 1, 1]](x) = x 4 3 2 (91875 x - 40775 x - 935 x - 617 x - 44)/((-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x)) and in Maple notation F[[6, 2, 1],[3, 1, 1, 1, 1, 1, 1]](x) = x^3*(91875*x^4-40775*x^3-935*x^2-617*x-\ 44)/(-1+x)/(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 44, 7525, 915324, 100901910, 10768686544, 1136896173875, 119591963030024, 12564802733284060, 1319572238835683844, 138564468197356338225, 14549597641988816130724, 1527719250334363127888210, 160410923728720200241715144, 16843161077285317770110900575, 1768532406120546866916548537424, 185695919897907755295644872074360, 19498072193214897056497348661500444, 2047297601425286902963705526511700925 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [2, 2, 2, 2, 1]](x) = 3 2 2 x (245 x + 601 x + 38) -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[2, 2, 2, 2, 1]](x) = 2*x^3*(245*x^2+601*x+38)/(-1+x)/(1+x)/(-1+5*x )/(1+3*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 76, 11994, 1404274, 152550750, 16196525506, 1706890275804, 179442410712844, 18849115753557540, 1979425346213055136, 207849048075933249414, 21824478583299806792014, 2291581749982928960632530, 240616486203983317873080166, 25264745137370400839117236824, 2652798732432217425627442329784, 278543884160674216510763436831720, 29247108440805991272359404288232596, 3070946407422361032453427877723342034 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [2, 2, 2, 1, 1, 1]](x) = 3 2 6 x (3225 x - 180 x - 13) - ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[2, 2, 2, 1, 1, 1]](x) = -6*x^3*(3225*x^2-180*x-13)/(1+x)/(-1+5*x)/ (1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 78, 13248, 1586124, 173645760, 18485257338, 1949845258368, 205045871538264, 21540753515571840, 2262162106541066598, 237540428629523393088, 24942214309844841821604, 2618948928813517218784320, 274990212454943622673616658, 28873992430448396178352305408, 3031769909493047121857534772144, 318335865147161583150206678995200, 33425266703216166098848236910171518, 3509653034034451825221239195994203328 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 3 3 2 9 x (4375 x - 1425 x + 140 x + 4) - ---------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -9*x^3*(4375*x^3-1425*x^2+140*x+4)/(-1+ x)/(1+x)/(-1+5*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 36, 7020, 874611, 97021395, 10374465861, 1095956877645, 115309081543986, 12115649108830770, 1272430288869043986, 133615234291499455770, 14029951554088898340861, 1473157232524280513127645, 154681940606719022003809611, 16241618855632879082759221395, 1705370508062194869614044825236, 179063921834305714178063472112020, 18801712439674972785030136505762736, 1974179828813433774488900353511174520 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[6, 2, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (26250 x - 3475 x + 810 x + 7) ----------------------------------------------------------------- (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x) and in Maple notation F[[6, 2, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(26250*x^3-3475*x^2+810*x+7)/(1+ x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 7, 1902, 251846, 28475660, 3064171197, 324383219482, 34153532405356, 3589396879313880, 377001491669512787, 39589177704844163462, 4157004432832753479666, 436490393071889902809700, 45831663747513348134181577, 4812330730234310461680625842, 505294937962516467736560498776, 53055975881168582902685805971120, 5570877726351764112956841772353567, 584942170325958591529373327934078622 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 4 F[[6, 2, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 5 x 3 2 (18375 x - 6490 x + 533 x - 42)/((-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 35 x) (-1 + 105 x)) and in Maple notation F[[6, 2, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 5*x^4*(18375*x^3-6490*x^2+533*x-\ 42)/(-1+x)/(1+x)/(-1+5*x)/(1+3*x)/(-1+15*x)/(-1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 0, 210, 30305, 3515845, 381455275, 40492497185, 4267243470630, 448606293669870, 47122793387999450, 4948563425598364660, 519622621090839149455, 54561196471764709758395, 5728954375160050799890125, 601541215512999223206442635, 63161862843471616038346946780, 6631996831081227773240545741420, 696359710401695804414805420847300, 73117771102015132127968553093571110 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(-1+x)/(1+x)/(-1+5*x)/( 1+3*x)/(-1+15*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 Regarding Lambda=, [6, 1, 1, 1] 11 10 9 8 F[[6, 1, 1, 1], [9]](x) = (4963728 x - 1301268 x - 11105188 x + 3777901 x 7 6 5 4 3 2 + 3308496 x - 1174958 x - 199782 x + 79674 x + 943 x - 1273 x + 78 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[9]](x) = (4963728*x^11-1301268*x^10-11105188*x^9+3777901*x^8+ 3308496*x^7-1174958*x^6-199782*x^5+79674*x^4+943*x^3-1273*x^2+78*x-1)/(1+x)/(-1 +x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+ 56*x) The first 20 term , starting with k=1 are 0, 1, 1, 45, 1622, 86741, 4777953, 266778730, 14928617824, 835869536211, 46806935215535, 2621165632731020, 146784973352688606, 8219954484190509061, 460317396886267461997, 25777773490505742032190, 1443555305435621868701468, 80839096966821043279875191, 4526989428247463044946055339, 253511407955682647472267078640 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [8, 1]](x) = - x 7 6 5 4 3 2 (522640 x - 212180 x - 63276 x + 31853 x + 546 x - 942 x + 70 x - 1)/ ((1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[8, 1]](x) = -x^2*(522640*x^7-212180*x^6-63276*x^5+31853*x^4+546 *x^3-942*x^2+70*x-1)/(1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x) /(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 7, 286, 12782, 688872, 38181244, 2133602962, 119421775774, 6686861596708, 374454289502216, 20969309408533278, 1174279581661924426, 65759633139694240384, 3682539138431510643748, 206222187428034438700234, 11548442436733447364060438, 646712775642145936784139900, 36215915424708999277137051040, 2028091263627925884270831287830 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [7, 2]](x) = - x ( 7 6 5 4 3 2 582120 x + 444714 x - 67339 x - 55791 x + 3908 x + 1385 x - 129 x + 2 )/((1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[7, 2]](x) = -x^2*(582120*x^7+444714*x^6-67339*x^5-55791*x^4+ 3908*x^3+1385*x^2-129*x+2)/(1+x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1 +4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 21, 870, 42375, 2313877, 128739321, 7199336615, 403029264855, 22567911861357, 1263780082408701, 70771378254802315, 3963193049021227515, 221938754673433176017, 12428569495957859129361, 695999881267703640599295, 38975993206251919691141055, 2182655617549637944145969557, 122228714555057246329208442501, 6844808014698219861057204180755 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [7, 1, 1]](x) = - x 7 6 5 4 3 2 (362208 x - 393456 x + 54246 x + 34269 x - 6448 x - 252 x + 54 x - 1) /((1 + x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[7, 1, 1]](x) = -x^2*(362208*x^7-393456*x^6+54246*x^5+34269*x^4-\ 6448*x^3-252*x^2+54*x-1)/(1+x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+ 4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 25, 874, 44046, 2398098, 133502166, 7465786966, 417953993482, 23403722739130, 1310586221707422, 73392532593299118, 4109977865163137058, 230158706951226519922, 12888886861987240360918, 721777654326016991041030, 40419548505638023881977274, 2263494714431758685931661674, 126755703982118947475216424654, 7098319422637301588024363423902 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [6, 3]](x) = - x 6 5 4 3 2 (177408 x + 37804 x - 37894 x + 602 x + 1369 x - 121 x + 2)/((1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[6, 3]](x) = -x^2*(177408*x^6+37804*x^5-37894*x^4+602*x^3+1369*x ^2-121*x+2)/(1+x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56* x) The first 20 term , starting with k=1 are 0, 2, 29, 1438, 74245, 4100032, 228710939, 12796842578, 716471931785, 40120420504132, 2246716148945719, 125815731527865238, 7045675846881743045, 394557776978422225352, 22095234537254735610419, 1237333120600506638672218, 69290654566386112643437825, 3880276653113510453111500492, 217295492538088270370688369839, 12168547581627273449404822556318 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 9 8 7 F[[6, 1, 1, 1], [6, 2, 1]](x) = x (1888656 x + 156828 x - 1453184 x 6 5 4 3 2 - 47647 x + 205911 x + 9858 x - 9594 x - 162 x + 86 x - 2)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[6, 2, 1]](x) = x^2*(1888656*x^9+156828*x^8-1453184*x^7-47647*x^ 6+205911*x^5+9858*x^4-9594*x^3-162*x^2+86*x-2)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x)/(-\ 1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 70, 3078, 162370, 8962232, 500240475, 27991941203, 1567266810340, 87763189230162, 4914688320621805, 275221866154344353, 15412415256476693910, 863095127821449767792, 48333325418845011227935, 2706666199462815179235903, 151573306837940768027483080, 8488105178320072433949436622, 475333889921933437706698790865, 26618697834737615837219111749853 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 9 8 7 F[[6, 1, 1, 1], [6, 1, 1, 1]](x) = - x (4971120 x - 2144748 x - 2094556 x 6 5 4 3 2 + 827747 x + 156720 x - 64314 x - 1482 x + 1239 x - 77 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[6, 1, 1, 1]](x) = -x*(4971120*x^9-2144748*x^8-2094556*x^7+ 827747*x^6+156720*x^5-64314*x^4-1482*x^3+1239*x^2-77*x+1)/(1+x)/(-1+x)/(1+2*x)/ (-1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 1, 45, 1622, 86741, 4777953, 266778730, 14928617824, 835869536211, 46806935215535, 2621165632731020, 146784973352688606, 8219954484190509061, 460317396886267461997, 25777773490505742032190, 1443555305435621868701468, 80839096966821043279875191, 4526989428247463044946055339, 253511407955682647472267078640, 14196638845155594115335056936410 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [5, 4]](x) = - x 7 6 5 4 3 2 (44688 x - 6636 x - 21502 x + 8818 x - 33 x - 356 x + 47 x - 1)/( (1 + x) (-1 + x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[5, 4]](x) = -x^2*(44688*x^7-6636*x^6-21502*x^5+8818*x^4-33*x^3-\ 356*x^2+47*x-1)/(1+x)/(-1+x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(1+6*x)/(1+4*x)/(-1+14* x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 22, 1201, 64423, 3580340, 200039490, 11196194605, 626900086261, 35105204139814, 1965874532626048, 110088737761890119, 6164966006584323939, 345238050074370136048, 19333330155930601489246, 1082666479657511407360993, 60629322733772146347710257, 3395242071312585604586846042, 190133555968603483637763049884, 10647479133893177615360603459227 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 2 x (28 x + 53 x - 2) F[[6, 1, 1, 1], [5, 3, 1]](x) = -------------------------------------------- (-1 + 4 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x) and in Maple notation F[[6, 1, 1, 1],[5, 3, 1]](x) = x^2*(28*x^2+53*x-2)/(-1+4*x)/(1+4*x)/(-1+14*x)/( -1+56*x) The first 20 term , starting with k=1 are 0, 2, 87, 4526, 247764, 13795160, 771400656, 43182641504, 2418006573888, 135405269249408, 7582651690056960, 424627887212713472, 23779153179829711872, 1331632459013316663296, 74571416037944783130624, 4175999274789694622228480, 233855959061529899354505216, 13095933702871972372061978624, 733372287296798624722773147648, 41068848087724277390914708963328 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[6, 1, 1, 1], [5, 2, 2]](x) = 2 4 3 2 x (588 x - 318 x + 442 x - 75 x + 2) --------------------------------------------------------------- (-1 + 2 x) (-1 + 3 x) (1 + x) (1 + 6 x) (-1 + 14 x) (-1 + 56 x) and in Maple notation F[[6, 1, 1, 1],[5, 2, 2]](x) = x^2*(588*x^4-318*x^3+442*x^2-75*x+2)/(-1+2*x)/(-\ 1+3*x)/(1+x)/(1+6*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 61, 3348, 183085, 10214342, 571334571, 31986189608, 1791102180145, 100300008984282, 5616776346412231, 314539138247472668, 17614187015490545805, 986394406735817171822, 55238085851137705564291, 3093332794700108408904528, 173226636321707530062209065, 9700691631474690698250768962, 543238731327009353796179690751, 30421368953814499007398385877188 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[6, 1, 1, 1], [5, 2, 1, 1]](x) = - x (349272 x - 584178 x - 110759 x 5 4 3 2 + 211628 x - 6280 x - 10372 x + 494 x + 47 x - 2)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[5, 2, 1, 1]](x) = -x^2*(349272*x^8-584178*x^7-110759*x^6+211628 *x^5-6280*x^4-10372*x^3+494*x^2+47*x-2)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+3*x)/ (-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 105, 5246, 288720, 16085540, 899832465, 50377549259, 2820974564310, 157972318949798, 8846419727861595, 495399096689912417, 27742343866878579120, 1553571180613662575276, 86999985070912184101845, 4871999149575335416857095, 272831952177034116438026250, 15278589319151135418538919474, 855601001834069264829501820815, 47913656102173484510822689101293 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[6, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (462 x - 1905 x - 231 x + 278 x + 22 x - 1) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation F[[6, 1, 1, 1],[5, 1, 1, 1, 1]](x) = -x^2*(462*x^5-1905*x^4-231*x^3+278*x^2+22* x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 41, 1962, 106995, 5957421, 333249126, 18657880522, 1044796692365, 58508123367591, 3276449505611436, 183481112871679932, 10274941666985321235, 575396726158904318761, 32222216585784260977346, 1804444127933657908224942, 101048871154711985156778105, 5658736784558569693770977931, 316889259934121586787147172856, 17745798556298067382731687991552 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 7 6 5 F[[6, 1, 1, 1], [4, 4, 1]](x) = x (1190112 x - 1153152 x + 165514 x 4 3 2 + 127813 x - 26579 x - 4370 x + 1003 x - 41)/((1 + x) (-1 + x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[4, 4, 1]](x) = x^3*(1190112*x^7-1153152*x^6+165514*x^5+127813*x ^4-26579*x^3-4370*x^2+1003*x-41)/(1+x)/(-1+x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11 *x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 41, 2277, 127818, 7144285, 399884394, 22389708537, 1253765139206, 70209933222165, 3931742655927042, 220177388018937217, 12329930815163466414, 690476083658902906365, 38666660084200744801610, 2165332956162390692001417, 121258645423791533340297942, 6790484142016837580764328485, 380267111928736671446344726098, 21294958267668280494247693558737 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [4, 3, 2]](x) = - x 7 6 5 4 3 2 (182224 x + 236668 x - 91356 x - 13403 x + 4134 x - 9 x + 3 x - 1)/( (1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[4, 3, 2]](x) = -x^2*(182224*x^7+236668*x^6-91356*x^5-13403*x^4+ 4134*x^3-9*x^2+3*x-1)/(1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x )/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 74, 4512, 254531, 14276688, 799589875, 44777105584, 2507498103527, 140419429951560, 7863479283724691, 440354692982971416, 24659860479726425863, 1380952151358668716192, 77333319946523183798147, 4330665909236085951050208, 242517290804531611961036039, 13580968283433067873333517784, 760534223849087964857060548243, 42589916535219420779373313665160 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[6, 1, 1, 1], [4, 3, 1, 1]](x) = - x (709632 x - 363816 x - 324994 x 5 4 3 2 + 137833 x - 2834 x - 6713 x + 1072 x - 29 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[4, 3, 1, 1]](x) = -x^2*(709632*x^8-363816*x^7-324994*x^6+137833 *x^5-2834*x^4-6713*x^3+1072*x^2-29*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+3*x)/ (-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 105, 5788, 327627, 18353821, 1028020002, 57569777938, 3223913068509, 180539044436071, 10110184199071704, 566170266936789868, 31705534122390288771, 1775509897467135866401, 99428554051663930165086, 5567999023786604590881478, 311807945286220244814383913, 17461244935360799498621421211, 977829716370575416291954058148, 54758464116614282255794230163168 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [4, 2, 2, 1]](x) = - x 7 6 5 4 3 2 (1058712 x - 618 x - 179265 x + 15989 x + 6994 x - 969 x + 59 x - 2)/ ((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[4, 2, 2, 1]](x) = -x^2*(1058712*x^7-618*x^6-179265*x^5+15989*x^ 4+6994*x^3-969*x^2+59*x-2)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+11*x)/(1+ 6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 93, 5797, 326670, 18346839, 1027883028, 57568091849, 3223888055670, 180538702619731, 10110179363106768, 566170199534993781, 31705533176948605650, 1775509884241825760303, 99428553866444242662588, 5567999021193920614286593, 311807945249920317521034510, 17461244934852614619259944555, 977829716363460743336320482888, 54758464116514677342169863170285 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[6, 1, 1, 1], [4, 2, 1, 1, 1]](x) = x (582120 x + 309162 x + 56629 x 5 4 3 2 - 39331 x - 9266 x + 1131 x + 515 x - 62 x + 2)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[4, 2, 1, 1, 1]](x) = x^2*(582120*x^8+309162*x^7+56629*x^6-39331 *x^5-9266*x^4+1131*x^3+515*x^2-62*x+2)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+3*x)/( -1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 90, 5115, 286323, 16055152, 899387433, 50371435130, 2820888266391, 157971114979872, 8846402847105021, 495398860510579390, 27742340559461379579, 1553571134315265803012, 86999984422701986547729, 4871999140500588599672370, 272831952049986485694205887, 15278589317372475641556841672, 855601001809167985637789863557, 47913656101824866856049533913670 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 F[[6, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = - x (1215984 x + 133860 x 6 5 4 3 2 - 554624 x + 91647 x + 19057 x - 5854 x + 882 x - 53 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(1215984*x^8+133860*x^7-554624*x^6 +91647*x^5+19057*x^4-5854*x^3+882*x^2-53*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x)/(-1 +2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 25, 1560, 84892, 4758876, 266473872, 14924587000, 835811719264, 46806134229756, 2621154368744824, 146784815959995720, 8219952278881868616, 460317366022842333916, 25777773058352533722256, 1443555299385868958760120, 80839096882122152351418448, 4526989427061692680419873756, 253511407939081777747234476168, 14196638844923182447030730230600 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[6, 1, 1, 1], [3, 3, 3]](x) = 3 4 3 2 x (462 x + 917 x - 48 x + 27 x + 17) ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation F[[6, 1, 1, 1],[3, 3, 3]](x) = x^3*(462*x^4+917*x^3-48*x^2+27*x+17)/(1+x)/(-1+x )/(1+2*x)/(-1+2*x)/(-1+11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 17, 1098, 63295, 3564645, 199841082, 11193537978, 626864913545, 35104729932495, 1965868118267152, 110088650287434588, 6164964808397788575, 345238033585623530625, 19333329928263613365302, 1082666476504918503706878, 60629322690019151903688085, 3395242070704253387315849235, 190133555960133179345284260732, 10647479133775102929676327842048 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 7 6 5 F[[6, 1, 1, 1], [3, 3, 2, 1]](x) = - x (428624 x + 680604 x - 341972 x 4 3 2 - 48503 x + 33292 x - 296 x - 1069 x + 70)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[3, 3, 2, 1]](x) = -x^3*(428624*x^7+680604*x^6-341972*x^5-48503* x^4+33292*x^3-296*x^2-1069*x+70)/(1+x)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+3*x )/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 70, 4391, 253162, 14255196, 799301237, 44772983997, 2507440856284, 140418625644782, 7863468040043539, 440354535469934943, 24659858275145705066, 1380952120490899237008, 77333319514396133464801, 4330665903186176458337729, 242517290719833661982385208, 13580968282247291868899693874, 760534223832487128994574002223, 42589916534987008907985917720355 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[6, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 3 3 2 x (84 x - 702 x + 584 x - 55) - --------------------------------------------------------------- (-1 + 2 x) (-1 + 3 x) (1 + x) (1 + 6 x) (-1 + 14 x) (-1 + 56 x) and in Maple notation F[[6, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -x^3*(84*x^3-702*x^2+584*x-55)/(-1+2*x)/(-\ 1+3*x)/(1+x)/(1+6*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 55, 3156, 181155, 10183130, 570923925, 31980286296, 1791020470735, 100298859465510, 5616760286777145, 314538913211134436, 17614183866191456115, 986394362638374943890, 55238085233817052931965, 3093332786057358066918576, 173226636200710592583950295, 9700691629780724169934286270, 543238731303293878822183664385, 30421368953482482019229962534716 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[6, 1, 1, 1], [3, 2, 2, 2]](x) = - x (258720 x - 726768 x + 25402 x 5 4 3 2 + 172329 x - 24863 x - 6064 x + 990 x - 47 x + 1)/((1 + x) (-1 + x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[3, 2, 2, 2]](x) = -x^2*(258720*x^8-726768*x^7+25402*x^6+172329* x^5-24863*x^4-6064*x^3+990*x^2-47*x+1)/(1+x)/(-1+x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/ (-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 33, 2198, 126250, 7124144, 399588168, 22385634286, 1253707614570, 70209130604732, 3931731402205948, 220177230566514794, 12329928610220531310, 690476052793312515640, 38666659652060642926848, 2165332950112559599853222, 121258645339093113321643570, 6790484140831064398018498868, 380267111912135818659505614468, 21294958267435868724429465481570 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[6, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 2 2 x (196 x - x + 1) - -------------------------------------------- (-1 + 4 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x) and in Maple notation F[[6, 1, 1, 1],[3, 2, 2, 1, 1]](x) = -x^2*(196*x^2-x+1)/(-1+4*x)/(1+4*x)/(-1+14 *x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 69, 4258, 243948, 13741480, 770648112, 43172101792, 2417859001536, 135403203170944, 7582622764696320, 424627482256615936, 23779147510440152064, 1331632379641846048768, 74571414926744127418368, 4175999259232885173821440, 233855958843734566003064832, 13095933699822837700846845952, 733372287254110739308581421056, 41068848087126646995047305314304 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[6, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = x (439824 x + 155484 x - 153232 x 5 4 3 2 + 5129 x + 2543 x + 540 x + 239 x - 28 x + 1)/((1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = x^2*(439824*x^8+155484*x^7-153232*x^6+ 5129*x^5+2543*x^4+540*x^3+239*x^2-28*x+1)/(1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+3 *x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 49, 2817, 158565, 8910196, 499506189, 27981703072, 1567123255705, 87761180916216, 4914660196037229, 275221472461394152, 15412409744476471545, 863095050655274371936, 48333324338507726956669, 2706666184338158724390432, 151573306626195186726990585, 8488105175355636650226786856, 475333889880431322643289350509, 26618697834156586311021765121912 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[6, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x ( 7 6 5 4 3 2 569184 x - 83328 x - 118350 x + 25043 x + 2786 x - 1029 x + 65 x - 1) /((1 + x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^2*(569184*x^7-83328*x^6-118350*x^ 5+25043*x^4+2786*x^3-1029*x^2+65*x-1)/(1+x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11*x )/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 14, 782, 42427, 2377752, 133205121, 7461709438, 417896455739, 23402920069268, 1310574967776613, 73392375140037834, 4109975660216846511, 230158676085622707424, 12888886429847084799065, 721777648276185684144470, 40419548420939603004329443, 2263494713245985499749858220, 126755703965518094674633417677, 7098319422404889818151159765346 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[6, 1, 1, 1], [2, 2, 2, 2, 1]](x) = x 6 5 4 3 2 (30576 x + 1260 x - 8106 x + 728 x + 712 x - 163 x + 18)/((1 + x) (-1 + x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[2, 2, 2, 2, 1]](x) = x^3*(30576*x^6+1260*x^5-8106*x^4+728*x^3+ 712*x^2-163*x+18)/(1+x)/(-1+x)/(-1+4*x)/(-1+2*x)/(-1+3*x)/(1+6*x)/(1+4*x)/(-1+ 14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 18, 1079, 63049, 3558828, 199750782, 11192072787, 626842838283, 35104399830746, 1965863288937856, 110088580248832185, 6164963802003538077, 345238019206600460304, 19333329723803550563490, 1082666473607601912865823, 60629322649074196363700431, 3395242070126809600136923302, 190133555952002647775228163684, 10647479133660765743973062406501 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[6, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 x 4 3 2 (25658 x - 2205 x - 1926 x + 275 x - 12)/((1 + x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = 2*x^3*(25658*x^4-2205*x^3-1926*x^2+275* x-12)/(1+x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 24, 1250, 72330, 4068870, 228300454, 12790939770, 716390223930, 40119270990110, 2246700089325054, 125815506491570610, 7045672697582784850, 394557732880980393270, 22095233919934084168374, 1237333111957756300262570, 69290654445389175175918890, 3880276651419543924827259150, 217295492514372795396789111214, 12168547581295256461236689604450 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[6, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - 15 x 6 5 4 3 2 (24024 x + 702 x + 845 x - 346 x - 228 x + 29 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -15*x^3*(24024*x^6+702*x^5+845*x^4-\ 346*x^3-228*x^2+29*x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4 *x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 15, 720, 40905, 2290380, 128430990, 7194907755, 402967977315, 22567049700390, 1263768037594920, 70771209477198945, 3963190687045511805, 221938721600345911980, 12428569032967347284130, 695999874785640794628615, 38975993115504216224527575, 2182655616279163046476940850, 122228714537270640092984777820, 6844808014449207119907980137965 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 8 7 F[[6, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x (1419264 x - 627728 x 6 5 4 3 2 - 44396 x - 24980 x + 29299 x + 1834 x - 2786 x + 249 x - 6)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(1419264*x^8-627728*x^7-44396 *x^6-24980*x^5+29299*x^4+1834*x^3-2786*x^2+249*x-6)/(1+x)/(-1+x)/(1+2*x)/(-1+4* x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 6, 219, 12236, 679202, 38059397, 2131821741, 119397315162, 6686516389464, 374449473562703, 20969241885275123, 1174278636943687028, 65759619910021887486, 3682538953237909535049, 206222184835193593003065, 11548442400434459875192334, 646712775133955412933722468, 36215915417594360165724338435, 2028091263528320767490094280767 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 8 7 F[[6, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x (1215984 x + 133860 x 6 5 4 3 2 - 554624 x + 91647 x + 19057 x - 5854 x + 882 x - 53 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 3 x) (-1 + 11 x) (1 + 6 x) (1 + 4 x) (-1 + 14 x) (-1 + 56 x)) and in Maple notation F[[6, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(1215984*x^8+133860*x^7-\ 554624*x^6+91647*x^5+19057*x^4-5854*x^3+882*x^2-53*x+1)/(1+x)/(-1+x)/(1+2*x)/(-\ 1+4*x)/(-1+2*x)/(-1+3*x)/(-1+11*x)/(1+6*x)/(1+4*x)/(-1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 1, 25, 1560, 84892, 4758876, 266473872, 14924587000, 835811719264, 46806134229756, 2621154368744824, 146784815959995720, 8219952278881868616, 460317366022842333916, 25777773058352533722256, 1443555299385868958760120, 80839096882122152351418448, 4526989427061692680419873756, 253511407939081777747234476168 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 Regarding Lambda=, [5, 4] 9 8 7 6 5 F[[5, 4], [9]](x) = (142422 x - 41131 x - 268241 x + 133435 x + 39063 x 4 3 2 - 21619 x - 643 x + 783 x - 60 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[9]](x) = (142422*x^9-41131*x^8-268241*x^7+133435*x^6+39063*x^5-21619* x^4-643*x^3+783*x^2-60*x+1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+ 6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 1, 18, 421, 15954, 646020, 26831267, 1122736917, 47097209772, 1977276893614, 83034367580781, 3487285871511663, 146463801357320630, 6151448787352548408, 258360416917683288135, 10851131460566068187209, 455747436644956438463328, 19141391153309759908146402, 803938411838142761417758529 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [8, 1]](x) = - x 7 6 5 4 3 2 (22344 x - 35620 x - 14550 x + 12017 x + 409 x - 645 x + 56 x - 1)/( (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[8, 1]](x) = -x^2*(22344*x^7-35620*x^6-14550*x^5+12017*x^4+409*x^3-645 *x^2+56*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14*x)/ (-1+42*x) The first 20 term , starting with k=1 are 0, 1, 4, 103, 3227, 125752, 5146220, 214348597, 8977756589, 376720015978, 15817410193046, 664263682560391, 27898129435203911, 1171708205765809804, 49211559430185180032, 2066882903196070496185, 86809045634587783403393, 3645979408461042349937230, 153131128040700947575427978, 6431507278104282517238903179 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 5 4 3 2 F[[5, 4], [7, 2]](x) = - x (6174 x - 4173 x + 320 x + 391 x - 48 x + 1)/( (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[7, 2]](x) = -x^2*(6174*x^5-4173*x^4+320*x^3+391*x^2-48*x+1)/(-1+x)/(1 +3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 10, 302, 10566, 420009, 17311922, 722644567, 30289103512, 1271278965797, 53381647867614, 2241860385467187, 94155773363431988, 3954509406420182065, 166088932048660451386, 6975728663916859104287, 292980513135709708716144, 12305180281222592458118013, 516817554024703265090622038, 21706337020024712353253905867 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [7, 1, 1]](x) = x 7 6 5 4 3 2 (2940 x - 22960 x + 20731 x - 5537 x - 1001 x + 547 x - 51 x + 1)/( (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[7, 1, 1]](x) = x^2*(2940*x^7-22960*x^6+20731*x^5-5537*x^4-1001*x^3+ 547*x^2-51*x+1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14* x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 9, 305, 10845, 434268, 17935848, 749175995, 31407698975, 1318318531730, 55358119764222, 2324883495185505, 97642901697433365, 4100971002687724712, 172240349967368265956, 7234088648689188989735, 303831638546334866339115, 12760927633168940370816414, 535958943992235890970254250, 22510275415261995550264411085 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[5, 4], [6, 3]](x) = 2 5 4 3 2 x (672 x - 2356 x + 344 x + 317 x - 44 x + 1) - ------------------------------------------------------------------------ (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x) and in Maple notation F[[5, 4],[6, 3]](x) = -x^2*(672*x^5-2356*x^4+344*x^3+317*x^2-44*x+1)/(1+x)/(-1+ x)/(1+3*x)/(-1+2*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 17, 499, 18419, 741393, 30706105, 1283716239, 53833589183, 2259859887805, 94898027439413, 3985492067787099, 167387516532178267, 7030231595168226537, 295269109639843326641, 12401293962060389967079, 520854225408282385945271, 21875875773179075037490389, 918786758757998782964234989, 38589043535818879468399408179 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [6, 2, 1]](x) = x 6 5 4 3 2 (20286 x + 7203 x - 5052 x - 961 x + 67 x + 28 x - 1)/((1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[6, 2, 1]](x) = x^2*(20286*x^6+7203*x^5-5052*x^4-961*x^3+67*x^2+28*x-1 )/(1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 31, 1039, 39710, 1614656, 67074266, 2806826674, 117742907320, 4943191636036, 207585915007526, 8718214656622934, 366159503254852580, 15378621967573425016, 645901042289274199186, 27127828651385917793194, 1139368591612214215640240, 47853477883273343998815596, 2009846029595350547238989246, 84413532661975057296611931454 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [6, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (15288 x - 12040 x - 16994 x + 8046 x + 801 x - 476 x + 46 x - 1)/( (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[6, 1, 1, 1]](x) = -x^2*(15288*x^7-12040*x^6-16994*x^5+8046*x^4+801*x^ 3-476*x^2+46*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+ 14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 14, 534, 20875, 857401, 35722162, 1496279940, 62786581305, 2636234545191, 110710610674840, 4649688221314786, 195284700851164015, 8201926571048412021, 344480483862078329598, 14468174272407849331872, 607663234743365761529605, 25521854673449312565952291, 1071917879684042004366695236, 45020550714318035067182359998 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[5, 4], [5, 4]](x) = - x 7 6 5 4 3 2 (111594 x - 88901 x - 24655 x + 18173 x + 461 x - 740 x + 59 x - 1)/( (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[5, 4]](x) = -x*(111594*x^7-88901*x^6-24655*x^5+18173*x^4+461*x^3-740* x^2+59*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14*x)/( -1+42*x) The first 20 term , starting with k=1 are 1, 1, 18, 421, 15954, 646020, 26831267, 1122736917, 47097209772, 1977276893614, 83034367580781, 3487285871511663, 146463801357320630, 6151448787352548408, 258360416917683288135, 10851131460566068187209, 455747436644956438463328, 19141391153309759908146402, 803938411838142761417758529, 33765413064790038138904268355 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [5, 3, 1]](x) = x 6 5 4 3 2 (6804 x + 7362 x - 9960 x - 1978 x + 426 x + 12 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[5, 3, 1]](x) = x^2*(6804*x^6+7362*x^5-9960*x^4-1978*x^3+426*x^2+12*x-\ 1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 45, 1528, 60330, 2477569, 103297821, 4327892668, 181623581100, 7626121862767, 320268179838387, 13450858512461338, 564930387652620210, 23726996906953711645, 996532758873901178793, 41854360315787868664888, 1757882915467124471497560, 73831079400480736348105003, 3100905292132282694516851839, 130238021671925340009514132918 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[5, 4], [5, 2, 2]](x) = 2 3 2 x (316 x - 85 x + 28 x - 1) - --------------------------------------------------------------- (1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 2 x) (-1 + 14 x) (-1 + 42 x) and in Maple notation F[[5, 4],[5, 2, 2]](x) = -x^2*(316*x^3-85*x^2+28*x-1)/(1+x)/(1+3*x)/(-1+3*x)/(-\ 1+2*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 29, 1105, 44275, 1830091, 76445609, 3204863285, 134522270975, 5648787582031, 237233008755589, 9963561392228665, 418466428811932475, 17575545914839027571, 738172311089527799969, 31003228423088219621245, 1302135472772297578686775, 54689688162472791367114711, 2296966879108365336924378749, 96472608590534457544777825025 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [5, 2, 1, 1]](x) = - x 6 5 4 3 2 (6174 x - 2625 x - 682 x - 91 x - 97 x - 15 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[5, 2, 1, 1]](x) = -x^2*(6174*x^6-2625*x^5-682*x^4-91*x^3-97*x^2-15*x+ 1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 42, 1686, 69100, 2874026, 120288959, 5046103315, 211850940420, 8896538444436, 373637766341521, 15692550111697229, 659083798442907110, 27681473239967024206, 1162621227910518136803, 48830082497630236348503, 2050863337854355200101320, 86136258411228009704864936, 3617722828370351799025506005, 151944358442937296007183678337 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[5, 4], [5, 1, 1, 1, 1]](x) = 3 2 x (147 x - 62 x + 13) ------------------------------------------------------ (-1 + 2 x) (1 + 3 x) (-1 + 3 x) (-1 + 6 x) (-1 + 42 x) and in Maple notation F[[5, 4],[5, 1, 1, 1, 1]](x) = x^3*(147*x^2-62*x+13)/(-1+2*x)/(1+3*x)/(-1+3*x)/ (-1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 13, 588, 25140, 1058370, 44468508, 1867778178, 78447307980, 3294790675890, 138381231011628, 5812011838186818, 244104498019675020, 10252388921720937810, 430600334741661728748, 18085214059326083340258, 759578990492753378365260, 31902317600701989130598130, 1339897339229521628124989868, 56275688247640136888828538498 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [4, 4, 1]](x) = - x 7 6 5 4 3 2 (12348 x - 45024 x + 10227 x + 8315 x - 1711 x - 181 x + 37 x - 1)/( (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[4, 4, 1]](x) = -x^2*(12348*x^7-45024*x^6+10227*x^5+8315*x^4-1711*x^3-\ 181*x^2+37*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14* x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 23, 779, 31049, 1281238, 53513404, 2243410765, 94165635923, 3954151548860, 166063107701810, 6974492983443811, 292926500223690697, 12302882140710693442, 516720617764642702616, 21702259896173456584217, 911494830940679054031071, 38282781713731376281686184, 1607876815375858278340057822, 67530826013374135501749028783 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [4, 3, 2]](x) = x 6 5 4 3 2 (1176 x + 140 x - 3074 x - 1039 x + 115 x + 18 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[4, 3, 2]](x) = x^2*(1176*x^6+140*x^5-3074*x^4-1039*x^3+115*x^2+18*x-1 )/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 39, 1497, 61290, 2551838, 106878863, 4484765447, 188302528740, 7907901116520, 332120589418497, 13948907216515217, 585851898010146950, 24605748847756197122, 1033441019460508419891, 43404516767400035731707, 1822989619532194608968520, 76565562834575033000518844, 3215753622451291847844618245, 135061651910542345474691997317 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [4, 3, 1, 1]](x) = - x 6 5 4 3 2 (8568 x - 408 x - 422 x + 354 x - 86 x - 12 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[4, 3, 1, 1]](x) = -x^2*(8568*x^6-408*x^5-422*x^4+354*x^3-86*x^2-12*x+ 1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 45, 1868, 78086, 3271539, 137287489, 5764354282, 242078558172, 10166956511417, 427007362019183, 17934241764932736, 753237209561388058, 31635949574932727935, 1328709696958916558877, 55805804679543024397430, 2343843760242009524671544, 98441437421977820342521893, 4134540364608436144399542571, 173650695213949343381326303564 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [4, 2, 2, 1]](x) = x 6 5 4 3 2 (3528 x - 3156 x + 1970 x + 409 x - 101 x + 16 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[4, 2, 2, 1]](x) = x^2*(3528*x^6-3156*x^5+1970*x^4+409*x^3-101*x^2+16* x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 41, 1827, 77430, 3262646, 137161729, 5762598493, 242053957340, 10166612178156, 427002541038447, 17934174272459699, 753236264661730330, 31635936346357648306, 1328709511758784905245, 55805802086741503352745, 2343843723942786941523000, 98441436913788709332337496, 4134540357493788569640986923, 173650695114344277417169634431 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [4, 2, 1, 1, 1]](x) = x 5 4 3 2 (9702 x - 1155 x - 1883 x + 1120 x + 242 x - 31)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[4, 2, 1, 1, 1]](x) = x^3*(9702*x^5-1155*x^4-1883*x^3+1120*x^2+242*x-\ 31)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 31, 1525, 66864, 2842637, 119849807, 5039953926, 211764853828, 8895333212359, 373620893170833, 15692313886992512, 659080491298298342, 27681426939937466241, 1162620579710124453859, 48830073422824644248458, 2050863210807077232806856, 86136256632566116874219483, 3617722803469085304550364885, 151944358094319565063915728564 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [4, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (15288 x + 8372 x - 9542 x + 1431 x + 450 x - 2 x - 7)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[4, 1, 1, 1, 1, 1]](x) = -x^3*(15288*x^6+8372*x^5-9542*x^4+1431*x^3+ 450*x^2-2*x-7)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14*x )/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 7, 422, 19396, 836413, 35429611, 1492179416, 62729193842, 2635431042491, 110699361952465, 4649530737944950, 195282496089022888, 8201895704358310409, 344480051728497451319, 14468168222537394940724, 607663150045180688598334, 25521853487674716391062967, 1071917863083197678534306173, 45020550481906214423065977938 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[5, 4], [3, 3, 3]](x) = 2 3 2 x (231 x - 229 x + 40 x - 1) - ---------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 42 x) and in Maple notation F[[5, 4],[3, 3, 3]](x) = -x^2*(231*x^3-229*x^2+40*x-1)/(-1+x)/(1+3*x)/(-1+2*x)/ (-1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 8, 368, 15131, 635444, 26683265, 1120681178, 47068467167, 1976874911792, 83028741615677, 3487207121072918, 146462698920511883, 6151433353685784620, 258360200848914052169, 10851128435619160932338, 455747394295793071762679, 19141390560422039826417128, 803938403537718054776011541, 33765412948584112601419799438 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [3, 3, 2, 1]](x) = x 6 5 4 3 2 (1176 x + 3920 x + 3514 x - 400 x - 240 x + 26 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[3, 3, 2, 1]](x) = x^2*(1176*x^6+3920*x^5+3514*x^4-400*x^3-240*x^2+26* x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 31, 1396, 59782, 2530997, 106585787, 4480667150, 188245132624, 7907097648943, 332109340556653, 13948749733705304, 585849693245770226, 24605717981075046089, 1033440587326891755679, 43404510717529724517058, 1822989534834008963396788, 76565561648800439116322435, 3215753605850447512849719665, 135061651678130524867226177612 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[5, 4], [3, 3, 1, 1, 1]](x) = 3 3 2 2 x (756 x - 365 x - 37 x + 9) --------------------------------------------------------------- (1 + x) (1 + 3 x) (-1 + 3 x) (-1 + 2 x) (-1 + 14 x) (-1 + 42 x) and in Maple notation F[[5, 4],[3, 3, 1, 1, 1]](x) = 2*x^3*(756*x^3-365*x^2-37*x+9)/(1+x)/(1+3*x)/(-1 +3*x)/(-1+2*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 18, 952, 42140, 1800210, 76027298, 3199006972, 134440282680, 5647639746070, 237216939052478, 9963336416385792, 418463279150133620, 17575501819573846330, 738171693755815268058, 31003219780416244185412, 1302135351774889922606960, 54689686468509084182040990, 2296966855392873436333434038, 96472608258517570936504773832 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[5, 4], [3, 2, 2, 2]](x) = - x 7 6 5 4 3 2 (44100 x - 32928 x - 17043 x + 10931 x + 846 x - 621 x + 46 x - 1)/( (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[3, 2, 2, 2]](x) = -x^2*(44100*x^7-32928*x^6-17043*x^5+10931*x^4+846*x ^3-621*x^2+46*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+ 14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 14, 679, 29530, 1260426, 53220181, 2239312993, 94108237580, 3953348089936, 166051858804843, 6974335500773367, 292924295458754050, 12302851274031778006, 516720185631017087825, 21702253846303181155501, 911494746242493265282840, 38282780527956782970130236, 1607876798775013941054466327, 67530825780962314903445718595 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [3, 2, 2, 1, 1]](x) = 3 x 5 4 3 2 (1764 x - 2910 x - 1778 x + 231 x + 36 x - 8)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[3, 2, 2, 1, 1]](x) = 3*x^3*(1764*x^5-2910*x^4-1778*x^3+231*x^2+36*x-8 )/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 24, 1260, 56463, 2423889, 102544458, 4317352956, 181475995641, 7624055784303, 320239254268032, 13450453556363802, 564924718259704959, 23726917535483097117, 996531647673191779446, 41854344758978420257848, 1757882697671790261063717, 73831076351346065132972331, 3100905249444397266581229900, 130238021074294944142110483894 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [3, 2, 1, 1, 1, 1]](x) = - x 5 4 3 2 (59094 x - 8715 x - 3777 x + 751 x - 57 x + 14)/((1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[3, 2, 1, 1, 1, 1]](x) = -x^3*(59094*x^5-8715*x^4-3777*x^3+751*x^2-57* x+14)/(1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 14, 769, 36000, 1562290, 66342584, 2796576804, 117599433350, 4941182901430, 207557793114804, 8717820948549064, 366153991348104050, 15378544800853768170, 645899961955299643424, 27127813526709871312924, 1139368379866751175436350, 47853474918836854993324510, 2009845988093239726931546444, 84413532080945505709227774384 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [3, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (34692 x - 19936 x + 1993 x - 977 x + 203 x + 12 x + 3)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[3, 1, 1, 1, 1, 1, 1]](x) = x^3*(34692*x^6-19936*x^5+1993*x^4-977*x^3+ 203*x^2+12*x+3)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14* x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 3, 192, 9377, 413251, 17643444, 745074946, 31350313739, 1317515020377, 55346871076970, 2324726011676200, 97640696935852161, 4100940135995387503, 172239917833796338256, 7234082598818698812654, 303831553848149936584343, 12760926447394343623286629, 535958927391391567428558102, 22510275182850174896985519508 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [2, 2, 2, 2, 1]](x) = x 5 4 3 2 (26754 x - 23635 x + 1300 x + 1009 x - 105 x + 7)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[2, 2, 2, 2, 1]](x) = x^3*(26754*x^5-23635*x^4+1300*x^3+1009*x^2-105*x +7)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 7, 315, 14435, 625158, 26538148, 1118638535, 47039813485, 1976473425696, 83023118718254, 3487128388700385, 146461596592941175, 6151417920671391914, 258359984784066613000, 10851125410695756950715, 455747351946770792847905, 19141389967535166023862612, 803938395237298426422685186, 33765412832378217531438099125 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[5, 4], [2, 2, 2, 1, 1, 1]](x) = 3 4 3 2 2 x (1176 x - 496 x - 154 x + 10 x - 3) ------------------------------------------------------------------------ (1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x) and in Maple notation F[[5, 4],[2, 2, 2, 1, 1, 1]](x) = 2*x^3*(1176*x^4-496*x^3-154*x^2+10*x-3)/(1+x) /(-1+x)/(1+3*x)/(-1+2*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 6, 346, 16284, 711512, 30287794, 1277859926, 53751600888, 2258712051844, 94881957736302, 3985267091944226, 167384366870379412, 7030187499903045296, 295268492306130794730, 12401285319388414531246, 520854104410874729865456, 21875874079215367852416668, 918786735042506882373290278, 38589043203801992860126356986 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [2, 2, 1, 1, 1, 1, 1]](x) = - x 4 3 2 (2394 x - 177 x + 437 x + 8 x + 3)/((-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[2, 2, 1, 1, 1, 1, 1]](x) = -x^3*(2394*x^4-177*x^3+437*x^2+8*x+3)/(-1+ x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+6*x)/(-1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 3, 182, 8986, 397513, 16998530, 718250967, 30227617752, 1270418066981, 53369595677662, 2241691653235507, 94153411118480948, 3954476334965703729, 166088469048398422074, 6975722181912788048927, 292980422387654324570224, 12305179010749810637656957, 516817536238084345374036566, 21706336771012047374142625227 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[5, 4], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (13944 x - 18920 x - 678 x + 10 x + 322 x - 9 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(13944*x^6-18920*x^5-678*x^4+10*x^3 +322*x^2-9*x+1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+14* x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 1, 51, 2600, 116712, 5020985, 212590581, 8953164410, 376375647594, 15812589351779, 664196189527431, 27897184537781780, 1171694977181779596, 49211374230089312333, 2066880310394406275001, 86809009335365772895310, 3645978900271929049059918, 153131120926053381979381847, 6431507178499216516431671691 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 4 F[[5, 4], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (26754 x - 23635 x + 1300 x + 1009 x - 105 x + 7)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (-1 + 6 x) (-1 + 14 x) (-1 + 42 x)) and in Maple notation F[[5, 4],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^4*(26754*x^5-23635*x^4+1300*x^3+ 1009*x^2-105*x+7)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(-1+6*x)/(-1+ 14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 0, 7, 315, 14435, 625158, 26538148, 1118638535, 47039813485, 1976473425696, 83023118718254, 3487128388700385, 146461596592941175, 6151417920671391914, 258359984784066613000, 10851125410695756950715, 455747351946770792847905, 19141389967535166023862612, 803938395237298426422685186 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 Regarding Lambda=, [5, 3, 1] 8 7 6 5 F[[5, 3, 1], [9]](x) = (825714 x + 492237 x - 1226880 x - 1021473 x 4 3 2 - 140358 x + 34935 x + 4812 x - 193 x + 1)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[9]](x) = (825714*x^8+492237*x^7-1226880*x^6-1021473*x^5-140358*x^4 +34935*x^3+4812*x^2-193*x+1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+ 36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 1, 15, 2074, 313443, 50027434, 8077188504, 1307525164849, 211783811224371, 34307707925220592, 5557802981847700518, 900362437788256468279, 145858655691928750125009, 23629100089820863440733750, 3827914137789202122135973332, 620122087558426747570746386989, 100459778084981869229450789364207, 16274484046185665315705726820089308, 2636466415353147471163099568033093346, 427107559282568399169101769877024734979 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [8, 1]](x) = 2 5 4 3 2 x (8262 x + 44037 x - 9918 x - 3204 x + 270 x - 2) - ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[8, 1]](x) = -x^2*(8262*x^5+44037*x^4-9918*x^3-3204*x^2+270*x-2)/(-\ 1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 2, 120, 16202, 2495766, 399785951, 64601966739, 10459641518372, 1694250339262932, 274460937971484305, 44462397739373253453, 7202898562150951717862, 1166869211689849339576398, 189032799500125989345107939, 30623313058449744028380057687, 4960976698888314553895265430472, 803678224623007374479269480529064, 130195872367438809668799068985136853, 21091731322751505306458450810119994241, 3416860474257894912690345042289374141002 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [7, 2]](x) = 2 4 3 2 x (59049 x - 14499 x - 5697 x + 597 x - 5) - ---------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[7, 2]](x) = -x^2*(59049*x^4-14499*x^3-5697*x^2+597*x-5)/(-1+x)/(1+ 3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 5, 383, 53780, 8391758, 1348143017, 217990824098, 35299820767991, 5718041999234456, 926303761403560229, 150060523817415124958, 24309780179352478689587, 3938183500608591468462644, 637985695114517809366254641, 103353681457125219576037703738, 16743296354602925624585748954143, 2712414007953424993056092384757872, 439411069234733886382981344535696253, 71184593214092934944325575846992138838, 11531904100613433093590933345296761574859 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[5, 3, 1], [7, 1, 1]](x) = x 6 5 4 3 2 (69012 x + 183276 x + 88497 x + 15714 x - 1119 x - 379 x + 4)/( (-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[7, 1, 1]](x) = x^2*(69012*x^6+183276*x^5+88497*x^4+15714*x^3-1119* x^2-379*x+4)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 4, 393, 55486, 8693337, 1397737375, 226052468838, 36606786146788, 5929805659841799, 960610743899007121, 155618300683851443328, 25210141676985653293030, 4084042122454939448255361, 661614793985897755279466467, 107181595551050548745547129738, 17363418440582252945509300743112, 2812873785981559282929065278812123, 455685553278873038841841172129421013, 73821059629372407952642324602834022068, 11959011659893349212097566028915095247434 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [6, 3]](x) = 2 3 2 3 x (6966 x + 462 x - 185 x + 2) ------------------------------------------------------ (-1 + x) (1 + 3 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[6, 3]](x) = 3*x^2*(6966*x^3+462*x^2-185*x+2)/(-1+x)/(1+3*x)/(-1+6* x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 6, 657, 94446, 14878917, 2395257264, 387487422567, 62753371044930, 10165340829636369, 1646759824398769572, 266774177512912211307, 43217383851665345541654, 7001214999374448568988181, 1134196787253228606298246680, 183739877999787480502251827727, 29765860180697091910570176677418, 4822069347283263612022101076142553, 781175234188260755158040761363152588, 126550387935919636135979837427409469427, 20501162845526151230842318051153928273022 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [6, 2, 1]](x) = 2 3 2 x (35478 x + 2907 x - 1158 x + 13) - ------------------------------------------------------- (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[6, 2, 1]](x) = -x^2*(35478*x^3+2907*x^2-1158*x+13)/(1+2*x)/(1+6*x) /(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 13, 1390, 205147, 32494930, 5237735728, 847560705040, 137270550280912, 22236594904881760, 3602283942123664768, 583568399054534755840, 94538023062339698321152, 15315157663057189081930240, 2481055466785758570665771008, 401930982932630669384405463040, 65112819138366328563099345620992, 10548276696933430991612080273285120, 1708820824777866908159524464501096448, 276828973609501878272503049267854704640, 44846293724576852089569268884409901842432 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [6, 1, 1, 1]](x) = 2 5 4 3 2 x (38394 x - 1215 x + 2889 x - 177 x - 452 x + 6) ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[6, 1, 1, 1]](x) = x^2*(38394*x^5-1215*x^4+2889*x^3-177*x^2-452*x+6 )/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 6, 718, 108627, 17302570, 2792451030, 451996093969, 73209654071133, 11859470264021020, 1921216409799674604, 311236418559774844015, 50420276772886096311219, 8168084007990811762817050, 1323229579442709100926790578, 214363190795053986760017661981, 34726836870110809904958422556585, 5625747571565185510360528407778360, 911371106543420487685777976317854552, 147642119258229094665360112272412141867, 23918023319768132459557849063378948834431 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[5, 3, 1], [5, 4]](x) = - x 6 5 4 3 2 (84564 x - 53730 x - 124398 x - 26952 x + 1780 x + 406 x - 5)/( (-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[5, 4]](x) = -x^2*(84564*x^6-53730*x^5-124398*x^4-26952*x^3+1780*x^ 2+406*x-5)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 5, 559, 82052, 12998074, 2095094081, 339024285565, 54908220105212, 8894637962079148, 1440913576849212527, 233427359621818441651, 37815209224935870245402, 6126063065222875796070322, 992422186714303427939672093, 160772393173052267759639855017, 26045127655346531425227982550072, 4219310678773372396645043695780696, 683528329911146763263809362624276779, 110731589443800751309001227324167668863, 17938517489830740835827707538529880106422 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [5, 3, 1]](x) = - x 6 5 4 3 2 (433998 x + 514755 x + 106704 x - 20439 x - 3990 x + 178 x - 1)/( (-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[5, 3, 1]](x) = -x*(433998*x^6+514755*x^5+106704*x^4-20439*x^3-3990 *x^2+178*x-1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x ) The first 20 term , starting with k=1 are 1, 15, 2074, 313443, 50027434, 8077188504, 1307525164849, 211783811224371, 34307707925220592, 5557802981847700518, 900362437788256468279, 145858655691928750125009, 23629100089820863440733750, 3827914137789202122135973332, 620122087558426747570746386989, 100459778084981869229450789364207, 16274484046185665315705726820089308, 2636466415353147471163099568033093346, 427107559282568399169101769877024734979, 69191424603608986983658936570874698355565 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 6 x (135 x - 2) F[[5, 3, 1], [5, 2, 2]](x) = - ---------------------------------- (1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[5, 2, 2]](x) = -6*x^2*(135*x-2)/(1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 12, 1494, 231120, 37017000, 5981664672, 968485317984, 156875031452160, 25413049811894400, 4116888679572997632, 666935052051003969024, 108043445526837952204800, 17503036990752406058035200, 2835491949856457782219579392, 459349694341510606829093412864, 74414650428056238377775440855040, 12055173367355445339703147312742400, 1952938085439954195042451500012797952, 316375969838693973397254153655681941504, 51252907113775593867168760205064017018880 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[5, 3, 1], [5, 2, 1, 1]](x) = - x 6 5 4 3 2 (144342 x + 46899 x - 70704 x - 9618 x + 6654 x + 778 x - 16)/( (-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[5, 2, 1, 1]](x) = -x^2*(144342*x^6+46899*x^5-70704*x^4-9618*x^3+ 6654*x^2+778*x-16)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+ 162*x) The first 20 term , starting with k=1 are 0, 16, 2310, 362200, 58239000, 9418852042, 1525282754079, 247075235796979, 40025447662385220, 6484095861826072468, 1050422569874399575953, 170168421768955771794253, 27567283082745739708084710, 4465899814627106196343407694, 723475768357593872726125838307, 117203074415898303455327762575447, 18986898053286376618409925415832520, 3075877484557183664693428393325182120, 498292152495556217170731913428220221141, 80723328704182635867312834917180543679361 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 2 x (666 x + 15 x + 5) F[[5, 3, 1], [5, 1, 1, 1, 1]](x) = ------------------------------------------- (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[5, 1, 1, 1, 1]](x) = x^2*(666*x^2+15*x+5)/(1+2*x)/(1+6*x)/(-1+6*x) /(-1+162*x) The first 20 term , starting with k=1 are 0, 5, 815, 132866, 21523160, 3486784640, 564859065680, 91507169828576, 14824161510552320, 2401514164752300800, 389045294689812273920, 63025337739751130241536, 10210104713839680922388480, 1654036963642028364934799360, 267953988110008595041073500160, 43408546073821392398652192874496, 7032184463959065568578834136432640, 1139213883161368622109843068403384320, 184552649072141716781794475521394278400, 29897529149686958118650707624244762771456 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[5, 3, 1], [4, 4, 1]](x) = x 6 5 4 3 2 (84564 x + 133596 x + 116433 x + 23292 x - 2394 x - 494 x + 8)/( (-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[4, 4, 1]](x) = x^2*(84564*x^6+133596*x^5+116433*x^4+23292*x^3-2394 *x^2-494*x+8)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x ) The first 20 term , starting with k=1 are 0, 8, 1050, 161768, 25912062, 4187164685, 677939728431, 109812521995478, 17789134868536164, 2881822075700342087, 466854536435710337937, 75630411868786539329408, 12252125893526684512735206, 1984844364899520446574114689, 321544786039057424790161027043, 52090255299639366864407544370058, 8438621357148811737792555758726088, 1367056659807967936529714780506288091, 221463178887085781378077920253981595349, 35877034979642915707018132097842802421428 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [4, 3, 2]](x) = 2 5 4 3 2 x (1458 x + 83079 x + 37296 x - 2817 x - 695 x + 14) ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[4, 3, 2]](x) = x^2*(1458*x^5+83079*x^4+37296*x^3-2817*x^2-695*x+14 )/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 14, 2035, 321194, 51740137, 8371305695, 1355770617670, 219621125768996, 35578128681576199, 5763639073402349201, 933708890063498665600, 151260817156487868427238, 24504251550134302109324161, 3969688721269954483516481507, 643089571771067738857080685450, 104180510588225037742754913381320, 16877242713899690420087791467868123, 2734113319601610283061525193103808213, 442926357773655841516231313730606231220, 71754069959267265449398981201234653485642 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [4, 3, 1, 1]](x) = 2 5 4 3 2 x (13122 x + 79461 x + 19863 x - 4836 x - 561 x + 16) ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[4, 3, 1, 1]](x) = x^2*(13122*x^5+79461*x^4+19863*x^3-4836*x^2-561* x+16)/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 16, 2559, 410953, 66450963, 10760515564, 1743040357338, 282366660369523, 45743187400053861, 7410388741804444162, 1200482701960560823992, 194478187845982793462473, 31505466075670616628472419, 5103885491465010270550837960, 826829449156760997905014547046, 133946370746814737681204735770303, 21699312060387087921114970344580737, 3515288553761219858223757218617205358, 569476745708544035172362087447402838900, 92255232804756284750966733263486388741013 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [4, 2, 2, 1]](x) = 2 5 4 3 2 x (39366 x - 891 x + 1890 x - 510 x - 425 x + 15) ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[4, 2, 2, 1]](x) = x^2*(39366*x^5-891*x^4+1890*x^3-510*x^2-425*x+15 )/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 15, 2500, 408975, 66378850, 10757924934, 1742947062001, 282363301933341, 45743066495175580, 7410384389235880428, 1200482545268050203247, 194478182205052665073587, 31505465872597130482724890, 5103885484154364778446412722, 826829448893577760134400323613, 133946370737340141121791753215913, 21699312060046002444974128195687480, 3515288553748940781082698749918659416, 569476745708101988395283911482285517099, 92255232804740371066991919355293983163519 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [4, 2, 1, 1, 1]](x) = 2 5 4 3 2 x (98415 x + 43659 x - 18306 x - 5610 x + 167 x + 10) ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[4, 2, 1, 1, 1]](x) = x^2*(98415*x^5+43659*x^4-18306*x^3-5610*x^2+ 167*x+10)/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 10, 2117, 355195, 57987092, 9409781917, 1524956237903, 247063481165350, 40025024495941124, 6484080627832320199, 1050422021450635078289, 170168402025700186384000, 27567282371988539014262246, 4465899789039846969080158081, 723475767436452540558362619875, 117203074382737215497206004261770, 18986898052092577451918035810929608, 3075877484514206894699717405382963163, 498292152494009053450958335635293378261, 80723328704126937973400986009997221588660 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [4, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (75330 x + 36963 x + 5148 x + 1098 x - 202 x - 2) - ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(75330*x^5+36963*x^4+5148*x^3+1098*x^ 2-202*x-2)/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 2, 592, 103940, 17134738, 2786403635, 451778420401, 73201817626730, 11859188153198236, 1921206253802999993, 311236052943936884335, 50420263610715675804800, 8168083534152678148333594, 1323229562384536281662900351, 214363190180959765321372528669, 34726836848003417932838068268150, 5625747570769319399366170430329912, 911371106514769307689969240468099109, 147642119257197652185511068873790613803, 23918023319731000530283283074476755816780 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [3, 3, 3]](x) = 2 4 3 2 3 x (1350 x - 45 x - 186 x + 7 x + 1) - -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[3, 3, 3]](x) = -3*x^2*(1350*x^4-45*x^3-186*x^2+7*x+1)/(-1+x)/(1+3* x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 3, 495, 79701, 12914148, 2092070025, 338915448690, 54904301869287, 8894496906666936, 1440908498850373797, 233427176813899454430, 37815202643850641874483, 6126062828303808988762164, 992422178185217017654891569, 160772392866005157040316690490, 26045127644292835439144297950239, 4219310678375439341147864701675632, 683528329896821173265904148382569341, 110731589443285030069076705624808477270, 17938517489812174871190424513610941879755 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [3, 3, 2, 1]](x) = 2 5 4 3 2 x (7290 x + 43713 x - 8919 x - 2871 x + 243 x - 11) - ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[3, 3, 2, 1]](x) = -x^2*(7290*x^5+43713*x^4-8919*x^3-2871*x^2+243*x -11)/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 11, 1902, 316550, 51572046, 8365259855, 1355552934771, 219613289380580, 35577846570417492, 5763628917407690129, 933708524447648612685, 151260803994317520480230, 24504251076296168059484238, 3969688704211781666864730083, 643089571156973517402762719319, 104180510566117645770728596089800, 16877242713103824309092869268438184, 2734113319572959103065719842585941717, 442926357772624399036382250019993369473, 71754069959230133520124415334204408470090 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [3, 3, 1, 1, 1]](x) = 2 2 7 x (810 x - 6 x + 1) - ------------------------------------------- (-1 + x) (1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[3, 3, 1, 1, 1]](x) = -7*x^2*(810*x^2-6*x+1)/(-1+x)/(1+6*x)/(-1+36* x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 7, 1309, 224455, 36777055, 5973026647, 968174349079, 156863836571575, 25412646796193335, 4116874171007759287, 666934529742655388599, 108043426723737403309495, 17503036313840786297804215, 2835491925487639470851263927, 459349693464233147619834056119, 74414650396474249846242104012215, 12055173366218493752567947186400695, 1952938085399023937905584295464496567, 316375969837220484140326934291943091639, 51252907113722548253919380307969418423735 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[5, 3, 1], [3, 2, 2, 2]](x) = - x 6 5 4 3 2 (37908 x + 37692 x + 46773 x + 318 x - 4400 x + 49 x - 5)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[3, 2, 2, 2]](x) = -x^2*(37908*x^6+37692*x^5+46773*x^4+318*x^3-4400 *x^2+49*x-5)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 5, 916, 157133, 25743910, 4181119226, 677722043215, 109804685621027, 17788852757293540, 2881811919706186772, 466854170819857261969, 75630398706616209521741, 12252125419688550354057190, 1984844347841347630575395918, 321544785424963203331924856803, 52090255277531974892404736319575, 8438621356352945626797492503817160, 1367056659779316756533910276321361064, 221463178886054338898228851465370965717, 35877034979605783777743566261280544275329 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [3, 2, 2, 1, 1]](x) = 2 4 3 2 x (41067 x - 1755 x - 243 x + 369 x + 7) ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[3, 2, 2, 1, 1]](x) = x^2*(41067*x^4-1755*x^3-243*x^2+369*x+7)/(-1+ x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 7, 1734, 301480, 49595334, 8061641233, 1306965413805, 211763660481340, 34306982496706668, 5557776866431783279, 900361497633219953331, 145858621846347816533530, 23629098871379947545799602, 3827914093925329163632111645, 620122085979327320982324916137, 100459778028134289872761310803000, 16274484044139152458861943426171736, 2636466415279473008316741138845100331, 427107559279916118506632759788301239423, 69191424603513504879810052847508382016950 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [3, 2, 1, 1, 1, 1]](x) = 2 2 3 x (405 x - 159 x - 1) --------------------------------------------- (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[3, 2, 1, 1, 1, 1]](x) = 3*x^2*(405*x^2-159*x-1)/(1+6*x)/(-1+6*x)/( -1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 3, 1071, 193455, 32075190, 5222618208, 847016515296, 137250959204880, 22235889627614880, 3602258552133237888, 583567485014932298496, 94537990156913692404480, 15315156478461854773624320, 2481055424140326524138631168, 401930981397395115777997111296, 65112819083097848632857233018880, 10548276694943765714125832690933760, 1708820824706238958170004740709122048, 276828973606923272072880428076306333696, 44846293724484022266382853988324386734080 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 3 F[[5, 3, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x 5 4 3 2 (53460 x + 92988 x - 16335 x - 12444 x + 934 x - 268)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^3*(53460*x^5+92988*x^4-16335*x^3-\ 12444*x^2+934*x-268)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-\ 1+162*x) The first 20 term , starting with k=1 are 0, 0, 268, 50790, 8525566, 1391689599, 225834797587, 36598949688420, 5929523549102932, 960600587901828753, 155617935068016506701, 25210128514815214647270, 4084041648616805942609998, 661614776927724935362543587, 107181594936956327310820200535, 17363418418474860973365437213640, 2812873785185693171934848356842664, 455685553250221858846031589946726101, 73821059628340965472793286282210261889, 11959011659856217282823000009544915360330 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[5, 3, 1], [2, 2, 2, 2, 1]](x) = - x 6 5 4 3 2 (102060 x + 208602 x + 59148 x - 9951 x - 4812 x - 40 x - 2)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[2, 2, 2, 2, 1]](x) = -x^2*(102060*x^6+208602*x^5+59148*x^4-9951*x^ 3-4812*x^2-40*x-2)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+ 162*x) The first 20 term , starting with k=1 are 0, 2, 426, 77408, 12829983, 2089048241, 338806602666, 54900383716796, 8894355850920441, 1440903420854553455, 233426994005968388736, 37815196062765522298394, 6126062591384741746230399, 992422169656130611287920669, 160772392558958046305321888886, 26045127633239139453201665258552, 4219310677977506285650121496350757, 683528329882495583268004012106410283, 110731589442769308829152163613554807116, 17938517489793608906553141671499635090870 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [2, 2, 2, 1, 1, 1]](x) = 2 3 2 x (6318 x - 954 x + 270 x + 1) ------------------------------------------------------ (-1 + x) (1 + 3 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[2, 2, 2, 1, 1, 1]](x) = x^2*(6318*x^3-954*x^2+270*x+1)/(-1+x)/(1+3 *x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 1, 472, 87781, 14638972, 2386619239, 387176453662, 62742176164345, 10164937813935304, 1646745315833531227, 266773655204563630882, 43217365048564796646349, 7001214322462828808757196, 1134196762884410294929931215, 183739877122510021292992470982, 29765860149115103379036839834593, 4822069346146312024886900949800848, 781175234147330498021173556814851203, 126550387934446146879052618063670619562, 20501162845473105617592938154059329677877 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 3 F[[5, 3, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 3 x 4 3 2 (56862 x + 16119 x + 42 x + 149 x + 83)/((-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x)) and in Maple notation F[[5, 3, 1],[2, 2, 1, 1, 1, 1, 1]](x) = 3*x^3*(56862*x^4+16119*x^3+42*x^2+149*x +83)/(-1+x)/(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 0, 249, 48753, 8211963, 1341663522, 217757603259, 35291424572544, 5717739737668641, 926292879978371694, 150060132086161248039, 24309766077027021668220, 3938182992924876920387709, 637985676837904074207430266, 103353680799167125178888708739, 16743296330916434225876973194856, 2712414007100711302705044928748217, 439411069204036193530328825292023238, 71184593212987818001630174019182617759, 11531904100573648883653898346305845061652 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (86994 x + 33345 x - 12285 x - 921 x - 68) - ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(86994*x^4+33345*x^3-12285*x^2-\ 921*x-68)/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 0, 68, 14181, 2423912, 397193766, 64508680733, 10456283026203, 1694129434720574, 274456585400905032, 44462241046874725943, 7202892921220750769565, 1166869008616363629185336, 189032792189480494628543898, 30623312795266506273438667073, 4960976689413717994388245879167, 803678224281921898338991553617298, 130195872355159732527737214954701964, 21091731322309458529380295156994006123, 3416860474241981228715531012225020561409 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[5, 3, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (41067 x - 1755 x - 243 x + 369 x + 7) ------------------------------------------------------------------------ (-1 + x) (1 + x) (1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 36 x) (-1 + 162 x) and in Maple notation F[[5, 3, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(41067*x^4-1755*x^3-243*x^2+ 369*x+7)/(-1+x)/(1+x)/(1+3*x)/(1+6*x)/(-1+6*x)/(-1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 0, 7, 1734, 301480, 49595334, 8061641233, 1306965413805, 211763660481340, 34306982496706668, 5557776866431783279, 900361497633219953331, 145858621846347816533530, 23629098871379947545799602, 3827914093925329163632111645, 620122085979327320982324916137, 100459778028134289872761310803000, 16274484044139152458861943426171736, 2636466415279473008316741138845100331, 427107559279916118506632759788301239423 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 Regarding Lambda=, [5, 2, 2] 9 8 7 6 F[[5, 2, 2], [9]](x) = (1734080 x - 241584 x - 2976188 x + 533387 x 5 4 3 2 + 462879 x - 88289 x - 15493 x + 3495 x - 148 x + 1)/((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[9]](x) = (1734080*x^9-241584*x^8-2976188*x^7+533387*x^6+462879*x^5 -88289*x^4-15493*x^3+3495*x^2-148*x+1)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/( 1+4*x)/(8*x-1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 7, 604, 68977, 8235686, 987561067, 118494013184, 14219022572597, 1706277590116906, 204753208911272527, 24570383035099891364, 2948445923556987439417, 353813510013979711948526, 42457621185422313851875187, 5094914541925588034670283144, 611389745024568894753010157437, 73366769402818234967548615778546, 8804012328335587555930856475353047, 1056481479400218493971256387148100524 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [8, 1]](x) = 2 5 4 3 2 x (2920 x + 11744 x - 911 x - 927 x + 95 x - 1) - ------------------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[8, 1]](x) = -x^2*(2920*x^5+11744*x^4-911*x^3-927*x^2+95*x-1)/(1+x) /(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 41, 4673, 550449, 65866761, 7900190081, 947946668113, 113752076184529, 13650218666689241, 1638025630476098721, 196563063466667022753, 23587567372190474753009, 2830508079786666672318121, 339660969476876190453466561, 40759316335274666666756676593, 4891117960193950476190115087889, 586934155222493866666668103105401, 70432098626683660190476184720869601, 8451851835201727146666666689616205633 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [7, 2]](x) = 2 5 4 3 2 x (18880 x - 46224 x - 11036 x + 4761 x - 299 x + 3) - ------------------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[7, 2]](x) = -x^2*(18880*x^5-46224*x^4-11036*x^3+4761*x^2-299*x+3)/ (1+x)/(-1+x)/(-1+4*x)/(1+4*x)/(8*x-1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 145, 15712, 1856457, 222268928, 26662492745, 3199306801152, 383912991527497, 46069482675109888, 5528336396257628745, 663400337067204411392, 79608039838480486142537, 9552964768426701040386048, 1146355771967390751079567945, 137562692631210668865913618432, 16507523115647756208068555477577, 1980902773875780266807407734161408, 237708332865054622477316378960630345, 28524999943805774506675673923187638272 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[5, 2, 2], [7, 1, 1]](x) = x 7 6 5 4 3 2 (78080 x - 521664 x + 75680 x + 59076 x - 7099 x - 603 x - 11 x + 1)/ ((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[7, 1, 1]](x) = x^2*(78080*x^7-521664*x^6+75680*x^5+59076*x^4-7099* x^3-603*x^2-11*x+1)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x-1)/(-1+ 20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 137, 16179, 1924067, 230486141, 27649755357, 3317795380439, 398131909703847, 47775758211031041, 5733089564354819777, 687970719288172952699, 82556485745772048819627, 9906778278115509740780741, 1188813393146310744569908197, 142657607173006219290162052959, 17118912860669724420987599463407, 2054269543278546488701238433175241, 246512345193389169776276568354028617, 29581481423205972195543545948967545219 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [6, 3]](x) = 2 4 3 2 x (1520 x + 1304 x + 576 x - 173 x + 3) -------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[6, 3]](x) = x^2*(1520*x^4+1304*x^3+576*x^2-173*x+3)/(1+x)/(-1+x)/( -1+3*x)/(1+4*x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 244, 27751, 3297768, 395098483, 47399110924, 5687639366111, 682511644441168, 81901295746047403, 9828153457777722804, 1179378374298412941271, 141525404103111110208568, 16983048476119365083181923, 2037965816809244444429797084, 244555898010607746031806481231, 29346707761142897777777541559968, 3521604931334547098412699374310043, 422592591760093639111111107314885764, 50711111011210196439365079380414929991 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 7 6 5 4 F[[5, 2, 2], [6, 2, 1]](x) = x (235200 x + 333360 x - 249356 x - 22371 x 3 2 + 20177 x - 264 x - 211 x + 5)/((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[6, 2, 1]](x) = x^2*(235200*x^7+333360*x^6-249356*x^5-22371*x^4+ 20177*x^3-264*x^2-211*x+5)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x-\ 1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 5, 529, 60558, 7211195, 864222800, 103684449869, 12441688931138, 1492993778126575, 179159075558330820, 21499085511133467809, 2579890190222400863318, 309586821404445875881955, 37150418540088900337253240, 4458050224241777869400151349, 534966026897635556288486749098, 64195923227488711116975114669335, 7703510787294094222269133507358060, 924421294475200284444819743469716489, 110930555337022213688891891269390300478 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [6, 1, 1, 1]](x) = 2 4 3 2 x (7000 x + 3520 x - 222 x - 127 x + 3) - --------------------------------------------------------------- (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[6, 1, 1, 1]](x) = -x^2*(7000*x^4+3520*x^3-222*x^2-127*x+3)/(1+x)/( -1+3*x)/(1+3*x)/(1+4*x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 278, 32223, 3844450, 460888883, 55297777878, 6635555555303, 796263111112810, 95551502222216763, 11466178844444472478, 1375941432888888788783, 165112971377777778233970, 19813556553955555553839043, 2377626786247111111118479078, 285315214345102222222193724663, 34237825721321244444444562951930, 4108539086556728888888888423627723, 493024690386771057777777779679477678, 59162962846411798755555555548027674943 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[5, 2, 2], [5, 4]](x) = x ( 7 6 5 4 3 2 397440 x - 113792 x + 72648 x - 36072 x - 6667 x + 3211 x - 231 x + 3 )/((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[5, 4]](x) = x^2*(397440*x^7-113792*x^6+72648*x^5-36072*x^4-6667*x^ 3+3211*x^2-231*x+3)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x-1)/(-1+ 20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 213, 24253, 2884611, 345690403, 41473788713, 4976675645273, 597197511811071, 71663630227851583, 8599634204489204613, 1031956076089247390293, 123834728561780641481531, 14860167416035578471412763, 1783220089696711294369841313, 213986410759054223688387255313, 25678369290995484456172681799991, 3081404314917637688982716270805943, 369768517790080113778528379570738813, 44372222134808885475561560393268488333 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [5, 3, 1]](x) = 2 3 2 x (2520 x - 496 x + 244 x - 7) - --------------------------------------------------- (1 + x) (-1 + x) (8 x - 1) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[5, 3, 1]](x) = -x^2*(2520*x^3-496*x^2+244*x-7)/(1+x)/(-1+x)/(8*x-1 )/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 792, 93079, 11119488, 1333253047, 159967967232, 19195702594999, 2303475197902848, 276416841126079927, 33170017279865782272, 3980402000456069115319, 477648238591991410065408, 57317788601782788423380407, 6878134631628799450244186112, 825376155783753138459096346039, 99045138693816319964815627911168, 11885416643253277256861382166146487, 1426249997190299647997748200186314752, 171149999662834085302839128458633375159 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 7 6 5 4 F[[5, 2, 2], [5, 2, 2]](x) = - x (1030720 x - 265296 x - 247300 x + 61349 x 3 2 + 11309 x - 3062 x + 141 x - 1)/((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[5, 2, 2]](x) = -x*(1030720*x^7-265296*x^6-247300*x^5+61349*x^4+ 11309*x^3-3062*x^2+141*x-1)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x -1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 1, 7, 604, 68977, 8235686, 987561067, 118494013184, 14219022572597, 1706277590116906, 204753208911272527, 24570383035099891364, 2948445923556987439417, 353813510013979711948526, 42457621185422313851875187, 5094914541925588034670283144, 611389745024568894753010157437, 73366769402818234967548615778546, 8804012328335587555930856475353047, 1056481479400218493971256387148100524, 126777777528025179022246241435473574657 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [5, 2, 1, 1]](x) = 2 3 2 x (17600 x - 2160 x - 124 x + 7) ------------------------------------------------------- (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[5, 2, 1, 1]](x) = x^2*(17600*x^3-2160*x^2-124*x+7)/(-1+4*x)/(1+4*x )/(8*x-1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 912, 108288, 12966400, 1555331072, 186626650112, 22394933198848, 2687386665615360, 322486293324890112, 38698353066599514112, 4643802325332795588608, 557256278186662371000320, 66870753365333298959613952, 8024490403498666391777574912, 962938848413013331134086381568, 115552661809425066649074301665280, 13866319417128277333192592265838592, 1663958330055338666665540763896512512, 199674999606639547733324326076812361728 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [5, 1, 1, 1, 1]](x) = 2 2 2 x (175 x - 86 x + 2) - ------------------------------------------- (-1 + 3 x) (1 + 3 x) (8 x - 1) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[5, 1, 1, 1, 1]](x) = -2*x^2*(175*x^2-86*x+2)/(-1+3*x)/(1+3*x)/(8*x -1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 4, 340, 40066, 4800500, 576004114, 69120032660, 8294400262306, 995328002096180, 119439360016778674, 14332723200134208980, 1719926784001073754946, 206391214080008589855860, 24766945689600068719594834, 2972033482752000549755105300, 356644017930240004398047573986, 42797282151628800035184365711540, 5135673858195456000281474986276594, 616280862983454720002251799756289620, 73953703558014566400018014398595575426 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[5, 2, 2], [4, 4, 1]](x) = - x ( 7 6 5 4 3 2 526080 x - 48384 x - 224688 x + 58664 x + 9012 x - 4465 x + 326 x - 5 )/((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[4, 4, 1]](x) = -x^2*(526080*x^7-48384*x^6-224688*x^5+58664*x^4+ 9012*x^3-4465*x^2+326*x-5)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x-\ 1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 5, 414, 48267, 5764748, 691291625, 82945799034, 9953315730167, 1194394312499928, 143327246233402485, 17199268124533775654, 2063912146489604624867, 247669457009783502055908, 29720334829795601366897345, 3566440179347911477577571474, 427972821517198225154225396367, 51356738581972764467900094784688, 6162808629834911289076538414080205, 739537035580152945779278968098434494, 88744444269617625315567565147087804667 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [4, 3, 2]](x) = 2 3 2 x (920 x - 936 x - 155 x + 7) - ----------------------------------------------------- (1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[4, 3, 2]](x) = -x^2*(920*x^3-936*x^2-155*x+7)/(1+x)/(-1+3*x)/(1+4* x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 811, 96219, 11524447, 1382488847, 165889777851, 19906595554979, 2388787911112567, 286654478222213847, 34398535964444470291, 4127824287288888762539, 495338913905777778214287, 59440669657315555553602847, 7132880358650311111118301931, 855945643033486222222191598899, 102713477163927324444444561357607, 12325617259669458488888888404495847, 1479074071160298609777777779665128771, 177488888539235104995555555547855488059 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [4, 3, 1, 1]](x) = 2 3 2 x (1160 x + 272 x + 192 x - 9) --------------------------------------------------- (1 + x) (-1 + x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[4, 3, 1, 1]](x) = x^2*(1160*x^3+272*x^2+192*x-9)/(1+x)/(-1+x)/(1+4 *x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 9, 1032, 123529, 14813312, 1777409609, 213285332992, 25594163810889, 3071298133327872, 368555745523831369, 44226688853333245952, 5307202650209524159049, 636864317781333331935232, 76423718128883809529401929, 9170846175368533333310963712, 1100501541042273523809613288009, 132060184925033813333332975419392, 15847222191003277409523810955465289, 1901666662920377685333333327606710272, 228199999550445010163809523832430301769 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [4, 2, 2, 1]](x) = 2 3 2 x (200 x + 1360 x + 63 x - 8) --------------------------------------------------- (1 + x) (-1 + x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[4, 2, 2, 1]](x) = x^2*(200*x^3+1360*x^2+63*x-8)/(1+x)/(-1+x)/(1+4* x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 8, 1025, 123328, 14809545, 1777333248, 213283809865, 25594133331968, 3071297523814985, 368555733333311488, 44226688609523896905, 5307202645333332983808, 636864317683809525207625, 76423718126933333327740928, 9170846175329523809546179145, 1100501541041493333333243854848, 132060184925018209523809881723465, 15847222191002965333333331901677568, 1901666662920371443809523815250432585, 228199999550444885333333333310426841088 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [4, 2, 1, 1, 1]](x) = 2 3 2 x (13120 x - 1936 x + 156 x - 7) - ------------------------------------------------------- (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[4, 2, 1, 1, 1]](x) = -x^2*(13120*x^3-1936*x^2+156*x-7)/(-1+4*x)/(1 +4*x)/(8*x-1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 880, 107648, 12953088, 1555064832, 186621317120, 22394826539008, 2687384532287488, 322486250658332672, 38698352213266268160, 4643802308266130669568, 557256277845329039065088, 66870753358506632320909312, 8024490403362133058466611200, 962938848410282664467867107328, 115552661809370453315741326245888, 13866319417127185066525932757450752, 1663958330055316821332207436289802240, 199674999606639110826657659524678156288 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [4, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (1640 x - 2512 x - 747 x + 15 x - 2) --------------------------------------------------------------- (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[4, 1, 1, 1, 1, 1]](x) = x^2*(1640*x^4-2512*x^3-747*x^2+15*x-2)/(1+ x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 2, 255, 31782, 3835547, 460711162, 55294222095, 6635484445102, 796261688886947, 95551473777786882, 11466178275555525335, 1375941421511111244022, 165112971150222221746347, 19813556549404444446456202, 2377626786156088888881343775, 285315214343281777777808932542, 34237825721284835555555435453747, 4108539086556000711111111600287122, 493024690386756494222222220306173415, 59162962846411507484444444452187558662 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 x (62 x + 1) F[[5, 2, 2], [3, 3, 3]](x) = - --------------------------------- (-1 + 3 x) (8 x - 1) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[3, 3, 3]](x) = -x^2*(62*x+1)/(-1+3*x)/(8*x-1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 193, 23939, 2879497, 345595931, 41471967313, 4976639738099, 597196797903577, 71663615983224971, 8599633919865788833, 1031956070398926277859, 123834728447991410124457, 14860167413759931280700411, 1783220089651199450244717553, 213986410758143995601955083219, 25678369290977279964815632694137, 3081404314917273599718525037638251, 369768517790072831997748200229361473, 44372222134808739839981985601619658179 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [3, 3, 2, 1]](x) = 2 3 2 2 x (980 x + 156 x + 20 x - 3) ----------------------------------------------------- (1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[3, 3, 2, 1]](x) = 2*x^2*(980*x^3+156*x^2+20*x-3)/(1+x)/(-1+3*x)/(1 +4*x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 6, 788, 95778, 11515544, 1382311126, 165886222068, 19906524444778, 2388786488886704, 286654449777783966, 34398535395555523148, 4127824275911111217778, 495338913678222221726664, 59440669652764444446220006, 7132880358559288888881166628, 855945643031665777777806806778, 102713477163890915555555433859424, 12325617259668730311111111581155246, 1479074071160284046222222220291824508, 177488888539234813724444444452015371778 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[5, 2, 2], [3, 3, 1, 1, 1]](x) = x 6 5 4 3 2 (75200 x - 20400 x - 11516 x - 373 x + 2481 x - 177 x + 5)/((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[3, 3, 1, 1, 1]](x) = x^2*(75200*x^6-20400*x^5-11516*x^4-373*x^3+ 2481*x^2-177*x+5)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x-1)/(-1+20 *x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 5, 563, 68335, 8222845, 987306985, 118488931543, 14218920983475, 1706275558334465, 204753168276322765, 24570382222400896123, 2948445907303018719415, 353813509688900337548485, 42457621178920726542831345, 5094914541795556288489406303, 611389745021968259832255932155, 73366769402766222269133531272905, 8804012328334547301962600598224725, 1056481479400197688891891269605534083, 126777777528024762920658939817629997695 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[5, 2, 2], [3, 2, 2, 2]](x) = - 2 x 7 6 5 4 3 2 (124800 x - 282240 x - 51624 x + 52244 x + 51 x - 1597 x + 98 x - 2)/ ((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[3, 2, 2, 2]](x) = -2*x^2*(124800*x^7-282240*x^6-51624*x^5+52244*x^ 4+51*x^3-1597*x^2+98*x-2)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x-1 )/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 4, 396, 47826, 5755930, 691113904, 82942244616, 9953244619966, 1194392890295910, 143327217788972604, 17199267555645178036, 2063912135111827080106, 247669456782227951160690, 29720334825244490259514504, 3566440179256889255429914656, 427972821515377780709840604246, 51356738581936355579012398942270, 6162808629834183111298761590739604, 739537035580138382223723431631622476, 88744444269617334044456454051247688386 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [3, 2, 2, 1, 1]](x) = 2 3 2 x (360 x - 2768 x + 143 x + 4) --------------------------------------------------- (1 + x) (-1 + x) (8 x - 1) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[3, 2, 2, 1, 1]](x) = x^2*(360*x^3-2768*x^2+143*x+4)/(1+x)/(-1+x)/( 8*x-1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 4, 735, 91936, 11096631, 1332795904, 159958824375, 19195519737856, 2303471540759991, 276416767983222784, 33170015817008639415, 3980401971198926258176, 477648238006848552922551, 57317788590079931280523264, 6878134631394742307387043255, 825376155779071995601953488896, 99045138693722697107672770768311, 11885416643251404799718525023289344, 1426249997190262198854891057329171895, 171149999662833336319981985601490518016 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[5, 2, 2], [3, 2, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (456000 x - 149040 x + 6812 x + 2179 x + 408 x + 214 x - 30 x - 3)/( (1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[3, 2, 1, 1, 1, 1]](x) = -x^2*(456000*x^7-149040*x^6+6812*x^5+2179* x^4+408*x^3+214*x^2-30*x-3)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x -1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 474, 59456, 7188980, 863778498, 103675561094, 12441511155636, 1492990222572840, 179159004447256118, 21499084088911274714, 2579890161777957001416, 309586820835556987459100, 37150418528711122568796138, 4458050224014222313852052334, 534966026893084445177524768796, 64195923227397688894753011751760, 7703510787292273777824691449006558, 924421294475163875555930856489701954, 110930555337021485511114113529790009776 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 3 F[[5, 2, 2], [3, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (354560 x - 5568 x - 45760 x + 13252 x + 1811 x - 1874 x + 119)/( (1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[3, 1, 1, 1, 1, 1, 1]](x) = x^3*(354560*x^6-5568*x^5-45760*x^4+ 13252*x^3+1811*x^2-1874*x+119)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/( 8*x-1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 119, 15738, 1915249, 230308420, 27646200939, 3317724270238, 398130487499829, 47775729766601160, 5733088995466222159, 687970707910395407938, 82556485518216497924409, 9906778273564398633397900, 1188813393055288522422251379, 142657607171185774845777260838, 17118912860633315532099903620989, 2054269543277818310923461609834640, 246512345193374606220721031887216599, 29581481423205680924432434853127428938 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[5, 2, 2], [2, 2, 2, 2, 1]](x) = - x 7 6 5 4 3 2 (17280 x + 334976 x - 8280 x - 25080 x - 3148 x + 835 x - 42 x - 1)/( (1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[2, 2, 2, 2, 1]](x) = -x^2*(17280*x^7+334976*x^6-8280*x^5-25080*x^4 -3148*x^3+835*x^2-42*x-1)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x-1 )/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 190, 23791, 2875708, 345512341, 41470232930, 4976604529611, 597196089585208, 71663601783334321, 8599633635600257470, 1031956064711468447431, 123834728334225084993908, 14860167411484467341660301, 1783220089605689072132706010, 213986410757233779243644549251, 25678369290959075567283554301808, 3081404314916909511204933720842281, 369768517790065550222972820197434550, 44372222134808594204450449205802403071 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [2, 2, 2, 1, 1, 1]](x) = 2 4 3 2 x (1520 x + 2024 x - 390 x + 75 x + 1) -------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[2, 2, 2, 1, 1, 1]](x) = x^2*(1520*x^4+2024*x^3-390*x^2+75*x+1)/(1+ x)/(-1+x)/(-1+3*x)/(1+4*x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 214, 27109, 3285098, 394844401, 47394032014, 5687537776989, 682509612702418, 81901255111097641, 9828152645079426614, 1179378358044444221269, 141525403778031746993338, 16983048469617777774138081, 2037965816679212698427877214, 244555898008007111111052255949, 29346707761090885079365320365858, 3521604931333506844444443497181721, 422592591760072834031746035585303814, 50711111011209780337777777762571353029 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [2, 2, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (57920 x - 10096 x - 16340 x + 2551 x - 120) ------------------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[2, 2, 1, 1, 1, 1, 1]](x) = x^3*(57920*x^4-10096*x^3-16340*x^2+2551 *x-120)/(1+x)/(-1+x)/(-1+4*x)/(1+4*x)/(8*x-1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 120, 15209, 1846912, 222078025, 26658682880, 3199230603849, 383911467712512, 46069452198810185, 5528335786733731840, 663400324876726473289, 79608039594670960934912, 9552964763550510536233545, 1146355771869866941533388800, 137562692629260192674990035529, 16507523115608746684258673754112, 1980902773875000076331210099692105, 237708332865039018667792563710197760, 28524999943805462430485197618178986569 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[5, 2, 2], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 2 x (5780 x + 976 x - 355 x + 76 x - 17) - ------------------------------------------------------------------------ (1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 20 x) (-1 + 120 x) and in Maple notation F[[5, 2, 2],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -2*x^3*(5780*x^4+976*x^3-355*x^2+76* x-17)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 34, 4472, 546682, 65790400, 7898666954, 947916189192, 113751466671642, 13650206476169360, 1638025386666749674, 196563058590475847512, 23587567274666668025402, 2830508077836190470657120, 339660969437866666688681994, 40759316334494476190387243432, 4891117960178346666667021391962, 586934155222181790476189049317680, 70432098626677418666666672364591914, 8451851835201602316190476167612744952 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 3 F[[5, 2, 2], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (75200 x - 20400 x - 11516 x - 373 x + 2481 x - 177 x + 5)/((1 + x) (-1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (1 + 4 x) (8 x - 1) (-1 + 20 x) (-1 + 120 x)) and in Maple notation F[[5, 2, 2],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(75200*x^6-20400*x^5-11516*x^ 4-373*x^3+2481*x^2-177*x+5)/(1+x)/(-1+x)/(-1+3*x)/(1+3*x)/(-1+4*x)/(1+4*x)/(8*x -1)/(-1+20*x)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 5, 563, 68335, 8222845, 987306985, 118488931543, 14218920983475, 1706275558334465, 204753168276322765, 24570382222400896123, 2948445907303018719415, 353813509688900337548485, 42457621178920726542831345, 5094914541795556288489406303, 611389745021968259832255932155, 73366769402766222269133531272905, 8804012328334547301962600598224725, 1056481479400197688891891269605534083 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 Regarding Lambda=, [5, 2, 1, 1] 7 6 5 4 F[[5, 2, 1, 1], [9]](x) = (546777 x + 74043 x - 1191880 x - 306882 x 3 2 + 39096 x + 2826 x - 205 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x)) and in Maple notation F[[5, 2, 1, 1],[9]](x) = (546777*x^7+74043*x^6-1191880*x^5-306882*x^4+39096*x^3 +2826*x^2-205*x+1)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 1, 18, 3556, 664838, 125615865, 23739552796, 4486749129920, 847994906903316, 160271024625755449, 30291223370575341614, 5725041211257606744804, 1082032788804413172501034, 204504197081466530389365953, 38651293248343027890415104072, 7305094423935697752414568099408, 1380662846123823022667570435371592, 260945277917402050689040927214450577, 49318657526388977064378037970201585770, 9321226272487516444371800743722399900932 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [8, 1]](x) = 2 4 3 2 2 x (58212 x - 5463 x - 1743 x + 131 x - 1) - ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[8, 1]](x) = -2*x^2*(58212*x^4-5463*x^3-1743*x^2+131*x-1)/(-1+x) /(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 2, 150, 28324, 5318516, 1004901678, 189916139866, 35893984580960, 6783959106920712, 1282168193583619834, 242329786896248467262, 45800329688587895250876, 8656262310404796413196988, 1636033576651086994142720870, 309210345986730723909511452738, 58440755391485297982291596268472, 11045302768990578222720554025689744, 2087562223339216280314192234577283186, 394549260211111813886607288802719918694, 74569810179900131499769516604440749496148 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [7, 2]](x) = 2 4 3 2 x (185058 x - 29817 x - 5353 x + 517 x - 5) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[7, 2]](x) = -x^2*(185058*x^4-29817*x^3-5353*x^2+517*x-5)/(1+x)/ (-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 5, 508, 95368, 17948742, 3391484553, 640966148510, 121142176652936, 22895861586722884, 4327317644468948101, 817863030594219768312, 154576112695130706837504, 29214885297535958011461626, 5521613321195726408972307449, 1043584917705180740353145743114, 197237549446262135284500870533872, 37277896845343185858198315680184168, 7045522503769854617438460258236999397, 1331603753212502364967450105898139516116, 251673109357162943666812087133454219626840 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [7, 1, 1]](x) = 2 4 3 2 4 x (29106 x + 3717 x + 337 x - 73 x + 1) ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[7, 1, 1]](x) = 4*x^2*(29106*x^4+3717*x^3+337*x^2-73*x+1)/(-1+x) /(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 4, 532, 98816, 18613464, 3517075948, 664705425356, 125628917384520, 23743856345857648, 4487588665677125012, 848154253896484421700, 160301153904915747160144, 30296918086309865747454152, 5726117518276547722162988796, 1082236210953510269315710775164, 204542643870197549002464131051288, 38658559691467002922269039405252576, 7306467781687256542929574467100516900, 1380922410738891339403413005092341119348, 260994335629650460055979015417484786011552 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [6, 3]](x) = 2 4 3 2 x (81081 x - 16740 x - 4714 x + 540 x - 7) - ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[6, 3]](x) = -x^2*(81081*x^4-16740*x^3-4714*x^2+540*x-7)/(-1+x)/ (1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 7, 902, 169309, 31906668, 6029238419, 1139494252210, 215363843397801, 40703753415758936, 7693009134554678431, 1453978720825901306718, 274801978119796009515893, 51937573862184131364561604, 9816201459901366302247347243, 1855262075920276280125776505226, 350644532348909516363378863502785, 66271816613943421658939131225079472, 12525373340035296680392737231302752055, 2367295561266670862291288294969874868534, 447418861079400788556989197363855928737677 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [6, 2, 1]](x) = 2 5 4 3 2 x (87318 x - 179676 x - 58699 x - 11 x + 681 x - 13) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[6, 2, 1]](x) = -x^2*(87318*x^5-179676*x^4-58699*x^3-11*x^2+681* x-13)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 13, 1984, 370006, 69794450, 13188861953, 2492642371254, 471108371846286, 89039459938533100, 16828457467024592893, 3180578451506341022324, 601129327130627884066766, 113613442823394148475335950, 21472940693531417958645688833, 4058385791075545283388010643194, 767034914513238324635096843783446, 144969598843001208807450085261643000, 27399254181327210940647271493993205773, 5178459040270842499762559548786078881864, 978728758611189224726896066682037049588326 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [6, 1, 1, 1]](x) = 2 5 4 3 2 2 x (14553 x - 94941 x - 23972 x + 2008 x + 83 x - 3) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[6, 1, 1, 1]](x) = -2*x^2*(14553*x^5-94941*x^4-23972*x^3+2008*x^ 2+83*x-3)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 6, 1064, 197154, 37223016, 7034008438, 1329408572800, 251257779378626, 47487711616408112, 8975177307849004230, 1696308507309907951416, 320602307799574660091458, 60593836172405607468576088, 11452235036548585157782497782, 2164472421906925975657778341712, 409085287740393110521877053126050, 77317119382933964125866546348171744, 14612935563374512209565701800263391494, 2761844821477782660406895092028895212488, 521988671259300919725535085460105337144802 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [5, 4]](x) = 2 3 2 x (24255 x + 952 x - 397 x + 6) - ------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[5, 4]](x) = -x^2*(24255*x^3+952*x^2-397*x+6)/(-1+x)/(1+3*x)/(1+ 11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 6, 785, 148049, 27916934, 5275550720, 997056853919, 188443349512403, 35615783964511388, 6731382986908226954, 1272231380601260432573, 240451730852263445909477, 45445377129357508484688962, 8589176277412568702202822308, 1623354316430218095378083082347, 306813965805295330040348668948871, 57987839537200483520827679308603256, 10959701672530884376281657990212098382, 2071383616108336999904765237604960796841, 391491503444475689890761196026500923488185 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [5, 3, 1]](x) = 2 3 2 3 x (6237 x + 1872 x + 169 x - 6) ------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[5, 3, 1]](x) = 3*x^2*(6237*x^3+1872*x^2+169*x-6)/(-1+x)/(1+3*x) /(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 18, 3039, 570423, 107675226, 20348367864, 3845787854385, 726852848178069, 137375165344010532, 25963905776397338190, 4907178181698746727411, 927456676131253270212195, 175289311784389448529396318, 33129679927156965498797829396, 6261509506230719652625999220517, 1183425296677565146176724250391201, 223667381072058954228749575932414984, 42273135022619124324699981587447991882, 7989622519275014118832871185357649564103, 1510038656142977660510391118697037123665487 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 x (441 x - 13) F[[5, 2, 1, 1], [5, 2, 2]](x) = - ---------------------------------- (1 + 3 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[5, 2, 2]](x) = -x^2*(441*x-13)/(1+3*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 13, 2250, 422343, 79757460, 15072800193, 2848730642910, 538409491161723, 101759381221902120, 19232522786179616373, 3634946801027986774770, 687004945277530334765103, 129843934655001290763265980, 24540503649743753161688192553, 4638155189800488040914862669830, 876611330872269532293381489696483, 165679541534858464747219020603881040, 31313433350088239823243563201097152733, 5918238903166677116299435979432935800090, 1118547152698501970564427853335823906099863 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [5, 2, 1, 1]](x) = 5 4 3 2 x (632016 x + 232704 x - 25991 x - 2691 x + 187 x - 1) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[5, 2, 1, 1]](x) = -x*(632016*x^5+232704*x^4-25991*x^3-2691*x^2+ 187*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 18, 3556, 664838, 125615865, 23739552796, 4486749129920, 847994906903316, 160271024625755449, 30291223370575341614, 5725041211257606744804, 1082032788804413172501034, 204504197081466530389365953, 38651293248343027890415104072, 7305094423935697752414568099408, 1380662846123823022667570435371592, 260945277917402050689040927214450577, 49318657526388977064378037970201585770, 9321226272487516444371800743722399900932, 1761711765500140603349155347059928533236590 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [5, 1, 1, 1, 1]](x) = 2 3 2 x (7623 x + 3120 x - 209 x - 6) ----------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[5, 1, 1, 1, 1]](x) = x^2*(7623*x^3+3120*x^2-209*x-6)/(-1+x)/(1+ 3*x)/(-1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 6, 1319, 246019, 46522410, 8792351516, 1661757682249, 314072157367569, 59359638153159620, 11218971605711985826, 2120385633530695791579, 400752884736680954943119, 75742295215239003517159630, 14315293795680097624209462936, 2705590527383539223057067065309, 511356609675488904283609422028669, 96646399228667403003786601026010440, 18266169454218139166648755300180142846, 3452306026847228302508072833939912719439, 652485839074126149173897225133114732960219 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [4, 4, 1]](x) = 2 5 4 3 2 x (218295 x + 49077 x - 2066 x - 334 x - 277 x + 9) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[4, 4, 1]](x) = x^2*(218295*x^5+49077*x^4-2066*x^3-334*x^2-277*x +9)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 9, 1568, 295681, 55829392, 10550966025, 1994111355600, 376886644586657, 71231566844430944, 13462765950424117321, 2544462760718314777792, 480903461694283526582913, 90890754258500752628947056, 17178352554820628731925962697, 3246708632860341610663283650544, 613627931610588672791676971353249, 115975679074400925320900959635272128, 21919403345061767876293243633356994953, 4142767232216673981409346603693661788256, 782983006888951379395102266688762605470465 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [4, 3, 2]](x) = 2 4 3 2 x (43659 x + 630 x - 1820 x - 374 x + 17) ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[4, 3, 2]](x) = x^2*(43659*x^4+630*x^3-1820*x^2-374*x+17)/(-1+x) /(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 17, 3128, 591021, 111653704, 21101804801, 3988220284272, 753773235688525, 142463132594982128, 26925531877568817105, 5088925520951278243336, 961806923378337157011149, 181781508516787100603098872, 34356705109636750405436838529, 6493417265720588624114443089920, 1227255863221175358488364425768493, 231951358148801808918964653747485536, 43838806690123534876339789706581798673, 8285534464433347944418256382510397281624, 1565966013777902758403787228023911543938157 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [4, 3, 1, 1]](x) = 2 5 4 3 2 x (168399 x - 47925 x - 42966 x + 6458 x + 279 x - 21) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[4, 3, 1, 1]](x) = -x^2*(168399*x^5-47925*x^4-42966*x^3+6458*x^2 +279*x-21)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 21, 4026, 759547, 143551708, 27130775465, 5127709858214, 969136970450271, 183166883843706456, 34618540965358203229, 6542904240808941685042, 1236608901477648005196515, 233719082378542717408009844, 44172906569529099489818910513, 8348679341640675745198291798110, 1577900395570080900283997119961479, 298223174762745147136487637996615472, 56364180030158829804193216102899700517, 10652830025700018769909184439179323679018, 2013384874857303546187936242201875454444363 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [4, 2, 2, 1]](x) = 2 4 3 2 4 x (48330 x + 4689 x - 1867 x - 21 x + 5) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[4, 2, 2, 1]](x) = 4*x^2*(48330*x^4+4689*x^3-1867*x^2-21*x+5)/(1 +x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 20, 4016, 759312, 143546856, 27130673332, 5127707714152, 969136925422784, 183166882898135792, 34618540945501199604, 6542904240391944667968, 1236608901468891067660816, 233719082378358821720291608, 44172906569525237680375233236, 8348679341640594647199979358264, 1577900395570079197226032544375808, 298223174762745111372270381952363104, 56364180030158829053144653725841260628, 10652830025700018754137164629261483861840, 2013384874857303545856723826193599656022160 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [4, 2, 1, 1, 1]](x) = 2 4 3 2 x (116424 x + 66303 x - 8309 x + 29 x + 17) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[4, 2, 1, 1, 1]](x) = x^2*(116424*x^4+66303*x^3-8309*x^2+29*x+17 )/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 17, 3514, 664036, 125598822, 23739195513, 4486741625156, 847994749308752, 160271021316253204, 30291223301075843689, 5725041209798117140758, 1082032788773763891258948, 204504197080822895481953546, 38651293248329511557363429345, 7305094423935413909420470972720, 1380662846123817061964694431583424, 260945277917401925514280531027282248, 49318657526388974435708069650593901281, 9321226272487516389169731409009669975442, 1761711765500140602189911891030964110454980 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [4, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (160083 x - 43533 x - 34358 x + 2050 x - 13 x - 5) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(160083*x^5-43533*x^4-34358*x^3+ 2050*x^2-13*x-5)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 5, 1038, 196615, 37211668, 7033770209, 1329403569746, 251257674315219, 47487709410074376, 8975177261516002333, 1696308506336914891894, 320602307779141805900543, 60593836171976517530389724, 11452235036539574269081115577, 2164472421906736746995047721082, 409085287740389136719959714875787, 77317119382933880676026282230567312, 14612935563374510457119056253836745141, 2761844821477782623605515535553806498910, 521988671259300918952706114774128861580151 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [3, 3, 3]](x) = 2 3 2 x (2079 x - 534 x - 37 x - 4) ----------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[3, 3, 3]](x) = x^2*(2079*x^3-534*x^2-37*x-4)/(-1+x)/(1+3*x)/(-1 +9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 4, 777, 147695, 27911782, 5275422702, 997054420439, 188443295967553, 35615782870094364, 6731382963623964200, 1272231380115865542901, 240451730842033157469411, 45445377129143100299590946, 8589176277408061612014213298, 1623354316430123497879999563363, 306813965805293342942785511066069, 57987839537200441797967251038564728, 10959701672530883500034751965292518796, 2071383616108336981504326536545141717025, 391491503444475689504343873787240640290327 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [3, 3, 2, 1]](x) = 2 4 3 2 2 x (43659 x + 7371 x + 17 x + 97 x - 8) - ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[3, 3, 2, 1]](x) = -2*x^2*(43659*x^4+7371*x^3+17*x^2+97*x-8)/(-1 +x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 16, 3102, 590482, 111642356, 21101566572, 3988215281218, 753773130625118, 142463130388648392, 26925531831235815208, 5088925519978285183814, 961806923357904302820234, 181781508516358010664912508, 34356705109627739516735456324, 6493417265720399395451712469290, 1227255863221171384686447087518230, 231951358148801725469124389629881104, 43838806690123533123893144160155152320, 8285534464433347907616876826035308568046, 1565966013777902757630958257337935068373506 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 x (63 x - 11) F[[5, 2, 1, 1], [3, 3, 1, 1, 1]](x) = - ---------------------------------- (1 + 3 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[3, 3, 1, 1, 1]](x) = -x^2*(63*x-11)/(1+3*x)/(-1+21*x)/(-1+189*x ) The first 20 term , starting with k=1 are 0, 11, 2214, 421569, 79741260, 15072459831, 2848723495794, 538409341070829, 101759378069997720, 19232522719989610851, 3634946799637996698174, 687004945248340543038489, 129843934654388305137361380, 24540503649730880463543133071, 4638155189800217714253819609354, 876611330872263855433499575860549, 165679541534858345533161500442024240, 31313433350088237319748355277612066491, 5918238903166677063726036613040007269334, 1118547152698501969460386466641571632113009 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [3, 2, 2, 2]](x) = 2 5 4 3 2 2 x (43659 x - 48636 x - 7826 x + 816 x - 49 x + 4) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[3, 2, 2, 2]](x) = 2*x^2*(43659*x^5-48636*x^4-7826*x^3+816*x^2-\ 49*x+4)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 8, 1542, 295142, 55818044, 10550727796, 1994106352546, 376886539523250, 71231564638097208, 13462765904091115424, 2544462759745321718270, 480903461673850672391998, 90890754258071662690760692, 17178352554811617843224580492, 3246708632860152382000553029914, 613627931610584698989759633102986, 115975679074400841871060695517667696, 21919403345061766123846598086930348600, 4142767232216673944607967047218573074678, 782983006888951378622273296002786129905814 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [3, 2, 2, 1, 1]](x) = 2 2 3 x (741 x + 6 x + 5) - ------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[3, 2, 2, 1, 1]](x) = -3*x^2*(741*x^2+6*x+5)/(-1+x)/(1+3*x)/(1+ 11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 15, 2973, 569034, 107646054, 20347755249, 3845774989467, 726852578014788, 137375159670581628, 25963905657255331203, 4907178179196764580681, 927456676078711645130862, 175289311783286074402688322, 33129679927133794642136961477, 6261509506230233064636120994215, 1183425296677554927828936807638856, 223667381072058739643446039634615736, 42273135022619119818408607325194207671, 7989622519275014024200752325850320095669, 1510038656142977658523116622647383204828370 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [3, 2, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (174636 x - 32823 x + 5195 x + 3513 x - 131 x + 10) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[3, 2, 1, 1, 1, 1]](x) = x^2*(174636*x^5-32823*x^4+5195*x^3+3513 *x^2-131*x+10)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 10, 1919, 368658, 69766080, 13188266380, 2492629863619, 471108109187768, 89039454422698760, 16828457351192088150, 3180578449073858373519, 601129327079545748589478, 113613442822321423629870040, 21472940693508890736892233320, 4058385791075072211731184091619, 767034914513228390130303498157788, 144969598843001000182849424967631920, 27399254181327206559530657627926589890, 5178459040270842407759110657598357097919, 978728758611189222794823639967095860676698 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = 2 3 2 x (7 x - 3) (2079 x + 963 x - 35 x + 1) - ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^2*(7*x-3)*(2079*x^3+963*x^2-35*x+ 1)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 3, 506, 98277, 18602116, 3516837719, 664700422302, 125628812321113, 23743854139523912, 4487588619344123115, 848154252923491362178, 160301153884482892969229, 30296918085880775809267788, 5726117518267536833461606591, 1082236210953321040652980154534, 204542643870193575200546792801025, 38658559691466919472428775287648144, 7306467781687254790482928920673870547, 1380922410738891302602033448617252405770, 260994335629650459283150044731508310446901 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [2, 2, 2, 2, 1]](x) = 2 3 2 x (4851 x - 2618 x + 27 x - 4) ------------------------------------------------------ (-1 + x) (1 + 3 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[2, 2, 2, 2, 1]](x) = x^2*(4851*x^3-2618*x^2+27*x-4)/(-1+x)/(1+3 *x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 4, 761, 147503, 27905606, 5275312430, 997051851047, 188443244448449, 35615781758179292, 6731382940575220136, 1272231379628267387813, 240451730831830591674275, 45445377128928418546635458, 8589176277403557813501041522, 1623354316430028866715353657459, 306813965805291356238431327111381, 57987839537200400070987415201760504, 10959701672530882623835012443753166988, 2071383616108336963103385681129968938385, 391491503444475689117932225340524157358167 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 4 3 2 x (43659 x + 21636 x - 5294 x + 164 x - 5) - ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[2, 2, 2, 1, 1, 1]](x) = -x^2*(43659*x^4+21636*x^3-5294*x^2+164* x-5)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 5, 866, 168535, 31890468, 6028898057, 1139487105094, 215363693306907, 40703750263854536, 7693009068364672909, 1453978719435911230122, 274801978090606217789279, 51937573861571145738657004, 9816201459888493604102287761, 1855262075920005953464733444750, 350644532348903839503496949666851, 66271816613943302444881611063222672, 12525373340035294176897529307817665813, 2367295561266670809717888928576946337778, 447418861079400787452947810669603654750823 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (168399 x - 25056 x + 11169 x - 4241 x + 132 x - 3) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -x^2*(168399*x^5-25056*x^4+11169*x^3 -4241*x^2+132*x-3)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 3, 483, 94781, 17936612, 3391229221, 640960788355, 121142064084219, 22895859222796224, 4327317594826439039, 817863029551727225627, 154576112673238362998257, 29214885297076218792166036, 5521613321186071885363114257, 1043584917704977995357364643499, 197237549446257877639589431569695, 37277896845343096447655175569553248, 7045522503769852739817054315590899675, 1331603753212502325537400581103539973171, 251673109357162942838781047112764723571333 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (51975 x - 12078 x + 2280 x - 66 x + 1) ----------------------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^2*(51975*x^4-12078*x^3+2280*x^2 -66*x+1)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 1, 140, 28089, 5313664, 1004799545, 189913995804, 35893939553473, 6783958161350048, 1282168173726616209, 242329786479251450188, 45800329679830957715177, 8656262310220900725478752, 1636033576647225184699043593, 309210345986649625911199012892, 58440755391483594924327020682801, 11045302768990542458503297981437376, 2087562223339215529265629857518843297, 394549260211111798114587478884880101516, 74569810179900131168557100596164951073945 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[5, 2, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (116424 x + 66303 x - 8309 x + 29 x + 17) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 21 x) (-1 + 189 x) and in Maple notation F[[5, 2, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(116424*x^4+66303*x^3-8309 *x^2+29*x+17)/(1+x)/(-1+x)/(1+3*x)/(-1+9*x)/(1+11*x)/(-1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 0, 17, 3514, 664036, 125598822, 23739195513, 4486741625156, 847994749308752, 160271021316253204, 30291223301075843689, 5725041209798117140758, 1082032788773763891258948, 204504197080822895481953546, 38651293248329511557363429345, 7305094423935413909420470972720, 1380662846123817061964694431583424, 260945277917401925514280531027282248, 49318657526388974435708069650593901281, 9321226272487516389169731409009669975442 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 Regarding Lambda=, [5, 1, 1, 1, 1] 8 7 6 5 F[[5, 1, 1, 1, 1], [9]](x) = (507200 x - 775680 x + 86864 x + 221728 x 4 3 2 - 80828 x + 5940 x + 701 x - 81 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[9]](x) = (507200*x^8-775680*x^7+86864*x^6+221728*x^5-80828*x ^4+5940*x^3+701*x^2-81*x+1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x) /(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 1, 1, 89, 4601, 325881, 22689721, 1588780089, 111203172921, 7784275823161, 544898185782841, 38142878497336889, 2670001383329386041, 186900097386394832441, 13083006805923302280761, 915810476470107695058489, 64106733351795954732731961, 4487471334631269555238833721, 314122993424077740786950311481, 21988609539685997308914142776889 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [8, 1]](x) = x 5 4 3 2 (8000 x + 320 x - 1232 x - 324 x + 68 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[8, 1]](x) = x^2*(8000*x^5+320*x^4-1232*x^3-324*x^2+68*x-1)/( -1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 1, 7, 599, 36887, 2600231, 181538887, 12709646279, 889627936967, 62274149710791, 4359185756250567, 305143022376853959, 21360011094137045447, 1495200778533955588551, 104664054450154340872647, 7326483811705246993117639, 512853866814645062046544327, 35899770677044598766704685511, 2512983947392649691363115037127, 175908876317487422839526534181319 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [7, 2]](x) = 2 4 3 2 2 x (6000 x - 1440 x - 360 x + 64 x - 1) - ------------------------------------------------------------------ (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[7, 2]](x) = -2*x^2*(6000*x^4-1440*x^3-360*x^2+64*x-1)/(1+2*x )/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 24, 1912, 124800, 8765472, 612726144, 42894023552, 3002497474560, 210175151514112, 14712252242417664, 1029857690121172992, 72090037474060369920, 5046302626511028887552, 353241183772399506653184, 24726882864401063706591232, 1730881800499739715572858880, 121161726035015104960103841792, 8481320822450223963093956296704, 593692457571519010445230404534272 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [7, 1, 1]](x) = - x 6 5 4 3 2 (56000 x - 21120 x - 3248 x + 3072 x - 604 x + 56 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[7, 1, 1]](x) = -x^2*(56000*x^6-21120*x^5-3248*x^4+3072*x^3-\ 604*x^2+56*x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-\ 1+70*x) The first 20 term , starting with k=1 are 0, 1, 25, 1929, 129657, 9086201, 635451705, 44482351929, 3113704604217, 217959384410681, 15257150837308985, 1068000564406492729, 74760038898784325177, 5233202723479124463161, 366324190582479680081465, 25642693340829445961846329, 1794988533851951984333524537, 125649197369642205732843523641, 8795443815874343357852578713145, 615681067111204591011307912924729 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [6, 3]](x) = 2 3 2 2 x (1000 x + 140 x - 50 x + 1) - ------------------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[6, 3]](x) = -2*x^2*(1000*x^3+140*x^2-50*x+1)/(1+2*x)/(-1+2*x )/(-1+10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 40, 3288, 221760, 15573152, 1089287040, 76255092608, 5337773148160, 373644620370432, 26155115092592640, 1830858106481481728, 128160066620370370560, 8971204668425925926912, 627984326706481481482240, 43958902869953703703707648, 3077123200888425925925928960, 215398624062239814814814830592, 15077903684355953703703703715840, 1055453257904921759259259259322368 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [6, 2, 1]](x) = 2 x 5 4 3 2 (28000 x - 14800 x + 1680 x - 152 x + 30 x - 1)/((1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[6, 2, 1]](x) = 2*x^2*(28000*x^5-14800*x^4+1680*x^3-152*x^2+ 30*x-1)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 100, 7064, 486480, 34054112, 2382954560, 166806802304, 11676392669440, 817347485660672, 57214315654763520, 4005002095780665344, 280350145870981222400, 19624510210966631800832, 1373715714684318324244480, 96160100027902206301405184, 6731207001944820645454479360, 471184490136137442392199790592, 32982914309528787617319979581440, 2308804001667015133111342555725824 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [6, 1, 1, 1]](x) = x 5 4 3 2 (11200 x - 480 x - 4656 x + 756 x + 12 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[6, 1, 1, 1]](x) = x^2*(11200*x^5-480*x^4-4656*x^3+756*x^2+12 *x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 1, 63, 3719, 260311, 18156711, 1270992583, 88963072199, 6227417751751, 435918603414471, 30514302515509703, 2136001112191668679, 149520077881174081991, 10466405445293214847431, 732648381173302489018823, 51285386681492284030153159, 3589977067704737654639129031, 251298394739267746914852827591, 17590887631748770061733485375943, 1231362134222407515432119126749639 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [5, 4]](x) = 2 3 2 2 x (1400 x - 12 x - 50 x + 1) - ------------------------------------------------------ (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[5, 4]](x) = -2*x^2*(1400*x^3-12*x^2-50*x+1)/(1+2*x)/(-1+4*x) /(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 36, 2856, 193968, 13623392, 953114176, 66722869376, 4670550175488, 326939008445952, 22885725569135616, 1602000839710877696, 112140058279005073408, 7849804084525878566912, 549486285866784886112256, 38464040011174783331106816, 2692482800777233884933193728, 188473796054456366271607734272, 13193165723811445605033235972096, 923521600666806192148702663540736 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 2 2 x (300 x + 8 x + 1) F[[5, 1, 1, 1, 1], [5, 3, 1]](x) = ------------------------------------------- (1 + 2 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[5, 3, 1]](x) = 2*x^2*(300*x^2+8*x+1)/(1+2*x)/(-1+6*x)/(10*x+ 1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 144, 10760, 749568, 52527392, 3676464384, 257357786240, 18014996717568, 1261050280305152, 88273514681831424, 6179146078090987520, 432540224968545927168, 30277815752811275558912, 2119447102646867653361664, 148361297185781205920153600, 10385290802999687235520954368, 726970356210028123413125660672, 50887924934701468740478754095104, 3562154745429107812442872524308480 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [5, 2, 2]](x) = 2 2 2 x (20 x - 20 x + 1) - ------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[5, 2, 2]](x) = -2*x^2*(20*x^2-20*x+1)/(1+2*x)/(-1+4*x)/(-1+6 *x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 116, 7960, 555920, 38905952, 2723361856, 190634986880, 13344446961920, 934111274378752, 65387789127812096, 4577145238470809600, 320400166690085048320, 22428011668288662167552, 1569960816780102358286336, 109897257174606540135301120, 7692808002222454055865221120, 538496560155571761373182820352, 37694759210890023160835507224576, 2638633144762301620446509795901440 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [5, 2, 1, 1]](x) = 2 4 3 2 2 x (8400 x - 4560 x + 768 x - 20 x - 1) - ------------------------------------------------------------------ (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[5, 2, 1, 1]](x) = -2*x^2*(8400*x^4-4560*x^3+768*x^2-20*x-1)/ (1+2*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 192, 12424, 876864, 61267872, 4289440512, 300249309824, 21017519192064, 1471225181819392, 102985769424248832, 7209003743212161024, 504630262692606296064, 35324118376822304448512, 2472688286444267160010752, 173088180049932269626753024, 12116172603501926951093796864, 848132082245018228373229535232, 59369245757151942703572710326272, 4155847203000624322888102928973824 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [5, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (164800 x - 185920 x + 60016 x - 4112 x - 708 x + 80 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[5, 1, 1, 1, 1]](x) = -x*(164800*x^6-185920*x^5+60016*x^4-\ 4112*x^3-708*x^2+80*x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/( 10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 1, 89, 4601, 325881, 22689721, 1588780089, 111203172921, 7784275823161, 544898185782841, 38142878497336889, 2670001383329386041, 186900097386394832441, 13083006805923302280761, 915810476470107695058489, 64106733351795954732731961, 4487471334631269555238833721, 314122993424077740786950311481, 21988609539685997308914142776889, 1539202667778008699902847210458681 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [4, 4, 1]](x) = - 2 x 5 4 3 2 (5600 x - 9840 x + 752 x + 376 x - 42 x + 1)/((1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[4, 4, 1]](x) = -2*x^2*(5600*x^5-9840*x^4+752*x^3+376*x^2-42* x+1)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 76, 5592, 388464, 27235552, 1906283456, 133444625792, 9341105899264, 653877905751552, 45771451693710336, 3204001668310177792, 224280116613563838464, 15699608167940638564352, 1098972571739125297954816, 76928080022238455431790592, 5384965601555023324944728064, 376947592108901621430195453952, 26386331447622946765614392016896, 1847043201333611273186263673864192 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [4, 3, 2]](x) = 2 2 2 x (260 x + 4 x + 1) - ------------------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[4, 3, 2]](x) = -2*x^2*(260*x^2+4*x+1)/(1+2*x)/(-1+2*x)/(-1+ 10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 148, 11088, 776992, 54464352, 3812587968, 266888657408, 18682214351872, 1307755754629632, 91542903657407488, 6408003331018518528, 448560233254629629952, 31399216335324074074112, 2197945143481018518519808, 153856160044421296296296448, 10769931203110324074074079232, 753895184217797685185185185792, 52772662895245921296296296316928, 3694086402667221990740740740743168 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [4, 3, 1, 1]](x) = 2 2 2 x (200 x - 30 x - 1) --------------------------------------------- (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[4, 3, 1, 1]](x) = 2*x^2*(200*x^2-30*x-1)/(-1+2*x)/(-1+10*x)/ (10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 204, 14208, 1000416, 70020832, 4902041664, 343142083328, 24020004166656, 1681400208333312, 117698020416666624, 8238861420833333248, 576720300041666666496, 40370421002083333332992, 2825929470204166666665984, 197815062914208333333331968, 13847054404000416666666663936, 969293808280020833333333327872, 67850566579602041666666666655744, 4749539660572142083333333333311488 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [4, 2, 2, 1]](x) = 2 2 2 x (200 x - 30 x - 1) --------------------------------------------- (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[4, 2, 2, 1]](x) = 2*x^2*(200*x^2-30*x-1)/(-1+2*x)/(-1+10*x)/ (10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 204, 14208, 1000416, 70020832, 4902041664, 343142083328, 24020004166656, 1681400208333312, 117698020416666624, 8238861420833333248, 576720300041666666496, 40370421002083333332992, 2825929470204166666665984, 197815062914208333333331968, 13847054404000416666666663936, 969293808280020833333333327872, 67850566579602041666666666655744, 4749539660572142083333333333311488 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [4, 2, 1, 1, 1]](x) = 2 4 3 2 2 x (8400 x - 4560 x + 768 x - 20 x - 1) - ------------------------------------------------------------------ (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[4, 2, 1, 1, 1]](x) = -2*x^2*(8400*x^4-4560*x^3+768*x^2-20*x-\ 1)/(1+2*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 192, 12424, 876864, 61267872, 4289440512, 300249309824, 21017519192064, 1471225181819392, 102985769424248832, 7209003743212161024, 504630262692606296064, 35324118376822304448512, 2472688286444267160010752, 173088180049932269626753024, 12116172603501926951093796864, 848132082245018228373229535232, 59369245757151942703572710326272, 4155847203000624322888102928973824 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (11200 x - 480 x - 4656 x + 756 x + 12 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = x^2*(11200*x^5-480*x^4-4656*x^3+756* x^2+12*x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 1, 63, 3719, 260311, 18156711, 1270992583, 88963072199, 6227417751751, 435918603414471, 30514302515509703, 2136001112191668679, 149520077881174081991, 10466405445293214847431, 732648381173302489018823, 51285386681492284030153159, 3589977067704737654639129031, 251298394739267746914852827591, 17590887631748770061733485375943, 1231362134222407515432119126749639 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [3, 3, 3]](x) = 3 2 32 x (160 x - 10 x - 1) - ------------------------------------------------------------------ (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[3, 3, 3]](x) = -32*x^3*(160*x^2-10*x-1)/(1+2*x)/(-1+2*x)/(-1 +10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 0, 32, 2752, 193920, 13615360, 953121792, 66722138112, 4670551347200, 326938937712640, 22885725700349952, 1602000832720617472, 112140058292620001280, 7849804083829794078720, 549486285868163949658112, 38464040011105279994232832, 2692482800777372420706140160, 188473796054449419709421977600, 13193165723811459481219494838272, 923521600666805497628057709576192 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [3, 3, 2, 1]](x) = 2 2 2 x (260 x + 4 x + 1) - ------------------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[3, 3, 2, 1]](x) = -2*x^2*(260*x^2+4*x+1)/(1+2*x)/(-1+2*x)/(-\ 1+10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 148, 11088, 776992, 54464352, 3812587968, 266888657408, 18682214351872, 1307755754629632, 91542903657407488, 6408003331018518528, 448560233254629629952, 31399216335324074074112, 2197945143481018518519808, 153856160044421296296296448, 10769931203110324074074079232, 753895184217797685185185185792, 52772662895245921296296296316928, 3694086402667221990740740740743168 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 2 2 2 x (20 x - 20 x + 1) - ------------------------------------------- (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -2*x^2*(20*x^2-20*x+1)/(1+2*x)/(-1+4*x) /(-1+6*x)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 116, 7960, 555920, 38905952, 2723361856, 190634986880, 13344446961920, 934111274378752, 65387789127812096, 4577145238470809600, 320400166690085048320, 22428011668288662167552, 1569960816780102358286336, 109897257174606540135301120, 7692808002222454055865221120, 538496560155571761373182820352, 37694759210890023160835507224576, 2638633144762301620446509795901440 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [3, 2, 2, 2]](x) = - 2 x 5 4 3 2 (5600 x - 9840 x + 752 x + 376 x - 42 x + 1)/((1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[3, 2, 2, 2]](x) = -2*x^2*(5600*x^5-9840*x^4+752*x^3+376*x^2-\ 42*x+1)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 76, 5592, 388464, 27235552, 1906283456, 133444625792, 9341105899264, 653877905751552, 45771451693710336, 3204001668310177792, 224280116613563838464, 15699608167940638564352, 1098972571739125297954816, 76928080022238455431790592, 5384965601555023324944728064, 376947592108901621430195453952, 26386331447622946765614392016896, 1847043201333611273186263673864192 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 2 2 2 x (300 x + 8 x + 1) ------------------------------------------- (1 + 2 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[3, 2, 2, 1, 1]](x) = 2*x^2*(300*x^2+8*x+1)/(1+2*x)/(-1+6*x)/ (10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 144, 10760, 749568, 52527392, 3676464384, 257357786240, 18014996717568, 1261050280305152, 88273514681831424, 6179146078090987520, 432540224968545927168, 30277815752811275558912, 2119447102646867653361664, 148361297185781205920153600, 10385290802999687235520954368, 726970356210028123413125660672, 50887924934701468740478754095104, 3562154745429107812442872524308480 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = 2 x 5 4 3 2 (28000 x - 14800 x + 1680 x - 152 x + 30 x - 1)/((1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = 2*x^2*(28000*x^5-14800*x^4+1680*x^3-\ 152*x^2+30*x-1)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 100, 7064, 486480, 34054112, 2382954560, 166806802304, 11676392669440, 817347485660672, 57214315654763520, 4005002095780665344, 280350145870981222400, 19624510210966631800832, 1373715714684318324244480, 96160100027902206301405184, 6731207001944820645454479360, 471184490136137442392199790592, 32982914309528787617319979581440, 2308804001667015133111342555725824 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (56000 x - 21120 x - 3248 x + 3072 x - 604 x + 56 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^2*(56000*x^6-21120*x^5-3248*x^ 4+3072*x^3-604*x^2+56*x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/ (10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 1, 25, 1929, 129657, 9086201, 635451705, 44482351929, 3113704604217, 217959384410681, 15257150837308985, 1068000564406492729, 74760038898784325177, 5233202723479124463161, 366324190582479680081465, 25642693340829445961846329, 1794988533851951984333524537, 125649197369642205732843523641, 8795443815874343357852578713145, 615681067111204591011307912924729 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [2, 2, 2, 2, 1]](x) = 2 3 2 2 x (1400 x - 12 x - 50 x + 1) - ------------------------------------------------------ (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[2, 2, 2, 2, 1]](x) = -2*x^2*(1400*x^3-12*x^2-50*x+1)/(1+2*x) /(-1+4*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 36, 2856, 193968, 13623392, 953114176, 66722869376, 4670550175488, 326939008445952, 22885725569135616, 1602000839710877696, 112140058279005073408, 7849804084525878566912, 549486285866784886112256, 38464040011174783331106816, 2692482800777233884933193728, 188473796054456366271607734272, 13193165723811445605033235972096, 923521600666806192148702663540736 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 3 2 2 x (1000 x + 140 x - 50 x + 1) - ------------------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = -2*x^2*(1000*x^3+140*x^2-50*x+1)/(1+ 2*x)/(-1+2*x)/(-1+10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 40, 3288, 221760, 15573152, 1089287040, 76255092608, 5337773148160, 373644620370432, 26155115092592640, 1830858106481481728, 128160066620370370560, 8971204668425925926912, 627984326706481481482240, 43958902869953703703707648, 3077123200888425925925928960, 215398624062239814814814830592, 15077903684355953703703703715840, 1055453257904921759259259259322368 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 F[[5, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 2 4 3 2 2 x (6000 x - 1440 x - 360 x + 64 x - 1) - ------------------------------------------------------------------ (1 + 2 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x) and in Maple notation F[[5, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -2*x^2*(6000*x^4-1440*x^3-360*x^2 +64*x-1)/(1+2*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 2, 24, 1912, 124800, 8765472, 612726144, 42894023552, 3002497474560, 210175151514112, 14712252242417664, 1029857690121172992, 72090037474060369920, 5046302626511028887552, 353241183772399506653184, 24726882864401063706591232, 1730881800499739715572858880, 121161726035015104960103841792, 8481320822450223963093956296704, 593692457571519010445230404534272 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (8000 x + 320 x - 1232 x - 324 x + 68 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^2*(8000*x^5+320*x^4-1232*x^3 -324*x^2+68*x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 1, 7, 599, 36887, 2600231, 181538887, 12709646279, 889627936967, 62274149710791, 4359185756250567, 305143022376853959, 21360011094137045447, 1495200778533955588551, 104664054450154340872647, 7326483811705246993117639, 512853866814645062046544327, 35899770677044598766704685511, 2512983947392649691363115037127, 175908876317487422839526534181319 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 2 F[[5, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (164800 x - 185920 x + 60016 x - 4112 x - 708 x + 80 x - 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation F[[5, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^2*(164800*x^6-185920*x^5 +60016*x^4-4112*x^3-708*x^2+80*x-1)/(-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/ (-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 0, 1, 1, 89, 4601, 325881, 22689721, 1588780089, 111203172921, 7784275823161, 544898185782841, 38142878497336889, 2670001383329386041, 186900097386394832441, 13083006805923302280761, 915810476470107695058489, 64106733351795954732731961, 4487471334631269555238833721, 314122993424077740786950311481, 21988609539685997308914142776889 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (230400 x - 396800 x + 18944 x + 174688 x - 67200 x + 5224 x + 668 x - 80 x + 1)/((-1 + x) (1 + 2 x) (-1 + 4 x) (-1 + 2 x) (-1 + 10 x) (-1 + 6 x) (10 x + 1) (-1 + 70 x)) and in Maple notation (230400*x^8-396800*x^7+18944*x^6+174688*x^5-67200*x^4+5224*x^3+668*x^2-80*x+1)/ (-1+x)/(1+2*x)/(-1+4*x)/(-1+2*x)/(-1+10*x)/(-1+6*x)/(10*x+1)/(-1+70*x) The first 20 term , starting with k=1 are 1, 49, 2473, 174329, 12133177, 849519289, 59459771193, 4162205486649, 291353735616057, 20394763692631609, 1427633393908563513, 99934337795055078969, 6995403639204893396537, 489678254766537982479929, 34277477833013054975807033, 2399423448313135119660191289, 167959641381855008247552642617, 11757174896730072765529608392249, 823002242771098648938836300893753, 57610156993976927646719356113751609 Regarding Lambda=, [4, 4, 1] 9 8 7 6 5 F[[4, 4, 1], [9]](x) = (174216 x - 69496 x - 362954 x + 180084 x + 76819 x 4 3 2 - 47363 x + 2234 x + 1240 x - 99 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[9]](x) = (174216*x^9-69496*x^8-362954*x^7+180084*x^6+76819*x^5-\ 47363*x^4+2234*x^3+1240*x^2-99*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2 *x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 2, 148, 11566, 968980, 81328828, 6830946102, 573788810282, 48198118792384, 4048639955604904, 340085728230526906, 28567200777152107798, 2399644859771840413188, 201570168143649837819380, 16931894122986359799498910, 1422279106315728899389656514, 119471444930309485959186857992, 10035601374143032360789436486656, 842990515427973216335049945387714 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [8, 1]](x) = - x 5 4 3 2 (23520 x - 10600 x - 1768 x + 1090 x - 86 x + 1)/((1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[8, 1]](x) = -x^2*(23520*x^5-10600*x^4-1768*x^3+1090*x^2-86*x+1)/(1 +6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 13, 1137, 92443, 7749673, 650611771, 54647257169, 4590306472051, 385584892326105, 32389118844756619, 2720685814573779361, 228537606059858896099, 19197158875969190921897, 1612561345118336559731707, 135455152983458717257571313, 11378232850519781488855339987, 955771559442391188498985974649, 80284810993143073117637792035435, 6743924123423769129800495288866625 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [7, 2]](x) = 2 4 3 2 2 x (7644 x - 1192 x - 637 x + 73 x - 1) - ----------------------------------------------------------------- (1 + 6 x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[7, 2]](x) = -2*x^2*(7644*x^4-1192*x^3-637*x^2+73*x-1)/(1+6*x)/(1+2 *x)/(-1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 46, 3786, 311872, 26150200, 2195765728, 184433688224, 15492273796864, 1301348859654528, 109313273999649280, 9182314594611937792, 771314420038909566976, 64790411200606842484736, 5442394539693370880253952, 457161141318038759807754240, 38401535870488381976599134208, 3225729013117847926278836813824, 270961237101854759126728202911744, 22760743916555177235782231047405568 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [7, 1, 1]](x) = x 6 5 4 3 2 (37632 x + 6776 x - 15224 x + 4406 x - 548 x + 52 x - 1)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[7, 1, 1]](x) = x^2*(37632*x^6+6776*x^5-15224*x^4+4406*x^3-548*x^2+ 52*x-1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 46, 3915, 323306, 27117565, 2277073232, 191264339799, 16066058501182, 1349546921028609, 113361913151664188, 9522400311593251963, 799881620658576907178, 67190056058173912923333, 5643964707806154005603464, 474093035440592985901088607, 39823814976798061005176491094, 3345200458048072714050946356937, 280996838475996605712915737907860, 23603734431983133851272923091052931 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [6, 3]](x) = - 2 x 6 5 4 3 2 (24192 x + 1680 x - 10616 x + 1688 x + 360 x - 56 x + 1)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[6, 3]](x) = -2*x^2*(24192*x^6+1680*x^5-10616*x^4+1688*x^3+360*x^2-\ 56*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 84, 6670, 554248, 46483114, 3903520012, 327881096342, 27541806647376, 2313508891106146, 194334706662749620, 16324114797319003534, 1371225635100085603384, 115182953238172245795098, 9675368070463114588210908, 812730917897295062280106246, 68269397103070291004183066272, 5734629356653669539303433125970, 481708865958848952546300520735876, 40463544740542481971824878000212478 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [6, 2, 1]](x) = x 6 5 4 3 2 (17640 x - 4536 x - 8122 x + 2672 x - 637 x + 117 x - 3)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[6, 2, 1]](x) = x^2*(17640*x^6-4536*x^5-8122*x^4+2672*x^3-637*x^2+ 117*x-3)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 177, 14560, 1211835, 101675688, 8538855017, 717238634850, 60247683988095, 5060800448871748, 425107167305364357, 35709001069946197590, 2999556076092300516155, 251962710198856829522808, 21164867654003016093761697, 1777848882898442396030063530, 149339306162939793197522889015, 12544501717679531563810639637868, 1053738144284976895924299715303037, 88514004119936606684714222042084670 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [6, 1, 1, 1]](x) = - x 6 5 4 3 2 (23520 x - 2296 x - 6492 x - 898 x + 923 x - 103 x + 2)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[6, 1, 1, 1]](x) = -x^2*(23520*x^6-2296*x^5-6492*x^4-898*x^3+923*x^ 2-103*x+2)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 93, 7755, 646035, 54223747, 4554006023, 382526595495, 32132088518595, 2699093439063867, 226723820686525503, 19044800544399749935, 1599763240215044841755, 134380112100912852686787, 11287929415396251016288983, 948186070878160977873456375, 79647629953553773270455257715, 6690400916095552538689118223307, 561993676951984911016363554215663, 47207468863966151496559372481847615 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [5, 4]](x) = x 7 6 5 4 3 2 (49392 x + 21000 x - 21116 x - 8590 x + 5032 x - 377 x - 24 x + 1)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[5, 4]](x) = x^2*(49392*x^7+21000*x^6-21116*x^5-8590*x^4+5032*x^3-\ 377*x^2-24*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14* x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 75, 5809, 484867, 40669619, 3415546477, 286895429017, 24099073751919, 2024320178635327, 170042866927728289, 14283600427945377365, 1199822430437120348131, 100785084079541536949275, 8465947061601213630231861, 711139553159376915348724753, 59735722465175917537757646103, 5017800687071812623973228839863, 421495257713990758379031871734793, 35405601647974642673829840100157581 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [5, 3, 1]](x) = 2 3 2 3 x (1512 x - 420 x - 6 x + 1) ------------------------------------------------------- (1 + 6 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[5, 3, 1]](x) = 3*x^2*(1512*x^3-420*x^2-6*x+1)/(1+6*x)/(-1+2*x)/(-1 +4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 276, 22344, 1869024, 156855744, 13174056000, 1106594048640, 92953533078528, 7808091602196480, 655879622339656704, 55093887263734880256, 4627886515982713331712, 388742467144104583643136, 32654367237326871680729088, 2742966847896564669139156992, 230409215222766947050207051776, 19354374078704800696505453248512, 1625767422611096538907211905892352, 136564463499330615191530739319963648 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 4 3 2 F[[4, 4, 1], [5, 2, 2]](x) = x (264 x + 1444 x - 832 x + 110 x - 3)/( (1 + x) (-1 + 2 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[5, 2, 2]](x) = x^2*(264*x^4+1444*x^3-832*x^2+110*x-3)/(1+x)/(-1+2* x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 202, 16545, 1384060, 116185003, 9758488122, 819698336565, 68854455196120, 5783771366465103, 485836754607106342, 40810286824550480185, 3428064085388058270180, 287957383062358621138803, 24188420175694789450505362, 2031827294736755632125249405, 170673492757584979572254680240, 14336573391632903374772894982103, 1204272164897104594751057458933182, 101158861851355955916871848612756225 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [5, 2, 1, 1]](x) = 2 4 3 2 2 x (588 x - 364 x - 65 x - 36 x + 2) ----------------------------------------------------------------- (1 + 6 x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[5, 2, 1, 1]](x) = 2*x^2*(588*x^4-364*x^3-65*x^2-36*x+2)/(1+6*x)/(1 +2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 4, 312, 26014, 2179240, 182983624, 15369506432, 1291023349984, 108445745335680, 9109439601199744, 765192884285412352, 64276201689623586304, 5399200933659320453120, 453532878311640297048064, 38096761776557240290639872, 3200127989208121436612583424, 268810751093164580900106895360, 22580103091821378150425349750784, 1896728659712933511412468794785792, 159325207415885543414663220668268544 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [5, 1, 1, 1, 1]](x) = 2 3 2 2 x (462 x - 40 x - 26 x + 1) - ----------------------------------------------------- (-1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 4 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[5, 1, 1, 1, 1]](x) = -2*x^2*(462*x^3-40*x^2-26*x+1)/(-1+3*x)/(1+2* x)/(1+6*x)/(-1+4*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 114, 9614, 806710, 67766082, 5692328054, 478155662434, 40165074903270, 3373866295926962, 283404768832111894, 23806000582046469954, 1999704048890988415430, 167975140106848437730642, 14109911768975235996166134, 1185232588593920019112392674, 99559537441889280428102625190, 8363001145118699563005979195122, 702492096189970763250156194882774, 59009336079957544113266903406718594 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [4, 4, 1]](x) = - x 7 6 5 4 3 2 (164472 x - 99848 x - 53062 x + 36572 x - 1723 x - 1189 x + 97 x - 1) /((-1 + x) (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[4, 4, 1]](x) = -x*(164472*x^7-99848*x^6-53062*x^5+36572*x^4-1723*x ^3-1189*x^2+97*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1 +14*x)/(-1+84*x) The first 20 term , starting with k=1 are 1, 2, 148, 11566, 968980, 81328828, 6830946102, 573788810282, 48198118792384, 4048639955604904, 340085728230526906, 28567200777152107798, 2399644859771840413188, 201570168143649837819380, 16931894122986359799498910, 1422279106315728899389656514, 119471444930309485959186857992, 10035601374143032360789436486656, 842990515427973216335049945387714, 70811203295949169141754436236790830 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [4, 3, 2]](x) = - x 6 5 4 3 2 (14112 x - 17304 x - 3216 x + 5934 x - 1032 x - 13 x + 3)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[4, 3, 2]](x) = -x^2*(14112*x^6-17304*x^5-3216*x^4+5934*x^3-1032*x^ 2-13*x+3)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 281, 23083, 1937031, 162647727, 13661740827, 1147575594147, 96396208726007, 8097279510358231, 680171450830989843, 57134401475595448491, 4799289718441097801103, 403140336271867522813215, 33863788245756645587782379, 2844558212628432906576043315, 238942889860576622566648462119, 20071202748285471835831207611879, 1685981030855938132238618020007235, 141622406591898222077654389678965819 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [4, 3, 1, 1]](x) = 2 5 4 3 2 x (20832 x - 2744 x - 3628 x + 850 x + 23 x - 4) - ------------------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[4, 3, 1, 1]](x) = -x^2*(20832*x^5-2744*x^4-3628*x^3+850*x^2+23*x-4 )/(1+x)/(-1+2*x)/(1+6*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 4, 357, 29641, 2489771, 209109853, 17564967763, 1475452589965, 123937957977267, 10410787597961677, 874506146244877619, 73458516115431082189, 6170515351336419027763, 518323289479173078506701, 43539156315787626559329075, 3657289130519678098401758413, 307212286963562215385185956659, 25805832104937955600538465848525, 2167689896814770483940583545844531, 182085951332440471637658623562796237 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [4, 2, 2, 1]](x) = 2 4 3 2 2 x (2352 x - 292 x - 88 x - 17 x + 2) ----------------------------------------------------------------- (1 + 6 x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[4, 2, 2, 1]](x) = 2*x^2*(2352*x^4-292*x^3-88*x^2-17*x+2)/(1+6*x)/( 1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 4, 350, 29616, 2489000, 209101664, 17564837600, 1475450861056, 123937933212800, 10410787254618624, 874506141417920000, 73458516047974608896, 6170515350391302809600, 518323289465945805021184, 43539156315602418609152000, 3657289130517085343827623936, 307212286963525915880778137600, 25805832104937447413118976262144, 2167689896814763369282857372876800, 182085951332440372032653577054846976 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [4, 2, 1, 1, 1]](x) = 2 5 4 3 2 x (19992 x - 1792 x - 2850 x - 178 x + 16 x + 3) ------------------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[4, 2, 1, 1, 1]](x) = x^2*(19992*x^5-1792*x^4-2850*x^3-178*x^2+16*x +3)/(1+x)/(-1+2*x)/(1+6*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 301, 25849, 2177029, 182952059, 15369068365, 1291017193875, 108445659289933, 9109438395720115, 765192867413735629, 64276201453389881651, 5399200930352229983437, 453532878265339942091571, 38096761775909041851583693, 3200127989199046619283755827, 268810751093037533692595752141, 22580103091819599488109638955827, 1896728659712908610148512173182157, 159325207415885194796917047988138803 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [4, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (51744 x - 15064 x - 9368 x + 2246 x + 46 x - 9 x + 1)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(51744*x^6-15064*x^5-9368*x^4+2246*x^ 3+46*x^2-9*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+ 84*x) The first 20 term , starting with k=1 are 0, 1, 89, 7627, 644671, 54202055, 4553717875, 382522468091, 32132031294767, 2699092634570959, 226723809443780011, 19044800386880380355, 1599763238010499261063, 134380112070044860991063, 11287929414964125253934147, 948186070872111060472155019, 79647629953469075367206996959, 6690400916094366762401422735967, 561993676951968310182189136238683, 47207468863965919084677810716745683 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [3, 3, 3]](x) = 2 3 2 x (756 x - 84 x - 14 x + 1) - ----------------------------------------------------- (-1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 4 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[3, 3, 3]](x) = -x^2*(756*x^3-84*x^2-14*x+1)/(-1+3*x)/(1+2*x)/(1+6* x)/(-1+4*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 69, 5759, 484057, 40659339, 3415398161, 286893387187, 24099044994009, 2024319777204107, 170042861301216673, 14283600349215520995, 1199822429334664525481, 100785084064108622914555, 8465947061385144192261105, 711139553156351995723977683, 59735722465133568350615108473, 5017800687071219737238228824683, 421495257713982457953474493339457, 35405601647974526467939811805005251 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [3, 3, 2, 1]](x) = x 6 5 4 3 2 (14112 x - 7560 x - 524 x - 90 x + 39 x + 24 x - 3)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[3, 3, 2, 1]](x) = x^2*(14112*x^6-7560*x^5-524*x^4-90*x^3+39*x^2+24 *x-3)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 270, 22998, 1935408, 162627590, 13661443348, 1147571522730, 96396151166256, 8097278707880862, 680171439576151116, 57134401318148638322, 4799289716236116863944, 403140336241002143256294, 33863788245324504152594724, 2844558212622383083211738874, 238942889860491924099178219872, 20071202748284286062928844013486, 1685981030855921531384131610696572, 141622406591897989665894699861865986 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [3, 3, 1, 1, 1]](x) = 2 x 6 5 4 3 2 (4032 x + 1764 x - 3462 x + 926 x - 15 x - 9 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[3, 3, 1, 1, 1]](x) = 2*x^2*(4032*x^6+1764*x^5-3462*x^4+926*x^3-15* x^2-9*x+1)/(-1+x)/(1+x)/(-1+2*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 192, 16392, 1381930, 116155122, 9758069832, 819692480252, 68854373207910, 5783770218629142, 485836738537403572, 40810286599574637312, 3428064082238396472690, 287957383018263355957562, 24188420175077455737978912, 2031827294728112960149813572, 170673492757463982164598622270, 14336573391631209411065709908382, 1204272164897080879259156868075852, 101158861851355623899985240339705032 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [3, 2, 2, 2]](x) = x 5 4 3 2 (12936 x + 3152 x + 38 x - 738 x + 63 x - 2)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[3, 2, 2, 2]](x) = x^2*(12936*x^5+3152*x^4+38*x^3-738*x^2+63*x-2)/( 1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 133, 11490, 967285, 81309072, 6830646263, 573784752830, 48198061148545, 4048639153631292, 340085716972664443, 28567200619723436970, 2399644857566750635205, 201570168112785111295112, 16931894122554214446096223, 1422279106309679099534593110, 119471444930224787350661093065, 10035601374141846588733405827732, 842990515427956615475485538134403, 70811203295948936730025214406560450 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [3, 2, 2, 1, 1]](x) = 2 3 2 3 x (1176 x - 284 x + 14 x - 1) - ------------------------------------------------------- (1 + 6 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[3, 2, 2, 1, 1]](x) = -3*x^2*(1176*x^3-284*x^2+14*x-1)/(1+6*x)/(-1+ 2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 252, 22104, 1864992, 156803136, 13173296064, 1106583549312, 92953385247744, 7808089537603584, 655879593405121536, 55093886858832783360, 4627886510312995577856, 388742467064735065423872, 32654367236215659243552768, 2742966847881007930111131648, 230409215222549151293259644928, 19354374078701751564371519078400, 1625767422611053851006556848783360, 136564463499330017561226248389656576 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [3, 2, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (38808 x + 20832 x - 6486 x - 9330 x + 1025 x + 533 x - 63 x - 1)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[3, 2, 1, 1, 1, 1]](x) = -x^2*(38808*x^7+20832*x^6-6486*x^5-9330*x^ 4+1025*x^3+533*x^2-63*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4 *x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 162, 14266, 1208260, 101622426, 8538128802, 717228351316, 60247540718520, 5060798438899126, 425107139190942142, 35709000676193123016, 2999556070580664465780, 251962710121688482869226, 21164867652922691892350682, 1777848882883317661299929116, 149339306162728048086763485040, 12544501717676567125207233333726, 1053738144284935393826168675722422, 88514004119936025655086487596765616 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 3 F[[4, 4, 1], [3, 1, 1, 1, 1, 1, 1]](x) = 2 x 6 5 4 3 2 (18816 x + 16828 x - 14072 x + 103 x + 1251 x - 289 x + 22)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[3, 1, 1, 1, 1, 1, 1]](x) = 2*x^3*(18816*x^6+16828*x^5-14072*x^4+ 103*x^3+1251*x^2-289*x+22)/(-1+x)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4 *x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 0, 44, 3778, 322008, 27095492, 2276787422, 191260198430, 16066001361356, 1349546116031944, 113361901911942090, 9522400154055743042, 799881618454140165944, 67190056027305268194956, 5643964707374032161458198, 474093035434543044990546214, 39823814976713363242983731172, 3345200458046886936916917930128, 280996838475980004883819317786146, 23603734431982901439360893339081546 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [2, 2, 2, 2, 1]](x) = - x 6 5 4 3 2 (7056 x - 3192 x - 1364 x - 1114 x + 592 x - 34 x + 1)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[2, 2, 2, 2, 1]](x) = -x^2*(7056*x^6-3192*x^5-1364*x^4-1114*x^3+592 *x^2-34*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84* x) The first 20 term , starting with k=1 are 0, 1, 64, 5723, 483244, 40649477, 3415248998, 286891357579, 24099016192168, 2024319376157873, 170042855672889562, 14283600270498566855, 1199822428232139410972, 100785084048676157390989, 8465947061169072195044206, 711139553153327091984414851, 59735722465091219070287403856, 5017800687070626851070865219625, 421495257713974157524545462424130, 35405601647974410262070150282970367 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[4, 4, 1], [2, 2, 2, 1, 1, 1]](x) = - x 5 4 3 2 (12000 x - 9568 x + 212 x + 504 x - 25 x + 1)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[2, 2, 2, 1, 1, 1]](x) = -x^2*(12000*x^5-9568*x^4+212*x^3+504*x^2-\ 25*x+1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 73, 6517, 552113, 46453233, 3903101701, 327875240029, 27541724659081, 2313507743270185, 194334690593046509, 16324114572343160661, 1371225631950423804529, 115182953194076980613857, 9675368069845780875678997, 812730917888652390304670413, 68269397102949293596526986457, 5734629356651975575596248052249, 481708865958825237054399929791165, 40463544740542149954938269727161285 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[4, 4, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 3 3 2 2 x (3108 x + 392 x - 195 x + 21) ----------------------------------------------------------------- (1 + 6 x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x) and in Maple notation F[[4, 4, 1],[2, 2, 1, 1, 1, 1, 1]](x) = 2*x^3*(3108*x^3+392*x^2-195*x+21)/(1+6* x)/(1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 0, 42, 3642, 310432, 26126808, 2195457824, 184429260960, 15492212515584, 1301347997517696, 109313261954930176, 9182314425834705408, 771314417676935315456, 64790411167533761009664, 5442394539230380391374848, 457161141311556697053044736, 38401535870397634273495810048, 3225729013116577451382615539712, 270961237101836972520497754275840, 22760743916554928223041104874962944 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 3 F[[4, 4, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x 4 3 2 (3360 x - 6664 x + 2220 x - 218 x + 13)/((1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(3360*x^4-6664*x^3+2220*x^2-218 *x+13)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(-1+4*x)/(-1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 0, 13, 1069, 91931, 7739929, 650490939, 54645472273, 4590282043507, 385584546967513, 32389114029892235, 2720685747044746657, 228537605115178034403, 19197158862739305297577, 1612561344933144282387451, 135455152980865868646439921, 11378232850483482548669502419, 955771559441882997694164499321, 80284810993135958480223610401387, 6743924123423669524673576832915265 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 3 F[[4, 4, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (12936 x + 3152 x + 38 x - 738 x + 63 x - 2)/((1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 14 x) (-1 + 84 x)) and in Maple notation F[[4, 4, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(12936*x^5+3152*x^4+38*x^3-\ 738*x^2+63*x-2)/(1+x)/(-1+2*x)/(1+6*x)/(-1+3*x)/(1+2*x)/(-1+4*x)/(-1+14*x)/(-1+ 84*x) The first 20 term , starting with k=1 are 0, 0, 2, 133, 11490, 967285, 81309072, 6830646263, 573784752830, 48198061148545, 4048639153631292, 340085716972664443, 28567200619723436970, 2399644857566750635205, 201570168112785111295112, 16931894122554214446096223, 1422279106309679099534593110, 119471444930224787350661093065, 10035601374141846588733405827732, 842990515427956615475485538134403 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 2 x) (-1 + 3 x) (1 + 6 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+2*x) /(-1+3*x)/(1+6*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 Regarding Lambda=, [4, 3, 2] 11 10 9 8 F[[4, 3, 2], [9]](x) = (12260472 x + 967570 x - 24034609 x + 85018 x 7 6 5 4 3 2 + 9021374 x - 92587 x - 1071843 x + 11427 x + 40135 x - 407 x - 167 x + 1)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[9]](x) = (12260472*x^11+967570*x^10-24034609*x^9+85018*x^8+9021374 *x^7-92587*x^6-1071843*x^5+11427*x^4+40135*x^3-407*x^2-167*x+1)/(1+x)/(-1+2*x)/ (-1+x)/(-1+3*x)/(1+2*x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 1, 12, 2226, 368496, 61963374, 10408761915, 1748684868407, 293778818157718, 49354844391888252, 8291613804562243353, 1392991119837415781673, 234022508120827286030820, 39315781364451743838344750, 6605051269225249908351040951, 1109648613229876707211868773899, 186420967022618697047152111330002, 31318722459799948985433823791831168, 5261545373246391297820837806278952309, 883939622705393739820750371900408619085 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 9 8 7 F[[4, 3, 2], [8, 1]](x) = - x (1881600 x - 1568216 x - 3014522 x 6 5 4 3 2 + 371185 x + 598196 x - 7786 x - 31867 x - 359 x + 235 x - 2)/( (1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[8, 1]](x) = -x^2*(1881600*x^9-1568216*x^8-3014522*x^7+371185*x^6+ 598196*x^5-7786*x^4-31867*x^3-359*x^2+235*x-2)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/( 1+2*x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 2, 99, 17708, 2948915, 495689519, 83270311772, 13989475093274, 2350230594557105, 394838754276791501, 66332910447708030830, 11143928958507873950600, 1872180064969163854162115, 314526250915571200209078923, 52840410153802576586086227248, 8877188905839004102771910577686, 1491367736180949707169187325193845, 250549779678399589746144792918318185, 42092362985971130412170342023767564626, 7071516981643149918087560918347709230532 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 5 F[[4, 3, 2], [7, 2]](x) = - x (1245384 x - 347898 x - 828549 x + 250877 x 4 3 2 + 99612 x - 27336 x - 1643 x + 337 x - 4)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[7, 2]](x) = -x^2*(1245384*x^8-347898*x^7-828549*x^6+250877*x^5+ 99612*x^4-27336*x^3-1643*x^2+337*x-4)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+2*x)/(1 +4*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 4, 343, 59545, 9955064, 1672906522, 281037870228, 47214468325294, 7932028386028432, 1332580793431117870, 223873572790441053788, 37610760234461512543438, 6318607719277610118131640, 1061526096839940580652484958, 178336384269085211441143731748, 29960512557206613765182264044702, 5033366109610705605024929597177888, 845605506414598609782758458535113486, 142061725077652565218784458210602456108, 23866369813045630972289607700187564528686 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [7, 1, 1]](x) = - x (364560 x + 101248 x - 658899 x 5 4 3 2 + 204544 x + 82303 x - 22948 x - 1654 x + 322 x - 4)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[7, 1, 1]](x) = -x^2*(364560*x^8+101248*x^7-658899*x^6+204544*x^5+ 82303*x^4-22948*x^3-1654*x^2+322*x-4)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+4*x)/(1 +3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 4, 354, 61760, 10323448, 1734868377, 291446611155, 48963152900582, 8225807200085626, 1381935637765609835, 232165186594199794441, 39003751354287679462584, 6552630227398279920792624, 1100841878204390119726450253, 184941435538310430482804671087, 31070161170436490040260516143466, 5219787076633324296022211254116742, 876924228874398558683494096681329231, 147323270450898956515419521420706519093, 24750309435751024712093757227752978074428 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[4, 3, 2], [6, 3]](x) = - x ( 7 6 5 4 3 2 193536 x - 62466 x - 124012 x + 74467 x - 11370 x - 920 x + 242 x - 5 )/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[6, 3]](x) = -x^2*(193536*x^7-62466*x^6-124012*x^5+74467*x^4-11370* x^3-920*x^2+242*x-5)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+3*x)/(1+15*x)/(-1+4*x)/( -1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 5, 623, 105589, 17701051, 2973998207, 499623601793, 83936819735305, 14101383961700063, 2369032518794310979, 397997462775928802413, 66863573749515626209901, 11233080389835347651783435, 1887157505493085197285237511, 317042460922820078070735848633, 53263133435033948177247318661777, 8948206417085699289350887384830967, 1503298678070397521378262380729150603, 252554177915826782709851168730506672453, 42429101889858899504697829055280756361333 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 5 F[[4, 3, 2], [6, 2, 1]](x) = x (1664040 x + 305382 x - 954315 x - 277212 x 4 3 2 + 35235 x + 7428 x + 730 x + 282 x - 10)/((1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[6, 2, 1]](x) = x^2*(1664040*x^8+305382*x^7-954315*x^6-277212*x^5+ 35235*x^4+7428*x^3+730*x^2+282*x-10)/(1+x)/(-1+2*x)/(-1+x)/(1+2*x)/(1+4*x)/(1+3 *x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 10, 1358, 230982, 38719920, 6505624395, 1092926388918, 183611794114297, 30846777364908680, 5182258635112026555, 870619449811320457578, 146264067577128845841837, 24572363352762444592895340, 4128157043266139564303009215, 693530383268668405621736965838, 116513104389136765448887367116977, 19574201537374967083460732374208400, 3288465858278994578927286653685727875, 552462264190871087153368283883769963698, 92813660384066342666742523122297566833717 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 9 8 7 F[[4, 3, 2], [6, 1, 1, 1]](x) = - x (526848 x - 538664 x - 701278 x 6 5 4 3 2 - 257703 x + 215999 x + 42146 x - 16191 x - 891 x + 284 x - 6)/( (1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[6, 1, 1, 1]](x) = -x^2*(526848*x^9-538664*x^8-701278*x^7-257703*x^ 6+215999*x^5+42146*x^4-16191*x^3-891*x^2+284*x-6)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x )/(1+2*x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 718, 123245, 20649310, 3469678686, 582893787805, 97926293070563, 16451614531656336, 2763871272726734096, 464330373218815852507, 78007502707956007127541, 13105260454803566606287822, 2201683756408643168910286226, 369882871076622469456690422249, 62140322340872949687217565018279, 10439574153266648960220852126876268, 1753848457748797110616218060346591476, 294646540901797913114906863179515681431, 49500618871502049422685784907627658360377 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[4, 3, 2], [5, 4]](x) = x 7 6 5 4 3 2 (8232 x - 2534 x - 53596 x + 18825 x + 4452 x - 1435 x + x + 3)/( (1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 4 x) (1 + 3 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[5, 4]](x) = x^2*(8232*x^7-2534*x^6-53596*x^5+18825*x^4+4452*x^3-\ 1435*x^2+x+3)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+4*x)/(1+3*x)/(-1+4*x)/(-1+14*x) /(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 553, 92247, 15490090, 2602218056, 437171030415, 73444710525791, 12338711052754660, 2072903452442993094, 348247779948555350017, 58505627030491131683525, 9828945341110383935666310, 1651262817306374734257612092, 277412153307468578620630206259, 46605241755654687933976143630699, 7829680614949987107067974722253640, 1315386343311597827465659438567077650, 220984905676348434922926141892881473541, 37125464153626537065773326823887612963313 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[4, 3, 2], [5, 3, 1]](x) = 2 5 4 3 2 x (252 x + 14391 x - 8681 x + 286 x + 278 x - 13) ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x) and in Maple notation F[[4, 3, 2],[5, 3, 1]](x) = x^2*(252*x^5+14391*x^4-8681*x^3+286*x^2+278*x-13)/( -1+x)/(1+x)/(-1+3*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 13, 2127, 355756, 59746497, 10037113465, 1686230928486, 283286737965244, 47592171166577799, 7995484744620743407, 1343241436937196255360, 225664561403221697471002, 37911646315710063544388481, 6369156581038854132125480989, 1070018305614521382593648468874, 179763075343239506713277036033080, 30200196657664235929956211267809243, 5073633038487591619462402737846413611, 852370350465915391834900290221254799028, 143198218878273785824976281578880960292278 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[4, 3, 2], [5, 2, 2]](x) = 2 3 2 x (1274 x + 192 x - 227 x + 10) - ----------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 14 x) (-1 + 168 x) and in Maple notation F[[4, 3, 2],[5, 2, 2]](x) = -x^2*(1274*x^3+192*x^2-227*x+10)/(1+x)/(-1+2*x)/(1+ 3*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 10, 1573, 263502, 44256295, 7434893930, 1249059877083, 209842027146902, 35253460109722615, 5922581292120362850, 994993656987837402643, 167158934372719317065102, 28082700974599522125352335, 4717893763732477193105727770, 792606152307052773106328161003, 133157833587584818347167311524102, 22370516042714248816838366091164455, 3758246695175993791912045114206404690, 631385444789566956910788373732198436163, 106072754724647248759186353910667514939902 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [5, 2, 1, 1]](x) = - x (927864 x + 125034 x - 432759 x 5 4 3 2 + 76527 x + 14931 x - 9824 x + 1387 x + 260 x - 16)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[5, 2, 1, 1]](x) = -x^2*(927864*x^8+125034*x^7-432759*x^6+76527*x^5 +14931*x^4-9824*x^3+1387*x^2+260*x-16)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+2*x)/( 1+4*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 16, 2460, 415181, 69699930, 11709997491, 1967268484503, 330501201896938, 55524199491107364, 9328065537190962461, 1567115009715584909481, 263275321637514477782760, 44230254034985311414213638, 7430682677878761641323487611, 1248354689883606131034476484219, 209723587900446113996455229022422, 35233562767274941444233084621847752, 5919238544902190227974688414561066041, 994432075543567957035898129498396774317, 167064588691319416797016876614089413540324 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[4, 3, 2], [5, 1, 1, 1, 1]](x) = 2 4 3 2 x (6125 x - 993 x - 1137 x + 19 x - 6) --------------------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x) and in Maple notation F[[4, 3, 2],[5, 1, 1, 1, 1]](x) = x^2*(6125*x^4-993*x^3-1137*x^2+19*x-6)/(-1+x) /(1+2*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 899, 153894, 25812030, 4337065416, 728617348494, 122407859325324, 20564518195444170, 3454839089405091576, 580412966531506282614, 97509378384621327636804, 16381575568506459092082210, 2752104695510733987291725136, 462353588845778576969818213134, 77675402926091171924366169597084, 13049467691583311318392000330798650, 2192310572185996384963378938500877096, 368308176127247391421744818978593094054, 61875773589377561777634672230987312146164 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 5 F[[4, 3, 2], [4, 4, 1]](x) = x (1192464 x + 205800 x - 674433 x - 70518 x 4 3 2 + 116360 x - 1713 x - 3871 x + 97 x + 6)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[4, 4, 1]](x) = x^2*(1192464*x^8+205800*x^7-674433*x^6-70518*x^5+ 116360*x^4-1713*x^3-3871*x^2+97*x+6)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+4*x)/(1+ 3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 1111, 184308, 30981529, 5204394073, 874342388808, 146889411883122, 24677422183597027, 4145806902882438435, 696495559915513348870, 117011254060543115310076, 19657890682225071586254945, 3302525634612652943710577277, 554824306614938157546365133172, 93210483511309354597438316285670, 15659361229899974445470538036387583, 2630772686623195650233176357050183799, 441969811352696869899130689951912402114, 74250928307253074130506765312435962913904 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 9 8 7 F[[4, 3, 2], [4, 3, 2]](x) = - x (8060808 x - 1185298 x - 4991183 x 6 5 4 3 2 + 225518 x + 747006 x - 11427 x - 32148 x + 186 x + 155 x - 1)/( (1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[4, 3, 2]](x) = -x*(8060808*x^9-1185298*x^8-4991183*x^7+225518*x^6+ 747006*x^5-11427*x^4-32148*x^3+186*x^2+155*x-1)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/ (1+2*x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 12, 2226, 368496, 61963374, 10408761915, 1748684868407, 293778818157718, 49354844391888252, 8291613804562243353, 1392991119837415781673, 234022508120827286030820, 39315781364451743838344750, 6605051269225249908351040951, 1109648613229876707211868773899, 186420967022618697047152111330002, 31318722459799948985433823791831168, 5261545373246391297820837806278952309, 883939622705393739820750371900408619085, 148501856614506148260435483502061663210464 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [4, 3, 1, 1]](x) = - x (516096 x - 953880 x + 48630 x 5 4 3 2 + 452521 x - 131528 x - 16716 x + 6855 x - 118 x - 16)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[4, 3, 1, 1]](x) = -x^2*(516096*x^8-953880*x^7+48630*x^6+452521*x^5 -131528*x^4-16716*x^3+6855*x^2-118*x-16)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+2*x) /(1+4*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 16, 2838, 473973, 79662906, 13382739134, 2248308177081, 377715633792499, 63456228296192028, 10660646322553051632, 1790988582602095721559, 300886081870185428870885, 50548861754284886725748730, 8492208774718304238946176850, 1426691074152696353148987947397, 239684100457652639174528975600631, 40266928876885648190086525531046712, 6764844051316788818013325590833213588, 1136493800621220522514000696295920218195, 190930958504365047764900900736698303189737 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 5 F[[4, 3, 2], [4, 2, 2, 1]](x) = x (451584 x + 508632 x - 4014 x - 339371 x 4 3 2 + 70037 x + 22622 x - 6216 x + 114 x + 16)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[4, 2, 2, 1]](x) = x^2*(451584*x^8+508632*x^7-4014*x^6-339371*x^5+ 70037*x^4+22622*x^3-6216*x^2+114*x+16)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+2*x)/( 1+4*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 16, 2834, 473932, 79662250, 13382730241, 2248308051321, 377715632036710, 63456228271591196, 10660646322208718371, 1790988582597274740823, 300886081870117936397848, 50548861754283941826091002, 8492208774718291010371097221, 1426691074152696167948856293765, 239684100457652636581727454555946, 40266928876885648153787302947898168, 6764844051316788817505136479823029191, 1136493800621220522506886048721161662547, 190930958504365047764801295670734146520604 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [4, 2, 1, 1, 1]](x) = x (342216 x - 637602 x + 111693 x 5 4 3 2 + 125658 x - 14107 x - 6148 x + 220 x - 101 x + 15)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[4, 2, 1, 1, 1]](x) = x^2*(342216*x^8-637602*x^7+111693*x^6+125658* x^5-14107*x^4-6148*x^3+220*x^2-101*x+15)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+2*x) /(1+4*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 15, 2449, 415020, 69697694, 11709966102, 1967268045351, 330501195747549, 55524199405020772, 9328065535985730384, 1567115009698711738793, 263275321637278253078043, 44230254034982004269604870, 7430682677878715341293929646, 1248354689883605482834082801275, 209723587900446104921649636922377, 35233562767274941317185806654553288, 5919238544902190226196026521730420588, 994432075543567957010996863003921633197, 167064588691319416796668258883146145590551 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 9 8 7 F[[4, 3, 2], [4, 1, 1, 1, 1, 1]](x) = - x (526848 x + 1094296 x + 713306 x 6 5 4 3 2 - 518685 x - 110098 x + 64088 x + 4018 x - 2356 x + 124 x - 5)/( (1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[4, 1, 1, 1, 1, 1]](x) = -x^2*(526848*x^9+1094296*x^8+713306*x^7-\ 518685*x^6-110098*x^5+64088*x^4+4018*x^3-2356*x^2+124*x-5)/(1+x)/(-1+2*x)/(-1+x )/(-1+3*x)/(1+2*x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 5, 711, 123133, 20647831, 3469657698, 582893495254, 97926288970039, 16451614474268873, 2763871271923231396, 464330373207567130132, 78007502707798523757705, 13105260454801361844146695, 2201683756408612302220184614, 369882871076622037323109543970, 62140322340872943637347110627131, 10439574153266648875522667053944997, 1753848457748797109430443464171702152, 294646540901797913098306018853683292368, 49500618871502049422453373086983541978317 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[4, 3, 2], [3, 3, 3]](x) = - x 6 5 4 3 2 (3591 x - 12840 x - 7996 x + 185 x + 1124 x - 93 x - 3)/((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[3, 3, 3]](x) = -x^2*(3591*x^6-12840*x^5-7996*x^4+185*x^3+1124*x^2-\ 93*x-3)/(1+x)/(-1+3*x)/(-1+x)/(1+2*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 558, 92062, 15491439, 2602176046, 437171358393, 73444701357856, 12338711130842367, 2072903450439453994, 348247779966957998853, 58505627030051983766020, 9828945341114687650588635, 1651262817306278209455200782, 277412153307469578925734926913, 46605241755654666663462208440904, 7829680614949987338402563314133943, 1315386343311597822767516919055746610, 220984905676348434976204548059030928573, 37125464153626537064733438488557512460108 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [3, 3, 2, 1]](x) = x (3007032 x - 2270486 x - 1596583 x 5 4 3 2 + 394591 x + 178241 x - 19805 x - 6003 x + 378 x + 11)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[3, 3, 2, 1]](x) = x^2*(3007032*x^8-2270486*x^7-1596583*x^6+394591* x^5+178241*x^4-19805*x^3-6003*x^2+378*x+11)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+2 *x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 11, 2215, 368390, 61961855, 10408741053, 1748684575288, 293778814059336, 49354844334491965, 8291613803758775435, 1392991119826166919146, 234022508120669803219542, 39315781364449539073965295, 6605051269225219041669884457, 1109648613229876275078252098764, 186420967022618690997281800093508, 31318722459799948900735638146215745, 5261545373246391296635063212394668519, 883939622705393739804149527565413545742, 148501856614506148260203071681454197041234 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[4, 3, 2], [3, 3, 1, 1, 1]](x) = 2 3 2 x (742 x - 36 x + 58 x - 9) ----------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 3 x) (-1 + 14 x) (-1 + 168 x) and in Maple notation F[[4, 3, 2],[3, 3, 1, 1, 1]](x) = x^2*(742*x^3-36*x^2+58*x-9)/(1+x)/(-1+2*x)/(1 +3*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 9, 1562, 263349, 44254160, 7434864049, 1249059458772, 209842021290589, 35253460027734320, 5922581290972526889, 994993656971767699532, 167158934372494341222229, 28082700974596372463553480, 4717893763732433097840546529, 792606152307052155772615629092, 133157833587584809704495336088269, 22370516042714248695840958435084640, 3758246695175993790218081407021330969, 631385444789566956887072881831607491452, 106072754724647248758854337024059241888709 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [3, 2, 2, 2]](x) = - x (77616 x + 279552 x - 440325 x 5 4 3 2 + 159960 x + 4611 x - 14298 x + 2450 x - 88 x - 6)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[3, 2, 2, 2]](x) = -x^2*(77616*x^8+279552*x^7-440325*x^6+159960*x^5 +4611*x^4-14298*x^3+2450*x^2-88*x-6)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+4*x)/(1+ 3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 1102, 184208, 30980010, 5204373261, 874342095585, 146889407785350, 24677422126198684, 4145806902078979511, 696495559904264451903, 117011254060385632639632, 19657890682222866821318298, 3302525634612622077031661841, 554824306614937725412739518381, 93210483511309348547568040856954, 15659361229899974360772352247639352, 2630772686623195649047401763738627851, 441969811352696869882529845614626810619, 74250928307253074130274353491837659603716 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[4, 3, 2], [3, 2, 2, 1, 1]](x) = 2 5 4 3 2 3 x (588 x + 2199 x - 732 x + 150 x - 38 x + 4) - ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x) and in Maple notation F[[4, 3, 2],[3, 2, 2, 1, 1]](x) = -3*x^2*(588*x^5+2199*x^4-732*x^3+150*x^2-38*x +4)/(-1+x)/(1+x)/(-1+3*x)/(1+4*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 12, 2106, 355488, 59742630, 10037059785, 1686230175123, 283286727425532, 47592171018992340, 7995484742554664943, 1343241436908270685005, 225664561402816741373466, 37911646315704394151473230, 6369156581038774760654866461, 1070018305614520271392939069527, 179763075343239491156467587626040, 30200196657664235712160877057375400, 5073633038487591616413268066631280939, 852370350465915391792212404793319177089, 143198218878273785824378651183013556643254 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [3, 2, 1, 1, 1, 1]](x) = - x (452760 x - 695982 x - 491763 x 5 4 3 2 - 60415 x + 426 x + 10852 x + 2688 x - 135 x + 9)/((1 + x) (-1 + 2 x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[3, 2, 1, 1, 1, 1]](x) = -x^2*(452760*x^8-695982*x^7-491763*x^6-\ 60415*x^5+426*x^4+10852*x^3+2688*x^2-135*x+9)/(1+x)/(-1+2*x)/(-1+x)/(1+2*x)/(1+ 4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 9, 1341, 230712, 38716210, 6505572029, 1092925657236, 183611783864427, 30846777221434710, 5182258633103291949, 870619449783198564856, 146264067576735137767967, 24572363352756932686146810, 4128157043266062397583352369, 693530383268667325287762410076, 116513104389136750324211320636707, 19574201537374966871715269334004510, 3288465858278994575962850164680236789, 552462264190871087111866173063462520896, 92813660384066342666161493570710182676647 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 6 F[[4, 3, 2], [3, 1, 1, 1, 1, 1, 1]](x) = x (905520 x + 21224 x - 115011 x 5 4 3 2 - 29926 x - 20651 x + 567 x + 2625 x - 159 x + 3)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[3, 1, 1, 1, 1, 1, 1]](x) = x^2*(905520*x^8+21224*x^7-115011*x^6-\ 29926*x^5-20651*x^4+567*x^3+2625*x^2-159*x+3)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1 +4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 348, 61647, 10321980, 1734847360, 291446318751, 48963148799533, 8225807142700390, 1381935636962098482, 232165186582951107189, 39003751354130195953279, 6552630227396075159211420, 1100841878204359253034113044, 184941435538309998349232743387, 31070161170436483990390025966385, 5219787076633324211324026324361970, 876924228874398557497719499933799446, 147323270450898956498818677097164822945, 24750309435751024711861345407099699182851 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[4, 3, 2], [2, 2, 2, 2, 1]](x) = x 7 6 5 4 3 2 (8232 x + 27706 x - 12952 x + 3126 x + 8 x - 64 x - 7 x + 3)/((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 4 x) (1 + 3 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[2, 2, 2, 2, 1]](x) = x^2*(8232*x^7+27706*x^6-12952*x^5+3126*x^4+8* x^3-64*x^2-7*x+3)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+4*x)/(1+3*x)/(-1+4*x)/(-1+ 14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 545, 92146, 15488582, 2602197215, 437170737339, 73444706427494, 12338710995358544, 2072903451639525517, 348247779937306488173, 58505627030333648873612, 9828945341108179171289586, 1651262817306343867576461059, 277412153307468146487013542047, 46605241755654681884105832416050, 7829680614949987022369789076681908, 1315386343311597826279884844682881241, 220984905676348434906325297557886574961, 37125464153626537065540915003280147143608 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[4, 3, 2], [2, 2, 2, 1, 1, 1]](x) = 2 x ( 7 6 5 4 3 2 84672 x - 229911 x + 127310 x - 13253 x - 4540 x + 1024 x - 40 x + 2) /((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[2, 2, 2, 1, 1, 1]](x) = 2*x^2*(84672*x^7-229911*x^6+127310*x^5-\ 13253*x^4-4540*x^3+1024*x^2-40*x+2)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/(1+3*x)/(1+ 15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 4, 612, 105436, 17698916, 2973968326, 499623183482, 83936813878992, 14101383879711768, 2369032517646475018, 397997462759859099302, 66863573749290650367028, 11233080389832197989984580, 1887157505493041102020056270, 317042460922819460737023316722, 53263133435033939534575343225944, 8948206417085699168353479728751152, 1503298678070397519684298673544076882, 252554177915826782686135676829915727742, 42429101889858899504365812168672483310140 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 8 7 F[[4, 3, 2], [2, 2, 1, 1, 1, 1, 1]](x) = - x (942984 x - 280578 x 6 5 4 3 2 - 462867 x + 161842 x + 37297 x - 5458 x - 2611 x + 174 x - 3)/( (1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[2, 2, 1, 1, 1, 1, 1]](x) = -x^2*(942984*x^8-280578*x^7-462867*x^6+ 161842*x^5+37297*x^4-5458*x^3-2611*x^2+174*x-3)/(1+x)/(-1+2*x)/(-1+x)/(-1+3*x)/ (1+2*x)/(1+4*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 336, 59425, 9953484, 1672884026, 281037556836, 47214463931694, 7932028324542672, 1332580792570219054, 223873572778388863836, 37610760234292780311758, 6318607719275247873180600, 1061526096839907509198006622, 178336384269084748440881702436, 29960512557206607283178192989342, 5033366109610705514276874213031968, 845605506414598608512285676714652430, 142061725077652565200997839290885870636, 23866369813045630972040595035208453248046 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 9 8 F[[4, 3, 2], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x (1021440 x - 1340008 x 7 6 5 4 3 2 - 444862 x + 23645 x + 16137 x - 7812 x + 858 x + 1216 x - 71 x + 1) /((1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^2*(1021440*x^9-1340008*x^8-444862* x^7+23645*x^6+16137*x^5-7812*x^4+858*x^3+1216*x^2-71*x+1)/(1+x)/(-1+2*x)/(-1+x) /(-1+3*x)/(1+2*x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 1, 96, 17656, 2948288, 495680479, 83270186537, 13989473335258, 2350230569964926, 394838753932423117, 66332910442887189563, 11143928958440380917640, 1872180064968218956739984, 314526250915557971625048715, 52840410153802391385990359549, 8877188905839001509970246356502, 1491367736180949670869965314685762, 250549779678399589237955679617440873, 42092362985971130405055694458171518495, 7071516981643149917987955852346901999044 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 3 8 7 F[[4, 3, 2], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x (3007032 x - 2270486 x 6 5 4 3 2 - 1596583 x + 394591 x + 178241 x - 19805 x - 6003 x + 378 x + 11)/( (1 + x) (-1 + 2 x) (-1 + x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 14 x) (-1 + 168 x)) and in Maple notation F[[4, 3, 2],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(3007032*x^8-2270486*x^7-\ 1596583*x^6+394591*x^5+178241*x^4-19805*x^3-6003*x^2+378*x+11)/(1+x)/(-1+2*x)/( -1+x)/(-1+3*x)/(1+2*x)/(1+4*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 0, 11, 2215, 368390, 61961855, 10408741053, 1748684575288, 293778814059336, 49354844334491965, 8291613803758775435, 1392991119826166919146, 234022508120669803219542, 39315781364449539073965295, 6605051269225219041669884457, 1109648613229876275078252098764, 186420967022618690997281800093508, 31318722459799948900735638146215745, 5261545373246391296635063212394668519, 883939622705393739804149527565413545742 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 Regarding Lambda=, [4, 3, 1, 1] 9 8 7 6 F[[4, 3, 1, 1], [9]](x) = (858438 x - 353295 x - 1761216 x + 827015 x 5 4 3 2 + 224121 x - 85676 x - 12381 x + 1787 x + 208 x - 1)/((1 + x) (-1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[9]](x) = (858438*x^9-353295*x^8-1761216*x^7+827015*x^6+224121*x ^5-85676*x^4-12381*x^3+1787*x^2+208*x-1)/(1+x)/(-1+x)/(-1+2*x)/(1+4*x)/(-1+3*x) /(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 1, 27, 6009, 1295651, 279872452, 60452351983, 13057708740740, 2820465082120407, 609220457786042208, 131591618881373140919, 28423789678380141747076, 6139538570530079461214143, 1326140331234497438236850144, 286446311546651444213923348335, 61872403294076711972007770507892, 13364439111520569785758419034260359, 2886718848088443073725568424250405760, 623531271187103703924707064330219916231, 134682754576414400047736867066253435717188 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [8, 1]](x) = - x 6 5 4 3 2 (165888 x + 87102 x - 44055 x - 9732 x + 1304 x + 195 x - 2)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[8, 1]](x) = -x^2*(165888*x^6+87102*x^5-44055*x^4-9732*x^3+1304* x^2+195*x-2)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x ) The first 20 term , starting with k=1 are 0, 2, 219, 48019, 10365436, 2238977763, 483618828260, 104461669829315, 22563720657710592, 4873763662282204219, 1052732951051036233396, 227390317427040695669931, 49116308564240639490271568, 10609122649875979472469520595, 2291570492373211554007060944852, 494979226352613695773429134164467, 106915512892164558286090864508894464, 23093750784707544589804336792352092491, 4988250169496829631397658403402113326228, 1077462036611315200381894919567471536042523 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [7, 2]](x) = - x 6 5 4 3 2 (196830 x + 98091 x - 68391 x - 13341 x + 2176 x + 290 x - 5)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[7, 2]](x) = -x^2*(196830*x^6+98091*x^5-68391*x^4-13341*x^3+2176 *x^2+290*x-5)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216* x) The first 20 term , starting with k=1 are 0, 5, 745, 162014, 34983679, 7556547269, 1632213566689, 352558135492583, 76152557221317943, 16448952360188882093, 3552973709797366334293, 767442321316261288303787, 165767541404312167711777867, 35805788943331430635111880177, 7734050411759588995530378959257, 1670554888940071223228529217133951, 360839856011055384215617681945569151, 77941408898387962990589088077188467221, 16835344322051800005967102044086072574781, 3636434373563188801288895309165857035708275 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [7, 1, 1]](x) = 2 4 3 2 x (7236 x - 927 x - 605 x - 33 x + 4) ------------------------------------------------------------- (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[7, 1, 1]](x) = x^2*(7236*x^4-927*x^3-605*x^2-33*x+4)/(1+4*x)/(1 +x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 4, 775, 167989, 36279480, 7836418679, 1692665924390, 365615844197349, 78973022303646280, 17058172817973644899, 3684565328678746969050, 795866110994641384169729, 171907079974842247442689100, 37131929274565928071699210239, 8020496723306240439754008212830, 1732427292234147935200477634484829, 374204295122575954001376450324506640, 80828127746476406064314654365147791899, 17458875593238903709891809120991463096930, 3771117128139603201336632176155210300609849 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [6, 3]](x) = 2 4 3 2 x (20736 x + 1026 x - 1358 x - 86 x + 7) ------------------------------------------------------------- (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[6, 3]](x) = x^2*(20736*x^4+1026*x^3-1358*x^2-86*x+7)/(1+4*x)/(1 +x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 7, 1328, 287975, 62193516, 13433858953, 2901713030914, 626770018443291, 135382323950847872, 29242581973652632589, 6316397706306575844990, 1364341904562240970828447, 294697851385443865530987868, 63654735899255876578311072465, 13749422954239269326342429662106, 2969875358115682174619902559726243, 641493077352987349716730933524053304, 138562504708245267538824350049012751381, 29929501016980977788385965445853859627062, 6464772219667891202291369382212220474589479 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [6, 2, 1]](x) = x 6 5 4 3 2 (194886 x - 41823 x - 9372 x - 1115 x - 603 x + 13 x + 14)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[6, 2, 1]](x) = x^2*(194886*x^6-41823*x^5-9372*x^4-1115*x^3-603* x^2+13*x+14)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x ) The first 20 term , starting with k=1 are 0, 14, 2911, 629904, 136048520, 29386565279, 6347497260216, 1371059415323479, 296148833642476240, 63968148067366057419, 13817119982545619396096, 2984497916229902293217879, 644651549905658454073529160, 139244734779622230032192054059, 30076862712398401651210852115176, 6496602345878054756982547406881479, 1403266106709659827505334998751253280, 303105479049286522741178392496125694699, 65470783474645888912094298260292087441456, 14141689230523512005012370534015808474888279 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [6, 1, 1, 1]](x) = - x 6 5 4 3 2 (41472 x - 106974 x - 2859 x + 9101 x + 124 x - 107 x - 7)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[6, 1, 1, 1]](x) = -x^2*(41472*x^6-106974*x^5-2859*x^4+9101*x^3+ 124*x^2-107*x-7)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+ 216*x) The first 20 term , starting with k=1 are 0, 7, 1556, 335933, 72559281, 15672834643, 3385331870767, 731231688199509, 157946044608970697, 34116345635932227839, 7369130657357626832763, 1591732221989281572927745, 343814159949684505551024373, 74263858549131856047417653595, 16040993446612480880368539796919, 3464854584468295870393210717592141, 748408590245151908002822483445959809, 161656255492952812128628682487649925911, 34917751186477807419783623873925115113235, 7542234256279206402673264301622981180776697 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [5, 4]](x) = 2 4 3 2 x (10962 x + 3048 x - 443 x - 118 x + 6) - -------------------------------------------------------------- (1 + x) (1 + 4 x) (-1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[5, 4]](x) = -x^2*(10962*x^4+3048*x^3-443*x^2-118*x+6)/(1+x)/(1+ 4*x)/(-1+3*x)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 6, 1166, 251955, 54419524, 11754625015, 2538998915250, 548423766023489, 118459533457985528, 25587259226937234579, 5526847993018331945614, 1193799166491960150778453, 257860619962263388581923412, 55697893911848891949981133223, 12030745084959360661052351875658, 2598640938351221902787895159343497, 561306442683863931002180184097198576, 121242191619714609096470940914681012347, 26188313389858355564837723051407048963782, 5656675692209404802004948179865489388769021 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 2 3 x (243 x + 19 x - 7) F[[4, 3, 1, 1], [5, 3, 1]](x) = - ----------------------------------------- (1 + x) (1 + 4 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[5, 3, 1]](x) = -3*x^2*(243*x^2+19*x-7)/(1+x)/(1+4*x)/(-1+6*x)/( -1+216*x) The first 20 term , starting with k=1 are 0, 21, 4500, 971781, 209904045, 45339266511, 9793281536985, 2115348811756311, 456915343338193545, 98693714161041887031, 21317842258785003690825, 4604653927897560519353271, 994605248425873070570788425, 214834733659988583233402504631, 46404302470557533978356536975945, 10023329333640427339324657871670711, 2165039136066332305294123990354022985, 467648453390327777943530769197852061111, 101012065932310800035802646070660566979145, 21818606241379132807733371550805284705365431 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 2 x (63 x - 8) F[[4, 3, 1, 1], [5, 2, 2]](x) = - --------------------------------- (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[5, 2, 2]](x) = -2*x^2*(63*x-8)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 16, 3330, 719856, 155484360, 33584642496, 7254282615840, 1566925045767936, 338455809879995520, 73106454934105906176, 15790994265766664102400, 3410854761405600413577216, 736744628463609681713940480, 159136839748139691285039661056, 34373557385598173317294292213760, 7424688395289205436536820944797696, 1603732693382468374291943450191626240, 346406261770613168847059830379149787136, 74823752542452444470964923006436505681920, 16161930549169728005728423371015244533989376 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [5, 2, 1, 1]](x) = - x 6 5 4 3 2 (118098 x - 112347 x + 11637 x + 14952 x + 464 x - 280 x - 24)/( (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 3, 1, 1],[5, 2, 1, 1]](x) = -x^2*(118098*x^6-112347*x^5+11637*x^4+14952*x ^3+464*x^2-280*x-24)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/( -1+216*x) The first 20 term , starting with k=1 are 0, 24, 5248, 1133752, 244887864, 52895812523, 11425495108832, 2467906947206991, 533067900559695268, 115142666521229302687, 24870815968582376576256, 5372096249213821755435095, 1160372789830185238517058212, 250640522603320013866642580391, 54138352882317122973895333269520, 11693884222580498562553119826455439, 2525878992077387689509741974908520196, 545589862288715740934119854855482381135, 117847410254362600041769748125633693250624, 25455040614942321609022266859884067152586023 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [5, 1, 1, 1, 1]](x) = 2 3 2 3 x (461 x + 276 x + 35 x + 3) --------------------------------------------------- (1 + x) (-1 + 2 x) (9 x + 1) (1 + 4 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[5, 1, 1, 1, 1]](x) = 3*x^2*(461*x^3+276*x^2+35*x+3)/(1+x)/(-1+2 *x)/(9*x+1)/(1+4*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 9, 1941, 419931, 90699000, 19591043439, 4231664839206, 914039610214341, 197432555761609800, 42645432044910683619, 9211413321697078006746, 1989665277486601528467201, 429767699937105635989965900, 92829823186414820021455820799, 20051241808265601100805080462686, 4331068230585369837988364848839261, 935510737806439885003556587829110800, 202070319366191015160785594995989053979, 43647188983097259274729532170712685757026, 9427792820349008003341580356010516109022521 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [4, 4, 1]](x) = 2 5 4 3 2 x (72252 x + 69021 x + 10900 x - 1356 x - 307 x - 10) ---------------------------------------------------------------------- (-1 + x) (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[4, 4, 1]](x) = x^2*(72252*x^5+69021*x^4+10900*x^3-1356*x^2-307* x-10)/(-1+x)/(1+4*x)/(1+x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 10, 2337, 503857, 108839354, 23509247459, 5077997849176, 1096847531889441, 236919066917243178, 51174518453863432303, 11053695986036758122860, 2387598332983919468558045, 515721239924526784478734222, 111395787823697783834672810667, 24061490169918721322685769725264, 5197281876702443805570580328766169, 1122612885367727862004407022893015986, 242484383239429218192941462642623752551, 52376626779716711129675449867029722718388, 11313351384418809604009896325878433095369813 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [4, 3, 2]](x) = x 6 5 4 3 2 (62208 x + 108810 x - 12693 x - 14672 x - 646 x + 322 x + 21)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[4, 3, 2]](x) = x^2*(62208*x^6+108810*x^5-12693*x^4-14672*x^3-\ 646*x^2+322*x+21)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+ 216*x) The first 20 term , starting with k=1 are 0, 21, 4669, 1007732, 217678600, 47018495126, 10155995698534, 2193695063752472, 473838133834796920, 102349036907723209826, 22107391972073551713394, 4775196665967838587381332, 1031442479849053572196738660, 222791575647395567639099334446, 48122980339837442645647037178574, 10394563753404887611138641926382512, 2245225770735455724008836832173857520, 484968766478858436385882718796106796186, 104753253559433422259350901596697606955274, 22626702768837619208019792634942325384919212 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 7 6 5 F[[4, 3, 1, 1], [4, 3, 1, 1]](x) = - x (821178 x - 447633 x - 140232 x 4 3 2 + 59197 x + 10065 x - 1395 x - 181 x + 1)/((1 + x) (-1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 3, 1, 1],[4, 3, 1, 1]](x) = -x*(821178*x^7-447633*x^6-140232*x^5+59197*x^ 4+10065*x^3-1395*x^2-181*x+1)/(1+x)/(-1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1 +6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 1, 27, 6009, 1295651, 279872452, 60452351983, 13057708740740, 2820465082120407, 609220457786042208, 131591618881373140919, 28423789678380141747076, 6139538570530079461214143, 1326140331234497438236850144, 286446311546651444213923348335, 61872403294076711972007770507892, 13364439111520569785758419034260359, 2886718848088443073725568424250405760, 623531271187103703924707064330219916231, 134682754576414400047736867066253435717188, 29091474988505510410311162016992031797443055 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [4, 2, 2, 1]](x) = - x 5 4 3 2 (39690 x - 3213 x + 4170 x + 1331 x - 200 x - 28)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 3, 1, 1],[4, 2, 2, 1]](x) = -x^2*(39690*x^5-3213*x^4+4170*x^3+1331*x^2-\ 200*x-28)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 28, 5996, 1295701, 279872081, 60452353879, 13057708728287, 2820465082188937, 609220457785604297, 131591618881375600915, 28423789678380126113963, 6139538570530079549557333, 1326140331234497437675854773, 286446311546651444217099306511, 61872403294076711971987600631399, 13364439111520569785758533290037889, 2886718848088443073725567698525051009, 623531271187103703924707068442093106667, 134682754576414400047736867040133340134595, 29091474988505510410311162017140037197502605 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [4, 2, 1, 1, 1]](x) = x ( 7 6 5 4 3 2 196830 x + 234819 x - 94203 x - 40605 x + 2348 x + 1881 x - 45 x - 25 )/((1 + x) (-1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 3, 1, 1],[4, 2, 1, 1, 1]](x) = x^2*(196830*x^7+234819*x^6-94203*x^5-40605 *x^4+2348*x^3+1881*x^2-45*x-25)/(1+x)/(-1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/ (1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 25, 5245, 1133779, 244887694, 52895813344, 11425495102084, 2467906947235258, 533067900559446058, 115142666521230311008, 24870815968582367538568, 5372096249213821791689182, 1160372789830185238190953342, 250640522603320013867947929892, 54138352882317122973883584314572, 11693884222580498562553166830645426, 2525878992077387689509741551824378546, 545589862288715740934119856547894279496, 117847410254362600041769748110400941930096, 25455040614942321609022266859944998835853990 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 7 6 F[[4, 3, 1, 1], [4, 1, 1, 1, 1, 1]](x) = - x (228096 x + 217458 x 5 4 3 2 - 133239 x - 21293 x + 10190 x - 103 x - 117 x + 8)/((1 + x) (-1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(228096*x^7+217458*x^6-133239*x^5-\ 21293*x^4+10190*x^3-103*x^2-117*x+8)/(1+x)/(-1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9* x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 8, 1547, 335977, 72558946, 15672836369, 3385331858538, 731231688263553, 157946044608526622, 34116345635934560185, 7369130657357610773434, 1591732221989281657286009, 343814159949684504971623278, 74263858549131856050460542081, 16040993446612480880347660183610, 3464854584468295870393320354643345, 748408590245151908002821731432728414, 161656255492952812128628686436047710057, 34917751186477807419783623846847009015066, 7542234256279206402673264301765146698063561 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [3, 3, 3]](x) = 2 3 2 x (1305 x + 797 x + 148 x + 5) --------------------------------------------------- (1 + x) (-1 + 2 x) (9 x + 1) (1 + 4 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[3, 3, 3]](x) = x^2*(1305*x^3+797*x^2+148*x+5)/(1+x)/(-1+2*x)/(9 *x+1)/(1+4*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 5, 1168, 251924, 54419719, 11754623144, 2538998929957, 548423765889998, 118459533459116023, 25587259226927047868, 5526847993018421108281, 1193799166491959348167082, 257860619962263395714896987, 55697893911848891885782437752, 12030745084959360661626878951965, 2598640938351221902782724388489126, 561306442683863931002226603563741311, 121242191619714609096470523139081106996, 26188313389858355564837726807157085646209, 5656675692209404802004948146063732944343330 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [3, 3, 2, 1]](x) = - x 6 5 4 3 2 (124416 x - 139482 x - 21945 x + 15481 x + 1075 x - 324 x - 21)/( (1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 3, 1, 1],[3, 3, 2, 1]](x) = -x^2*(124416*x^6-139482*x^5-21945*x^4+15481*x ^3+1075*x^2-324*x-21)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/ (-1+216*x) The first 20 term , starting with k=1 are 0, 21, 4671, 1007717, 217678676, 47018494437, 10155995701432, 2193695063725753, 473838133834905072, 102349036907722221173, 22107391972073555679488, 4775196665967838551396009, 1031442479849053572340783288, 222791575647395567637797871829, 48122980339837442645652243959064, 10394563753404887611138594980662585, 2245225770735455724008837019965107424, 484968766478858436385882717104596938005, 104753253559433422259350901603463721719760, 22626702768837619208019792634881407832967081 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [3, 3, 1, 1, 1]](x) = 2 2 5 x (198 x - 22 x - 3) - ----------------------------------------- (1 + x) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[3, 3, 1, 1, 1]](x) = -5*x^2*(198*x^2-22*x-3)/(1+x)/(1+6*x)/(-1+ 6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 15, 3335, 719815, 155484545, 33584641015, 7254282622505, 1566925045714615, 338455809880235465, 73106454934103986615, 15790994265766672740425, 3410854761405600344473015, 736744628463609682024909385, 159136839748139691282551909815, 34373557385598173317305487094345, 7424688395289205436536731385753015, 1603732693382468374291943853207327305, 346406261770613168847059827155024178615, 74823752542452444470964923020945070920265, 16161930549169728005728423370899176012082615 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [3, 2, 2, 2]](x) = 2 4 3 2 x (6156 x - 1845 x - 307 x + 110 x + 11) ------------------------------------------------------------- (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[3, 2, 2, 2]](x) = x^2*(6156*x^4-1845*x^3-307*x^2+110*x+11)/(1+4 *x)/(1+x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 11, 2332, 503878, 108839199, 23509248080, 5077997844101, 1096847531909742, 236919066917068663, 51174518453864130364, 11053695986036751971385, 2387598332983919493163946, 515721239924526784259378267, 111395787823697783835550234488, 24061490169918721322677906465309, 5197281876702443805570611781805990, 1122612885367727862004406740352528511, 242484383239429218192941463772785702452, 52376626779716711129675449856866855103873, 11313351384418809604009896325919084565827874 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 3 x (27 x - 7) F[[4, 3, 1, 1], [3, 2, 2, 1, 1]](x) = - --------------------------------- (1 + 4 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[3, 2, 2, 1, 1]](x) = -3*x^2*(27*x-7)/(1+4*x)/(-1+6*x)/(-1+216*x ) The first 20 term , starting with k=1 are 0, 21, 4497, 971778, 209903964, 45339266280, 9793281534576, 2115348811745952, 456915343338115008, 98693714161041481344, 21317842258785000994560, 4604653927897560504224256, 994605248425873070475820032, 214834733659988583232849471488, 46404302470557533978353151668224, 10023329333640427339324637828259840, 2165039136066332305294123869019815936, 467648453390327777943530768474141786112, 101012065932310800035802646066301125459968, 21818606241379132807733371550779196775727104 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [3, 2, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (43254 x - 1593 x + 2541 x + 1421 x - 399 x + 12 x + 14)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[3, 2, 1, 1, 1, 1]](x) = x^2*(43254*x^6-1593*x^5+2541*x^4+1421*x ^3-399*x^2+12*x+14)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-\ 1+216*x) The first 20 term , starting with k=1 are 0, 14, 2910, 629901, 136048440, 29386565214, 6347497256730, 1371059415322296, 296148833642342280, 63968148067366037214, 13817119982545614449550, 2984497916229902292883116, 644651549905658453893573020, 139244734779622230032186594514, 30076862712398401651204344536370, 6496602345878054756982547318598736, 1403266106709659827505334764020262160, 303105479049286522741178392494704800614, 65470783474645888912094298251834516627190, 14141689230523512005012370534015785665251156 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (11124 x + 3375 x - 193 x - 35 x + 4) ------------------------------------------------------------- (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = x^2*(11124*x^4+3375*x^3-193*x^2-35*x +4)/(1+4*x)/(1+x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 4, 773, 167997, 36279376, 7836419095, 1692665920134, 365615844214373, 78973022303484872, 17058172817974290531, 3684565328678741027290, 795866110994641407936769, 171907079974842247226688588, 37131929274565928072563212287, 8020496723306240439746198639966, 1732427292234147935200508872776285, 374204295122575954001376168643012624, 80828127746476406064314655491873767963, 17458875593238903709891809110842339377762, 3771117128139603201336632176195806795486521 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [2, 2, 2, 2, 1]](x) = 2 4 3 2 x (9774 x + 2256 x - 14 x - 96 x - 5) -------------------------------------------------------------- (1 + x) (1 + 4 x) (-1 + 3 x) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[2, 2, 2, 2, 1]](x) = x^2*(9774*x^4+2256*x^3-14*x^2-96*x-5)/(1+x )/(1+4*x)/(-1+3*x)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 5, 1166, 251933, 54419580, 11754624265, 2538998917966, 548423765996223, 118459533458092040, 25587259226936241005, 5526847993018335896946, 1193799166491960114748843, 257860619962263388725835180, 55697893911848891948679272025, 12030745084959360661057557460406, 2598640938351221902787848210036343, 561306442683863931002180371877686800, 121242191619714609096470939223138869125, 26188313389858355564837723058173066873146, 5656675692209404802004948179804571546251523 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 3, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 4 3 2 x (10368 x + 2286 x - 602 x - 121 x - 6) - ------------------------------------------------------------- (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 3, 1, 1],[2, 2, 2, 1, 1, 1]](x) = -x^2*(10368*x^4+2286*x^3-602*x^2-121*x-\ 6)/(1+4*x)/(1+x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 6, 1333, 287934, 62193701, 13433857472, 2901713037579, 626770018389970, 135382323951087817, 29242581973650713028, 6316397706306584483015, 1364341904562240901724246, 294697851385443865841956773, 63654735899255876575823321224, 13749422954239269326353624542691, 2969875358115682174619813000681562, 641493077352987349716731336539754369, 138562504708245267538824346824887142860, 29929501016980977788385965460362424865407, 6464772219667891202291369382096151952682718 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (118098 x - 405 x - 39636 x - 4737 x + 1150 x + 127 x + 3)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = x^2*(118098*x^6-405*x^5-39636*x^4-\ 4737*x^3+1150*x^2+127*x+3)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+ 6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 3, 748, 161971, 34983819, 7556546012, 1632213571847, 352558135450680, 76152557221501723, 16448952360187415656, 3552973709797372885431, 767442321316261236081824, 165767541404312167946269787, 35805788943331430633240075760, 7734050411759588995538796293575, 1670554888940071223228461954784728, 360839856011055384215617984554497211, 77941408898387962990589085657630320024, 16835344322051800005967102054973126271479, 3636434373563188801288895309078782447220592 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 3, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (165888 x - 17874 x - 6633 x + 1329 x - 44 x - 15 x - 1)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) ) and in Maple notation F[[4, 3, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^2*(165888*x^6-17874*x^5-6633*x ^4+1329*x^3-44*x^2-15*x-1)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+ 6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 1, 222, 47993, 10365545, 2238976971, 483618832027, 104461669802713, 22563720657842385, 4873763662281273311, 1052732951051040905327, 227390317427040662551053, 49116308564240639657195845, 10609122649875979471283231971, 2291570492373211554013049044947, 494979226352613695773386520839713, 106915512892164558286091079733345625, 23093750784707544589804335259736342151, 4988250169496829631397658411144563800087, 1077462036611315200381894919512320566087893 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 3 F[[4, 3, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x 5 4 3 2 (39690 x - 3213 x + 4170 x + 1331 x - 200 x - 28)/((1 + x) (-1 + 2 x) (1 + 4 x) (-1 + 3 x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 3, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(39690*x^5-3213*x^4+4170* x^3+1331*x^2-200*x-28)/(1+x)/(-1+2*x)/(1+4*x)/(-1+3*x)/(9*x+1)/(1+6*x)/(-1+6*x) /(-1+216*x) The first 20 term , starting with k=1 are 0, 0, 28, 5996, 1295701, 279872081, 60452353879, 13057708728287, 2820465082188937, 609220457785604297, 131591618881375600915, 28423789678380126113963, 6139538570530079549557333, 1326140331234497437675854773, 286446311546651444217099306511, 61872403294076711971987600631399, 13364439111520569785758533290037889, 2886718848088443073725567698525051009, 623531271187103703924707068442093106667, 134682754576414400047736867040133340134595 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 Regarding Lambda=, [4, 2, 2, 1] 10 9 8 7 F[[4, 2, 2, 1], [9]](x) = (3433752 x - 110754 x - 7435203 x + 351697 x 6 5 4 3 2 + 1882547 x + 85979 x - 124644 x - 10038 x + 2209 x + 206 x - 1)/( (-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[9]](x) = (3433752*x^10-110754*x^9-7435203*x^8+351697*x^7+ 1882547*x^6+85979*x^5-124644*x^4-10038*x^3+2209*x^2+206*x-1)/(-1+x)/(1+x)/(-1+2 *x)/(-1+4*x)/(1+3*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 1, 28, 6009, 1295701, 279872452, 60452353879, 13057708740740, 2820465082188937, 609220457786042208, 131591618881375600915, 28423789678380141747076, 6139538570530079549557333, 1326140331234497438236850144, 286446311546651444217099306511, 61872403294076711972007770507892, 13364439111520569785758533290037889, 2886718848088443073725568424250405760, 623531271187103703924707068442093106667, 134682754576414400047736867066253435717188 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 5 F[[4, 2, 2, 1], [8, 1]](x) = - x (663552 x - 458568 x - 593802 x + 44265 x 4 3 2 + 74681 x + 3443 x - 2133 x - 190 x + 2)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[8, 1]](x) = -x^2*(663552*x^8-458568*x^7-593802*x^6+44265*x^5+ 74681*x^4+3443*x^3-2133*x^2-190*x+2)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x)/(9* x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 2, 222, 48019, 10365545, 2238977763, 483618832027, 104461669829315, 22563720657842385, 4873763662282204219, 1052732951051040905327, 227390317427040695669931, 49116308564240639657195845, 10609122649875979472469520595, 2291570492373211554013049044947, 494979226352613695773429134164467, 106915512892164558286091079733345625, 23093750784707544589804336792352092491, 4988250169496829631397658411144563800087, 1077462036611315200381894919567471536042523 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 5 F[[4, 2, 2, 1], [7, 2]](x) = - x (787320 x + 201366 x - 802899 x - 24819 x 4 3 2 + 104248 x + 6935 x - 3124 x - 282 x + 5)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[7, 2]](x) = -x^2*(787320*x^8+201366*x^7-802899*x^6-24819*x^5+ 104248*x^4+6935*x^3-3124*x^2-282*x+5)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x)/(9 *x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 5, 748, 162014, 34983819, 7556547269, 1632213571847, 352558135492583, 76152557221501723, 16448952360188882093, 3552973709797372885431, 767442321316261288303787, 165767541404312167946269787, 35805788943331430635111880177, 7734050411759588995538796293575, 1670554888940071223228529217133951, 360839856011055384215617984554497211, 77941408898387962990589088077188467221, 16835344322051800005967102054973126271479, 3636434373563188801288895309165857035708275 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [7, 1, 1]](x) = 2 5 4 3 2 x (28944 x - 14832 x - 5795 x + 61 x + 51 x - 4) ------------------------------------------------------------------------ (1 + x) (-1 + 4 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[7, 1, 1]](x) = x^2*(28944*x^5-14832*x^4-5795*x^3+61*x^2+51*x-4) /(1+x)/(-1+4*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 4, 773, 167989, 36279376, 7836418679, 1692665920134, 365615844197349, 78973022303484872, 17058172817973644899, 3684565328678741027290, 795866110994641384169729, 171907079974842247226688588, 37131929274565928071699210239, 8020496723306240439746198639966, 1732427292234147935200477634484829, 374204295122575954001376168643012624, 80828127746476406064314654365147791899, 17458875593238903709891809110842339377762, 3771117128139603201336632176155210300609849 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [6, 3]](x) = 2 5 4 3 2 x (20736 x + 19170 x + 11476 x + 2287 x + 88 x - 7) ---------------------------------------------------------------------- (-1 + x) (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[6, 3]](x) = x^2*(20736*x^5+19170*x^4+11476*x^3+2287*x^2+88*x-7) /(-1+x)/(1+4*x)/(1+x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 7, 1333, 287975, 62193701, 13433858953, 2901713037579, 626770018443291, 135382323951087817, 29242581973652632589, 6316397706306584483015, 1364341904562240970828447, 294697851385443865841956773, 63654735899255876578311072465, 13749422954239269326353624542691, 2969875358115682174619902559726243, 641493077352987349716731336539754369, 138562504708245267538824350049012751381, 29929501016980977788385965460362424865407, 6464772219667891202291369382212220474589479 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 5 F[[4, 2, 2, 1], [6, 2, 1]](x) = x (779544 x - 143802 x - 250185 x - 9682 x 4 3 2 + 59 x - 140 x + 496 x - 26 x - 14)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[6, 2, 1]](x) = x^2*(779544*x^8-143802*x^7-250185*x^6-9682*x^5+ 59*x^4-140*x^3+496*x^2-26*x-14)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x)/(9*x+1)/ (-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 14, 2910, 629904, 136048440, 29386565279, 6347497256730, 1371059415323479, 296148833642342280, 63968148067366057419, 13817119982545614449550, 2984497916229902293217879, 644651549905658453893573020, 139244734779622230032192054059, 30076862712398401651204344536370, 6496602345878054756982547406881479, 1403266106709659827505334764020262160, 303105479049286522741178392496125694699, 65470783474645888912094298251834516627190, 14141689230523512005012370534015808474888279 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [6, 1, 1, 1]](x) = - x (165888 x + 1294920 x + 68754 x 5 4 3 2 - 231875 x - 29720 x + 9390 x + 1781 x + 105 x + 7)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[6, 1, 1, 1]](x) = -x^2*(165888*x^8+1294920*x^7+68754*x^6-231875 *x^5-29720*x^4+9390*x^3+1781*x^2+105*x+7)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x )/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 7, 1547, 335933, 72558946, 15672834643, 3385331858538, 731231688199509, 157946044608526622, 34116345635932227839, 7369130657357610773434, 1591732221989281572927745, 343814159949684504971623278, 74263858549131856047417653595, 16040993446612480880347660183610, 3464854584468295870393210717592141, 748408590245151908002821731432728414, 161656255492952812128628682487649925911, 34917751186477807419783623846847009015066, 7542234256279206402673264301622981180776697 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [5, 4]](x) = 2 5 4 3 2 x (43848 x - 2802 x - 4024 x + 491 x + 118 x - 6) - ------------------------------------------------------------------------- (-1 + x) (-1 + 4 x) (1 + 3 x) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[5, 4]](x) = -x^2*(43848*x^5-2802*x^4-4024*x^3+491*x^2+118*x-6)/ (-1+x)/(-1+4*x)/(1+3*x)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 6, 1166, 251955, 54419580, 11754625015, 2538998917966, 548423766023489, 118459533458092040, 25587259226937234579, 5526847993018335896946, 1193799166491960150778453, 257860619962263388725835180, 55697893911848891949981133223, 12030745084959360661057557460406, 2598640938351221902787895159343497, 561306442683863931002180371877686800, 121242191619714609096470940914681012347, 26188313389858355564837723058173066873146, 5656675692209404802004948179865489388769021 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [5, 3, 1]](x) = 2 4 3 2 3 x (972 x + 121 x - 54 x + 29 x + 7) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 4 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[5, 3, 1]](x) = -3*x^2*(972*x^4+121*x^3-54*x^2+29*x+7)/(-1+x)/(1 +x)/(-1+4*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 21, 4497, 971781, 209903964, 45339266511, 9793281534576, 2115348811756311, 456915343338115008, 98693714161041887031, 21317842258785000994560, 4604653927897560519353271, 994605248425873070475820032, 214834733659988583233402504631, 46404302470557533978353151668224, 10023329333640427339324657871670711, 2165039136066332305294123869019815936, 467648453390327777943530769197852061111, 101012065932310800035802646066301125459968, 21818606241379132807733371550805284705365431 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [5, 2, 2]](x) = 2 3 2 x (126 x - 1096 x - 121 x + 16) - -------------------------------------------------- (-1 + x) (1 + x) (-1 + 6 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[5, 2, 2]](x) = -x^2*(126*x^3-1096*x^2-121*x+16)/(-1+x)/(1+x)/(-\ 1+6*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 16, 3335, 719856, 155484545, 33584642496, 7254282622505, 1566925045767936, 338455809880235465, 73106454934105906176, 15790994265766672740425, 3410854761405600413577216, 736744628463609682024909385, 159136839748139691285039661056, 34373557385598173317305487094345, 7424688395289205436536820944797696, 1603732693382468374291943853207327305, 346406261770613168847059830379149787136, 74823752542452444470964923020945070920265, 16161930549169728005728423371015244533989376 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [5, 2, 1, 1]](x) = - x (472392 x - 1343142 x + 175311 x 5 4 3 2 + 202335 x - 22729 x - 11484 x + 242 x + 301 x + 24)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[5, 2, 1, 1]](x) = -x^2*(472392*x^8-1343142*x^7+175311*x^6+ 202335*x^5-22729*x^4-11484*x^3+242*x^2+301*x+24)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x) /(1+3*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 24, 5245, 1133752, 244887694, 52895812523, 11425495102084, 2467906947206991, 533067900559446058, 115142666521229302687, 24870815968582367538568, 5372096249213821755435095, 1160372789830185238190953342, 250640522603320013866642580391, 54138352882317122973883584314572, 11693884222580498562553119826455439, 2525878992077387689509741551824378546, 545589862288715740934119854855482381135, 117847410254362600041769748110400941930096, 25455040614942321609022266859884067152586023 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [5, 1, 1, 1, 1]](x) = 2 3 2 3 x (461 x + 276 x + 35 x + 3) --------------------------------------------------- (1 + x) (-1 + 2 x) (9 x + 1) (1 + 4 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[5, 1, 1, 1, 1]](x) = 3*x^2*(461*x^3+276*x^2+35*x+3)/(1+x)/(-1+2 *x)/(9*x+1)/(1+4*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 9, 1941, 419931, 90699000, 19591043439, 4231664839206, 914039610214341, 197432555761609800, 42645432044910683619, 9211413321697078006746, 1989665277486601528467201, 429767699937105635989965900, 92829823186414820021455820799, 20051241808265601100805080462686, 4331068230585369837988364848839261, 935510737806439885003556587829110800, 202070319366191015160785594995989053979, 43647188983097259274729532170712685757026, 9427792820349008003341580356010516109022521 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 2, 2, 1], [4, 4, 1]](x) = x 6 5 4 3 2 (289008 x - 10008 x - 48191 x - 6494 x + 1163 x + 262 x + 10)/((1 + x) (-1 + x) (-1 + 4 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[4, 4, 1]](x) = x^2*(289008*x^6-10008*x^5-48191*x^4-6494*x^3+ 1163*x^2+262*x+10)/(1+x)/(-1+x)/(-1+4*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+ 216*x) The first 20 term , starting with k=1 are 0, 10, 2332, 503857, 108839199, 23509247459, 5077997844101, 1096847531889441, 236919066917068663, 51174518453863432303, 11053695986036751971385, 2387598332983919468558045, 515721239924526784259378267, 111395787823697783834672810667, 24061490169918721322677906465309, 5197281876702443805570580328766169, 1122612885367727862004406740352528511, 242484383239429218192941462642623752551, 52376626779716711129675449856866855103873, 11313351384418809604009896325878433095369813 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [4, 3, 2]](x) = x (248832 x + 803304 x - 425406 x 5 4 3 2 - 295549 x + 22041 x + 22490 x + 904 x - 345 x - 21)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[4, 3, 2]](x) = x^2*(248832*x^8+803304*x^7-425406*x^6-295549*x^5 +22041*x^4+22490*x^3+904*x^2-345*x-21)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x)/( 9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 21, 4671, 1007732, 217678676, 47018495126, 10155995701432, 2193695063752472, 473838133834905072, 102349036907723209826, 22107391972073555679488, 4775196665967838587381332, 1031442479849053572340783288, 222791575647395567639099334446, 48122980339837442645652243959064, 10394563753404887611138641926382512, 2245225770735455724008837019965107424, 484968766478858436385882718796106796186, 104753253559433422259350901603463721719760, 22626702768837619208019792634942325384919212 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 2, 2, 1], [4, 3, 1, 1]](x) = - x ( 7 6 5 4 3 2 158760 x - 233766 x - 25431 x - 11726 x - 7353 x + 805 x + 434 x + 27 )/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[4, 3, 1, 1]](x) = -x^2*(158760*x^7-233766*x^6-25431*x^5-11726*x ^4-7353*x^3+805*x^2+434*x+27)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x)/(9*x+1)/(-\ 1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 27, 5996, 1295651, 279872081, 60452351983, 13057708728287, 2820465082120407, 609220457785604297, 131591618881373140919, 28423789678380126113963, 6139538570530079461214143, 1326140331234497437675854773, 286446311546651444213923348335, 61872403294076711971987600631399, 13364439111520569785758419034260359, 2886718848088443073725567698525051009, 623531271187103703924707064330219916231, 134682754576414400047736867040133340134595, 29091474988505510410311162016992031797443055 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 8 7 6 F[[4, 2, 2, 1], [4, 2, 2, 1]](x) = - x (3284712 x - 293598 x - 1132221 x 5 4 3 2 - 29497 x + 90817 x + 6183 x - 1969 x - 178 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[4, 2, 2, 1]](x) = -x*(3284712*x^8-293598*x^7-1132221*x^6-29497* x^5+90817*x^4+6183*x^3-1969*x^2-178*x+1)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x) /(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 1, 28, 6009, 1295701, 279872452, 60452353879, 13057708740740, 2820465082188937, 609220457786042208, 131591618881375600915, 28423789678380141747076, 6139538570530079549557333, 1326140331234497438236850144, 286446311546651444217099306511, 61872403294076711972007770507892, 13364439111520569785758533290037889, 2886718848088443073725568424250405760, 623531271187103703924707068442093106667, 134682754576414400047736867066253435717188, 29091474988505510410311162017140037197502605 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [4, 2, 1, 1, 1]](x) = x (787320 x + 649134 x + 95499 x 5 4 3 2 - 227766 x - 40663 x + 13290 x + 2559 x - 98 x - 25)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[4, 2, 1, 1, 1]](x) = x^2*(787320*x^8+649134*x^7+95499*x^6-\ 227766*x^5-40663*x^4+13290*x^3+2559*x^2-98*x-25)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x) /(1+3*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 25, 5248, 1133779, 244887864, 52895813344, 11425495108832, 2467906947235258, 533067900559695268, 115142666521230311008, 24870815968582376576256, 5372096249213821791689182, 1160372789830185238517058212, 250640522603320013867947929892, 54138352882317122973895333269520, 11693884222580498562553166830645426, 2525878992077387689509741974908520196, 545589862288715740934119856547894279496, 117847410254362600041769748125633693250624, 25455040614942321609022266859944998835853990 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [4, 1, 1, 1, 1, 1]](x) = - x (912384 x - 569592 x - 10326 x 5 4 3 2 + 181307 x + 24313 x - 9013 x - 2239 x - 92 x + 8)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(912384*x^8-569592*x^7-10326*x^6+ 181307*x^5+24313*x^4-9013*x^3-2239*x^2-92*x+8)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/( 1+3*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 8, 1556, 335977, 72559281, 15672836369, 3385331870767, 731231688263553, 157946044608970697, 34116345635934560185, 7369130657357626832763, 1591732221989281657286009, 343814159949684505551024373, 74263858549131856050460542081, 16040993446612480880368539796919, 3464854584468295870393320354643345, 748408590245151908002822483445959809, 161656255492952812128628686436047710057, 34917751186477807419783623873925115113235, 7542234256279206402673264301765146698063561 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [3, 3, 3]](x) = 2 3 2 x (1305 x + 797 x + 148 x + 5) --------------------------------------------------- (1 + x) (-1 + 2 x) (9 x + 1) (1 + 4 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[3, 3, 3]](x) = x^2*(1305*x^3+797*x^2+148*x+5)/(1+x)/(-1+2*x)/(9 *x+1)/(1+4*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 5, 1168, 251924, 54419719, 11754623144, 2538998929957, 548423765889998, 118459533459116023, 25587259226927047868, 5526847993018421108281, 1193799166491959348167082, 257860619962263395714896987, 55697893911848891885782437752, 12030745084959360661626878951965, 2598640938351221902782724388489126, 561306442683863931002226603563741311, 121242191619714609096470523139081106996, 26188313389858355564837726807157085646209, 5656675692209404802004948146063732944343330 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [3, 3, 2, 1]](x) = - x (497664 x - 335016 x + 27078 x 5 4 3 2 + 196079 x + 5644 x - 15056 x - 507 x + 343 x + 21)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[3, 3, 2, 1]](x) = -x^2*(497664*x^8-335016*x^7+27078*x^6+196079* x^5+5644*x^4-15056*x^3-507*x^2+343*x+21)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3*x) /(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 21, 4669, 1007717, 217678600, 47018494437, 10155995698534, 2193695063725753, 473838133834796920, 102349036907722221173, 22107391972073551713394, 4775196665967838551396009, 1031442479849053572196738660, 222791575647395567637797871829, 48122980339837442645647037178574, 10394563753404887611138594980662585, 2245225770735455724008836832173857520, 484968766478858436385882717104596938005, 104753253559433422259350901596697606955274, 22626702768837619208019792634881407832967081 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [3, 3, 1, 1, 1]](x) = 2 3 2 5 x (198 x - 4 x + 18 x + 3) - -------------------------------------------------- (-1 + x) (1 + x) (-1 + 6 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[3, 3, 1, 1, 1]](x) = -5*x^2*(198*x^3-4*x^2+18*x+3)/(-1+x)/(1+x) /(-1+6*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 15, 3330, 719815, 155484360, 33584641015, 7254282615840, 1566925045714615, 338455809879995520, 73106454934103986615, 15790994265766664102400, 3410854761405600344473015, 736744628463609681713940480, 159136839748139691282551909815, 34373557385598173317294292213760, 7424688395289205436536731385753015, 1603732693382468374291943450191626240, 346406261770613168847059827155024178615, 74823752542452444470964923006436505681920, 16161930549169728005728423370899176012082615 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 F[[4, 2, 2, 1], [3, 2, 2, 2]](x) = x 6 5 4 3 2 (24624 x + 175680 x + 36923 x - 9700 x - 1848 x + 60 x + 11)/((1 + x) (-1 + x) (-1 + 4 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[3, 2, 2, 2]](x) = x^2*(24624*x^6+175680*x^5+36923*x^4-9700*x^3-\ 1848*x^2+60*x+11)/(1+x)/(-1+x)/(-1+4*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+ 216*x) The first 20 term , starting with k=1 are 0, 11, 2337, 503878, 108839354, 23509248080, 5077997849176, 1096847531909742, 236919066917243178, 51174518453864130364, 11053695986036758122860, 2387598332983919493163946, 515721239924526784478734222, 111395787823697783835550234488, 24061490169918721322685769725264, 5197281876702443805570611781805990, 1122612885367727862004407022893015986, 242484383239429218192941463772785702452, 52376626779716711129675449867029722718388, 11313351384418809604009896325919084565827874 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [3, 2, 2, 1, 1]](x) = 2 4 3 2 3 x (108 x - 955 x - 265 x + 30 x + 7) - ------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 4 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[3, 2, 2, 1, 1]](x) = -3*x^2*(108*x^4-955*x^3-265*x^2+30*x+7)/(-\ 1+x)/(1+x)/(-1+4*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 21, 4500, 971778, 209904045, 45339266280, 9793281536985, 2115348811745952, 456915343338193545, 98693714161041481344, 21317842258785003690825, 4604653927897560504224256, 994605248425873070570788425, 214834733659988583232849471488, 46404302470557533978356536975945, 10023329333640427339324637828259840, 2165039136066332305294123990354022985, 467648453390327777943530768474141786112, 101012065932310800035802646070660566979145, 21818606241379132807733371550779196775727104 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [3, 2, 1, 1, 1, 1]](x) = x (173016 x - 585522 x - 19815 x 5 4 3 2 + 54277 x - 242 x + 1372 x + 705 x - 27 x - 14)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[3, 2, 1, 1, 1, 1]](x) = x^2*(173016*x^8-585522*x^7-19815*x^6+ 54277*x^5-242*x^4+1372*x^3+705*x^2-27*x-14)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/(1+3 *x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 14, 2911, 629901, 136048520, 29386565214, 6347497260216, 1371059415322296, 296148833642476240, 63968148067366037214, 13817119982545619396096, 2984497916229902292883116, 644651549905658454073529160, 139244734779622230032186594514, 30076862712398401651210852115176, 6496602345878054756982547318598736, 1403266106709659827505334998751253280, 303105479049286522741178392494704800614, 65470783474645888912094298260292087441456, 14141689230523512005012370534015785665251156 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [3, 1, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (44496 x + 6264 x + 155 x + 465 x + 49 x - 4) ------------------------------------------------------------------------ (1 + x) (-1 + 4 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[3, 1, 1, 1, 1, 1, 1]](x) = x^2*(44496*x^5+6264*x^4+155*x^3+465* x^2+49*x-4)/(1+x)/(-1+4*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 4, 775, 167997, 36279480, 7836419095, 1692665924390, 365615844214373, 78973022303646280, 17058172817974290531, 3684565328678746969050, 795866110994641407936769, 171907079974842247442689100, 37131929274565928072563212287, 8020496723306240439754008212830, 1732427292234147935200508872776285, 374204295122575954001376450324506640, 80828127746476406064314655491873767963, 17458875593238903709891809120991463096930, 3771117128139603201336632176195806795486521 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [2, 2, 2, 2, 1]](x) = 2 5 4 3 2 x (39096 x + 1554 x - 3100 x - 26 x + 96 x + 5) ------------------------------------------------------------------------- (-1 + x) (-1 + 4 x) (1 + 3 x) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[2, 2, 2, 2, 1]](x) = x^2*(39096*x^5+1554*x^4-3100*x^3-26*x^2+96 *x+5)/(-1+x)/(-1+4*x)/(1+3*x)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 5, 1166, 251933, 54419524, 11754624265, 2538998915250, 548423765996223, 118459533457985528, 25587259226936241005, 5526847993018331945614, 1193799166491960114748843, 257860619962263388581923412, 55697893911848891948679272025, 12030745084959360661052351875658, 2598640938351221902787848210036343, 561306442683863931002180184097198576, 121242191619714609096470939223138869125, 26188313389858355564837723051407048963782, 5656675692209404802004948179804571546251523 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 F[[4, 2, 2, 1], [2, 2, 2, 1, 1, 1]](x) = 2 5 4 3 2 2 x (5184 x + 15399 x + 5486 x + 748 x + 55 x + 3) - ---------------------------------------------------------------------- (-1 + x) (1 + 4 x) (1 + x) (9 x + 1) (1 + 6 x) (-1 + 6 x) (-1 + 216 x) and in Maple notation F[[4, 2, 2, 1],[2, 2, 2, 1, 1, 1]](x) = -2*x^2*(5184*x^5+15399*x^4+5486*x^3+748 *x^2+55*x+3)/(-1+x)/(1+4*x)/(1+x)/(9*x+1)/(1+6*x)/(-1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 6, 1328, 287934, 62193516, 13433857472, 2901713030914, 626770018389970, 135382323950847872, 29242581973650713028, 6316397706306575844990, 1364341904562240901724246, 294697851385443865530987868, 63654735899255876575823321224, 13749422954239269326342429662106, 2969875358115682174619813000681562, 641493077352987349716730933524053304, 138562504708245267538824346824887142860, 29929501016980977788385965445853859627062, 6464772219667891202291369382096151952682718 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 6 F[[4, 2, 2, 1], [2, 2, 1, 1, 1, 1, 1]](x) = x (472392 x - 8910 x - 344331 x 5 4 3 2 + 96438 x + 57013 x - 1851 x - 1871 x - 127 x - 3)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[2, 2, 1, 1, 1, 1, 1]](x) = x^2*(472392*x^8-8910*x^7-344331*x^6+ 96438*x^5+57013*x^4-1851*x^3-1871*x^2-127*x-3)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/( 1+3*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 3, 745, 161971, 34983679, 7556546012, 1632213566689, 352558135450680, 76152557221317943, 16448952360187415656, 3552973709797366334293, 767442321316261236081824, 165767541404312167711777867, 35805788943331430633240075760, 7734050411759588995530378959257, 1670554888940071223228461954784728, 360839856011055384215617681945569151, 77941408898387962990589085657630320024, 16835344322051800005967102044086072574781, 3636434373563188801288895309078782447220592 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 2 8 7 F[[4, 2, 2, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x (663552 x - 38664 x 6 5 4 3 2 - 249714 x - 108753 x - 3458 x + 5104 x + 669 x + 13 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^2*(663552*x^8-38664*x^7-249714 *x^6-108753*x^5-3458*x^4+5104*x^3+669*x^2+13*x+1)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x )/(1+3*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 1, 219, 47993, 10365436, 2238976971, 483618828260, 104461669802713, 22563720657710592, 4873763662281273311, 1052732951051036233396, 227390317427040662551053, 49116308564240639490271568, 10609122649875979471283231971, 2291570492373211554007060944852, 494979226352613695773386520839713, 106915512892164558286090864508894464, 23093750784707544589804335259736342151, 4988250169496829631397658403402113326228, 1077462036611315200381894919512320566087893 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 3 F[[4, 2, 2, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x ( 7 6 5 4 3 2 158760 x - 233766 x - 25431 x - 11726 x - 7353 x + 805 x + 434 x + 27 )/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 4 x) (1 + 3 x) (9 x + 1) (-1 + 6 x) (1 + 4 x) (1 + 6 x) (-1 + 216 x)) and in Maple notation F[[4, 2, 2, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(158760*x^7-233766*x^6-\ 25431*x^5-11726*x^4-7353*x^3+805*x^2+434*x+27)/(-1+x)/(1+x)/(-1+2*x)/(-1+4*x)/( 1+3*x)/(9*x+1)/(-1+6*x)/(1+4*x)/(1+6*x)/(-1+216*x) The first 20 term , starting with k=1 are 0, 0, 27, 5996, 1295651, 279872081, 60452351983, 13057708728287, 2820465082120407, 609220457785604297, 131591618881373140919, 28423789678380126113963, 6139538570530079461214143, 1326140331234497437675854773, 286446311546651444213923348335, 61872403294076711971987600631399, 13364439111520569785758419034260359, 2886718848088443073725567698525051009, 623531271187103703924707064330219916231, 134682754576414400047736867040133340134595 ---------------------------------- Their sum is 5 4 3 2 3711 x + 1662 x - 6837 x - 2635 x - 202 x + 1 ------------------------------------------------- (-1 + 216 x) (9 x + 1) (1 + x) (1 + 4 x) (-1 + x) and in Maple notation (3711*x^5+1662*x^4-6837*x^3-2635*x^2-202*x+1)/(-1+216*x)/(9*x+1)/(1+x)/(1+4*x)/ (-1+x) The first 20 term , starting with k=1 are 1, 341, 72732, 15716652, 3394741748, 733264697248, 158385170349424, 34211196833536004, 7389618515702209260, 1596157599394747402920, 344770041469237822825256, 74470328957355618213625996, 16085591054788811298042789412, 3474487667834383260501146701232, 750489336252226784087136548341728, 162105696630480985364451478699652628, 35014830472183892838706849604781530204, 7563203381991720853160811542525992992184, 1633651930510211704282734104935599477970840, 352868816990205728125070577360335527781279900 Regarding Lambda=, [4, 2, 1, 1, 1] 8 7 6 5 F[[4, 2, 1, 1, 1], [9]](x) = (1640331 x + 144315 x - 3666243 x - 79378 x 4 3 2 + 433300 x + 10205 x - 4413 x - 166 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[9]](x) = (1640331*x^8+144315*x^7-3666243*x^6-79378*x^5+ 433300*x^4+10205*x^3-4413*x^2-166*x+1)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/( 1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 1, 17, 3556, 664036, 125615865, 23739195513, 4486749129920, 847994749308752, 160271024625755449, 30291223301075843689, 5725041211257606744804, 1082032788773763891258948, 204504197081466530389365953, 38651293248329511557363429345, 7305094423935697752414568099408, 1380662846123817061964694431583424, 260945277917402050689040927214450577, 49318657526388974435708069650593901281, 9321226272487516444371800743722399900932 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [8, 1]](x) = 2 5 4 3 2 2 x (58212 x - 31302 x - 12039 x + 1575 x + 99 x - 1) - ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[8, 1]](x) = -2*x^2*(58212*x^5-31302*x^4-12039*x^3+1575*x^2+ 99*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 2, 140, 28324, 5313664, 1004901678, 189913995804, 35893984580960, 6783958161350048, 1282168193583619834, 242329786479251450188, 45800329688587895250876, 8656262310220900725478752, 1636033576651086994142720870, 309210345986649625911199012892, 58440755391485297982291596268472, 11045302768990542458503297981437376, 2087562223339216280314192234577283186, 394549260211111798114587478884880101516, 74569810179900131499769516604440749496148 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 4 3 2 F[[4, 2, 1, 1, 1], [7, 2]](x) = x (206631 x + 31860 x - 6528 x - 352 x + 5) /((-1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[7, 2]](x) = x^2*(206631*x^4+31860*x^3-6528*x^2-352*x+5)/(-1+ x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 5, 483, 95368, 17936612, 3391484553, 640960788355, 121142176652936, 22895859222796224, 4327317644468948101, 817863029551727225627, 154576112695130706837504, 29214885297076218792166036, 5521613321195726408972307449, 1043584917704977995357364643499, 197237549446262135284500870533872, 37277896845343096447655175569553248, 7045522503769854617438460258236999397, 1331603753212502325537400581103539973171, 251673109357162943666812087133454219626840 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [7, 1, 1]](x) = 2 x 6 5 4 3 2 (174636 x - 13041 x - 31044 x + 3296 x - 1418 x - 79 x + 2)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[7, 1, 1]](x) = 2*x^2*(174636*x^6-13041*x^5-31044*x^4+3296*x^ 3-1418*x^2-79*x+2)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1 +189*x) The first 20 term , starting with k=1 are 0, 4, 506, 98816, 18602116, 3517075948, 664700422302, 125628917384520, 23743854139523912, 4487588665677125012, 848154252923491362178, 160301153904915747160144, 30296918085880775809267788, 5726117518276547722162988796, 1082236210953321040652980154534, 204542643870197549002464131051288, 38658559691466919472428775287648144, 7306467781687256542929574467100516900, 1380922410738891302602033448617252405770, 260994335629650460055979015417484786011552 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [6, 3]](x) = 2 4 3 2 x (81081 x - 52380 x + 4070 x + 324 x - 7) - ----------------------------------------------------------------- (-1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[6, 3]](x) = -x^2*(81081*x^4-52380*x^3+4070*x^2+324*x-7)/(-1+ x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 7, 866, 169309, 31890468, 6029238419, 1139487105094, 215363843397801, 40703750263854536, 7693009134554678431, 1453978719435911230122, 274801978119796009515893, 51937573861571145738657004, 9816201459901366302247347243, 1855262075920005953464733444750, 350644532348909516363378863502785, 66271816613943302444881611063222672, 12525373340035296680392737231302752055, 2367295561266670809717888928576946337778, 447418861079400788556989197363855928737677 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [6, 2, 1]](x) = - x 5 4 3 2 (261954 x - 21357 x - 14898 x + 5678 x + 252 x - 13)/((-1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[6, 2, 1]](x) = -x^2*(261954*x^5-21357*x^4-14898*x^3+5678*x^2 +252*x-13)/(-1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 13, 1919, 370006, 69766080, 13188861953, 2492629863619, 471108371846286, 89039454422698760, 16828457467024592893, 3180578449073858373519, 601129327130627884066766, 113613442822321423629870040, 21472940693531417958645688833, 4058385791075072211731184091619, 767034914513238324635096843783446, 144969598843001000182849424967631920, 27399254181327210940647271493993205773, 5178459040270842407759110657598357097919, 978728758611189224726896066682037049588326 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [6, 1, 1, 1]](x) = 2 5 4 3 2 2 x (14553 x - 47124 x + 15187 x + 855 x - 12 x - 3) - ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[6, 1, 1, 1]](x) = -2*x^2*(14553*x^5-47124*x^4+15187*x^3+855* x^2-12*x-3)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 6, 1038, 197154, 37211668, 7034008438, 1329403569746, 251257779378626, 47487709410074376, 8975177307849004230, 1696308506336914891894, 320602307799574660091458, 60593836171976517530389724, 11452235036548585157782497782, 2164472421906736746995047721082, 409085287740393110521877053126050, 77317119382933880676026282230567312, 14612935563374512209565701800263391494, 2761844821477782623605515535553806498910, 521988671259300919725535085460105337144802 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [5, 4]](x) = 2 5 4 3 2 x (72765 x + 8400 x - 24968 x - 6390 x - 181 x + 6) - ------------------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[5, 4]](x) = -x^2*(72765*x^5+8400*x^4-24968*x^3-6390*x^2-181* x+6)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 6, 761, 148049, 27905606, 5275550720, 997051851047, 188443349512403, 35615781758179292, 6731382986908226954, 1272231379628267387813, 240451730852263445909477, 45445377128928418546635458, 8589176277412568702202822308, 1623354316430028866715353657459, 306813965805295330040348668948871, 57987839537200400070987415201760504, 10959701672530884376281657990212098382, 2071383616108336963103385681129968938385, 391491503444475689890761196026500923488185 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [5, 3, 1]](x) = 2 4 3 2 3 x (6237 x + 2169 x - 207 x + 67 x + 6) ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[5, 3, 1]](x) = 3*x^2*(6237*x^4+2169*x^3-207*x^2+67*x+6)/(-1+ x)/(1+x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 18, 2973, 570423, 107646054, 20348367864, 3845774989467, 726852848178069, 137375159670581628, 25963905776397338190, 4907178179196764580681, 927456676131253270212195, 175289311783286074402688322, 33129679927156965498797829396, 6261509506230233064636120994215, 1183425296677565146176724250391201, 223667381072058739643446039634615736, 42273135022619124324699981587447991882, 7989622519275014024200752325850320095669, 1510038656142977660510391118697037123665487 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [5, 2, 2]](x) = 2 x (147 x + 13) (-1 + 9 x) - -------------------------------------------- (-1 + 3 x) (1 + 3 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[5, 2, 2]](x) = -x^2*(147*x+13)*(-1+9*x)/(-1+3*x)/(1+3*x)/(1+ 21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 13, 2214, 422343, 79741260, 15072800193, 2848723495794, 538409491161723, 101759378069997720, 19232522786179616373, 3634946799637996698174, 687004945277530334765103, 129843934654388305137361380, 24540503649743753161688192553, 4638155189800217714253819609354, 876611330872269532293381489696483, 165679541534858345533161500442024240, 31313433350088239823243563201097152733, 5918238903166677063726036613040007269334, 1118547152698501970564427853335823906099863 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [5, 2, 1, 1]](x) = 2 x 5 4 3 2 (174636 x - 98496 x - 44355 x + 1031 x + 263 x + 9)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[5, 2, 1, 1]](x) = 2*x^2*(174636*x^5-98496*x^4-44355*x^3+1031 *x^2+263*x+9)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189* x) The first 20 term , starting with k=1 are 0, 18, 3514, 664838, 125598822, 23739552796, 4486741625156, 847994906903316, 160271021316253204, 30291223370575341614, 5725041209798117140758, 1082032788804413172501034, 204504197080822895481953546, 38651293248343027890415104072, 7305094423935413909420470972720, 1380662846123823022667570435371592, 260945277917401925514280531027282248, 49318657526388977064378037970201585770, 9321226272487516389169731409009669975442, 1761711765500140603349155347059928533236590 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 2 3 2 x (7623 x + 3120 x - 209 x - 6) ----------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[5, 1, 1, 1, 1]](x) = x^2*(7623*x^3+3120*x^2-209*x-6)/(-1+x)/ (1+3*x)/(-1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 6, 1319, 246019, 46522410, 8792351516, 1661757682249, 314072157367569, 59359638153159620, 11218971605711985826, 2120385633530695791579, 400752884736680954943119, 75742295215239003517159630, 14315293795680097624209462936, 2705590527383539223057067065309, 511356609675488904283609422028669, 96646399228667403003786601026010440, 18266169454218139166648755300180142846, 3452306026847228302508072833939912719439, 652485839074126149173897225133114732960219 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [4, 4, 1]](x) = x 6 5 4 3 2 (654885 x - 345492 x - 66525 x + 21796 x - 17 x + 48 x + 9)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[4, 4, 1]](x) = x^2*(654885*x^6-345492*x^5-66525*x^4+21796*x^ 3-17*x^2+48*x+9)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+ 189*x) The first 20 term , starting with k=1 are 0, 9, 1542, 295681, 55818044, 10550966025, 1994106352546, 376886644586657, 71231564638097208, 13462765950424117321, 2544462759745321718270, 480903461694283526582913, 90890754258071662690760692, 17178352554820628731925962697, 3246708632860152382000553029914, 613627931610588672791676971353249, 115975679074400841871060695517667696, 21919403345061767876293243633356994953, 4142767232216673944607967047218573074678, 782983006888951379395102266688762605470465 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [4, 3, 2]](x) = 2 5 4 3 2 x (43659 x - 92925 x + 15568 x + 364 x + 229 x + 17) ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[4, 3, 2]](x) = x^2*(43659*x^5-92925*x^4+15568*x^3+364*x^2+ 229*x+17)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 17, 3102, 591021, 111642356, 21101804801, 3988215281218, 753773235688525, 142463130388648392, 26925531877568817105, 5088925519978285183814, 961806923378337157011149, 181781508516358010664912508, 34356705109636750405436838529, 6493417265720399395451712469290, 1227255863221175358488364425768493, 231951358148801725469124389629881104, 43838806690123534876339789706581798673, 8285534464433347907616876826035308568046, 1565966013777902758403787228023911543938157 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [4, 3, 1, 1]](x) = 2 5 4 3 2 x (168399 x - 245727 x + 43524 x + 1204 x - 467 x - 21) - ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[4, 3, 1, 1]](x) = -x^2*(168399*x^5-245727*x^4+43524*x^3+1204 *x^2-467*x-21)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 21, 4016, 759547, 143546856, 27130775465, 5127707714152, 969136970450271, 183166882898135792, 34618540965358203229, 6542904240391944667968, 1236608901477648005196515, 233719082378358821720291608, 44172906569529099489818910513, 8348679341640594647199979358264, 1577900395570080900283997119961479, 298223174762745111372270381952363104, 56364180030158829804193216102899700517, 10652830025700018754137164629261483861840, 2013384874857303546187936242201875454444363 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [4, 2, 2, 1]](x) = 2 4 3 2 2 x (45873 x - 30051 x + 389 x + 323 x + 10) ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[4, 2, 2, 1]](x) = 2*x^2*(45873*x^4-30051*x^3+389*x^2+323*x+ 10)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 20, 4026, 759312, 143551708, 27130673332, 5127709858214, 969136925422784, 183166883843706456, 34618540945501199604, 6542904240808941685042, 1236608901468891067660816, 233719082378542717408009844, 44172906569525237680375233236, 8348679341640675745198291798110, 1577900395570079197226032544375808, 298223174762745147136487637996615472, 56364180030158829053144653725841260628, 10652830025700018769909184439179323679018, 2013384874857303545856723826193599656022160 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [4, 2, 1, 1, 1]](x) = - x 6 5 4 3 2 (1896048 x - 66069 x - 302703 x - 9056 x + 3680 x + 149 x - 1)/( (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[4, 2, 1, 1, 1]](x) = -x*(1896048*x^6-66069*x^5-302703*x^4-\ 9056*x^3+3680*x^2+149*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+ 21*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 17, 3556, 664036, 125615865, 23739195513, 4486749129920, 847994749308752, 160271024625755449, 30291223301075843689, 5725041211257606744804, 1082032788773763891258948, 204504197081466530389365953, 38651293248329511557363429345, 7305094423935697752414568099408, 1380662846123817061964694431583424, 260945277917402050689040927214450577, 49318657526388974435708069650593901281, 9321226272487516444371800743722399900932, 1761711765500140602189911891030964110454980 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (160083 x - 247275 x + 51592 x + 2736 x - 219 x - 5) - ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(160083*x^5-247275*x^4+51592*x^ 3+2736*x^2-219*x-5)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 5, 1064, 196615, 37223016, 7033770209, 1329408572800, 251257674315219, 47487711616408112, 8975177261516002333, 1696308507309907951416, 320602307779141805900543, 60593836172405607468576088, 11452235036539574269081115577, 2164472421906925975657778341712, 409085287740389136719959714875787, 77317119382933964125866546348171744, 14612935563374510457119056253836745141, 2761844821477782660406895092028895212488, 521988671259300918952706114774128861580151 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [3, 3, 3]](x) = 2 3 2 x (2079 x - 534 x - 37 x - 4) ----------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[3, 3, 3]](x) = x^2*(2079*x^3-534*x^2-37*x-4)/(-1+x)/(1+3*x)/ (-1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 4, 777, 147695, 27911782, 5275422702, 997054420439, 188443295967553, 35615782870094364, 6731382963623964200, 1272231380115865542901, 240451730842033157469411, 45445377129143100299590946, 8589176277408061612014213298, 1623354316430123497879999563363, 306813965805293342942785511066069, 57987839537200441797967251038564728, 10959701672530883500034751965292518796, 2071383616108336981504326536545141717025, 391491503444475689504343873787240640290327 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [3, 3, 2, 1]](x) = 2 5 4 3 2 2 x (43659 x - 30051 x + 2825 x + 331 x - 212 x - 8) - ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[3, 3, 2, 1]](x) = -2*x^2*(43659*x^5-30051*x^4+2825*x^3+331*x ^2-212*x-8)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 16, 3128, 590482, 111653704, 21101566572, 3988220284272, 753773130625118, 142463132594982128, 26925531831235815208, 5088925520951278243336, 961806923357904302820234, 181781508516787100603098872, 34356705109627739516735456324, 6493417265720588624114443089920, 1227255863221171384686447087518230, 231951358148801808918964653747485536, 43838806690123533123893144160155152320, 8285534464433347944418256382510397281624, 1565966013777902757630958257337935068373506 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 2 2 x (189 x - 402 x - 11) - -------------------------------------------- (-1 + 3 x) (1 + 3 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -x^2*(189*x^2-402*x-11)/(-1+3*x)/(1+3*x )/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 11, 2250, 421569, 79757460, 15072459831, 2848730642910, 538409341070829, 101759381221902120, 19232522719989610851, 3634946801027986774770, 687004945248340543038489, 129843934655001290763265980, 24540503649730880463543133071, 4638155189800488040914862669830, 876611330872263855433499575860549, 165679541534858464747219020603881040, 31313433350088237319748355277612066491, 5918238903166677116299435979432935800090, 1118547152698501969460386466641571632113009 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [3, 2, 2, 2]](x) = 2 x 6 5 4 3 2 (130977 x - 8694 x + 11424 x - 1250 x - 229 x + 120 x + 4)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[3, 2, 2, 2]](x) = 2*x^2*(130977*x^6-8694*x^5+11424*x^4-1250* x^3-229*x^2+120*x+4)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/( -1+189*x) The first 20 term , starting with k=1 are 0, 8, 1568, 295142, 55829392, 10550727796, 1994111355600, 376886539523250, 71231566844430944, 13462765904091115424, 2544462760718314777792, 480903461673850672391998, 90890754258500752628947056, 17178352554811617843224580492, 3246708632860341610663283650544, 613627931610584698989759633102986, 115975679074400925320900959635272128, 21919403345061766123846598086930348600, 4142767232216673981409346603693661788256, 782983006888951378622273296002786129905814 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 2 3 2 3 x (5793 x + 2231 x + 243 x + 5) ------------------------------------------------------------- (-1 + x) (1 + x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[3, 2, 2, 1, 1]](x) = 3*x^2*(5793*x^3+2231*x^2+243*x+5)/(-1+x )/(1+x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 15, 3039, 569034, 107675226, 20347755249, 3845787854385, 726852578014788, 137375165344010532, 25963905657255331203, 4907178181698746727411, 927456676078711645130862, 175289311784389448529396318, 33129679927133794642136961477, 6261509506230719652625999220517, 1183425296677554927828936807638856, 223667381072058954228749575932414984, 42273135022619119818408607325194207671, 7989622519275014118832871185357649564103, 1510038656142977658523116622647383204828370 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = 2 x 5 4 3 2 (261954 x + 47439 x - 25737 x - 2570 x + 157 x + 5)/((-1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = 2*x^2*(261954*x^5+47439*x^4-25737*x^ 3-2570*x^2+157*x+5)/(-1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189* x) The first 20 term , starting with k=1 are 0, 10, 1984, 368658, 69794450, 13188266380, 2492642371254, 471108109187768, 89039459938533100, 16828457351192088150, 3180578451506341022324, 601129327079545748589478, 113613442823394148475335950, 21472940693508890736892233320, 4058385791075545283388010643194, 767034914513228390130303498157788, 144969598843001208807450085261643000, 27399254181327206559530657627926589890, 5178459040270842499762559548786078881864, 978728758611189222794823639967095860676698 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (43659 x - 302022 x - 27285 x + 17704 x + 3277 x - 34 x - 3)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^2*(43659*x^6-302022*x^5-27285* x^4+17704*x^3+3277*x^2-34*x-3)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/ (1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 3, 532, 98277, 18613464, 3516837719, 664705425356, 125628812321113, 23743856345857648, 4487588619344123115, 848154253896484421700, 160301153884482892969229, 30296918086309865747454152, 5726117518267536833461606591, 1082236210953510269315710775164, 204542643870193575200546792801025, 38658559691467002922269039405252576, 7306467781687254790482928920673870547, 1380922410738891339403413005092341119348, 260994335629650459283150044731508310446901 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [2, 2, 2, 2, 1]](x) = 2 5 4 3 2 x (14553 x - 46662 x - 16412 x - 950 x - 157 x - 4) ------------------------------------------------------------------------ (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[2, 2, 2, 2, 1]](x) = x^2*(14553*x^5-46662*x^4-16412*x^3-950* x^2-157*x-4)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 4, 785, 147503, 27916934, 5275312430, 997056853919, 188443244448449, 35615783964511388, 6731382940575220136, 1272231380601260432573, 240451730831830591674275, 45445377129357508484688962, 8589176277403557813501041522, 1623354316430218095378083082347, 306813965805291356238431327111381, 57987839537200483520827679308603256, 10959701672530882623835012443753166988, 2071383616108336999904765237604960796841, 391491503444475689117932225340524157358167 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 4 3 2 x (43659 x - 14004 x + 3490 x - 52 x - 5) - ----------------------------------------------------------------- (-1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = -x^2*(43659*x^4-14004*x^3+3490*x^2-\ 52*x-5)/(-1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 5, 902, 168535, 31906668, 6028898057, 1139494252210, 215363693306907, 40703753415758936, 7693009068364672909, 1453978720825901306718, 274801978090606217789279, 51937573862184131364561604, 9816201459888493604102287761, 1855262075920276280125776505226, 350644532348903839503496949666851, 66271816613943421658939131225079472, 12525373340035294176897529307817665813, 2367295561266670862291288294969874868534, 447418861079400787452947810669603654750823 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 2 F[[4, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - x 5 4 3 2 (505197 x + 61155 x - 6642 x + 2796 x - 7 x - 3)/((-1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -x^2*(505197*x^5+61155*x^4-6642*x ^3+2796*x^2-7*x-3)/(-1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1+21*x)/(-1+189*x ) The first 20 term , starting with k=1 are 0, 3, 508, 94781, 17948742, 3391229221, 640966148510, 121142064084219, 22895861586722884, 4327317594826439039, 817863030594219768312, 154576112673238362998257, 29214885297535958011461626, 5521613321186071885363114257, 1043584917705180740353145743114, 197237549446257877639589431569695, 37277896845343185858198315680184168, 7045522503769852739817054315590899675, 1331603753212502364967450105898139516116, 251673109357162942838781047112764723571333 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 F[[4, 2, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (51975 x - 91377 x + 7500 x - 1168 x - 19 x + 1) ------------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x) and in Maple notation F[[4, 2, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^2*(51975*x^5-91377*x^4+7500* x^3-1168*x^2-19*x+1)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+11*x)/(1+21*x)/(-1+189*x ) The first 20 term , starting with k=1 are 0, 1, 150, 28089, 5318516, 1004799545, 189916139866, 35893939553473, 6783959106920712, 1282168173726616209, 242329786896248467262, 45800329679830957715177, 8656262310404796413196988, 1636033576647225184699043593, 309210345986730723909511452738, 58440755391483594924327020682801, 11045302768990578222720554025689744, 2087562223339215529265629857518843297, 394549260211111813886607288802719918694, 74569810179900131168557100596164951073945 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 3 F[[4, 2, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 2 x 5 4 3 2 (174636 x - 98496 x - 44355 x + 1031 x + 263 x + 9)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 9 x) (1 + 3 x) (1 + 11 x) (1 + 21 x) (-1 + 189 x)) and in Maple notation F[[4, 2, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 2*x^3*(174636*x^5-98496*x^4 -44355*x^3+1031*x^2+263*x+9)/(-1+x)/(1+x)/(-1+3*x)/(-1+9*x)/(1+3*x)/(1+11*x)/(1 +21*x)/(-1+189*x) The first 20 term , starting with k=1 are 0, 0, 18, 3514, 664838, 125598822, 23739552796, 4486741625156, 847994906903316, 160271021316253204, 30291223370575341614, 5725041209798117140758, 1082032788804413172501034, 204504197080822895481953546, 38651293248343027890415104072, 7305094423935413909420470972720, 1380662846123823022667570435371592, 260945277917401925514280531027282248, 49318657526388977064378037970201585770, 9321226272487516389169731409009669975442 ---------------------------------- Their sum is 6 5 4 3 2 26037 x + 8187 x - 49631 x - 15810 x + 955 x + 183 x - 1 ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 9 x) (1 + 11 x) (-1 + 189 x) and in Maple notation (26037*x^6+8187*x^5-49631*x^4-15810*x^3+955*x^2+183*x-1)/(1+x)/(-1+x)/(1+3*x)/( -1+9*x)/(1+11*x)/(-1+189*x) The first 20 term , starting with k=1 are 1, 268, 48697, 9213696, 1741187817, 329085869612, 62197206794513, 11755272264994960, 2221746455489157553, 419910080110541665836, 79363005140589302754249, 14999607971574270351336704, 2834925906627501308038302809, 535800996352598105116813443340, 101266388310641037608234829393505, 19139347390711156151905927506869328, 3617336656844408512200695763821929185, 683676628143593208811301553354430584524, 129214882719139116465274810547065342449881, 24421612833917293011937593194814335925231232 Regarding Lambda=, [4, 1, 1, 1, 1, 1] 10 9 8 F[[4, 1, 1, 1, 1, 1], [9]](x) = (1240932 x + 331812 x - 2739121 x 7 6 5 4 3 2 - 1018589 x + 711668 x + 275954 x - 23712 x - 10029 x + 409 x + 52 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[9]](x) = (1240932*x^10+331812*x^9-2739121*x^8-1018589*x^7 +711668*x^6+275954*x^5-23712*x^4-10029*x^3+409*x^2+52*x-1)/(1+x)/(-1+x)/(1+2*x) /(1+4*x)/(-1+2*x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 1, 45, 1560, 86741, 4758876, 266778730, 14924587000, 835869536211, 46806134229756, 2621165632731020, 146784815959995720, 8219954484190509061, 460317366022842333916, 25777773490505742032190, 1443555299385868958760120, 80839096966821043279875191, 4526989427061692680419873756, 253511407955682647472267078640 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [8, 1]](x) = - x 7 6 5 4 3 2 (33860 x - 125540 x - 35897 x + 19949 x + 4984 x - 436 x - 46 x + 1)/ ((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[8, 1]](x) = -x^2*(33860*x^7-125540*x^6-35897*x^5+19949*x^ 4+4984*x^3-436*x^2-46*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1+3*x)/(-1+11 *x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 6, 286, 12236, 688872, 38059397, 2133602962, 119397315162, 6686861596708, 374449473562703, 20969309408533278, 1174278636943687028, 65759633139694240384, 3682538953237909535049, 206222187428034438700234, 11548442400434459875192334, 646712775642145936784139900, 36215915417594360165724338435, 2028091263627925884270831287830 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [7, 2]](x) = - x 7 6 5 4 3 2 (145530 x + 206895 x + 9167 x - 45325 x - 7875 x + 900 x + 85 x - 2)/ ((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[7, 2]](x) = -x^2*(145530*x^7+206895*x^6+9167*x^5-45325*x^ 4-7875*x^3+900*x^2+85*x-2)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(1+3*x)/(-1+11*x)/(-1+6 *x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 15, 870, 40905, 2313877, 128430990, 7199336615, 402967977315, 22567911861357, 1263768037594920, 70771378254802315, 3963190687045511805, 221938754673433176017, 12428569032967347284130, 695999881267703640599295, 38975993115504216224527575, 2182655617549637944145969557, 122228714537270640092984777820, 6844808014698219861057204180755 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[4, 1, 1, 1, 1, 1], [7, 1, 1]](x) = - x (181104 x - 16548 x - 67464 x 5 4 3 2 - 28163 x - 1197 x + 1319 x - 264 x - 38 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[7, 1, 1]](x) = -x^2*(181104*x^8-16548*x^7-67464*x^6-28163 *x^5-1197*x^4+1319*x^3-264*x^2-38*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1 +3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 14, 874, 42427, 2398098, 133205121, 7465786966, 417896455739, 23403722739130, 1310574967776613, 73392532593299118, 4109975660216846511, 230158706951226519922, 12888886429847084799065, 721777654326016991041030, 40419548420939603004329443, 2263494714431758685931661674, 126755703965518094674633417677, 7098319422637301588024363423902 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[4, 1, 1, 1, 1, 1], [6, 3]](x) = - 2 x (177408 x + 15980 x - 156300 x 5 4 3 2 - 26031 x + 19735 x + 3937 x - 315 x - 40 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[6, 3]](x) = -2*x^2*(177408*x^8+15980*x^7-156300*x^6-26031 *x^5+19735*x^4+3937*x^3-315*x^2-40*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/( 1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 24, 1438, 72330, 4100032, 228300454, 12796842578, 716390223930, 40120420504132, 2246700089325054, 125815731527865238, 7045672697582784850, 394557776978422225352, 22095233919934084168374, 1237333120600506638672218, 69290654445389175175918890, 3880276653113510453111500492, 217295492514372795396789111214, 12168547581627273449404822556318 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 6 F[[4, 1, 1, 1, 1, 1], [6, 2, 1]](x) = x (472164 x + 146160 x - 285065 x 5 4 3 2 - 97348 x + 2950 x + 1421 x + 290 x + 55 x - 2)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[6, 2, 1]](x) = x^2*(472164*x^8+146160*x^7-285065*x^6-\ 97348*x^5+2950*x^4+1421*x^3+290*x^2+55*x-2)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2* x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 49, 3078, 158565, 8962232, 499506189, 27991941203, 1567123255705, 87763189230162, 4914660196037229, 275221866154344353, 15412409744476471545, 863095127821449767792, 48333324338507726956669, 2706666199462815179235903, 151573306626195186726990585, 8488105178320072433949436622, 475333889880431322643289350509, 26618697834737615837219111749853 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [6, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (303996 x + 10200 x - 117263 x - 25322 x + 378 x - 88 x - 27 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[6, 1, 1, 1]](x) = -x^2*(303996*x^7+10200*x^6-117263*x^5-\ 25322*x^4+378*x^3-88*x^2-27*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1+3*x)/ (-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 25, 1622, 84892, 4777953, 266473872, 14928617824, 835811719264, 46806935215535, 2621154368744824, 146784973352688606, 8219952278881868616, 460317396886267461997, 25777773058352533722256, 1443555305435621868701468, 80839096882122152351418448, 4526989428247463044946055339, 253511407939081777747234476168, 14196638845155594115335056936410 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [5, 4]](x) = - x 7 6 5 4 3 2 (22344 x - 1680 x - 12230 x - 3060 x + 1081 x + 398 x + 23 x - 1)/( (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[5, 4]](x) = -x^2*(22344*x^7-1680*x^6-12230*x^5-3060*x^4+ 1081*x^3+398*x^2+23*x-1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1+3*x)/(-1+6*x) /(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 18, 1201, 63049, 3580340, 199750782, 11196194605, 626842838283, 35105204139814, 1965863288937856, 110088737761890119, 6164963802003538077, 345238050074370136048, 19333329723803550563490, 1082666479657511407360993, 60629322649074196363700431, 3395242071312585604586846042, 190133555952002647775228163684, 10647479133893177615360603459227 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 x (7 x - 2) F[[4, 1, 1, 1, 1, 1], [5, 3, 1]](x) = -------------------------------- (1 + 4 x) (1 + 14 x) (-1 + 56 x) and in Maple notation F[[4, 1, 1, 1, 1, 1],[5, 3, 1]](x) = x^2*(7*x-2)/(1+4*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 69, 4526, 243948, 13795160, 770648112, 43182641504, 2417859001536, 135405269249408, 7582622764696320, 424627887212713472, 23779147510440152064, 1331632459013316663296, 74571414926744127418368, 4175999274789694622228480, 233855958843734566003064832, 13095933702871972372061978624, 733372287254110739308581421056, 41068848087724277390914708963328 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[4, 1, 1, 1, 1, 1], [5, 2, 2]](x) = 2 5 4 3 2 x (1176 x + 528 x + 158 x - 452 x - 37 x + 2) ------------------------------------------------------------------------- (-1 + 2 x) (1 + 2 x) (1 + 3 x) (-1 + x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x) and in Maple notation F[[4, 1, 1, 1, 1, 1],[5, 2, 2]](x) = x^2*(1176*x^5+528*x^4+158*x^3-452*x^2-37*x +2)/(-1+2*x)/(1+2*x)/(1+3*x)/(-1+x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 55, 3348, 181155, 10214342, 570923925, 31986189608, 1791020470735, 100300008984282, 5616760286777145, 314539138247472668, 17614183866191456115, 986394406735817171822, 55238085233817052931965, 3093332794700108408904528, 173226636200710592583950295, 9700691631474690698250768962, 543238731303293878822183664385, 30421368953814499007398385877188 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [5, 2, 1, 1]](x) = - x 7 6 5 4 3 2 (87318 x + 21777 x - 16181 x + 6172 x + 3757 x + 274 x + 10 x - 2)/( (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[5, 2, 1, 1]](x) = -x^2*(87318*x^7+21777*x^6-16181*x^5+ 6172*x^4+3757*x^3+274*x^2+10*x-2)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(1+3*x)/(-1+11*x )/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 90, 5246, 286323, 16085540, 899387433, 50377549259, 2820888266391, 157972318949798, 8846402847105021, 495399096689912417, 27742340559461379579, 1553571180613662575276, 86999984422701986547729, 4871999149575335416857095, 272831952049986485694205887, 15278589319151135418538919474, 855601001809167985637789863557, 47913656102173484510822689101293 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[4, 1, 1, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 2 5 4 3 2 x (462 x - 1905 x - 231 x + 278 x + 22 x - 1) - ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 1, 1, 1, 1, 1],[5, 1, 1, 1, 1]](x) = -x^2*(462*x^5-1905*x^4-231*x^3+278*x ^2+22*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(-1+11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 41, 1962, 106995, 5957421, 333249126, 18657880522, 1044796692365, 58508123367591, 3276449505611436, 183481112871679932, 10274941666985321235, 575396726158904318761, 32222216585784260977346, 1804444127933657908224942, 101048871154711985156778105, 5658736784558569693770977931, 316889259934121586787147172856, 17745798556298067382731687991552 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 7 6 5 F[[4, 1, 1, 1, 1, 1], [4, 4, 1]](x) = x (595056 x + 479556 x - 108832 x 4 3 2 - 71265 x + 21645 x + 5684 x - 561 x - 33)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[4, 4, 1]](x) = x^3*(595056*x^7+479556*x^6-108832*x^5-\ 71265*x^4+21645*x^3+5684*x^2-561*x-33)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1 +3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 33, 2277, 126250, 7144285, 399588168, 22389708537, 1253707614570, 70209933222165, 3931731402205948, 220177388018937217, 12329928610220531310, 690476083658902906365, 38666659652060642926848, 2165332956162390692001417, 121258645339093113321643570, 6790484142016837580764328485, 380267111912135818659505614468, 21294958267668280494247693558737 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [4, 3, 2]](x) = - x 7 6 5 4 3 2 (107156 x + 70060 x + 2187 x - 7927 x - 82 x + 462 x + 18 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[4, 3, 2]](x) = -x^2*(107156*x^7+70060*x^6+2187*x^5-7927*x ^4-82*x^3+462*x^2+18*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1+3*x)/(-1+11* x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 70, 4512, 253162, 14276688, 799301237, 44777105584, 2507440856284, 140419429951560, 7863468040043539, 440354692982971416, 24659858275145705066, 1380952151358668716192, 77333319514396133464801, 4330665909236085951050208, 242517290719833661982385208, 13580968283433067873333517784, 760534223832487128994574002223, 42589916535219420779373313665160 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [4, 3, 1, 1]](x) = - x ( 7 6 5 4 3 2 177408 x + 420678 x + 241631 x + 46730 x + 1100 x - 628 x - 43 x - 1) /((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[4, 3, 1, 1]](x) = -x^2*(177408*x^7+420678*x^6+241631*x^5+ 46730*x^4+1100*x^3-628*x^2-43*x-1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(1+3*x)/(-1+11* x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 93, 5788, 326670, 18353821, 1027883028, 57569777938, 3223888055670, 180539044436071, 10110179363106768, 566170266936789868, 31705533176948605650, 1775509897467135866401, 99428553866444242662588, 5567999023786604590881478, 311807945249920317521034510, 17461244935360799498621421211, 977829716363460743336320482888, 54758464116614282255794230163168 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [4, 2, 2, 1]](x) = x 6 5 4 3 2 (118122 x + 144555 x + 44817 x + 2347 x - 473 x + 5 x + 2)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[4, 2, 2, 1]](x) = x^2*(118122*x^6+144555*x^5+44817*x^4+ 2347*x^3-473*x^2+5*x+2)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(1+3*x)/(-1+11*x)/(-1+6*x) /(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 105, 5797, 327627, 18346839, 1028020002, 57568091849, 3223913068509, 180538702619731, 10110184199071704, 566170199534993781, 31705534122390288771, 1775509884241825760303, 99428554051663930165086, 5567999021193920614286593, 311807945286220244814383913, 17461244934852614619259944555, 977829716370575416291954058148, 54758464116514677342169863170285 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 F[[4, 1, 1, 1, 1, 1], [4, 2, 1, 1, 1]](x) = x ( 7 6 5 4 3 2 145530 x + 287385 x + 205018 x + 46300 x - 2460 x - 1155 x + 5 x + 2) /((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[4, 2, 1, 1, 1]](x) = x^2*(145530*x^7+287385*x^6+205018*x^ 5+46300*x^4-2460*x^3-1155*x^2+5*x+2)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(1+3*x)/(-1+ 11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 2, 105, 5115, 288720, 16055152, 899832465, 50371435130, 2820974564310, 157971114979872, 8846419727861595, 495398860510579390, 27742343866878579120, 1553571134315265803012, 86999985070912184101845, 4871999140500588599672370, 272831952177034116438026250, 15278589317372475641556841672, 855601001834069264829501820815, 47913656101824866856049533913670 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 8 7 F[[4, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = - x (1242780 x + 610332 x 6 5 4 3 2 - 453919 x - 198371 x + 21380 x + 8890 x - 417 x - 51 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -x*(1242780*x^8+610332*x^7-453919 *x^6-198371*x^5+21380*x^4+8890*x^3-417*x^2-51*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x) /(-1+2*x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 1, 45, 1560, 86741, 4758876, 266778730, 14924587000, 835869536211, 46806134229756, 2621165632731020, 146784815959995720, 8219954484190509061, 460317366022842333916, 25777773490505742032190, 1443555299385868958760120, 80839096966821043279875191, 4526989427061692680419873756, 253511407955682647472267078640, 14196638844923182447030730230600 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[4, 1, 1, 1, 1, 1], [3, 3, 3]](x) = 3 4 3 2 x (462 x + 917 x - 48 x + 27 x + 17) ----------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation F[[4, 1, 1, 1, 1, 1],[3, 3, 3]](x) = x^3*(462*x^4+917*x^3-48*x^2+27*x+17)/(1+x) /(-1+x)/(1+2*x)/(-1+2*x)/(-1+11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 17, 1098, 63295, 3564645, 199841082, 11193537978, 626864913545, 35104729932495, 1965868118267152, 110088650287434588, 6164964808397788575, 345238033585623530625, 19333329928263613365302, 1082666476504918503706878, 60629322690019151903688085, 3395242070704253387315849235, 190133555960133179345284260732, 10647479133775102929676327842048 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[4, 1, 1, 1, 1, 1], [3, 3, 2, 1]](x) = - x 6 5 4 3 2 (45556 x + 224600 x + 8799 x - 34806 x - 4141 x + 543 x + 74)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[3, 3, 2, 1]](x) = -x^3*(45556*x^6+224600*x^5+8799*x^4-\ 34806*x^3-4141*x^2+543*x+74)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1+3*x)/(-1+ 11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 74, 4391, 254531, 14255196, 799589875, 44772983997, 2507498103527, 140418625644782, 7863479283724691, 440354535469934943, 24659860479726425863, 1380952120490899237008, 77333319946523183798147, 4330665903186176458337729, 242517290804531611961036039, 13580968282247291868899693874, 760534223849087964857060548243, 42589916534987008907985917720355 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 F[[4, 1, 1, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 3 4 3 2 x (168 x + 792 x + 826 x - 350 x - 61) - ------------------------------------------------------------------------- (-1 + 2 x) (1 + 2 x) (1 + 3 x) (-1 + x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x) and in Maple notation F[[4, 1, 1, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -x^3*(168*x^4+792*x^3+826*x^2-350*x-\ 61)/(-1+2*x)/(1+2*x)/(1+3*x)/(-1+x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 61, 3156, 183085, 10183130, 571334571, 31980286296, 1791102180145, 100298859465510, 5616776346412231, 314538913211134436, 17614187015490545805, 986394362638374943890, 55238085851137705564291, 3093332786057358066918576, 173226636321707530062209065, 9700691629780724169934286270, 543238731327009353796179690751, 30421368953482482019229962534716 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 F[[4, 1, 1, 1, 1, 1], [3, 2, 2, 2]](x) = - x (129360 x + 121716 x 6 5 4 3 2 - 223912 x - 116189 x + 13837 x + 6792 x - 344 x - 11 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[3, 2, 2, 2]](x) = -x^2*(129360*x^8+121716*x^7-223912*x^6-\ 116189*x^5+13837*x^4+6792*x^3-344*x^2-11*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+ 2*x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 41, 2198, 127818, 7124144, 399884394, 22385634286, 1253765139206, 70209130604732, 3931742655927042, 220177230566514794, 12329930815163466414, 690476052793312515640, 38666660084200744801610, 2165332950112559599853222, 121258645423791533340297942, 6790484140831064398018498868, 380267111928736671446344726098, 21294958267435868724429465481570 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 x (49 x + 1) F[[4, 1, 1, 1, 1, 1], [3, 2, 2, 1, 1]](x) = - -------------------------------- (1 + 4 x) (1 + 14 x) (-1 + 56 x) and in Maple notation F[[4, 1, 1, 1, 1, 1],[3, 2, 2, 1, 1]](x) = -x^2*(49*x+1)/(1+4*x)/(1+14*x)/(-1+ 56*x) The first 20 term , starting with k=1 are 0, 1, 87, 4258, 247764, 13741480, 771400656, 43172101792, 2418006573888, 135403203170944, 7582651690056960, 424627482256615936, 23779153179829711872, 1331632379641846048768, 74571416037944783130624, 4175999259232885173821440, 233855959061529899354505216, 13095933699822837700846845952, 733372287296798624722773147648, 41068848087126646995047305314304 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 F[[4, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = x (109956 x + 244692 x 6 5 4 3 2 - 6877 x - 138129 x - 41715 x + 2734 x + 1233 x - 18 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = x^2*(109956*x^8+244692*x^7-6877*x ^6-138129*x^5-41715*x^4+2734*x^3+1233*x^2-18*x-1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/ (-1+2*x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 70, 2817, 162370, 8910196, 500240475, 27981703072, 1567266810340, 87761180916216, 4914688320621805, 275221472461394152, 15412415256476693910, 863095050655274371936, 48333325418845011227935, 2706666184338158724390432, 151573306837940768027483080, 8488105175355636650226786856, 475333889921933437706698790865, 26618697834156586311021765121912 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 2 8 7 F[[4, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x (284592 x - 116508 x 6 5 4 3 2 - 283104 x + 1143 x + 42869 x + 3212 x - 928 x - 27 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^2*(284592*x^8-116508*x^7-\ 283104*x^6+1143*x^5+42869*x^4+3212*x^3-928*x^2-27*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+ 4*x)/(-1+2*x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 1, 25, 782, 44046, 2377752, 133502166, 7461709438, 417953993482, 23402920069268, 1310586221707422, 73392375140037834, 4109977865163137058, 230158676085622707424, 12888886861987240360918, 721777648276185684144470, 40419548505638023881977274, 2263494713245985499749858220, 126755703982118947475216424654, 7098319422404889818151159765346 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[4, 1, 1, 1, 1, 1], [2, 2, 2, 2, 1]](x) = x 6 5 4 3 2 (15288 x - 6384 x - 5334 x + 1864 x + 1242 x + 177 x + 22)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[2, 2, 2, 2, 1]](x) = x^3*(15288*x^6-6384*x^5-5334*x^4+ 1864*x^3+1242*x^2+177*x+22)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1+3*x)/(-1+6 *x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 22, 1079, 64423, 3558828, 200039490, 11192072787, 626900086261, 35104399830746, 1965874532626048, 110088580248832185, 6164966006584323939, 345238019206600460304, 19333330155930601489246, 1082666473607601912865823, 60629322733772146347710257, 3395242070126809600136923302, 190133555968603483637763049884, 10647479133660765743973062406501 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[4, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = x 6 5 4 3 2 (67432 x - 24616 x + 11010 x + 12050 x + 2645 x + 258 x - 29)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = x^3*(67432*x^6-24616*x^5+11010*x^ 4+12050*x^3+2645*x^2+258*x-29)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+2*x)/(1+3*x)/(-\ 1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 29, 1250, 74245, 4068870, 228710939, 12790939770, 716471931785, 40119270990110, 2246716148945719, 125815506491570610, 7045675846881743045, 394557732880980393270, 22095234537254735610419, 1237333111957756300262570, 69290654566386112643437825, 3880276651419543924827259150, 217295492538088270370688369839, 12168547581295256461236689604450 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[4, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - 3 x 6 5 4 3 2 (30030 x + 7527 x - 6500 x + 1770 x + 1445 x + 110 x - 7)/((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -3*x^3*(30030*x^6+7527*x^5-\ 6500*x^4+1770*x^3+1445*x^2+110*x-7)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(1+3*x)/(-1+11 *x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 21, 720, 42375, 2290380, 128739321, 7194907755, 403029264855, 22567049700390, 1263780082408701, 70771209477198945, 3963193049021227515, 221938721600345911980, 12428569495957859129361, 695999874785640794628615, 38975993206251919691141055, 2182655616279163046476940850, 122228714555057246329208442501, 6844808014449207119907980137965 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[4, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x 7 6 5 4 3 2 (354816 x + 3932 x - 57136 x + 1457 x + 4692 x + 1476 x + 145 x - 7)/ ((1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(354816*x^7+3932*x^6-\ 57136*x^5+1457*x^4+4692*x^3+1476*x^2+145*x-7)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1+ 2*x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 7, 219, 12782, 679202, 38181244, 2131821741, 119421775774, 6686516389464, 374454289502216, 20969241885275123, 1174279581661924426, 65759619910021887486, 3682539138431510643748, 206222184835193593003065, 11548442436733447364060438, 646712775133955412933722468, 36215915424708999277137051040, 2028091263528320767490094280767 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 3 F[[4, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (303996 x + 10200 x - 117263 x - 25322 x + 378 x - 88 x - 27 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (1 + 4 x) (-1 + 2 x) (1 + 3 x) (-1 + 11 x) (-1 + 6 x) (1 + 14 x) (-1 + 56 x)) and in Maple notation F[[4, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(303996*x^7+10200*x ^6-117263*x^5-25322*x^4+378*x^3-88*x^2-27*x+1)/(1+x)/(-1+x)/(1+2*x)/(1+4*x)/(-1 +2*x)/(1+3*x)/(-1+11*x)/(-1+6*x)/(1+14*x)/(-1+56*x) The first 20 term , starting with k=1 are 0, 0, 1, 25, 1622, 84892, 4777953, 266473872, 14928617824, 835811719264, 46806935215535, 2621154368744824, 146784973352688606, 8219952278881868616, 460317396886267461997, 25777773058352533722256, 1443555305435621868701468, 80839096882122152351418448, 4526989428247463044946055339, 253511407939081777747234476168 ---------------------------------- Their sum is 6 5 4 3 2 1474 x - 1023 x - 3659 x + 1734 x + 437 x - 64 x + 1 ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (-1 + 11 x) (1 + 4 x) (-1 + 56 x) and in Maple notation (1474*x^6-1023*x^5-3659*x^4+1734*x^3+437*x^2-64*x+1)/(1+x)/(-1+x)/(-1+2*x)/(-1+ 11*x)/(1+4*x)/(-1+56*x) The first 20 term , starting with k=1 are 1, 29, 1313, 71538, 3981911, 222734984, 12470340073, 698308241198, 39104921889611, 2189871893247204, 122632784950834973, 6867435505516888298, 384576383339712909151, 21536277412363180028864, 1206031534491066639175313, 67537765924885759014865638, 3782114891720848751922433331, 211798433935567239035032140764, 11860712300382962183325429245893, 664199888821349047040574061994018 Regarding Lambda=, [3, 3, 3] F[[3, 3, 3], [9]](x) = 6 5 4 3 2 4688 x + 4904 x - 1724 x - 1600 x + 81 x + 41 x - 1 ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[9]](x) = (4688*x^6+4904*x^5-1724*x^4-1600*x^3+81*x^2+41*x-1)/(1+x) /(-1+2*x)/(1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 1, 23, 385, 15567, 636177, 26698351, 1120708625, 47069006063, 1976875909393, 83028760983279, 3487207157072145, 146462699617472239, 6151433354982527249, 258360200874002247407, 10851128435665850601745, 455747394296696225722095, 19141390560423720716931345, 803938403537750568130244335 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 4 64 x F[[3, 3, 3], [8, 1]](x) = - ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[8, 1]](x) = -64*x^4/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 0, 64, 2816, 120832, 5080064, 213455872, 8965332992, 376547344384, 15814995181568, 664229918556160, 27897656821219328, 1171701590844768256, 49211466824187379712, 2066881606772598243328, 86809027484762582810624, 3645979154365670697730048, 153131124483369453744029696, 6431507228301720177162059776 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 2 3 2 x (72 x + 148 x + 32 x - 1) F[[3, 3, 3], [7, 2]](x) = - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[7, 2]](x) = -x^2*(72*x^3+148*x^2+32*x-1)/(1+x)/(-1+2*x)/(1+2*x)/(1 +6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 3, 249, 9631, 408977, 17149871, 720455569, 30258166767, 1270848798993, 53375614784239, 2241776029546769, 94154591989292783, 3954492871060197649, 166088700539468820207, 6975725422928046821649, 292980467761355827965679, 12305179645986677603766545, 516817545131382062381723375, 21706336895518397001882997009 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 3 8 x (-1 + 14 x) F[[3, 3, 3], [7, 1, 1]](x) = ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[7, 1, 1]](x) = 8*x^3*(-1+14*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 8, 240, 10176, 423552, 17791872, 747118848, 31379085312, 1317916545024, 55352498251776, 2324804745179136, 97641799418462208, 4100955569045078016, 172240133904246865920, 7234085623743275925504, 303831596197374317297664, 12760927040281257996976128, 535958935691818478093205504, 22510275299056071400045412352 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 2 3 2 x (216 x - 116 x - 30 x + 1) F[[3, 3, 3], [6, 3]](x) = - --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[6, 3]](x) = -x^2*(216*x^3-116*x^2-30*x+1)/(1+x)/(-1+2*x)/(-1+6*x)/ (1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 13, 439, 17409, 727055, 30499025, 1280809839, 53792669713, 2259286753519, 94889995275537, 3985379608082159, 167385941798027537, 7030209548551458543, 295268800976469938449, 12401289640760972111599, 520854164909703940804881, 21875874926198537962188527, 918786746900257346446758161, 38589043369810483558902853359 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [6, 2, 1]](x) = 3 2 8 x (42 x - 27 x - 2) ----------------------------------------------------- (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[6, 2, 1]](x) = 8*x^3*(42*x^2-27*x-2)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+ 6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 16, 888, 37600, 1587840, 66699136, 2801678208, 117670835200, 4942186414080, 207571841990656, 8718017771747328, 366156746866892800, 15378583383102750720, 645900502106641432576, 27127821089007898165248, 1139368485738919454310400, 47853476401053659572469760, 2009846008844274860368592896, 84413532371460229665186643968 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [6, 1, 1, 1]](x) = 2 3 2 x (280 x - 52 x - 30 x + 1) - --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[6, 1, 1, 1]](x) = -x^2*(280*x^3-52*x^2-30*x+1)/(1+x)/(-1+2*x)/(-1+ 6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 13, 503, 20225, 847887, 35579089, 1494265711, 62758002705, 2635834097903, 110704990457105, 4649609526638319, 195283598619246865, 8201911139396226799, 344480267800657318161, 14468171247533570354927, 607663192394466523615505, 25521854080564208659918575, 1071917871383626800190787857, 45020550598112203736064913135 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 3 4 x (42 x + 1) F[[3, 3, 3], [5, 4]](x) = - ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[5, 4]](x) = -4*x^3*(42*x+1)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 4, 344, 14944, 634688, 26676160, 1120654976, 47068208128, 1976873977856, 83028732261376, 3487207087536128, 146462698583498752, 6151433352479227904, 258360200836779261952, 10851128435575731814400, 455747394295356198486016, 19141390560420476440543232, 803938403537702327150706688, 33765412948584056320089915392 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 3 36 x F[[3, 3, 3], [5, 3, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[5, 3, 1]](x) = 36*x^3/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 36, 1368, 58464, 2449728, 102923712, 4322586240, 181549882368, 7625087502336, 320253720450048, 13450655986808832, 564927553078566912, 23726957219504308224, 996532203277954105344, 41854352537321433759744, 1757882806569616050487296, 73831077875911179126177792, 3100905270788345693267558400, 130238021373110062097433624576 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 2 2 x (30 x - 1) F[[3, 3, 3], [5, 2, 2]](x) = - -------------------------------- (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[5, 2, 2]](x) = -2*x^2*(30*x-1)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 2, 24, 1080, 43200, 1816992, 76235904, 3202001280, 134481254400, 5648216044032, 237224973072384, 9963448989972480, 418464853950873600, 17575523870290255872, 738172002421583806464, 31003224101863248199680, 1302135412273554574540800, 54689687315494934350528512, 2296966867250617976127750144, 96472608424526158117278842880 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [5, 2, 1, 1]](x) = 2 2 x (40 x - 4 x - 1) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[5, 2, 1, 1]](x) = -x^2*(40*x^2-4*x-1)/(1+x)/(-1+2*x)/(1+2*x)/(1+6* x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 39, 1617, 68095, 2858705, 120073583, 5043041809, 211808049135, 8895936301329, 373629335234287, 15692432016355601, 659082145067859695, 27681450090564505873, 1162620903817422925551, 48830077960249480581393, 2050863274330971878452975, 86136257521897856729944337, 3617722815919727755649281775, 151944358268628459099316621585 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [5, 1, 1, 1, 1]](x) = 3 2 8 x (112 x - 34 x - 1) ----------------------------------------------------- (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[5, 1, 1, 1, 1]](x) = 8*x^3*(112*x^2-34*x-1)/(-1+2*x)/(1+2*x)/(-1+6 *x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 8, 608, 24960, 1059200, 44462208, 1867808768, 78447083520, 3294791782400, 138381222955008, 5812011878064128, 244104497729863680, 10252388923156889600, 430600334731230609408, 18085214059377780850688, 759578990492377877053440, 31902317600703850269900800, 1339897339229508110249361408, 56275688247640203890102632448 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [4, 4, 1]](x) = 2 3 2 x (336 x + 60 x + 24 x - 1) --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[4, 4, 1]](x) = x^2*(336*x^3+60*x^2+24*x-1)/(1+x)/(-1+2*x)/(-1+6*x) /(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 19, 753, 30335, 1271825, 53368623, 2241398545, 94137004015, 3953751146769, 166057485685487, 6974414289957137, 292925397928869615, 12302866709094338833, 516720401700985974511, 21702256871300355526929, 911494788591699785412335, 38282781120846312989856017, 1607876807075440200286138095, 67530825897168305604097282321 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 3 32 x F[[3, 3, 3], [4, 3, 2]](x) = - ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[4, 3, 2]](x) = -32*x^3/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 32, 1408, 60416, 2540032, 106727936, 4482666496, 188273672192, 7907497590784, 332114959278080, 13948828410609664, 585850795422384128, 24605733412093689856, 1033440803386299121664, 43404513742381291405312, 1822989577182835348865024, 76565562241684726872014848, 3215753614150860088581029888, 135061651794336326840316067840 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 2 2 x (36 x - 8 x - 1) F[[3, 3, 3], [4, 3, 1, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[4, 3, 1, 1]](x) = -x^2*(36*x^2-8*x-1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 0, 1, 45, 1847, 77825, 3267087, 137226961, 5763476335, 242066341905, 10166784344303, 427004954553617, 17934208018691823, 753236737220411665, 31635942960645148399, 1328709604362769060113, 55805803383142263516911, 2343843742092539289669905, 98441437167883264834203375, 4134540361051117435027788049, 173650695164146810399218921199 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 2 2 x (36 x - 8 x - 1) F[[3, 3, 3], [4, 2, 2, 1]](x) = - ---------------------------------------- (1 + x) (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[4, 2, 2, 1]](x) = -x^2*(36*x^2-8*x-1)/(1+x)/(-1+2*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 0, 1, 45, 1847, 77825, 3267087, 137226961, 5763476335, 242066341905, 10166784344303, 427004954553617, 17934208018691823, 753236737220411665, 31635942960645148399, 1328709604362769060113, 55805803383142263516911, 2343843742092539289669905, 98441437167883264834203375, 4134540361051117435027788049, 173650695164146810399218921199 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [4, 2, 1, 1, 1]](x) = 2 2 x (40 x - 4 x - 1) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[4, 2, 1, 1, 1]](x) = -x^2*(40*x^2-4*x-1)/(1+x)/(-1+2*x)/(1+2*x)/(1 +6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 39, 1617, 68095, 2858705, 120073583, 5043041809, 211808049135, 8895936301329, 373629335234287, 15692432016355601, 659082145067859695, 27681450090564505873, 1162620903817422925551, 48830077960249480581393, 2050863274330971878452975, 86136257521897856729944337, 3617722815919727755649281775, 151944358268628459099316621585 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [4, 1, 1, 1, 1, 1]](x) = 2 3 2 x (280 x - 52 x - 30 x + 1) - --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[4, 1, 1, 1, 1, 1]](x) = -x^2*(280*x^3-52*x^2-30*x+1)/(1+x)/(-1+2*x )/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 13, 503, 20225, 847887, 35579089, 1494265711, 62758002705, 2635834097903, 110704990457105, 4649609526638319, 195283598619246865, 8201911139396226799, 344480267800657318161, 14468171247533570354927, 607663192394466523615505, 25521854080564208659918575, 1071917871383626800190787857, 45020550598112203736064913135 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [3, 3, 3]](x) = 4 3 2 x (1360 x + 1000 x - 100 x - 40 x + 1) - ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[3, 3, 3]](x) = -x*(1360*x^4+1000*x^3-100*x^2-40*x+1)/(1+x)/(-1+2*x )/(1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 1, 23, 385, 15567, 636177, 26698351, 1120708625, 47069006063, 1976875909393, 83028760983279, 3487207157072145, 146462699617472239, 6151433354982527249, 258360200874002247407, 10851128435665850601745, 455747394296696225722095, 19141390560423720716931345, 803938403537750568130244335, 33765412948584173114040062225 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 3 32 x F[[3, 3, 3], [3, 3, 2, 1]](x) = - ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[3, 3, 2, 1]](x) = -32*x^3/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 32, 1408, 60416, 2540032, 106727936, 4482666496, 188273672192, 7907497590784, 332114959278080, 13948828410609664, 585850795422384128, 24605733412093689856, 1033440803386299121664, 43404513742381291405312, 1822989577182835348865024, 76565562241684726872014848, 3215753614150860088581029888, 135061651794336326840316067840 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 2 2 x (30 x - 1) F[[3, 3, 3], [3, 3, 1, 1, 1]](x) = - -------------------------------- (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[3, 3, 1, 1, 1]](x) = -2*x^2*(30*x-1)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 2, 24, 1080, 43200, 1816992, 76235904, 3202001280, 134481254400, 5648216044032, 237224973072384, 9963448989972480, 418464853950873600, 17575523870290255872, 738172002421583806464, 31003224101863248199680, 1302135412273554574540800, 54689687315494934350528512, 2296966867250617976127750144, 96472608424526158117278842880 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [3, 2, 2, 2]](x) = 2 3 2 x (336 x + 60 x + 24 x - 1) --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[3, 2, 2, 2]](x) = x^2*(336*x^3+60*x^2+24*x-1)/(1+x)/(-1+2*x)/(-1+6 *x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 19, 753, 30335, 1271825, 53368623, 2241398545, 94137004015, 3953751146769, 166057485685487, 6974414289957137, 292925397928869615, 12302866709094338833, 516720401700985974511, 21702256871300355526929, 911494788591699785412335, 38282781120846312989856017, 1607876807075440200286138095, 67530825897168305604097282321 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 3 36 x F[[3, 3, 3], [3, 2, 2, 1, 1]](x) = -------------------------------- (-1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[3, 2, 2, 1, 1]](x) = 36*x^3/(-1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 36, 1368, 58464, 2449728, 102923712, 4322586240, 181549882368, 7625087502336, 320253720450048, 13450655986808832, 564927553078566912, 23726957219504308224, 996532203277954105344, 41854352537321433759744, 1757882806569616050487296, 73831077875911179126177792, 3100905270788345693267558400, 130238021373110062097433624576 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [3, 2, 1, 1, 1, 1]](x) = 3 2 8 x (42 x - 27 x - 2) ----------------------------------------------------- (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[3, 2, 1, 1, 1, 1]](x) = 8*x^3*(42*x^2-27*x-2)/(-1+2*x)/(1+2*x)/(-1 +6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 16, 888, 37600, 1587840, 66699136, 2801678208, 117670835200, 4942186414080, 207571841990656, 8718017771747328, 366156746866892800, 15378583383102750720, 645900502106641432576, 27127821089007898165248, 1139368485738919454310400, 47853476401053659572469760, 2009846008844274860368592896, 84413532371460229665186643968 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [3, 1, 1, 1, 1, 1, 1]](x) = 3 8 x (-1 + 14 x) ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[3, 1, 1, 1, 1, 1, 1]](x) = 8*x^3*(-1+14*x)/(-1+2*x)/(-1+6*x)/(1+6* x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 8, 240, 10176, 423552, 17791872, 747118848, 31379085312, 1317916545024, 55352498251776, 2324804745179136, 97641799418462208, 4100955569045078016, 172240133904246865920, 7234085623743275925504, 303831596197374317297664, 12760927040281257996976128, 535958935691818478093205504, 22510275299056071400045412352 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [2, 2, 2, 2, 1]](x) = 3 4 x (42 x + 1) - ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[2, 2, 2, 2, 1]](x) = -4*x^3*(42*x+1)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1 +42*x) The first 20 term , starting with k=1 are 0, 0, 4, 344, 14944, 634688, 26676160, 1120654976, 47068208128, 1976873977856, 83028732261376, 3487207087536128, 146462698583498752, 6151433352479227904, 258360200836779261952, 10851128435575731814400, 455747394295356198486016, 19141390560420476440543232, 803938403537702327150706688, 33765412948584056320089915392 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [2, 2, 2, 1, 1, 1]](x) = 2 3 2 x (216 x - 116 x - 30 x + 1) - --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[2, 2, 2, 1, 1, 1]](x) = -x^2*(216*x^3-116*x^2-30*x+1)/(1+x)/(-1+2* x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 13, 439, 17409, 727055, 30499025, 1280809839, 53792669713, 2259286753519, 94889995275537, 3985379608082159, 167385941798027537, 7030209548551458543, 295268800976469938449, 12401289640760972111599, 520854164909703940804881, 21875874926198537962188527, 918786746900257346446758161, 38589043369810483558902853359 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [2, 2, 1, 1, 1, 1, 1]](x) = 2 3 2 x (72 x + 148 x + 32 x - 1) - -------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[2, 2, 1, 1, 1, 1, 1]](x) = -x^2*(72*x^3+148*x^2+32*x-1)/(1+x)/(-1+ 2*x)/(1+2*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 3, 249, 9631, 408977, 17149871, 720455569, 30258166767, 1270848798993, 53375614784239, 2241776029546769, 94154591989292783, 3954492871060197649, 166088700539468820207, 6975725422928046821649, 292980467761355827965679, 12305179645986677603766545, 516817545131382062381723375, 21706336895518397001882997009 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 4 64 x - ------------------------------------------- (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -64*x^4/(-1+2*x)/(-1+6*x)/(1+6*x)/(-\ 1+42*x) The first 20 term , starting with k=1 are 0, 0, 0, 64, 2816, 120832, 5080064, 213455872, 8965332992, 376547344384, 15814995181568, 664229918556160, 27897656821219328, 1171701590844768256, 49211466824187379712, 2066881606772598243328, 86809027484762582810624, 3645979154365670697730048, 153131124483369453744029696, 6431507228301720177162059776 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 F[[3, 3, 3], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (1360 x + 1000 x - 100 x - 40 x + 1) - ------------------------------------------------------------- (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation F[[3, 3, 3],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^2*(1360*x^4+1000*x^3-100*x^2-\ 40*x+1)/(1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 1, 23, 385, 15567, 636177, 26698351, 1120708625, 47069006063, 1976875909393, 83028760983279, 3487207157072145, 146462699617472239, 6151433354982527249, 258360200874002247407, 10851128435665850601745, 455747394296696225722095, 19141390560423720716931345, 803938403537750568130244335 ---------------------------------- Their sum is 5 4 3 2 1216 x + 184 x - 1356 x + 20 x + 42 x - 1 --------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 42 x) and in Maple notation (1216*x^5+184*x^4-1356*x^3+20*x^2+42*x-1)/(1+x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(-1+ 42*x) The first 20 term , starting with k=1 are 1, 19, 537, 22639, 943569, 39636911, 1664496785, 69909133551, 2936174516497, 123319339426543, 5179411928709393, 217535301356465903, 9136482645192691985, 383732271110718271215, 16116755386226129932561, 676903726221951968800495, 28429956501306717349417233, 1194058173054898491110911727, 50150443268305187074444169489, 2106318617268818446174395494127 Regarding Lambda=, [3, 3, 2, 1] 9 8 7 6 F[[3, 3, 2, 1], [9]](x) = (1532559 x + 305873 x - 2758495 x - 1002545 x 5 4 3 2 + 512079 x + 148058 x - 23564 x - 4401 x - 141 x + 1)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[9]](x) = (1532559*x^9+305873*x^8-2758495*x^7-1002545*x^6+512079 *x^5+148058*x^4-23564*x^3-4401*x^2-141*x+1)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4 *x)/(1+3*x)/(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 1, 11, 2226, 368390, 61963374, 10408741053, 1748684868407, 293778814059336, 49354844391888252, 8291613803758775435, 1392991119837415781673, 234022508120669803219542, 39315781364451743838344750, 6605051269225219041669884457, 1109648613229876707211868773899, 186420967022618690997281800093508, 31318722459799948985433823791831168, 5261545373246391296635063212394668519, 883939622705393739820750371900408619085 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 7 6 5 4 F[[3, 3, 2, 1], [8, 1]](x) = - x (235200 x + 256013 x - 261492 x - 84201 x 3 2 + 18000 x + 4632 x + 186 x - 2)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[8, 1]](x) = -x^2*(235200*x^7+256013*x^6-261492*x^5-84201*x^4+ 18000*x^3+4632*x^2+186*x-2)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+ 15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 2, 96, 17708, 2948288, 495689519, 83270186537, 13989475093274, 2350230569964926, 394838754276791501, 66332910442887189563, 11143928958507873950600, 1872180064968218956739984, 314526250915571200209078923, 52840410153802391385990359549, 8877188905839004102771910577686, 1491367736180949670869965314685762, 250549779678399589746144792918318185, 42092362985971130405055694458171518495, 7071516981643149918087560918347709230532 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [7, 2]](x) = - x 6 5 4 3 2 (155673 x + 13080 x - 84540 x + 1012 x + 4719 x + 240 x - 4)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[7, 2]](x) = -x^2*(155673*x^6+13080*x^5-84540*x^4+1012*x^3+4719* x^2+240*x-4)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+15*x)/(1+14*x)/(-1+168*x ) The first 20 term , starting with k=1 are 0, 4, 336, 59545, 9953484, 1672906522, 281037556836, 47214468325294, 7932028324542672, 1332580793431117870, 223873572778388863836, 37610760234461512543438, 6318607719275247873180600, 1061526096839940580652484958, 178336384269084748440881702436, 29960512557206613765182264044702, 5033366109610705514276874213031968, 845605506414598609782758458535113486, 142061725077652565200997839290885870636, 23866369813045630972289607700187564528686 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [7, 1, 1]](x) = - x ( 7 6 5 4 3 2 91140 x - 86093 x - 176618 x - 53785 x + 11812 x + 4916 x + 216 x - 4 )/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[7, 1, 1]](x) = -x^2*(91140*x^7-86093*x^6-176618*x^5-53785*x^4+ 11812*x^3+4916*x^2+216*x-4)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+ 15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 4, 348, 61760, 10321980, 1734868377, 291446318751, 48963152900582, 8225807142700390, 1381935637765609835, 232165186582951107189, 39003751354287679462584, 6552630227396075159211420, 1100841878204390119726450253, 184941435538309998349232743387, 31070161170436490040260516143466, 5219787076633324211324026324361970, 876924228874398558683494096681329231, 147323270450898956498818677097164822945, 24750309435751024712093757227752978074428 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [6, 3]](x) = - x 7 6 5 4 3 2 (193536 x + 404958 x + 25088 x - 65590 x + 327 x + 2713 x + 93 x - 5) /((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[6, 3]](x) = -x^2*(193536*x^7+404958*x^6+25088*x^5-65590*x^4+327 *x^3+2713*x^2+93*x-5)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+15*x)/( 1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 5, 612, 105589, 17698916, 2973998207, 499623183482, 83936819735305, 14101383879711768, 2369032518794310979, 397997462759859099302, 66863573749515626209901, 11233080389832197989984580, 1887157505493085197285237511, 317042460922819460737023316722, 53263133435033948177247318661777, 8948206417085699168353479728751152, 1503298678070397521378262380729150603, 252554177915826782686135676829915727742, 42429101889858899504697829055280756361333 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [6, 2, 1]](x) = x 5 4 3 2 (208005 x + 46074 x - 5805 x + 2187 x + 49 x - 10)/((-1 + x) (1 + 2 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[6, 2, 1]](x) = x^2*(208005*x^5+46074*x^4-5805*x^3+2187*x^2+49*x -10)/(-1+x)/(1+2*x)/(-1+4*x)/(1+3*x)/(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 10, 1341, 230982, 38716210, 6505624395, 1092925657236, 183611794114297, 30846777221434710, 5182258635112026555, 870619449783198564856, 146264067577128845841837, 24572363352756932686146810, 4128157043266139564303009215, 693530383268667325287762410076, 116513104389136765448887367116977, 19574201537374966871715269334004510, 3288465858278994578927286653685727875, 552462264190871087111866173063462520896, 92813660384066342666742523122297566833717 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [6, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (65856 x + 162659 x - 20737 x - 59161 x + 140 x + 3418 x + 135 x - 6) /((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[6, 1, 1, 1]](x) = -x^2*(65856*x^7+162659*x^6-20737*x^5-59161*x^ 4+140*x^3+3418*x^2+135*x-6)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+ 15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 711, 123245, 20647831, 3469678686, 582893495254, 97926293070563, 16451614474268873, 2763871272726734096, 464330373207567130132, 78007502707956007127541, 13105260454801361844146695, 2201683756408643168910286226, 369882871076622037323109543970, 62140322340872949687217565018279, 10439574153266648875522667053944997, 1753848457748797110616218060346591476, 294646540901797913098306018853683292368, 49500618871502049422685784907627658360377 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [5, 4]](x) = x 6 5 4 3 2 (2058 x + 8764 x - 10695 x - 3252 x + 1041 x + 77 x + 3)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[5, 4]](x) = x^2*(2058*x^6+8764*x^5-10695*x^4-3252*x^3+1041*x^2+ 77*x+3)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 545, 92247, 15488582, 2602218056, 437170737339, 73444710525791, 12338710995358544, 2072903452442993094, 348247779937306488173, 58505627030491131683525, 9828945341108179171289586, 1651262817306374734257612092, 277412153307468146487013542047, 46605241755654687933976143630699, 7829680614949987022369789076681908, 1315386343311597827465659438567077650, 220984905676348434906325297557886574961, 37125464153626537065773326823887612963313 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[3, 3, 2, 1], [5, 3, 1]](x) = 2 4 3 2 x (63 x - 1863 x + 271 x + 13 x + 13) -------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 14 x) (-1 + 168 x) and in Maple notation F[[3, 3, 2, 1],[5, 3, 1]](x) = x^2*(63*x^4-1863*x^3+271*x^2+13*x+13)/(1+x)/(-1+ x)/(-1+3*x)/(-1+4*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 13, 2106, 355756, 59742630, 10037113465, 1686230175123, 283286737965244, 47592171018992340, 7995484744620743407, 1343241436908270685005, 225664561403221697471002, 37911646315704394151473230, 6369156581038854132125480989, 1070018305614520271392939069527, 179763075343239506713277036033080, 30200196657664235712160877057375400, 5073633038487591619462402737846413611, 852370350465915391792212404793319177089, 143198218878273785824976281578880960292278 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[3, 3, 2, 1], [5, 2, 2]](x) = 2 3 2 2 x (637 x + 234 x - 31 x - 5) - ---------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 3 x) (1 + 14 x) (-1 + 168 x) and in Maple notation F[[3, 3, 2, 1],[5, 2, 2]](x) = -2*x^2*(637*x^3+234*x^2-31*x-5)/(-1+x)/(1+2*x)/( 1+3*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 10, 1562, 263502, 44254160, 7434893930, 1249059458772, 209842027146902, 35253460027734320, 5922581292120362850, 994993656971767699532, 167158934372719317065102, 28082700974596372463553480, 4717893763732477193105727770, 792606152307052155772615629092, 133157833587584818347167311524102, 22370516042714248695840958435084640, 3758246695175993791912045114206404690, 631385444789566956887072881831607491452, 106072754724647248759186353910667514939902 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [5, 2, 1, 1]](x) = - x 6 5 4 3 2 (115983 x + 70644 x - 20421 x - 4716 x + 995 x - 145 x - 16)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[5, 2, 1, 1]](x) = -x^2*(115983*x^6+70644*x^5-20421*x^4-4716*x^3 +995*x^2-145*x-16)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+15*x)/(1+14*x)/(-1 +168*x) The first 20 term , starting with k=1 are 0, 16, 2449, 415181, 69697694, 11709997491, 1967268045351, 330501201896938, 55524199405020772, 9328065537190962461, 1567115009698711738793, 263275321637514477782760, 44230254034982004269604870, 7430682677878761641323487611, 1248354689883605482834082801275, 209723587900446113996455229022422, 35233562767274941317185806654553288, 5919238544902190227974688414561066041, 994432075543567957010996863003921633197, 167064588691319416797016876614089413540324 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[3, 3, 2, 1], [5, 1, 1, 1, 1]](x) = 2 4 3 2 x (6125 x - 993 x - 1137 x + 19 x - 6) --------------------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x) and in Maple notation F[[3, 3, 2, 1],[5, 1, 1, 1, 1]](x) = x^2*(6125*x^4-993*x^3-1137*x^2+19*x-6)/(-1 +x)/(1+2*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 899, 153894, 25812030, 4337065416, 728617348494, 122407859325324, 20564518195444170, 3454839089405091576, 580412966531506282614, 97509378384621327636804, 16381575568506459092082210, 2752104695510733987291725136, 462353588845778576969818213134, 77675402926091171924366169597084, 13049467691583311318392000330798650, 2192310572185996384963378938500877096, 368308176127247391421744818978593094054, 61875773589377561777634672230987312146164 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [4, 4, 1]](x) = x ( 7 6 5 4 3 2 298116 x + 34125 x - 102864 x - 24711 x + 974 x + 2514 x + 256 x + 6) /((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[4, 4, 1]](x) = x^2*(298116*x^7+34125*x^6-102864*x^5-24711*x^4+ 974*x^3+2514*x^2+256*x+6)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+15* x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 1102, 184308, 30980010, 5204394073, 874342095585, 146889411883122, 24677422126198684, 4145806902882438435, 696495559904264451903, 117011254060543115310076, 19657890682222866821318298, 3302525634612652943710577277, 554824306614937725412739518381, 93210483511309354597438316285670, 15659361229899974360772352247639352, 2630772686623195650233176357050183799, 441969811352696869882529845614626810619, 74250928307253074130506765312435962913904 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [4, 3, 2]](x) = x 6 5 4 3 2 (375879 x + 281006 x - 63922 x - 27719 x + 3357 x + 523 x + 12)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[4, 3, 2]](x) = x^2*(375879*x^6+281006*x^5-63922*x^4-27719*x^3+ 3357*x^2+523*x+12)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+15*x)/(1+ 14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 12, 2215, 368496, 61961855, 10408761915, 1748684575288, 293778818157718, 49354844334491965, 8291613804562243353, 1392991119826166919146, 234022508120827286030820, 39315781364449539073965295, 6605051269225249908351040951, 1109648613229876275078252098764, 186420967022618697047152111330002, 31318722459799948900735638146215745, 5261545373246391297820837806278952309, 883939622705393739804149527565413545742, 148501856614506148260435483502061663210464 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [4, 3, 1, 1]](x) = - x 6 5 4 3 2 (64512 x - 79851 x - 25020 x + 25226 x - 2357 x - 530 x - 16)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[4, 3, 1, 1]](x) = -x^2*(64512*x^6-79851*x^5-25020*x^4+25226*x^3 -2357*x^2-530*x-16)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+15*x)/(1+14*x)/(-\ 1+168*x) The first 20 term , starting with k=1 are 0, 16, 2834, 473973, 79662250, 13382739134, 2248308051321, 377715633792499, 63456228271591196, 10660646322553051632, 1790988582597274740823, 300886081870185428870885, 50548861754283941826091002, 8492208774718304238946176850, 1426691074152696167948856293765, 239684100457652639174528975600631, 40266928876885648153787302947898168, 6764844051316788818013325590833213588, 1136493800621220522506886048721161662547, 190930958504365047764900900736698303189737 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [4, 2, 2, 1]](x) = x 6 5 4 3 2 (56448 x + 99795 x + 38337 x - 34546 x + 1740 x + 534 x + 16)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[4, 2, 2, 1]](x) = x^2*(56448*x^6+99795*x^5+38337*x^4-34546*x^3+ 1740*x^2+534*x+16)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+15*x)/(1+14*x)/(-1 +168*x) The first 20 term , starting with k=1 are 0, 16, 2838, 473932, 79662906, 13382730241, 2248308177081, 377715632036710, 63456228296192028, 10660646322208718371, 1790988582602095721559, 300886081870117936397848, 50548861754284886725748730, 8492208774718291010371097221, 1426691074152696353148987947397, 239684100457652636581727454555946, 40266928876885648190086525531046712, 6764844051316788817505136479823029191, 1136493800621220522514000696295920218195, 190930958504365047764801295670734146520604 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [4, 2, 1, 1, 1]](x) = 3 x 6 5 4 3 2 (14259 x - 17815 x - 2301 x - 670 x + 410 x + 100 x + 5)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[4, 2, 1, 1, 1]](x) = 3*x^2*(14259*x^6-17815*x^5-2301*x^4-670*x^ 3+410*x^2+100*x+5)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+15*x)/(1+14*x)/(-1 +168*x) The first 20 term , starting with k=1 are 0, 15, 2460, 415020, 69699930, 11709966102, 1967268484503, 330501195747549, 55524199491107364, 9328065535985730384, 1567115009715584909481, 263275321637278253078043, 44230254034985311414213638, 7430682677878715341293929646, 1248354689883606131034476484219, 209723587900446104921649636922377, 35233562767274941444233084621847752, 5919238544902190226196026521730420588, 994432075543567957035898129498396774317, 167064588691319416796668258883146145590551 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [4, 1, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (65856 x - 41461 x + 80270 x - 6835 x - 9751 x + 115 x - 13 x - 5)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(65856*x^7-41461*x^6+80270*x^5-\ 6835*x^4-9751*x^3+115*x^2-13*x-5)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x )/(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 5, 718, 123133, 20649310, 3469657698, 582893787805, 97926288970039, 16451614531656336, 2763871271923231396, 464330373218815852507, 78007502707798523757705, 13105260454803566606287822, 2201683756408612302220184614, 369882871076622469456690422249, 62140322340872943637347110627131, 10439574153266648960220852126876268, 1753848457748797109430443464171702152, 294646540901797913114906863179515681431, 49500618871502049422453373086983541978317 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [3, 3, 3]](x) = - x 6 5 4 3 2 (3591 x - 12840 x - 7996 x + 185 x + 1124 x - 93 x - 3)/((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[3, 3, 3]](x) = -x^2*(3591*x^6-12840*x^5-7996*x^4+185*x^3+1124*x ^2-93*x-3)/(1+x)/(-1+3*x)/(-1+x)/(1+2*x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 558, 92062, 15491439, 2602176046, 437171358393, 73444701357856, 12338711130842367, 2072903450439453994, 348247779966957998853, 58505627030051983766020, 9828945341114687650588635, 1651262817306278209455200782, 277412153307469578925734926913, 46605241755654666663462208440904, 7829680614949987338402563314133943, 1315386343311597822767516919055746610, 220984905676348434976204548059030928573, 37125464153626537064733438488557512460108 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 7 6 5 F[[3, 3, 2, 1], [3, 3, 2, 1]](x) = - x (1007601 x + 664831 x - 287501 x 4 3 2 - 115543 x + 17332 x + 3727 x + 130 x - 1)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[3, 3, 2, 1]](x) = -x*(1007601*x^7+664831*x^6-287501*x^5-115543* x^4+17332*x^3+3727*x^2+130*x-1)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/ (1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 11, 2226, 368390, 61963374, 10408741053, 1748684868407, 293778814059336, 49354844391888252, 8291613803758775435, 1392991119837415781673, 234022508120669803219542, 39315781364451743838344750, 6605051269225219041669884457, 1109648613229876707211868773899, 186420967022618690997281800093508, 31318722459799948985433823791831168, 5261545373246391296635063212394668519, 883939622705393739820750371900408619085, 148501856614506148260203071681454197041234 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[3, 3, 2, 1], [3, 3, 1, 1, 1]](x) = 2 3 2 x (742 x + 696 x + 223 x + 9) ---------------------------------------------------- (-1 + x) (1 + 2 x) (1 + 3 x) (1 + 14 x) (-1 + 168 x) and in Maple notation F[[3, 3, 2, 1],[3, 3, 1, 1, 1]](x) = x^2*(742*x^3+696*x^2+223*x+9)/(-1+x)/(1+2* x)/(1+3*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 9, 1573, 263349, 44256295, 7434864049, 1249059877083, 209842021290589, 35253460109722615, 5922581290972526889, 994993656987837402643, 167158934372494341222229, 28082700974599522125352335, 4717893763732433097840546529, 792606152307052773106328161003, 133157833587584809704495336088269, 22370516042714248816838366091164455, 3758246695175993790218081407021330969, 631385444789566956910788373732198436163, 106072754724647248758854337024059241888709 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 7 6 5 F[[3, 3, 2, 1], [3, 2, 2, 2]](x) = - x (19404 x - 196287 x - 147678 x 4 3 2 + 30408 x + 23025 x - 1145 x - 265 x - 6)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[3, 2, 2, 2]](x) = -x^2*(19404*x^7-196287*x^6-147678*x^5+30408*x ^4+23025*x^3-1145*x^2-265*x-6)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/( 1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 6, 1111, 184208, 30981529, 5204373261, 874342388808, 146889407785350, 24677422183597027, 4145806902078979511, 696495559915513348870, 117011254060385632639632, 19657890682225071586254945, 3302525634612622077031661841, 554824306614938157546365133172, 93210483511309348547568040856954, 15659361229899974445470538036387583, 2630772686623195649047401763738627851, 441969811352696869899130689951912402114, 74250928307253074130274353491837659603716 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 F[[3, 3, 2, 1], [3, 2, 2, 1, 1]](x) = 2 4 3 2 3 x (147 x - 1284 x + 705 x - 65 x - 4) - -------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 14 x) (-1 + 168 x) and in Maple notation F[[3, 3, 2, 1],[3, 2, 2, 1, 1]](x) = -3*x^2*(147*x^4-1284*x^3+705*x^2-65*x-4)/( 1+x)/(-1+x)/(-1+3*x)/(-1+4*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 12, 2127, 355488, 59746497, 10037059785, 1686230928486, 283286727425532, 47592171166577799, 7995484742554664943, 1343241436937196255360, 225664561402816741373466, 37911646315710063544388481, 6369156581038774760654866461, 1070018305614521382593648468874, 179763075343239491156467587626040, 30200196657664235929956211267809243, 5073633038487591616413268066631280939, 852370350465915391834900290221254799028, 143198218878273785824378651183013556643254 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [3, 2, 1, 1, 1, 1]](x) = - x 5 4 3 2 (56595 x + 12831 x + 741 x - 143 x + 107 x + 9)/((-1 + x) (1 + 2 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[3, 2, 1, 1, 1, 1]](x) = -x^2*(56595*x^5+12831*x^4+741*x^3-143*x ^2+107*x+9)/(-1+x)/(1+2*x)/(-1+4*x)/(1+3*x)/(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 9, 1358, 230712, 38719920, 6505572029, 1092926388918, 183611783864427, 30846777364908680, 5182258633103291949, 870619449811320457578, 146264067576735137767967, 24572363352762444592895340, 4128157043266062397583352369, 693530383268668405621736965838, 116513104389136750324211320636707, 19574201537374967083460732374208400, 3288465858278994575962850164680236789, 552462264190871087153368283883769963698, 92813660384066342666161493570710182676647 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [3, 1, 1, 1, 1, 1, 1]](x) = x 7 6 5 4 3 2 (226380 x + 14651 x - 376 x + 30817 x + 2611 x - 1473 x - 69 x + 3)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[3, 1, 1, 1, 1, 1, 1]](x) = x^2*(226380*x^7+14651*x^6-376*x^5+ 30817*x^4+2611*x^3-1473*x^2-69*x+3)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3 *x)/(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 354, 61647, 10323448, 1734847360, 291446611155, 48963148799533, 8225807200085626, 1381935636962098482, 232165186594199794441, 39003751354130195953279, 6552630227398279920792624, 1100841878204359253034113044, 184941435538310430482804671087, 31070161170436483990390025966385, 5219787076633324296022211254116742, 876924228874398557497719499933799446, 147323270450898956515419521420706519093, 24750309435751024711861345407099699182851 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [2, 2, 2, 2, 1]](x) = x 6 5 4 3 2 (2058 x + 1204 x + 1446 x - 2484 x - 308 x + 85 x + 3)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[2, 2, 2, 2, 1]](x) = x^2*(2058*x^6+1204*x^5+1446*x^4-2484*x^3-\ 308*x^2+85*x+3)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x)/(1+14*x)/(-1+168 *x) The first 20 term , starting with k=1 are 0, 3, 553, 92146, 15490090, 2602197215, 437171030415, 73444706427494, 12338711052754660, 2072903451639525517, 348247779948555350017, 58505627030333648873612, 9828945341110383935666310, 1651262817306343867576461059, 277412153307468578620630206259, 46605241755654681884105832416050, 7829680614949987107067974722253640, 1315386343311597826279884844682881241, 220984905676348434922926141892881473541, 37125464153626537065540915003280147143608 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [2, 2, 2, 1, 1, 1]](x) = x 7 6 5 4 3 2 (169344 x - 20046 x - 66200 x - 1379 x - 1607 x - 15 x + 59 x + 4)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[2, 2, 2, 1, 1, 1]](x) = x^2*(169344*x^7-20046*x^6-66200*x^5-\ 1379*x^4-1607*x^3-15*x^2+59*x+4)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+3*x) /(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 4, 623, 105436, 17701051, 2973968326, 499623601793, 83936813878992, 14101383961700063, 2369032517646475018, 397997462775928802413, 66863573749290650367028, 11233080389835347651783435, 1887157505493041102020056270, 317042460922820078070735848633, 53263133435033939534575343225944, 8948206417085699289350887384830967, 1503298678070397519684298673544076882, 252554177915826782709851168730506672453, 42429101889858899504365812168672483310140 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (117873 x + 15825 x - 43899 x - 1582 x + 1877 x + 89 x - 3)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -x^2*(117873*x^6+15825*x^5-43899*x^4 -1582*x^3+1877*x^2+89*x-3)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+15*x)/(1+ 14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 3, 343, 59425, 9955064, 1672884026, 281037870228, 47214463931694, 7932028386028432, 1332580792570219054, 223873572790441053788, 37610760234292780311758, 6318607719277610118131640, 1061526096839907509198006622, 178336384269085211441143731748, 29960512557206607283178192989342, 5033366109610705605024929597177888, 845605506414598608512285676714652430, 142061725077652565218784458210602456108, 23866369813045630972040595035208453248046 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 2 F[[3, 3, 2, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x 7 6 5 4 3 2 (127680 x + 242299 x + 45825 x - 6429 x + 187 x - 705 x - 42 x + 1)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^2*(127680*x^7+242299*x^6+45825* x^5-6429*x^4+187*x^3-705*x^2-42*x+1)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/(1+ 3*x)/(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 1, 99, 17656, 2948915, 495680479, 83270311772, 13989473335258, 2350230594557105, 394838753932423117, 66332910447708030830, 11143928958440380917640, 1872180064969163854162115, 314526250915557971625048715, 52840410153802576586086227248, 8877188905839001509970246356502, 1491367736180949707169187325193845, 250549779678399589237955679617440873, 42092362985971130412170342023767564626, 7071516981643149917987955852346901999044 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 3 F[[3, 3, 2, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (375879 x + 281006 x - 63922 x - 27719 x + 3357 x + 523 x + 12)/( (-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (-1 + 4 x) (1 + 3 x) (1 + 15 x) (1 + 14 x) (-1 + 168 x)) and in Maple notation F[[3, 3, 2, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(375879*x^6+281006*x^5-\ 63922*x^4-27719*x^3+3357*x^2+523*x+12)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(-1+4*x)/( 1+3*x)/(1+15*x)/(1+14*x)/(-1+168*x) The first 20 term , starting with k=1 are 0, 0, 12, 2215, 368496, 61961855, 10408761915, 1748684575288, 293778818157718, 49354844334491965, 8291613804562243353, 1392991119826166919146, 234022508120827286030820, 39315781364449539073965295, 6605051269225249908351040951, 1109648613229876275078252098764, 186420967022618697047152111330002, 31318722459799948900735638146215745, 5261545373246391297820837806278952309, 883939622705393739804149527565413545742 ---------------------------------- Their sum is 7 6 5 4 3 2 (56703 x - 15430 x - 76877 x + 16678 x + 10254 x - 1861 x - 156 x + 1)/( (1 + x) (-1 + 3 x) (-1 + x) (1 + 3 x) (1 + 15 x) (-1 + 4 x) (-1 + 168 x)) and in Maple notation (56703*x^7-15430*x^6-76877*x^5+16678*x^4+10254*x^3-1861*x^2-156*x+1)/(1+x)/(-1+ 3*x)/(-1+x)/(1+3*x)/(1+15*x)/(-1+4*x)/(-1+168*x) The first 20 term , starting with k=1 are 1, 214, 34120, 5753231, 966212574, 162328592085, 27271130073184, 4581550952286907, 769700543481290738, 129309691552386971921, 21724028177088012739668, 3649636733806480919333463, 613138971278653372032649222, 103007347174826297834685095437, 17305234325370630066217576136072, 2907279366662268670674648492261299, 488422933599261094380089328215329626, 82055052844675864490253780676418361033, 13785248877905545224846653547679359871996, 2315921811488131597916977520087772668036815 Regarding Lambda=, [3, 3, 1, 1, 1] 9 8 7 6 F[[3, 3, 1, 1, 1], [9]](x) = (1734080 x - 151120 x - 2984764 x + 245333 x 5 4 3 2 + 500471 x - 40207 x - 21905 x + 1625 x + 108 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) ) and in Maple notation F[[3, 3, 1, 1, 1],[9]](x) = (1734080*x^9-151120*x^8-2984764*x^7+245333*x^6+ 500471*x^5-40207*x^4-21905*x^3+1625*x^2+108*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/ (1+4*x)/(-1+4*x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 5, 604, 68335, 8235686, 987306985, 118494013184, 14218920983475, 1706277590116906, 204753168276322765, 24570383035099891364, 2948445907303018719415, 353813510013979711948526, 42457621178920726542831345, 5094914541925588034670283144, 611389745021968259832255932155, 73366769402818234967548615778546, 8804012328334547301962600598224725, 1056481479400218493971256387148100524 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [8, 1]](x) = 2 3 2 x (2920 x - 872 x - 70 x + 1) - ------------------------------------------------------- (-1 + 3 x) (1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[8, 1]](x) = -x^2*(2920*x^3-872*x^2-70*x+1)/(-1+3*x)/(1+3*x)/ (-1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 34, 4673, 546682, 65866761, 7898666954, 947946668113, 113751466671642, 13650218666689241, 1638025386666749674, 196563063466667022753, 23587567274666668025402, 2830508079786666672318121, 339660969437866666688681994, 40759316335274666666756676593, 4891117960178346666667021391962, 586934155222493866666668103105401, 70432098626677418666666672364591914, 8451851835201727146666666689616205633 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [7, 2]](x) = 2 3 2 x (18880 x - 2096 x - 204 x + 3) - ------------------------------------------------------ (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[7, 2]](x) = -x^2*(18880*x^3-2096*x^2-204*x+3)/(1+4*x)/(-1+4* x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 120, 15712, 1846912, 222268928, 26658682880, 3199306801152, 383911467712512, 46069482675109888, 5528335786733731840, 663400337067204411392, 79608039594670960934912, 9552964768426701040386048, 1146355771869866941533388800, 137562692631210668865913618432, 16507523115608746684258673754112, 1980902773875780266807407734161408, 237708332865039018667792563710197760, 28524999943805774506675673923187638272 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[3, 3, 1, 1, 1], [7, 1, 1]](x) = x 6 5 4 3 2 (78080 x - 25152 x + 1824 x + 5260 x - 1691 x - 10 x - 1)/((-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) ) and in Maple notation F[[3, 3, 1, 1, 1],[7, 1, 1]](x) = x^2*(78080*x^6-25152*x^5+1824*x^4+5260*x^3-\ 1691*x^2-10*x-1)/(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/(8*x-1)/(20*x+1)/(-1+ 120*x) The first 20 term , starting with k=1 are 0, 1, 119, 16179, 1915249, 230486141, 27646200939, 3317795380439, 398130487499829, 47775758211031041, 5733088995466222159, 687970719288172952699, 82556485518216497924409, 9906778278115509740780741, 1188813393055288522422251379, 142657607173006219290162052959, 17118912860633315532099903620989, 2054269543278546488701238433175241, 246512345193374606220721031887216599, 29581481423205972195543545948967545219 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [6, 3]](x) = 2 3 2 x (1520 x - 104 x - 110 x + 3) ------------------------------------------------------ (-1 + x) (-1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[6, 3]](x) = x^2*(1520*x^3-104*x^2-110*x+3)/(-1+x)/(-1+3*x)/( -1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 214, 27751, 3285098, 395098483, 47394032014, 5687639366111, 682509612702418, 81901295746047403, 9828152645079426614, 1179378374298412941271, 141525403778031746993338, 16983048476119365083181923, 2037965816679212698427877214, 244555898010607746031806481231, 29346707761090885079365320365858, 3521604931334547098412699374310043, 422592591760072834031746035585303814, 50711111011210196439365079380414929991 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[3, 3, 1, 1, 1], [6, 2, 1]](x) = x 6 5 4 3 2 (235200 x - 189360 x + 436 x + 13275 x - 1307 x + 71 x - 5)/((-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) ) and in Maple notation F[[3, 3, 1, 1, 1],[6, 2, 1]](x) = x^2*(235200*x^6-189360*x^5+436*x^4+13275*x^3-\ 1307*x^2+71*x-5)/(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/(8*x-1)/(20*x+1)/(-1+ 120*x) The first 20 term , starting with k=1 are 0, 5, 474, 60558, 7188980, 864222800, 103675561094, 12441688931138, 1492990222572840, 179159075558330820, 21499084088911274714, 2579890190222400863318, 309586820835556987459100, 37150418540088900337253240, 4458050224014222313852052334, 534966026897635556288486749098, 64195923227397688894753011751760, 7703510787294094222269133507358060, 924421294475163875555930856489701954, 110930555337022213688891891269390300478 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [6, 1, 1, 1]](x) = 2 4 3 2 x (7000 x - 3992 x + 267 x + 60 x - 3) - ---------------------------------------------------------------- (-1 + 3 x) (-1 + x) (1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[6, 1, 1, 1]](x) = -x^2*(7000*x^4-3992*x^3+267*x^2+60*x-3)/(-\ 1+3*x)/(-1+x)/(1+3*x)/(-1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 255, 32223, 3835547, 460888883, 55294222095, 6635555555303, 796261688886947, 95551502222216763, 11466178275555525335, 1375941432888888788783, 165112971150222221746347, 19813556553955555553839043, 2377626786156088888881343775, 285315214345102222222193724663, 34237825721284835555555435453747, 4108539086556728888888888423627723, 493024690386756494222222220306173415, 59162962846411798755555555548027674943 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 7 6 5 4 F[[3, 3, 1, 1, 1], [5, 4]](x) = x (397440 x - 371264 x + 88024 x + 14612 x 3 2 - 13468 x + 1145 x + 134 x - 3)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x)) and in Maple notation F[[3, 3, 1, 1, 1],[5, 4]](x) = x^2*(397440*x^7-371264*x^6+88024*x^5+14612*x^4-\ 13468*x^3+1145*x^2+134*x-3)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/(8*x -1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 190, 24253, 2875708, 345690403, 41470232930, 4976675645273, 597196089585208, 71663630227851583, 8599633635600257470, 1031956076089247390293, 123834728334225084993908, 14860167416035578471412763, 1783220089605689072132706010, 213986410759054223688387255313, 25678369290959075567283554301808, 3081404314917637688982716270805943, 369768517790065550222972820197434550, 44372222134808885475561560393268488333 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 2 7 x (360 x - 4 x + 1) F[[3, 3, 1, 1, 1], [5, 3, 1]](x) = - ------------------------------------------ (-1 + x) (8 x - 1) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[5, 3, 1]](x) = -7*x^2*(360*x^2-4*x+1)/(-1+x)/(8*x-1)/(20*x+1 )/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 735, 93079, 11096631, 1333253047, 159958824375, 19195702594999, 2303471540759991, 276416841126079927, 33170015817008639415, 3980402000456069115319, 477648238006848552922551, 57317788601782788423380407, 6878134631394742307387043255, 825376155783753138459096346039, 99045138693722697107672770768311, 11885416643253277256861382166146487, 1426249997190262198854891057329171895, 171149999662834085302839128458633375159 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[3, 3, 1, 1, 1], [5, 2, 2]](x) = x 5 4 3 2 (75200 x - 5136 x - 14956 x + 3009 x + 200 x - 7)/((-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x)) and in Maple notation F[[3, 3, 1, 1, 1],[5, 2, 2]](x) = x^2*(75200*x^5-5136*x^4-14956*x^3+3009*x^2+ 200*x-7)/(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 563, 68977, 8222845, 987561067, 118488931543, 14219022572597, 1706275558334465, 204753208911272527, 24570382222400896123, 2948445923556987439417, 353813509688900337548485, 42457621185422313851875187, 5094914541795556288489406303, 611389745024568894753010157437, 73366769402766222269133531272905, 8804012328335587555930856475353047, 1056481479400197688891891269605534083, 126777777528025179022246241435473574657 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [5, 2, 1, 1]](x) = 2 3 2 x (17600 x - 1936 x - 124 x - 7) ------------------------------------------------------ (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[5, 2, 1, 1]](x) = x^2*(17600*x^3-1936*x^2-124*x-7)/(1+4*x)/( -1+4*x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 880, 108288, 12953088, 1555331072, 186621317120, 22394933198848, 2687384532287488, 322486293324890112, 38698352213266268160, 4643802325332795588608, 557256277845329039065088, 66870753365333298959613952, 8024490403362133058466611200, 962938848413013331134086381568, 115552661809370453315741326245888, 13866319417128277333192592265838592, 1663958330055316821332207436289802240, 199674999606639547733324326076812361728 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 2 2 2 x (175 x - 86 x + 2) - ------------------------------------------- (-1 + 3 x) (1 + 3 x) (8 x - 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[5, 1, 1, 1, 1]](x) = -2*x^2*(175*x^2-86*x+2)/(-1+3*x)/(1+3*x )/(8*x-1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 4, 340, 40066, 4800500, 576004114, 69120032660, 8294400262306, 995328002096180, 119439360016778674, 14332723200134208980, 1719926784001073754946, 206391214080008589855860, 24766945689600068719594834, 2972033482752000549755105300, 356644017930240004398047573986, 42797282151628800035184365711540, 5135673858195456000281474986276594, 616280862983454720002251799756289620, 73953703558014566400018014398595575426 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 7 6 5 F[[3, 3, 1, 1, 1], [4, 4, 1]](x) = - x (526080 x - 562560 x - 179664 x 4 3 2 + 93056 x + 9238 x - 2631 x - 144 x + 5)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x)) and in Maple notation F[[3, 3, 1, 1, 1],[4, 4, 1]](x) = -x^2*(526080*x^7-562560*x^6-179664*x^5+93056* x^4+9238*x^3-2631*x^2-144*x+5)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/( 8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 5, 396, 48267, 5755930, 691291625, 82942244616, 9953315730167, 1194392890295910, 143327246233402485, 17199267555645178036, 2063912146489604624867, 247669456782227951160690, 29720334829795601366897345, 3566440179256889255429914656, 427972821517198225154225396367, 51356738581936355579012398942270, 6162808629834911289076538414080205, 739537035580138382223723431631622476, 88744444269617625315567565147087804667 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [4, 3, 2]](x) = 2 3 2 x (920 x - 48 x - 32 x - 7) - ------------------------------------------------------ (-1 + x) (-1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[4, 3, 2]](x) = -x^2*(920*x^3-48*x^2-32*x-7)/(-1+x)/(-1+3*x)/ (-1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 788, 96219, 11515544, 1382488847, 165886222068, 19906595554979, 2388786488886704, 286654478222213847, 34398535395555523148, 4127824287288888762539, 495338913678222221726664, 59440669657315555553602847, 7132880358559288888881166628, 855945643033486222222191598899, 102713477163890915555555433859424, 12325617259669458488888888404495847, 1479074071160284046222222220291824508, 177488888539235104995555555547855488059 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [4, 3, 1, 1]](x) = 2 2 x (1160 x - 80 x - 9) ------------------------------------------- (-1 + x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[4, 3, 1, 1]](x) = x^2*(1160*x^2-80*x-9)/(-1+x)/(-1+4*x)/(20* x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 9, 1025, 123529, 14809545, 1777409609, 213283809865, 25594163810889, 3071297523814985, 368555745523831369, 44226688609523896905, 5307202650209524159049, 636864317683809525207625, 76423718128883809529401929, 9170846175329523809546179145, 1100501541042273523809613288009, 132060184925018209523809881723465, 15847222191003277409523810955465289, 1901666662920371443809523815250432585, 228199999550445010163809523832430301769 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 8 x (25 x + 1) F[[3, 3, 1, 1, 1], [4, 2, 2, 1]](x) = ---------------------------------- (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[4, 2, 2, 1]](x) = 8*x^2*(25*x+1)/(-1+4*x)/(20*x+1)/(-1+120*x ) The first 20 term , starting with k=1 are 0, 8, 1032, 123328, 14813312, 1777333248, 213285332992, 25594133331968, 3071298133327872, 368555733333311488, 44226688853333245952, 5307202645333332983808, 636864317781333331935232, 76423718126933333327740928, 9170846175368533333310963712, 1100501541041493333333243854848, 132060184925033813333332975419392, 15847222191002965333333331901677568, 1901666662920377685333333327606710272, 228199999550444885333333333310426841088 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [4, 2, 1, 1, 1]](x) = 2 3 2 x (13120 x - 2160 x + 156 x + 7) - ------------------------------------------------------ (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[4, 2, 1, 1, 1]](x) = -x^2*(13120*x^3-2160*x^2+156*x+7)/(1+4* x)/(-1+4*x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 7, 912, 107648, 12966400, 1555064832, 186626650112, 22394826539008, 2687386665615360, 322486250658332672, 38698353066599514112, 4643802308266130669568, 557256278186662371000320, 66870753358506632320909312, 8024490403498666391777574912, 962938848410282664467867107328, 115552661809425066649074301665280, 13866319417127185066525932757450752, 1663958330055338666665540763896512512, 199674999606639110826657659524678156288 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = 2 2 2 2 x (164 x - 44 x - 1) (5 x + 10 x - 1) ---------------------------------------------------------------- (-1 + 3 x) (-1 + x) (1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = 2*x^2*(164*x^2-44*x-1)*(5*x^2+10*x-1 )/(-1+3*x)/(-1+x)/(1+3*x)/(-1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 2, 278, 31782, 3844450, 460711162, 55297777878, 6635484445102, 796263111112810, 95551473777786882, 11466178844444472478, 1375941421511111244022, 165112971377777778233970, 19813556549404444446456202, 2377626786247111111118479078, 285315214343281777777808932542, 34237825721321244444444562951930, 4108539086556000711111111600287122, 493024690386771057777777779679477678, 59162962846411507484444444452187558662 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 x (62 x + 1) F[[3, 3, 1, 1, 1], [3, 3, 3]](x) = - --------------------------------- (-1 + 3 x) (8 x - 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[3, 3, 3]](x) = -x^2*(62*x+1)/(-1+3*x)/(8*x-1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 193, 23939, 2879497, 345595931, 41471967313, 4976639738099, 597196797903577, 71663615983224971, 8599633919865788833, 1031956070398926277859, 123834728447991410124457, 14860167413759931280700411, 1783220089651199450244717553, 213986410758143995601955083219, 25678369290977279964815632694137, 3081404314917273599718525037638251, 369768517790072831997748200229361473, 44372222134808739839981985601619658179 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [3, 3, 2, 1]](x) = 2 3 2 x (1960 x - 1296 x + 163 x + 6) ------------------------------------------------------ (-1 + x) (-1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[3, 3, 2, 1]](x) = x^2*(1960*x^3-1296*x^2+163*x+6)/(-1+x)/(-1 +3*x)/(-1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 6, 811, 95778, 11524447, 1382311126, 165889777851, 19906524444778, 2388787911112567, 286654449777783966, 34398535964444470291, 4127824275911111217778, 495338913905777778214287, 59440669652764444446220006, 7132880358650311111118301931, 855945643031665777777806806778, 102713477163927324444444561357607, 12325617259668730311111111581155246, 1479074071160298609777777779665128771, 177488888539234813724444444452015371778 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 7 6 5 F[[3, 3, 1, 1, 1], [3, 3, 1, 1, 1]](x) = - x (1030720 x - 79280 x - 275588 x 4 3 2 + 25211 x + 16981 x - 1562 x - 103 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x)) and in Maple notation F[[3, 3, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -x*(1030720*x^7-79280*x^6-275588*x^5+ 25211*x^4+16981*x^3-1562*x^2-103*x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1 +4*x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 1, 5, 604, 68335, 8235686, 987306985, 118494013184, 14218920983475, 1706277590116906, 204753168276322765, 24570383035099891364, 2948445907303018719415, 353813510013979711948526, 42457621178920726542831345, 5094914541925588034670283144, 611389745021968259832255932155, 73366769402818234967548615778546, 8804012328334547301962600598224725, 1056481479400218493971256387148100524, 126777777528024762920658939817629997695 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[3, 3, 1, 1, 1], [3, 2, 2, 2]](x) = - 2 x 6 5 4 3 2 (124800 x - 111168 x + 8328 x + 8888 x - 1684 x - 11 x + 2)/((-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) ) and in Maple notation F[[3, 3, 1, 1, 1],[3, 2, 2, 2]](x) = -2*x^2*(124800*x^6-111168*x^5+8328*x^4+ 8888*x^3-1684*x^2-11*x+2)/(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/(8*x-1)/(20* x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 4, 414, 47826, 5764748, 691113904, 82945799034, 9953244619966, 1194394312499928, 143327217788972604, 17199268124533775654, 2063912135111827080106, 247669457009783502055908, 29720334825244490259514504, 3566440179347911477577571474, 427972821515377780709840604246, 51356738581972764467900094784688, 6162808629834183111298761590739604, 739537035580152945779278968098434494, 88744444269617334044456454051247688386 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 4 x (90 x + 1) F[[3, 3, 1, 1, 1], [3, 2, 2, 1, 1]](x) = --------------------------------- (8 x - 1) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[3, 2, 2, 1, 1]](x) = 4*x^2*(90*x+1)/(8*x-1)/(20*x+1)/(-1+120 *x) The first 20 term , starting with k=1 are 0, 4, 792, 91936, 11119488, 1332795904, 159967967232, 19195519737856, 2303475197902848, 276416767983222784, 33170017279865782272, 3980401971198926258176, 477648238591991410065408, 57317788590079931280523264, 6878134631628799450244186112, 825376155779071995601953488896, 99045138693816319964815627911168, 11885416643251404799718525023289344, 1426249997190299647997748200186314752, 171149999662833336319981985601490518016 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[3, 3, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = - x ( 7 6 5 4 3 2 456000 x - 414480 x + 20108 x + 61521 x - 4183 x - 2554 x + 205 x + 3 )/((-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x)) and in Maple notation F[[3, 3, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = -x^2*(456000*x^7-414480*x^6+20108*x^ 5+61521*x^4-4183*x^3-2554*x^2+205*x+3)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-\ 1+4*x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 3, 529, 59456, 7211195, 863778498, 103684449869, 12441511155636, 1492993778126575, 179159004447256118, 21499085511133467809, 2579890161777957001416, 309586821404445875881955, 37150418528711122568796138, 4458050224241777869400151349, 534966026893084445177524768796, 64195923227488711116975114669335, 7703510787292273777824691449006558, 924421294475200284444819743469716489, 110930555337021485511114113529790009776 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 3 F[[3, 3, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (354560 x - 536896 x + 2688 x + 65708 x - 1601 x - 942 x - 137)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x)) and in Maple notation F[[3, 3, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = x^3*(354560*x^6-536896*x^5+2688*x ^4+65708*x^3-1601*x^2-942*x-137)/(-1+x)/(1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x) /(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 137, 15738, 1924067, 230308420, 27649755357, 3317724270238, 398131909703847, 47775729766601160, 5733089564354819777, 687970707910395407938, 82556485745772048819627, 9906778273564398633397900, 1188813393146310744569908197, 142657607171185774845777260838, 17118912860669724420987599463407, 2054269543277818310923461609834640, 246512345193389169776276568354028617, 29581481423205680924432434853127428938 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 2 F[[3, 3, 1, 1, 1], [2, 2, 2, 2, 1]](x) = - x 6 5 4 3 2 (17280 x + 22208 x + 27864 x - 8204 x - 943 x + 104 x + 1)/((-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) ) and in Maple notation F[[3, 3, 1, 1, 1],[2, 2, 2, 2, 1]](x) = -x^2*(17280*x^6+22208*x^5+27864*x^4-\ 8204*x^3-943*x^2+104*x+1)/(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/(8*x-1)/(20* x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 213, 23791, 2884611, 345512341, 41473788713, 4976604529611, 597197511811071, 71663601783334321, 8599634204489204613, 1031956064711468447431, 123834728561780641481531, 14860167411484467341660301, 1783220089696711294369841313, 213986410757233779243644549251, 25678369290995484456172681799991, 3081404314916909511204933720842281, 369768517790080113778528379570738813, 44372222134808594204450449205802403071 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 3 2 x (1520 x - 824 x + 136 x + 1) ------------------------------------------------------ (-1 + x) (-1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = x^2*(1520*x^3-824*x^2+136*x+1)/(-1+x )/(-1+3*x)/(-1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 1, 244, 27109, 3297768, 394844401, 47399110924, 5687537776989, 682511644441168, 81901255111097641, 9828153457777722804, 1179378358044444221269, 141525404103111110208568, 16983048469617777774138081, 2037965816809244444429797084, 244555898008007111111052255949, 29346707761142897777777541559968, 3521604931333506844444443497181721, 422592591760093639111111107314885764, 50711111011209780337777777762571353029 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 3 3 2 x (57920 x - 19984 x - 596 x + 145) --------------------------------------------------------------- (-1 + x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = x^3*(57920*x^3-19984*x^2-596*x+ 145)/(-1+x)/(1+4*x)/(-1+4*x)/(8*x-1)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 145, 15209, 1856457, 222078025, 26662492745, 3199230603849, 383912991527497, 46069452198810185, 5528336396257628745, 663400324876726473289, 79608039838480486142537, 9552964763550510536233545, 1146355771967390751079567945, 137562692629260192674990035529, 16507523115647756208068555477577, 1980902773875000076331210099692105, 237708332865054622477316378960630345, 28524999943805462430485197618178986569 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 F[[3, 3, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (11560 x - 2784 x - 167 x - 41) - ---------------------------------------------------------------- (-1 + 3 x) (-1 + x) (1 + 3 x) (-1 + 4 x) (20 x + 1) (-1 + 120 x) and in Maple notation F[[3, 3, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(11560*x^3-2784*x^2-167*x -41)/(-1+3*x)/(-1+x)/(1+3*x)/(-1+4*x)/(20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 41, 4472, 550449, 65790400, 7900190081, 947916189192, 113752076184529, 13650206476169360, 1638025630476098721, 196563058590475847512, 23587567372190474753009, 2830508077836190470657120, 339660969476876190453466561, 40759316334494476190387243432, 4891117960193950476190115087889, 586934155222181790476189049317680, 70432098626683660190476184720869601, 8451851835201602316190476167612744952 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 3 F[[3, 3, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (75200 x - 5136 x - 14956 x + 3009 x + 200 x - 7)/((-1 + x) (-1 + 3 x) (1 + 3 x) (1 + 4 x) (-1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 120 x)) and in Maple notation F[[3, 3, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(75200*x^5-5136*x^4-\ 14956*x^3+3009*x^2+200*x-7)/(-1+x)/(-1+3*x)/(1+3*x)/(1+4*x)/(-1+4*x)/(8*x-1)/( 20*x+1)/(-1+120*x) The first 20 term , starting with k=1 are 0, 0, 7, 563, 68977, 8222845, 987561067, 118488931543, 14219022572597, 1706275558334465, 204753208911272527, 24570382222400896123, 2948445923556987439417, 353813509688900337548485, 42457621185422313851875187, 5094914541795556288489406303, 611389745024568894753010157437, 73366769402766222269133531272905, 8804012328335587555930856475353047, 1056481479400197688891891269605534083 ---------------------------------- Their sum is 5 4 3 2 5260 x - 6981 x - 162 x + 1058 x - 128 x + 1 ---------------------------------------------------- (-1 + 120 x) (8 x - 1) (1 + 3 x) (-1 + 3 x) (-1 + x) and in Maple notation (5260*x^5-6981*x^4-162*x^3+1058*x^2-128*x+1)/(-1+120*x)/(8*x-1)/(1+3*x)/(-1+3*x )/(-1+x) The first 20 term , starting with k=1 are 1, 108, 12490, 1497288, 179658146, 21558865488, 2587062922490, 310447543382808, 37253705147049746, 4470444617176423968, 536453354057411277290, 64374402486859290452328, 7724928298422874322588546, 926991395810742994582814448, 111238967497289143956653244890, 13348676099674697151653244913848, 1601841131960963657213225875874546, 192220935835315638857705807177588928, 23066512300237876662861646456669785290, 2767981476028545199542893171654893615368 Regarding Lambda=, [3, 2, 2, 2] 10 9 8 7 F[[3, 2, 2, 2], [9]](x) = (1045296 x - 567432 x - 2116244 x + 1124902 x 6 5 4 3 2 + 467666 x - 308027 x + 12985 x + 10474 x - 714 x - 77 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[9]](x) = (1045296*x^10-567432*x^9-2116244*x^8+1124902*x^7+ 467666*x^6-308027*x^5+12985*x^4+10474*x^3-714*x^2-77*x+1)/(1+x)/(-1+x)/(1+2*x)/ (-1+6*x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 2, 148, 11490, 968980, 81309072, 6830946102, 573784752830, 48198118792384, 4048639153631292, 340085728230526906, 28567200619723436970, 2399644859771840413188, 201570168112785111295112, 16931894122986359799498910, 1422279106309679099534593110, 119471444930309485959186857992, 10035601374141846588733405827732, 842990515427973216335049945387714 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [8, 1]](x) = - x 5 4 3 2 (70560 x - 16632 x - 4168 x + 702 x + 62 x - 1)/((-1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[8, 1]](x) = -x^2*(70560*x^5-16632*x^4-4168*x^3+702*x^2+62*x-1)/ (-1+6*x)/(-1+3*x)/(1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 13, 1137, 91931, 7749673, 650490939, 54647257169, 4590282043507, 385584892326105, 32389114029892235, 2720685814573779361, 228537605115178034403, 19197158875969190921897, 1612561344933144282387451, 135455152983458717257571313, 11378232850483482548669502419, 955771559442391188498985974649, 80284810993135958480223610401387, 6743924123423769129800495288866625 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [7, 2]](x) = - 2 x 5 4 3 2 (45864 x - 14196 x - 3142 x + 597 x + 53 x - 1)/((1 + 2 x) (-1 + 6 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[7, 2]](x) = -2*x^2*(45864*x^5-14196*x^4-3142*x^3+597*x^2+53*x-1 )/(1+2*x)/(-1+6*x)/(-1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 42, 3786, 310432, 26150200, 2195457824, 184433688224, 15492212515584, 1301348859654528, 109313261954930176, 9182314594611937792, 771314417676935315456, 64790411200606842484736, 5442394539230380391374848, 457161141318038759807754240, 38401535870397634273495810048, 3225729013117847926278836813824, 270961237101836972520497754275840, 22760743916555177235782231047405568 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 8 7 6 F[[3, 2, 2, 2], [7, 1, 1]](x) = x (225792 x + 196560 x - 333416 x 5 4 3 2 + 84356 x + 1586 x - 358 x - 188 x - 33 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[7, 1, 1]](x) = x^2*(225792*x^8+196560*x^7-333416*x^6+84356*x^5+ 1586*x^4-358*x^3-188*x^2-33*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(-1+2*x )/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 44, 3915, 322008, 27117565, 2276787422, 191264339799, 16066001361356, 1349546921028609, 113361901911942090, 9522400311593251963, 799881618454140165944, 67190056058173912923333, 5643964707374032161458198, 474093035440592985901088607, 39823814976713363242983731172, 3345200458048072714050946356937, 280996838475980004883819317786146, 23603734431983133851272923091052931 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [6, 3]](x) = 2 5 4 3 2 x (24192 x - 14736 x - 1664 x + 858 x + 67 x - 2) - ------------------------------------------------------------------------- (-1 + 3 x) (-1 + x) (1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[6, 3]](x) = -x^2*(24192*x^5-14736*x^4-1664*x^3+858*x^2+67*x-2)/ (-1+3*x)/(-1+x)/(1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 73, 6670, 552113, 46483114, 3903101701, 327881096342, 27541724659081, 2313508891106146, 194334690593046509, 16324114797319003534, 1371225631950423804529, 115182953238172245795098, 9675368069845780875678997, 812730917897295062280106246, 68269397102949293596526986457, 5734629356653669539303433125970, 481708865958825237054399929791165, 40463544740542481971824878000212478 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 8 7 6 5 F[[3, 2, 2, 2], [6, 2, 1]](x) = x (105840 x - 296856 x + 64692 x + 57018 x 4 3 2 - 20676 x + 2957 x - 59 x - 69 x + 3)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[6, 2, 1]](x) = x^2*(105840*x^8-296856*x^7+64692*x^6+57018*x^5-\ 20676*x^4+2957*x^3-59*x^2-69*x+3)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(-1+2* x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 162, 14560, 1208260, 101675688, 8538128802, 717238634850, 60247540718520, 5060800448871748, 425107139190942142, 35709001069946197590, 2999556070580664465780, 251962710198856829522808, 21164867652922691892350682, 1777848882898442396030063530, 149339306162728048086763485040, 12544501717679531563810639637868, 1053738144284935393826168675722422, 88514004119936606684714222042084670 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [6, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (70560 x - 124488 x - 31708 x + 12926 x + 3699 x - 650 x - 61 x + 2)/ ((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[6, 1, 1, 1]](x) = -x^2*(70560*x^7-124488*x^6-31708*x^5+12926*x^ 4+3699*x^3-650*x^2-61*x+2)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(1+6*x)/(-1+4 *x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 89, 7755, 644671, 54223747, 4553717875, 382526595495, 32132031294767, 2699093439063867, 226723809443780011, 19044800544399749935, 1599763238010499261063, 134380112100912852686787, 11287929414964125253934147, 948186070878160977873456375, 79647629953469075367206996959, 6690400916095552538689118223307, 561993676951968310182189136238683, 47207468863966151496559372481847615 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [5, 4]](x) = x 7 6 5 4 3 2 (49392 x - 27384 x + 2692 x + 878 x - 940 x + 280 x - 19 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[5, 4]](x) = x^2*(49392*x^7-27384*x^6+2692*x^5+878*x^4-940*x^3+ 280*x^2-19*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(-1+2*x)/(-1+4*x)/(1+14* x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 64, 5809, 483244, 40669619, 3415248998, 286895429017, 24099016192168, 2024320178635327, 170042855672889562, 14283600427945377365, 1199822428232139410972, 100785084079541536949275, 8465947061169072195044206, 711139553159376915348724753, 59735722465091219070287403856, 5017800687071812623973228839863, 421495257713974157524545462424130, 35405601647974642673829840100157581 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [5, 3, 1]](x) = 2 3 2 3 x (1512 x - 284 x - 6 x - 1) ------------------------------------------------------ (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[5, 3, 1]](x) = 3*x^2*(1512*x^3-284*x^2-6*x-1)/(1+2*x)/(-1+6*x)/ (-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 252, 22344, 1864992, 156855744, 13173296064, 1106594048640, 92953385247744, 7808091602196480, 655879593405121536, 55093887263734880256, 4627886510312995577856, 388742467144104583643136, 32654367236215659243552768, 2742966847896564669139156992, 230409215222549151293259644928, 19354374078704800696505453248512, 1625767422611053851006556848783360, 136564463499330615191530739319963648 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [5, 2, 2]](x) = x 5 4 3 2 (8904 x - 3308 x - 928 x + 276 x + 39 x - 3)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[5, 2, 2]](x) = x^2*(8904*x^5-3308*x^4-928*x^3+276*x^2+39*x-3)/( 1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 192, 16545, 1381930, 116185003, 9758069832, 819698336565, 68854373207910, 5783771366465103, 485836738537403572, 40810286824550480185, 3428064082238396472690, 287957383062358621138803, 24188420175077455737978912, 2031827294736755632125249405, 170673492757463982164598622270, 14336573391632903374772894982103, 1204272164897080879259156868075852, 101158861851355955916871848612756225 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [5, 2, 1, 1]](x) = x 7 6 5 4 3 2 (7056 x - 112056 x + 12652 x + 14822 x - 908 x + 8 x - 5 x - 4)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[5, 2, 1, 1]](x) = x^2*(7056*x^7-112056*x^6+12652*x^5+14822*x^4-\ 908*x^3+8*x^2-5*x-4)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(1+6*x)/(-1+4*x)/(1 +14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 4, 301, 26014, 2177029, 182983624, 15369068365, 1291023349984, 108445659289933, 9109439601199744, 765192867413735629, 64276201689623586304, 5399200930352229983437, 453532878311640297048064, 38096761775909041851583693, 3200127989208121436612583424, 268810751093037533692595752141, 22580103091821378150425349750784, 1896728659712908610148512173182157, 159325207415885543414663220668268544 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [5, 1, 1, 1, 1]](x) = 2 3 2 2 x (462 x - 40 x - 26 x + 1) - ----------------------------------------------------- (-1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 4 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[5, 1, 1, 1, 1]](x) = -2*x^2*(462*x^3-40*x^2-26*x+1)/(-1+3*x)/(1 +2*x)/(1+6*x)/(-1+4*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 114, 9614, 806710, 67766082, 5692328054, 478155662434, 40165074903270, 3373866295926962, 283404768832111894, 23806000582046469954, 1999704048890988415430, 167975140106848437730642, 14109911768975235996166134, 1185232588593920019112392674, 99559537441889280428102625190, 8363001145118699563005979195122, 702492096189970763250156194882774, 59009336079957544113266903406718594 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [4, 4, 1]](x) = x 7 6 5 4 3 2 (77616 x + 65016 x - 84116 x + 8622 x + 2706 x - 105 x - 21 x + 2)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[4, 4, 1]](x) = x^2*(77616*x^7+65016*x^6-84116*x^5+8622*x^4+2706 *x^3-105*x^2-21*x+2)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(-\ 1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 2, 133, 11566, 967285, 81328828, 6830646263, 573788810282, 48198061148545, 4048639955604904, 340085716972664443, 28567200777152107798, 2399644857566750635205, 201570168143649837819380, 16931894122554214446096223, 1422279106315728899389656514, 119471444930224787350661093065, 10035601374143032360789436486656, 842990515427956615475485538134403, 70811203295949169141754436236790830 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [4, 3, 2]](x) = - x 6 5 4 3 2 (42336 x + 504 x + 3600 x - 3106 x + 196 x + 42 x + 3)/((-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) ) and in Maple notation F[[3, 2, 2, 2],[4, 3, 2]](x) = -x^2*(42336*x^6+504*x^5+3600*x^4-3106*x^3+196*x^ 2+42*x+3)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 270, 23083, 1935408, 162647727, 13661443348, 1147575594147, 96396151166256, 8097279510358231, 680171439576151116, 57134401475595448491, 4799289716236116863944, 403140336271867522813215, 33863788245324504152594724, 2844558212628432906576043315, 238942889860491924099178219872, 20071202748285471835831207611879, 1685981030855921531384131610696572, 141622406591898222077654389678965819 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [4, 3, 1, 1]](x) = 2 5 4 3 2 x (31248 x - 6252 x + 1216 x - 9 x - 54 x - 4) - ----------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 6 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[4, 3, 1, 1]](x) = -x^2*(31248*x^5-6252*x^4+1216*x^3-9*x^2-54*x-\ 4)/(-1+x)/(1+x)/(-1+6*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 4, 350, 29641, 2489000, 209109853, 17564837600, 1475452589965, 123937933212800, 10410787597961677, 874506141417920000, 73458516115431082189, 6170515350391302809600, 518323289479173078506701, 43539156315602418609152000, 3657289130519678098401758413, 307212286963525915880778137600, 25805832104937955600538465848525, 2167689896814763369282857372876800, 182085951332440471637658623562796237 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [4, 2, 2, 1]](x) = x 6 5 4 3 2 (14112 x + 64680 x + 6176 x - 6194 x - 412 x + 69 x + 4)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[4, 2, 2, 1]](x) = x^2*(14112*x^6+64680*x^5+6176*x^4-6194*x^3-\ 412*x^2+69*x+4)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84* x) The first 20 term , starting with k=1 are 0, 4, 357, 29616, 2489771, 209101664, 17564967763, 1475450861056, 123937957977267, 10410787254618624, 874506146244877619, 73458516047974608896, 6170515351336419027763, 518323289465945805021184, 43539156315787626559329075, 3657289130517085343827623936, 307212286963562215385185956659, 25805832104937447413118976262144, 2167689896814770483940583545844531, 182085951332440372032653577054846976 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [4, 2, 1, 1, 1]](x) = x 7 6 5 4 3 2 (119952 x - 26040 x - 25348 x + 7198 x + 2716 x + 50 x - 90 x - 3)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[4, 2, 1, 1, 1]](x) = x^2*(119952*x^7-26040*x^6-25348*x^5+7198*x ^4+2716*x^3+50*x^2-90*x-3)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+2*x)/(1+6*x)/(-1+4 *x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 312, 25849, 2179240, 182952059, 15369506432, 1291017193875, 108445745335680, 9109438395720115, 765192884285412352, 64276201453389881651, 5399200933659320453120, 453532878265339942091571, 38096761776557240290639872, 3200127989199046619283755827, 268810751093164580900106895360, 22580103091819599488109638955827, 1896728659712933511412468794785792, 159325207415885194796917047988138803 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [4, 1, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (155232 x - 45864 x - 45424 x + 3586 x + 2384 x - 213 x + 18 x + 1)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[4, 1, 1, 1, 1, 1]](x) = -x^2*(155232*x^7-45864*x^6-45424*x^5+ 3586*x^4+2384*x^3-213*x^2+18*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(1+6*x )/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 93, 7627, 646035, 54202055, 4554006023, 382522468091, 32132088518595, 2699092634570959, 226723820686525503, 19044800386880380355, 1599763240215044841755, 134380112070044860991063, 11287929415396251016288983, 948186070872111060472155019, 79647629953553773270455257715, 6690400916094366762401422735967, 561993676951984911016363554215663, 47207468863965919084677810716745683 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [3, 3, 3]](x) = 2 3 2 x (756 x - 84 x - 14 x + 1) - ----------------------------------------------------- (-1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 4 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[3, 3, 3]](x) = -x^2*(756*x^3-84*x^2-14*x+1)/(-1+3*x)/(1+2*x)/(1 +6*x)/(-1+4*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 69, 5759, 484057, 40659339, 3415398161, 286893387187, 24099044994009, 2024319777204107, 170042861301216673, 14283600349215520995, 1199822429334664525481, 100785084064108622914555, 8465947061385144192261105, 711139553156351995723977683, 59735722465133568350615108473, 5017800687071219737238228824683, 421495257713982457953474493339457, 35405601647974526467939811805005251 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [3, 3, 2, 1]](x) = x 6 5 4 3 2 (42336 x + 29736 x - 32868 x + 3702 x + 725 x - 53 x - 3)/((-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) ) and in Maple notation F[[3, 2, 2, 2],[3, 3, 2, 1]](x) = x^2*(42336*x^6+29736*x^5-32868*x^4+3702*x^3+ 725*x^2-53*x-3)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+ 84*x) The first 20 term , starting with k=1 are 0, 3, 281, 22998, 1937031, 162627590, 13661740827, 1147571522730, 96396208726007, 8097278707880862, 680171450830989843, 57134401318148638322, 4799289718441097801103, 403140336241002143256294, 33863788245756645587782379, 2844558212622383083211738874, 238942889860576622566648462119, 20071202748284286062928844013486, 1685981030855938132238618020007235, 141622406591897989665894699861865986 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [3, 3, 1, 1, 1]](x) = 2 x 6 5 4 3 2 (4032 x - 1980 x - 906 x - 136 x + 260 x - 24 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[3, 3, 1, 1, 1]](x) = 2*x^2*(4032*x^6-1980*x^5-906*x^4-136*x^3+ 260*x^2-24*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+4*x)/(1+14*x)/(-1+84 *x) The first 20 term , starting with k=1 are 0, 2, 202, 16392, 1384060, 116155122, 9758488122, 819692480252, 68854455196120, 5783770218629142, 485836754607106342, 40810286599574637312, 3428064085388058270180, 287957383018263355957562, 24188420175694789450505362, 2031827294728112960149813572, 170673492757584979572254680240, 14336573391631209411065709908382, 1204272164897104594751057458933182, 101158861851355623899985240339705032 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 8 7 6 F[[3, 2, 2, 2], [3, 2, 2, 2]](x) = - x (986832 x - 497016 x - 383308 x 5 4 3 2 + 246314 x - 13234 x - 9213 x + 721 x + 75 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[3, 2, 2, 2]](x) = -x*(986832*x^8-497016*x^7-383308*x^6+246314*x ^5-13234*x^4-9213*x^3+721*x^2+75*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(-\ 1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 1, 2, 148, 11490, 968980, 81309072, 6830946102, 573784752830, 48198118792384, 4048639153631292, 340085728230526906, 28567200619723436970, 2399644859771840413188, 201570168112785111295112, 16931894122986359799498910, 1422279106309679099534593110, 119471444930309485959186857992, 10035601374141846588733405827732, 842990515427973216335049945387714, 70811203295948936730025214406560450 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [3, 2, 2, 1, 1]](x) = 2 3 2 3 x (1176 x - 420 x + 14 x + 1) - ------------------------------------------------------ (1 + 2 x) (-1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[3, 2, 2, 1, 1]](x) = -3*x^2*(1176*x^3-420*x^2+14*x+1)/(1+2*x)/( -1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 3, 276, 22104, 1869024, 156803136, 13174056000, 1106583549312, 92953533078528, 7808089537603584, 655879622339656704, 55093886858832783360, 4627886515982713331712, 388742467064735065423872, 32654367237326871680729088, 2742966847881007930111131648, 230409215222766947050207051776, 19354374078701751564371519078400, 1625767422611096538907211905892352, 136564463499330017561226248389656576 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 8 7 6 F[[3, 2, 2, 2], [3, 2, 1, 1, 1, 1]](x) = - x (232848 x - 41832 x + 39420 x 5 4 3 2 - 66946 x + 8180 x + 2653 x + 78 x - 100 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[3, 2, 1, 1, 1, 1]](x) = -x^2*(232848*x^8-41832*x^7+39420*x^6-\ 66946*x^5+8180*x^4+2653*x^3+78*x^2-100*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3 *x)/(-1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 177, 14266, 1211835, 101622426, 8538855017, 717228351316, 60247683988095, 5060798438899126, 425107167305364357, 35709000676193123016, 2999556076092300516155, 251962710121688482869226, 21164867654003016093761697, 1777848882883317661299929116, 149339306162939793197522889015, 12544501717676567125207233333726, 1053738144284976895924299715303037, 88514004119936025655086487596765616 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 3 F[[3, 2, 2, 2], [3, 1, 1, 1, 1, 1, 1]](x) = 2 x ( 7 6 5 4 3 2 112896 x - 95256 x + 13004 x + 20578 x - 7543 x - 245 x + 118 x + 23) /((1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[3, 1, 1, 1, 1, 1, 1]](x) = 2*x^3*(112896*x^7-95256*x^6+13004*x^ 5+20578*x^4-7543*x^3-245*x^2+118*x+23)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/( -1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 0, 46, 3778, 323306, 27095492, 2277073232, 191260198430, 16066058501182, 1349546116031944, 113361913151664188, 9522400154055743042, 799881620658576907178, 67190056027305268194956, 5643964707806154005603464, 474093035434543044990546214, 39823814976798061005176491094, 3345200458046886936916917930128, 280996838475996605712915737907860, 23603734431982901439360893339081546 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 2 F[[3, 2, 2, 2], [2, 2, 2, 2, 1]](x) = - x 7 6 5 4 3 2 (7056 x - 50568 x + 30820 x + 4950 x - 5434 x + 719 x + 8 x - 1)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[2, 2, 2, 2, 1]](x) = -x^2*(7056*x^7-50568*x^6+30820*x^5+4950*x^ 4-5434*x^3+719*x^2+8*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/(-1+2*x)/(-1+4 *x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 75, 5723, 484867, 40649477, 3415546477, 286891357579, 24099073751919, 2024319376157873, 170042866927728289, 14283600270498566855, 1199822430437120348131, 100785084048676157390989, 8465947061601213630231861, 711139553153327091984414851, 59735722465175917537757646103, 5017800687070626851070865219625, 421495257713990758379031871734793, 35405601647974410262070150282970367 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [2, 2, 2, 1, 1, 1]](x) = 2 4 3 2 x (10416 x - 1144 x - 572 x + 14 x + 1) ------------------------------------------------------------------------- (-1 + 3 x) (-1 + x) (1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[2, 2, 2, 1, 1, 1]](x) = x^2*(10416*x^4-1144*x^3-572*x^2+14*x+1) /(-1+3*x)/(-1+x)/(1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 1, 84, 6517, 554248, 46453233, 3903520012, 327875240029, 27541806647376, 2313507743270185, 194334706662749620, 16324114572343160661, 1371225635100085603384, 115182953194076980613857, 9675368070463114588210908, 812730917888652390304670413, 68269397103070291004183066272, 5734629356651975575596248052249, 481708865958848952546300520735876, 40463544740542149954938269727161285 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 3 F[[3, 2, 2, 2], [2, 2, 1, 1, 1, 1, 1]](x) = 2 x 4 3 2 (18648 x - 4044 x - 346 x + 119 x + 23)/((1 + 2 x) (-1 + 6 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[2, 2, 1, 1, 1, 1, 1]](x) = 2*x^3*(18648*x^4-4044*x^3-346*x^2+ 119*x+23)/(1+2*x)/(-1+6*x)/(-1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 0, 46, 3642, 311872, 26126808, 2195765728, 184429260960, 15492273796864, 1301347997517696, 109313273999649280, 9182314425834705408, 771314420038909566976, 64790411167533761009664, 5442394539693370880253952, 457161141311556697053044736, 38401535870488381976599134208, 3225729013116577451382615539712, 270961237101854759126728202911744, 22760743916554928223041104874962944 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 F[[3, 2, 2, 2], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 x (5040 x - 900 x - 68 x - 13) - ----------------------------------------------------------------- (-1 + 6 x) (-1 + 3 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x) and in Maple notation F[[3, 2, 2, 2],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(5040*x^3-900*x^2-68*x-13)/( -1+6*x)/(-1+3*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 0, 13, 1069, 92443, 7739929, 650611771, 54645472273, 4590306472051, 385584546967513, 32389118844756619, 2720685747044746657, 228537606059858896099, 19197158862739305297577, 1612561345118336559731707, 135455152980865868646439921, 11378232850519781488855339987, 955771559441882997694164499321, 80284810993143073117637792035435, 6743924123423669524673576832915265 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 3 F[[3, 2, 2, 2], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 7 6 5 4 3 2 (77616 x + 65016 x - 84116 x + 8622 x + 2706 x - 105 x - 21 x + 2)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 4 x) (1 + 14 x) (-1 + 84 x)) and in Maple notation F[[3, 2, 2, 2],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(77616*x^7+65016*x^6-84116 *x^5+8622*x^4+2706*x^3-105*x^2-21*x+2)/(1+x)/(-1+x)/(1+2*x)/(-1+6*x)/(-1+3*x)/( -1+2*x)/(1+6*x)/(-1+4*x)/(1+14*x)/(-1+84*x) The first 20 term , starting with k=1 are 0, 0, 2, 133, 11566, 967285, 81328828, 6830646263, 573788810282, 48198061148545, 4048639955604904, 340085716972664443, 28567200777152107798, 2399644857566750635205, 201570168143649837819380, 16931894122554214446096223, 1422279106315728899389656514, 119471444930224787350661093065, 10035601374143032360789436486656, 842990515427956615475485538134403 ---------------------------------- Their sum is 6 5 4 3 2 4584 x + 6800 x - 1702 x - 2075 x + 225 x + 81 x - 1 ------------------------------------------------------------- (-1 + 84 x) (-1 + 4 x) (1 + 6 x) (-1 + 3 x) (1 + 2 x) (1 + x) and in Maple notation (4584*x^6+6800*x^5-1702*x^4-2075*x^3+225*x^2+81*x-1)/(-1+84*x)/(-1+4*x)/(1+6*x) /(-1+3*x)/(1+2*x)/(1+x) The first 20 term , starting with k=1 are 1, 56, 4282, 359550, 30194838, 2536380134, 213055742654, 17896683121198, 1503321376300630, 126278995639043382, 10607435633479991406, 891024593213436477566, 74846065829921647263302, 6287069529713459243359750, 528113840495930326373295838, 44361562601658148896695035854, 3726371258539284498358787097654, 313015185717299897915619614228438, 26293275600253191424589964894620750, 2208635150421268079667484792237791262 Regarding Lambda=, [3, 2, 2, 1, 1] 9 8 7 6 F[[3, 2, 2, 1, 1], [9]](x) = (1651428 x + 529956 x - 2835441 x - 994341 x 5 4 3 2 + 684872 x + 249756 x - 12569 x - 6250 x - 123 x + 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[9]](x) = (1651428*x^9+529956*x^8-2835441*x^7-994341*x^6+ 684872*x^5+249756*x^4-12569*x^3-6250*x^2-123*x+1)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/ (-1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 1, 7, 2074, 301480, 50027434, 8061641233, 1307525164849, 211763660481340, 34307707925220592, 5557776866431783279, 900362437788256468279, 145858621846347816533530, 23629100089820863440733750, 3827914093925329163632111645, 620122087558426747570746386989, 100459778028134289872761310803000, 16274484046185665315705726820089308, 2636466415279473008316741138845100331, 427107559282568399169101769877024734979 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [8, 1]](x) = 2 4 3 2 x (1377 x + 1476 x - 3516 x - 190 x + 2) - ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[8, 1]](x) = -x^2*(1377*x^4+1476*x^3-3516*x^2-190*x+2)/(1+x)/ (-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 2, 68, 16202, 2423912, 399785951, 64508680733, 10459641518372, 1694129434720574, 274460937971484305, 44462241046874725943, 7202898562150951717862, 1166869008616363629185336, 189032799500125989345107939, 30623312795266506273438667073, 4960976698888314553895265430472, 803678224281921898338991553617298, 130195872367438809668799068985136853, 21091731322309458529380295156994006123, 3416860474257894912690345042289374141002 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 7 6 5 F[[3, 2, 2, 1, 1], [7, 2]](x) = - x (236196 x - 81324 x - 226341 x 4 3 2 - 65961 x + 21741 x + 8102 x + 366 x - 5)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[7, 2]](x) = -x^2*(236196*x^7-81324*x^6-226341*x^5-65961*x^4+ 21741*x^3+8102*x^2+366*x-5)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+2*x)/(-1+6*x)/(1+6 *x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 5, 249, 53780, 8211963, 1348143017, 217757603259, 35299820767991, 5717739737668641, 926303761403560229, 150060132086161248039, 24309780179352478689587, 3938182992924876920387709, 637985695114517809366254641, 103353680799167125178888708739, 16743296354602925624585748954143, 2712414007100711302705044928748217, 439411069234733886382981344535696253, 71184593212987818001630174019182617759, 11531904100613433093590933345296761574859 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[3, 2, 2, 1, 1], [7, 1, 1]](x) = x 6 5 4 3 2 (69012 x - 60264 x - 27945 x - 20256 x - 2018 x - 232 x + 4)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[7, 1, 1]](x) = x^2*(69012*x^6-60264*x^5-27945*x^4-20256*x^3-\ 2018*x^2-232*x+4)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x)/(-1+ 162*x) The first 20 term , starting with k=1 are 0, 4, 268, 55486, 8525566, 1397737375, 225834797587, 36606786146788, 5929523549102932, 960610743899007121, 155617935068016506701, 25210141676985653293030, 4084041648616805942609998, 661614793985897755279466467, 107181594936956327310820200535, 17363418440582252945509300743112, 2812873785185693171934848356842664, 455685553278873038841841172129421013, 73821059628340965472793286282210261889, 11959011659893349212097566028915095247434 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [6, 3]](x) = 2 4 3 2 2 x (10449 x + 4230 x - 360 x + 151 x - 3) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[6, 3]](x) = 2*x^2*(10449*x^4+4230*x^3-360*x^2+151*x-3)/(1+x) /(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 6, 472, 94446, 14638972, 2395257264, 387176453662, 62753371044930, 10164937813935304, 1646759824398769572, 266773655204563630882, 43217383851665345541654, 7001214322462828808757196, 1134196787253228606298246680, 183739877122510021292992470982, 29765860180697091910570176677418, 4822069346146312024886900949800848, 781175234188260755158040761363152588, 126550387934446146879052618063670619562, 20501162845526151230842318051153928273022 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [6, 2, 1]](x) = 2 4 3 2 x (70956 x + 3276 x - 6135 x - 567 x + 13) - ----------------------------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[6, 2, 1]](x) = -x^2*(70956*x^4+3276*x^3-6135*x^2-567*x+13)/( 1+2*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 13, 1071, 205147, 32075190, 5237735728, 847016515296, 137270550280912, 22235889627614880, 3602283942123664768, 583567485014932298496, 94538023062339698321152, 15315156478461854773624320, 2481055466785758570665771008, 401930981397395115777997111296, 65112819138366328563099345620992, 10548276694943765714125832690933760, 1708820824777866908159524464501096448, 276828973606923272072880428076306333696, 44846293724576852089569268884409901842432 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [6, 1, 1, 1]](x) = 2 4 3 2 x (6399 x - 1197 x + 579 x + 182 x - 6) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[6, 1, 1, 1]](x) = x^2*(6399*x^4-1197*x^3+579*x^2+182*x-6)/(1 +x)/(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 6, 592, 108627, 17134738, 2792451030, 451778420401, 73209654071133, 11859188153198236, 1921216409799674604, 311236052943936884335, 50420276772886096311219, 8168083534152678148333594, 1323229579442709100926790578, 214363190180959765321372528669, 34726836870110809904958422556585, 5625747570769319399366170430329912, 911371106543420487685777976317854552, 147642119257197652185511068873790613803, 23918023319768132459557849063378948834431 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[3, 2, 2, 1, 1], [5, 4]](x) = - x 6 5 4 3 2 (84564 x + 199314 x + 85068 x + 8478 x - 1959 x - 179 x + 5)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[5, 4]](x) = -x^2*(84564*x^6+199314*x^5+85068*x^4+8478*x^3-\ 1959*x^2-179*x+5)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x)/(-1+ 162*x) The first 20 term , starting with k=1 are 0, 5, 426, 82052, 12829983, 2095094081, 338806602666, 54908220105212, 8894355850920441, 1440913576849212527, 233426994005968388736, 37815209224935870245402, 6126062591384741746230399, 992422186714303427939672093, 160772392558958046305321888886, 26045127655346531425227982550072, 4219310677977506285650121496350757, 683528329911146763263809362624276779, 110731589442769308829152163613554807116, 17938517489830740835827707538529880106422 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[3, 2, 2, 1, 1], [5, 3, 1]](x) = 3 x 6 5 4 3 2 (54756 x - 92448 x - 47121 x + 5272 x + 2132 x - 37 x + 5)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[5, 3, 1]](x) = 3*x^2*(54756*x^6-92448*x^5-47121*x^4+5272*x^3 +2132*x^2-37*x+5)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(1+36* x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 15, 1734, 313443, 49595334, 8077188504, 1306965413805, 211783811224371, 34306982496706668, 5557802981847700518, 900361497633219953331, 145858655691928750125009, 23629098871379947545799602, 3827914137789202122135973332, 620122085979327320982324916137, 100459778084981869229450789364207, 16274484044139152458861943426171736, 2636466415353147471163099568033093346, 427107559279916118506632759788301239423, 69191424603608986983658936570874698355565 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [5, 2, 2]](x) = 2 3 2 x (810 x + 5028 x + 131 x - 12) - -------------------------------------------------- (1 + x) (-1 + x) (1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[5, 2, 2]](x) = -x^2*(810*x^3+5028*x^2+131*x-12)/(1+x)/(-1+x) /(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 12, 1309, 231120, 36777055, 5981664672, 968174349079, 156875031452160, 25412646796193335, 4116888679572997632, 666934529742655388599, 108043445526837952204800, 17503036313840786297804215, 2835491949856457782219579392, 459349693464233147619834056119, 74414650428056238377775440855040, 12055173366218493752567947186400695, 1952938085439954195042451500012797952, 316375969837220484140326934291943091639, 51252907113775593867168760205064017018880 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 7 6 5 F[[3, 2, 2, 1, 1], [5, 2, 1, 1]](x) = - x (288684 x + 45684 x - 162207 x 4 3 2 - 41229 x - 3877 x - 1793 x - 149 x - 16)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[5, 2, 1, 1]](x) = -x^2*(288684*x^7+45684*x^6-162207*x^5-\ 41229*x^4-3877*x^3-1793*x^2-149*x-16)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+2*x)/(-1 +6*x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 16, 2117, 362200, 57987092, 9418852042, 1524956237903, 247075235796979, 40025024495941124, 6484095861826072468, 1050422021450635078289, 170168421768955771794253, 27567282371988539014262246, 4465899814627106196343407694, 723475767436452540558362619875, 117203074415898303455327762575447, 18986898052092577451918035810929608, 3075877484557183664693428393325182120, 498292152494009053450958335635293378261, 80723328704182635867312834917180543679361 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [5, 1, 1, 1, 1]](x) = 2 2 x (666 x + 15 x + 5) ------------------------------------------- (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[5, 1, 1, 1, 1]](x) = x^2*(666*x^2+15*x+5)/(1+2*x)/(1+6*x)/(-\ 1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 5, 815, 132866, 21523160, 3486784640, 564859065680, 91507169828576, 14824161510552320, 2401514164752300800, 389045294689812273920, 63025337739751130241536, 10210104713839680922388480, 1654036963642028364934799360, 267953988110008595041073500160, 43408546073821392398652192874496, 7032184463959065568578834136432640, 1139213883161368622109843068403384320, 184552649072141716781794475521394278400, 29897529149686958118650707624244762771456 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[3, 2, 2, 1, 1], [4, 4, 1]](x) = x 6 5 4 3 2 (84564 x - 19656 x + 38619 x + 22386 x - 740 x - 84 x + 8)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[4, 4, 1]](x) = x^2*(84564*x^6-19656*x^5+38619*x^4+22386*x^3-\ 740*x^2-84*x+8)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x)/(-1+162 *x) The first 20 term , starting with k=1 are 0, 8, 916, 161768, 25743910, 4187164685, 677722043215, 109812521995478, 17788852757293540, 2881822075700342087, 466854170819857261969, 75630411868786539329408, 12252125419688550354057190, 1984844364899520446574114689, 321544785424963203331924856803, 52090255299639366864407544370058, 8438621356352945626797492503817160, 1367056659807967936529714780506288091, 221463178886054338898228851465370965717, 35877034979642915707018132097842802421428 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [4, 3, 2]](x) = 2 4 3 2 x (243 x - 6876 x + 786 x - 96 x - 14) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[4, 3, 2]](x) = x^2*(243*x^4-6876*x^3+786*x^2-96*x-14)/(1+x)/ (-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 14, 1902, 321194, 51572046, 8371305695, 1355552934771, 219621125768996, 35577846570417492, 5763639073402349201, 933708524447648612685, 151260817156487868427238, 24504251076296168059484238, 3969688721269954483516481507, 643089571156973517402762719319, 104180510588225037742754913381320, 16877242713103824309092869268438184, 2734113319601610283061525193103808213, 442926357772624399036382250019993369473, 71754069959267265449398981201234653485642 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [4, 3, 1, 1]](x) = 2 4 3 2 x (2187 x - 1701 x - 885 x - 436 x - 16) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[4, 3, 1, 1]](x) = x^2*(2187*x^4-1701*x^3-885*x^2-436*x-16)/( 1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 16, 2500, 410953, 66378850, 10760515564, 1742947062001, 282366660369523, 45743066495175580, 7410388741804444162, 1200482545268050203247, 194478187845982793462473, 31505465872597130482724890, 5103885491465010270550837960, 826829448893577760134400323613, 133946370746814737681204735770303, 21699312060046002444974128195687480, 3515288553761219858223757218617205358, 569476745708101988395283911482285517099, 92255232804756284750966733263486388741013 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [4, 2, 2, 1]](x) = 2 4 3 2 3 x (2187 x + 4608 x + 1077 x - 208 x - 5) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[4, 2, 2, 1]](x) = 3*x^2*(2187*x^4+4608*x^3+1077*x^2-208*x-5) /(1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 15, 2559, 408975, 66450963, 10757924934, 1743040357338, 282363301933341, 45743187400053861, 7410384389235880428, 1200482701960560823992, 194478182205052665073587, 31505466075670616628472419, 5103885484154364778446412722, 826829449156760997905014547046, 133946370737340141121791753215913, 21699312060387087921114970344580737, 3515288553748940781082698749918659416, 569476745708544035172362087447402838900, 92255232804740371066991919355293983163519 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 7 6 5 F[[3, 2, 2, 1, 1], [4, 2, 1, 1, 1]](x) = x (393660 x + 276696 x - 132867 x 4 3 2 - 157518 x - 14325 x + 8555 x + 1080 x + 10)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x) ) and in Maple notation F[[3, 2, 2, 1, 1],[4, 2, 1, 1, 1]](x) = x^2*(393660*x^7+276696*x^6-132867*x^5-\ 157518*x^4-14325*x^3+8555*x^2+1080*x+10)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+2*x)/ (-1+6*x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 10, 2310, 355195, 58239000, 9409781917, 1525282754079, 247063481165350, 40025447662385220, 6484080627832320199, 1050422569874399575953, 170168402025700186384000, 27567283082745739708084710, 4465899789039846969080158081, 723475768357593872726125838307, 117203074382737215497206004261770, 18986898053286376618409925415832520, 3075877484514206894699717405382963163, 498292152495556217170731913428220221141, 80723328704126937973400986009997221588660 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [4, 1, 1, 1, 1, 1]](x) = 2 4 3 2 x (12555 x + 4482 x + 372 x + 460 x + 2) - ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -x^2*(12555*x^4+4482*x^3+372*x^2+460 *x+2)/(1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 2, 718, 103940, 17302570, 2786403635, 451996093969, 73201817626730, 11859470264021020, 1921206253802999993, 311236418559774844015, 50420263610715675804800, 8168084007990811762817050, 1323229562384536281662900351, 214363190795053986760017661981, 34726836848003417932838068268150, 5625747571565185510360528407778360, 911371106514769307689969240468099109, 147642119258229094665360112272412141867, 23918023319731000530283283074476755816780 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [3, 3, 3]](x) = 2 4 3 2 3 x (1350 x - 45 x - 186 x + 7 x + 1) - -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[3, 3, 3]](x) = -3*x^2*(1350*x^4-45*x^3-186*x^2+7*x+1)/(-1+x) /(1+3*x)/(1+2*x)/(1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 3, 495, 79701, 12914148, 2092070025, 338915448690, 54904301869287, 8894496906666936, 1440908498850373797, 233427176813899454430, 37815202643850641874483, 6126062828303808988762164, 992422178185217017654891569, 160772392866005157040316690490, 26045127644292835439144297950239, 4219310678375439341147864701675632, 683528329896821173265904148382569341, 110731589443285030069076705624808477270, 17938517489812174871190424513610941879755 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [3, 3, 2, 1]](x) = 2 4 3 2 x (1215 x - 13545 x - 6168 x + 616 x + 11) - ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[3, 3, 2, 1]](x) = -x^2*(1215*x^4-13545*x^3-6168*x^2+616*x+11 )/(1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 11, 2035, 316550, 51740137, 8365259855, 1355770617670, 219613289380580, 35578128681576199, 5763628917407690129, 933708890063498665600, 151260803994317520480230, 24504251550134302109324161, 3969688704211781666864730083, 643089571771067738857080685450, 104180510566117645770728596089800, 16877242713899690420087791467868123, 2734113319572959103065719842585941717, 442926357773655841516231313730606231220, 71754069959230133520124415334204408470090 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [3, 3, 1, 1, 1]](x) = 2 3 2 x (5670 x + 948 x - 654 x - 7) - -------------------------------------------------- (1 + x) (-1 + x) (1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[3, 3, 1, 1, 1]](x) = -x^2*(5670*x^3+948*x^2-654*x-7)/(1+x)/( -1+x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 7, 1494, 224455, 37017000, 5973026647, 968485317984, 156863836571575, 25413049811894400, 4116874171007759287, 666935052051003969024, 108043426723737403309495, 17503036990752406058035200, 2835491925487639470851263927, 459349694341510606829093412864, 74414650396474249846242104012215, 12055173367355445339703147312742400, 1952938085399023937905584295464496567, 316375969838693973397254153655681941504, 51252907113722548253919380307969418423735 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[3, 2, 2, 1, 1], [3, 2, 2, 2]](x) = - x 6 5 4 3 2 (37908 x - 71496 x + 38727 x + 32868 x + 4122 x - 425 x - 5)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[3, 2, 2, 2]](x) = -x^2*(37908*x^6-71496*x^5+38727*x^4+32868* x^3+4122*x^2-425*x-5)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x)/( -1+162*x) The first 20 term , starting with k=1 are 0, 5, 1050, 157133, 25912062, 4181119226, 677939728431, 109804685621027, 17789134868536164, 2881811919706186772, 466854536435710337937, 75630398706616209521741, 12252125893526684512735206, 1984844347841347630575395918, 321544786039057424790161027043, 52090255277531974892404736319575, 8438621357148811737792555758726088, 1367056659779316756533910276321361064, 221463178887085781378077920253981595349, 35877034979605783777743566261280544275329 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 7 6 5 F[[3, 2, 2, 1, 1], [3, 2, 2, 1, 1]](x) = - x (867996 x + 364284 x - 366111 x 4 3 2 - 148443 x + 9832 x + 5038 x + 116 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[3, 2, 2, 1, 1]](x) = -x*(867996*x^7+364284*x^6-366111*x^5-\ 148443*x^4+9832*x^3+5038*x^2+116*x-1)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+2*x)/(-1 +6*x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 7, 2074, 301480, 50027434, 8061641233, 1307525164849, 211763660481340, 34307707925220592, 5557776866431783279, 900362437788256468279, 145858621846347816533530, 23629100089820863440733750, 3827914093925329163632111645, 620122087558426747570746386989, 100459778028134289872761310803000, 16274484046185665315705726820089308, 2636466415279473008316741138845100331, 427107559282568399169101769877024734979, 69191424603513504879810052847508382016950 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [3, 2, 1, 1, 1, 1]](x) = 2 4 3 2 x (4860 x + 27360 x - 699 x - 1012 x - 3) ----------------------------------------------------------------- (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[3, 2, 1, 1, 1, 1]](x) = x^2*(4860*x^4+27360*x^3-699*x^2-1012 *x-3)/(1+2*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 3, 1390, 193455, 32494930, 5222618208, 847560705040, 137250959204880, 22236594904881760, 3602258552133237888, 583568399054534755840, 94537990156913692404480, 15315157663057189081930240, 2481055424140326524138631168, 401930982932630669384405463040, 65112819083097848632857233018880, 10548276696933430991612080273285120, 1708820824706238958170004740709122048, 276828973609501878272503049267854704640, 44846293724484022266382853988324386734080 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 3 F[[3, 2, 2, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - 3 x 5 4 3 2 (17820 x - 45288 x - 18147 x + 4602 x - 555 x - 131)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -3*x^3*(17820*x^5-45288*x^4-18147 *x^3+4602*x^2-555*x-131)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(-1+6*x)/(1+6*x)/(1+36*x )/(-1+162*x) The first 20 term , starting with k=1 are 0, 0, 393, 50790, 8693337, 1391689599, 226052468838, 36598949688420, 5929805659841799, 960600587901828753, 155618300683851443328, 25210128514815214647270, 4084042122454939448255361, 661614776927724935362543587, 107181595551050548745547129738, 17363418418474860973365437213640, 2812873785981559282929065278812123, 455685553250221858846031589946726101, 73821059629372407952642324602834022068, 11959011659856217282823000009544915360330 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 2 F[[3, 2, 2, 1, 1], [2, 2, 2, 2, 1]](x) = - x 6 5 4 3 2 (102060 x - 34074 x - 141426 x - 48759 x - 3217 x + 317 x + 2)/( (1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[2, 2, 2, 2, 1]](x) = -x^2*(102060*x^6-34074*x^5-141426*x^4-\ 48759*x^3-3217*x^2+317*x+2)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+6*x)/(1+6*x)/(1+36 *x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 2, 559, 77408, 12998074, 2089048241, 339024285565, 54900383716796, 8894637962079148, 1440903420854553455, 233427359621818441651, 37815196062765522298394, 6126063065222875796070322, 992422169656130611287920669, 160772393173052267759639855017, 26045127633239139453201665258552, 4219310678773372396645043695780696, 683528329882495583268004012106410283, 110731589443800751309001227324167668863, 17938517489793608906553141671499635090870 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 2 4 3 2 x (6318 x + 20700 x + 2445 x - 528 x - 1) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[2, 2, 2, 1, 1, 1]](x) = x^2*(6318*x^4+20700*x^3+2445*x^2-528 *x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 1, 657, 87781, 14878917, 2386619239, 387487422567, 62742176164345, 10165340829636369, 1646745315833531227, 266774177512912211307, 43217365048564796646349, 7001214999374448568988181, 1134196762884410294929931215, 183739877999787480502251827727, 29765860149115103379036839834593, 4822069347283263612022101076142553, 781175234147330498021173556814851203, 126550387935919636135979837427409469427, 20501162845473105617592938154059329677877 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 3 F[[3, 2, 2, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (341172 x + 13608 x - 203373 x - 47214 x + 1006 x + 1644 x + 383)/( (1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = x^3*(341172*x^6+13608*x^5-203373* x^4-47214*x^3+1006*x^2+1644*x+383)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+2*x)/(-1+6* x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 0, 383, 48753, 8391758, 1341663522, 217990824098, 35291424572544, 5718041999234456, 926292879978371694, 150060523817415124958, 24309766077027021668220, 3938183500608591468462644, 637985676837904074207430266, 103353681457125219576037703738, 16743296330916434225876973194856, 2712414007953424993056092384757872, 439411069204036193530328825292023238, 71184593214092934944325575846992138838, 11531904100573648883653898346305845061652 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 F[[3, 2, 2, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 3 3 2 3 x (4833 x + 3219 x - 433 x + 40) - ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 6 x) (1 + 36 x) (-1 + 162 x) and in Maple notation F[[3, 2, 2, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -3*x^3*(4833*x^3+3219*x^2-433* x+40)/(1+x)/(-1+x)/(1+3*x)/(-1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 0, 120, 14181, 2495766, 397193766, 64601966739, 10456283026203, 1694250339262932, 274456585400905032, 44462397739373253453, 7202892921220750769565, 1166869211689849339576398, 189032792189480494628543898, 30623313058449744028380057687, 4960976689413717994388245879167, 803678224623007374479269480529064, 130195872355159732527737214954701964, 21091731322751505306458450810119994241, 3416860474241981228715531012225020561409 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 3 F[[3, 2, 2, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 3 x 6 5 4 3 2 (54756 x - 92448 x - 47121 x + 5272 x + 2132 x - 37 x + 5)/((1 + x) (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 2 x) (-1 + 6 x) (1 + 6 x) (1 + 36 x) (-1 + 162 x)) and in Maple notation F[[3, 2, 2, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = 3*x^3*(54756*x^6-92448*x^5-\ 47121*x^4+5272*x^3+2132*x^2-37*x+5)/(1+x)/(-1+x)/(1+3*x)/(1+2*x)/(-1+2*x)/(-1+6 *x)/(1+6*x)/(1+36*x)/(-1+162*x) The first 20 term , starting with k=1 are 0, 0, 15, 1734, 313443, 49595334, 8077188504, 1306965413805, 211783811224371, 34306982496706668, 5557802981847700518, 900361497633219953331, 145858655691928750125009, 23629098871379947545799602, 3827914137789202122135973332, 620122085979327320982324916137, 100459778084981869229450789364207, 16274484044139152458861943426171736, 2636466415353147471163099568033093346, 427107559279916118506632759788301239423 ---------------------------------- Their sum is 6 5 4 3 2 17712 x - 6786 x - 19887 x - 4883 x + 647 x + 157 x - 1 -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (1 + 6 x) (-1 + 6 x) (-1 + 162 x) and in Maple notation (17712*x^6-6786*x^5-19887*x^4-4883*x^3+647*x^2+157*x-1)/(-1+x)/(1+3*x)/(1+2*x)/ (1+6*x)/(-1+6*x)/(-1+162*x) The first 20 term , starting with k=1 are 1, 194, 30698, 4972889, 805588790, 130505363063, 21141868165652, 3424982641972493, 554847187976437634, 89885244452150929187, 14561409601247621308736, 2358948355402113493542377, 382149633575142356124555758, 61908240639173061650405901791, 10029134983546035986292100940300, 1624719867334457829777815958347141, 263204618508182168423967535453268762, 42639148198325511284682686579897680475, 6907542008128732828118593834566685473944, 1119021805316854718155212199249867870588385 Regarding Lambda=, [3, 2, 1, 1, 1, 1] 8 7 6 F[[3, 2, 1, 1, 1, 1], [9]](x) = (1876875 x + 360875 x - 4220800 x 5 4 3 2 - 1192765 x + 237885 x + 49300 x - 2905 x - 82 x + 1)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) ) and in Maple notation F[[3, 2, 1, 1, 1, 1],[9]](x) = (1876875*x^8+360875*x^7-4220800*x^6-1192765*x^5+ 237885*x^4+49300*x^3-2905*x^2-82*x+1)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-\ 1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 0, 511, 30305, 3880646, 381455275, 40939299486, 4267243470630, 449153624534671, 47122793387999450, 4949233905858916961, 519622621090839149455, 54562017810082665623196, 5728954375160050799890125, 601542221652438688623244936, 63161862843471616038346946780, 6631998063602041117613189106221, 696359710401695804414805420847300, 73117772611853128474805967729123411 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [8, 1]](x) = - x 6 5 4 3 2 (459375 x + 283625 x - 89900 x - 26950 x + 2687 x + 157 x - 2)/( (1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[8, 1]](x) = -x^2*(459375*x^6+283625*x^5-89900*x^4-26950*x ^3+2687*x^2+157*x-2)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/( -1+105*x) The first 20 term , starting with k=1 are 0, 2, 7, 3699, 251846, 30664332, 3064171197, 327064030029, 34153532405356, 3592680864421302, 377001491669512787, 39593200586405442759, 4157004432832753479666, 436495321101797587135872, 45831663747513348134181577, 4812336767070947252909873889, 505294937962516467736560498776, 53055983276293462968889877016042, 5570877726351764112956841772353567, 584942179384986569610397021018795419 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [7, 2]](x) = 2 3 2 x (15 x - 2) (275 x + 2110 x + 137 x - 2) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[7, 2]](x) = x^2*(15*x-2)*(275*x^3+2110*x^2+137*x-2)/(1+x) /(-1+x)/(1+5*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 4, 36, 11499, 874611, 102492749, 10374465861, 1102658895874, 115309081543986, 12123859071395874, 1272430288869043986, 133625291495397567749, 14029951554088898340861, 1473169552599049596786499, 154681940606719022003809611, 16241633947724471057653427124, 1705370508062194869614044825236, 179063940322117914343494176864624, 18801712439674972785030136505762736, 1974179851461003719691457599401473999 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [7, 1, 1]](x) = x 6 5 4 3 2 (459375 x - 85750 x + 59600 x + 24850 x - 3523 x - 284 x + 4)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) ) and in Maple notation F[[3, 2, 1, 1, 1, 1],[7, 1, 1]](x) = x^2*(459375*x^6-85750*x^5+59600*x^4+24850* x^3-3523*x^2-284*x+4)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/ (-1+105*x) The first 20 term , starting with k=1 are 0, 4, 44, 11709, 915324, 106008594, 10768686544, 1143151393059, 119591963030024, 12572465365065744, 1319572238835683844, 138573854920995932409, 14549597641988816130724, 1527730749070814306544894, 160410923728720200241715144, 16843175163237470280859869759, 1768532406120546866916548537424, 185695937153199142116734722606044, 19498072193214897056497348661500444, 2047297622563018851819426152495045109 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [6, 3]](x) = 2 5 4 3 2 x (118125 x + 29475 x - 31410 x - 62 x + 357 x - 5) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[6, 3]](x) = -x^2*(118125*x^5+29475*x^4-31410*x^3-62*x^2+ 357*x-5)/(1+x)/(-1+x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 5, 78, 19203, 1586124, 180940465, 18485257338, 1958781271823, 205045871538264, 21551700132054045, 2262162106541066598, 237553838234714094043, 24942214309844841821604, 2618965355579875827454025, 274990212454943622673616658, 28874012553237185473972693863, 3031769909493047121857534772144, 318335889797577850037341654852405, 33425266703216166098848236910171518, 3509653064231211752157979541419279283 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [6, 2, 1]](x) = x 5 4 3 2 (459375 x + 67250 x + 9425 x - 2635 x - 528 x + 9)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[6, 2, 1]](x) = x^2*(459375*x^5+67250*x^4+9425*x^3-2635*x^ 2-528*x+9)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 9, 210, 40739, 3515845, 394221334, 40492497185, 4282881502314, 448606293669870, 47141949967046759, 4948563425598364660, 519646087899927962389, 54561196471764709758395, 5728983122001178492218684, 601541215512999223206442635, 63161898058351997308861540964, 6631996831081227773240545741420, 696359753539924271467371101457109, 73117771102015132127968553093571110, 7677366071400234762478925867051718039 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [6, 1, 1, 1]](x) = - x 6 5 4 3 2 (459375 x + 539000 x - 67175 x - 27340 x + 4413 x + 276 x - 5)/( (1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[6, 1, 1, 1]](x) = -x^2*(459375*x^6+539000*x^5-67175*x^4-\ 27340*x^3+4413*x^2+276*x-5)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+ 35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 5, 134, 21105, 1900444, 209416125, 21626021634, 2283164491305, 239293232052344, 25141097011367925, 2639278537682828734, 277143015939558257505, 29099359543532077316244, 3055455748651765730263725, 320822048683503735690203834, 33686343283471495935653319705, 3537065058744845877185393692144, 371391865678746432940027460823525, 38996144688397301014119678037586934, 4094595234557170343687352869353357905 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [5, 4]](x) = 2 3 2 x (2450 x + 545 x + 128 x - 3) --------------------------------------------------------------- (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[5, 4]](x) = x^2*(2450*x^3+545*x^2+128*x-3)/(-1+x)/(1+5*x) /(1+3*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 3, 76, 16152, 1404274, 157656783, 16196525506, 1713145478712, 179442410712844, 18856778384932323, 1979425346213055136, 207858434799562671072, 21824478583299806792014, 2291593248719379884976063, 240616486203983317873080166, 25264759223322553343508377232, 2652798732432217425627442329784, 278543901415965603331694341644003, 29247108440805991272359404288232596, 3070946428560092981309144530063701192 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [5, 3, 1]](x) = 2 4 3 2 3 x (4375 x + 450 x - 230 x + 88 x - 3) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[5, 3, 1]](x) = 3*x^2*(4375*x^4+450*x^3-230*x^2+88*x-3)/(1 +x)/(-1+x)/(1+5*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 9, 366, 59619, 5510166, 604822119, 62587201416, 6603825640869, 692275754888916, 72728362078765869, 7635098744600982666, 801733642761692047119, 84180342879723819732666, 8838995138729889621734619, 928092419794077230167388916, 97449776520417750570774078369, 10232223999172476358797550201416, 1074383608654608578465902805328369, 112810275802781451111143648624420166, 11845079068000388103642310202121734619 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [5, 2, 2]](x) = 2 2 x (55 x - 3) - ------------------------------------------ (-1 + x) (1 + 3 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[5, 2, 2]](x) = -2*x^2*(55*x-3)/(-1+x)/(1+3*x)/(1+35*x)/(-\ 1+105*x) The first 20 term , starting with k=1 are 0, 6, 298, 43172, 4116300, 446800666, 46403441318, 4890233363112, 512848982191480, 53871036363050126, 5655692554966567938, 593874537481870858252, 62355887763233091581060, 6547401068672191831756386, 687475962336935039732324158, 72185016290955757763120464592, 7579425301955139314434264637040, 795839706006122161789867609463446, 83563167405113688305836651071077978, 8774132637930457125986329052151078132 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [5, 2, 1, 1]](x) = 2 4 3 2 x (28875 x + 1725 x + 1930 x + 239 x - 9) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[5, 2, 1, 1]](x) = -x^2*(28875*x^4+1725*x^3+1930*x^2+239*x -9)/(1+x)/(-1+x)/(1+5*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 9, 526, 66639, 6541001, 701843514, 73153151001, 7699782518514, 807819406729126, 84844011187596639, 8907816382151260376, 935348877053191502889, 98210646435948938369751, 10312152371254170134862264, 1082774791603413164758682251, 113691395376050629653533299764, 11937595035457876947399377822876, 1253447530488914292643966277440389, 131611988889529850901960521936416626, 13819258896813821878131210555632909139 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 2 4 3 2 x (2625 x - 475 x + 160 x - 129 x + 3) -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[5, 1, 1, 1, 1]](x) = x^2*(2625*x^4-475*x^3+160*x^2-129*x+ 3)/(1+x)/(-1+x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 3, 237, 23677, 2465440, 258558693, 27143931017, 2850041652587, 299253306039960, 31421581118216383, 3299265777151739197, 346422902996919009297, 36374404760615774751080, 3819312499053743012864273, 401027812388479303518576777, 42107920300607871113619277807, 4421331631561089630273974726800, 464239821313873358627339213718363, 48745181237956086867591241966795757, 5118244029985379884272850056575402117 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [4, 4, 1]](x) = - x 6 5 4 3 2 (826875 x + 687750 x - 50800 x - 29190 x - 2055 x + 128 x - 4)/( (1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[4, 4, 1]](x) = -x^2*(826875*x^6+687750*x^5-50800*x^4-\ 29190*x^3-2055*x^2+128*x-4)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+ 35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 4, 200, 30079, 2879340, 312728734, 32481935500, 3423156232129, 358994180778440, 37709723852250184, 3958984764450302400, 415712175876901100179, 43649121428857086955540, 4583180747989442770437634, 481233173634638156402771300, 50529511403650784854149162229, 5305597711368323836435206050640, 557087794204281407997653237371084, 58494217183579520235257638771502200, 6141892846551319064508004519835938279 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [4, 3, 2]](x) = x 5 4 3 2 (23625 x - 98775 x - 950 x + 6162 x + 45 x + 5)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[4, 3, 2]](x) = x^2*(23625*x^5-98775*x^4-950*x^3+6162*x^2+ 45*x+5)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 5, 455, 58002, 5830534, 622888615, 65052992445, 6843181304132, 718097774289644, 75415615587943785, 7918103612937964435, 831419658211785200662, 87298407122674686037554, 9166355746570114373012555, 962466548496556019168695625, 101059015764316370667350819592, 10611195669240673499885137618264, 1114175579780915069957245713582925, 116988434669126608950468116357922015, 12283785682533771692746938599415952922 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [4, 3, 1, 1]](x) = 2 5 4 3 2 2 x (39375 x - 2250 x - 11925 x + 935 x + 70 x + 3) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[4, 3, 1, 1]](x) = 2*x^2*(39375*x^5-2250*x^4-11925*x^3+935 *x^2+70*x+3)/(1+x)/(-1+x)/(1+5*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 6, 650, 73026, 7562500, 798722526, 83716968750, 8795707360026, 923362578125000, 96959653088297526, 10180533911582031250, 1068964109722861735026, 112240949967846679687500, 11785309603413539072672526, 1237457163407275427246093750, 129933014231601403631846110026, 13642966071742045959472656250000, 1432511452323201533173529307047526, 150413701976277973588050842285156250, 15793438725627251496049198309580485026 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [4, 2, 2, 1]](x) = 2 4 3 2 x (53625 x - 2350 x + 1420 x - 274 x - 5) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[4, 2, 2, 1]](x) = -x^2*(53625*x^4-2350*x^3+1420*x^2-274*x -5)/(1+x)/(-1+x)/(1+5*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 5, 699, 71250, 7624974, 796534375, 83793561849, 8793026562500, 923456406233724, 96956369103515625, 10180648851054280599, 1068960086841308593750, 112241090768701161702474, 11785304675383631591796875, 1237457335888322192128499349, 129933008194764766845703125000, 13642966283031328247063954671224, 1432511444928076653107452392578125, 150413702235107344390365441640218099, 15793438716568223517968177795410156250 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [4, 2, 1, 1, 1]](x) = 2 5 4 3 2 x (275625 x - 212625 x + 12400 x + 10145 x - 365 x - 4) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[4, 2, 1, 1, 1]](x) = -x^2*(275625*x^5-212625*x^4+12400*x^ 3+10145*x^2-365*x-4)/(1+x)/(-1+x)/(1+5*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x ) The first 20 term , starting with k=1 are 0, 4, 705, 60384, 6759855, 694184009, 73421231730, 7690399702759, 808147805231730, 84832517240249634, 8908218670307184855, 935334796967740249634, 98211139238939701716105, 10312135123149493570327759, 1082775395287076843754450480, 113691374247122400892496109009, 11937595774970364954016606012980, 1253447504605977212412458658218384, 131611989795432648710062811772028605, 13819258865107223954847632795572280884 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (367500 x + 429625 x + 151325 x - 33190 x - 6890 x + 173 x + 1)/( (1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = x^2*(367500*x^6+429625*x^5+151325 *x^4-33190*x^3-6890*x^2+173*x+1)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x )/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 255, 16926, 2046390, 204309571, 21804743205, 2276909275376, 239512164414540, 25133434379667621, 2639546729787456555, 277133629215920697826, 29099688078859269834690, 3055444249915314602469671, 320822451139279522111237905, 33686329197519343426175916276, 3537065551753171214940808866840, 371391848423455046118969399435721, 38996145292332499552854802837527255, 4094595213419438394831633038098610726 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [3, 3, 3]](x) = 2 3 2 x (2790 x - 603 x - 2 x - 1) - -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[3, 3, 3]](x) = -x^2*(2790*x^3-603*x^2-2*x-1)/(1+x)/(-1+x) /(1+3*x)/(-1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 124, 13932, 1475066, 155072081, 16285409994, 1710010756672, 179551770065596, 18852945467399241, 1979559418237247264, 207853741077340463612, 21824642845557280163526, 2291587499270062936324201, 240616687430654838529691134, 25264752180328231511912350752, 2652798978936106410807763720856, 278543892788315804665990684870961, 29247108742773528962898234483269604, 3070946417991226083198860784500834092 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [3, 3, 2, 1]](x) = 2 x 5 4 3 2 (47250 x - 11925 x - 1150 x + 564 x + 204 x + 1)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[3, 3, 2, 1]](x) = 2*x^2*(47250*x^5-11925*x^4-1150*x^3+564 *x^2+204*x+1)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105* x) The first 20 term , starting with k=1 are 0, 2, 572, 53844, 5976376, 617782582, 65231711412, 6836926101224, 718316706586736, 75407952956569002, 7918371805040964652, 831410271488155779004, 87298735658001837865896, 9166344247833663448669022, 962466950952331804572477092, 101059001678364218162959679184, 10611196162248998837615121477856, 1114175562525623683136314808770642, 116988435273061807489202605374984732, 12283785661396039743891221947075593764 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 2 2 x (1575 x + 400 x + 1) ------------------------------------------ (-1 + x) (1 + 3 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[3, 3, 1, 1, 1]](x) = x^2*(1575*x^2+400*x+1)/(-1+x)/(1+3*x )/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 468, 37217, 4324720, 439505961, 46658755988, 4881297349657, 513161742662400, 53860089746567921, 5656075686543445108, 593861127876680157297, 62356357099414766114480, 6547384641905833223086681, 687476537273757591035763828, 72184996168166968467500076137, 7579426006252746939780978232960, 795839681355705894902732633606241, 83563168267878257646886375226080148, 8774132607733697199049588706726002177 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 2 F[[3, 2, 1, 1, 1, 1], [3, 2, 2, 2]](x) = - x 5 4 3 2 (546000 x - 64075 x - 30310 x + 2892 x - 234 x - 1)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[3, 2, 2, 2]](x) = -x^2*(546000*x^5-64075*x^4-30310*x^3+ 2892*x^2-234*x-1)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+ 105*x) The first 20 term , starting with k=1 are 0, 1, 316, 25926, 3025156, 307622831, 32660653816, 3416901032476, 359213113059256, 37702061220956781, 3959252956552895716, 415702789153273713026, 43649449964184228611356, 4583169249252991896956731, 481233576090413941552239616, 50529497317698632351029587576, 5305598204376649174158832081456, 557087776948990021176754121702681, 58494217787514718773991968842845516, 6141892825413587115652288662224176126 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 2 3 2 x (13025 x + 1615 x - 599 x - 1) ------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[3, 2, 2, 1, 1]](x) = x^2*(13025*x^3+1615*x^2-599*x-1)/(1+ x)/(-1+x)/(1+5*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 1, 669, 48916, 5885244, 591692041, 63046765869, 6587740826416, 692838723687744, 72708658169342041, 7635788381438140869, 801709505472354888916, 84181187684850803375244, 8838965570550444278717041, 928093454680357821750640869, 97449740299397929842472076416, 10232225266908170084402561187744, 1074383564283859298069155216217041, 112810277355757675925032675266265869, 11845079013646220235156179964542388916 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (2257500 x + 820375 x - 150950 x - 37785 x + 2395 x + 82 x - 1)/( (1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = -x*(2257500*x^6+820375*x^5-150950 *x^4-37785*x^3+2395*x^2+82*x-1)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x) /(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 0, 511, 30305, 3880646, 381455275, 40939299486, 4267243470630, 449153624534671, 47122793387999450, 4949233905858916961, 519622621090839149455, 54562017810082665623196, 5728954375160050799890125, 601542221652438688623244936, 63161862843471616038346946780, 6631998063602041117613189106221, 696359710401695804414805420847300, 73117772611853128474805967729123411, 7677366018555904890339628275736342605 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 3 F[[3, 2, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (459375 x + 812000 x + 205400 x - 38150 x - 6087 x + 166)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) ) and in Maple notation F[[3, 2, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = x^3*(459375*x^5+812000*x^4+ 205400*x^3-38150*x^2-6087*x+166)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x )/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 166, 7525, 1061296, 100901910, 10947408766, 1136896173875, 119810895408496, 12564802733284060, 1319840430940718566, 138564468197356338225, 14549926177316018821696, 1527719250334363127888210, 160411326184495986917062366, 16843161077285317770110900575, 1768532899128872204678321540896, 185695919897907755295644872074360, 19498072797150095595232632407160166, 2047297601425286902963705526511700925 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [2, 2, 2, 2, 1]](x) = 3 2 x (7175 x + 1130 x - 193) --------------------------------------------------------------- (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[2, 2, 2, 2, 1]](x) = x^3*(7175*x^2+1130*x-193)/(-1+x)/(1+ 5*x)/(1+3*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 193, 11994, 1550116, 152550750, 16375244473, 1706890275804, 179661343009936, 18849115753557540, 1979693538316055353, 207849048075933249414, 21824807118626958620356, 2291581749982928960632530, 240616888659759103276861633, 25264745137370400839117236824, 2652799225440542763357426189376, 278543884160674216510763436831720, 29247109044741189811093893305295313, 3070946407422361032453427877723342034 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 3 3 2 8 x (11925 x + 3645 x - 1041 x + 31) ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = 8*x^3*(11925*x^3+3645*x^2-1041*x+ 31)/(1+x)/(-1+x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 248, 13248, 1794544, 173645760, 18740572008, 1949845258368, 205358632009184, 21540753515571840, 2262545238117943768, 237540428629523393088, 24942683646026516355024, 2618948928813517218784320, 274990787391766173977056328, 28873992430448396178352305408, 3031770613790654747204248368064, 318335865147161583150206678995200, 33425267565980735439897961065173688, 3509653034034451825221239195994203328 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 F[[3, 2, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 3 4 3 2 x (196875 x - 294750 x + 5775 x + 7090 x - 166) - ------------------------------------------------------------------------- (1 + x) (-1 + x) (1 + 5 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x) and in Maple notation F[[3, 2, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -x^3*(196875*x^4-294750*x^3+ 5775*x^2+7090*x-166)/(1+x)/(-1+x)/(1+5*x)/(-1+15*x)/(-1+5*x)/(1+35*x)/(-1+105*x ) The first 20 term , starting with k=1 are 0, 0, 166, 7020, 1030991, 97021395, 10565953491, 1095956877645, 115543651937866, 12115649108830770, 1272717637552719116, 133615234291499455770, 14030303556225179672241, 1473157232524280513127645, 154682371809335936117172241, 16241618855632879082759221395, 1705371036285400588639974594116, 179063921834305714178063472112020, 18801713086748399790817826986312866, 1974179828813433774488900353511174520 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 3 F[[3, 2, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 2 x 5 4 3 2 (65625 x + 194000 x + 64300 x - 7950 x - 1509 x + 30)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = 2*x^3*(65625*x^5+194000*x^4 +64300*x^3-7950*x^2-1509*x+30)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/ (1+35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 60, 1902, 314424, 28475660, 3140766900, 324383219482, 34247360579184, 3589396879313880, 377116431143389740, 39589177704844163462, 4157145233687276184744, 436490393071889902809700, 45831836228560114033839780, 4812330730234310461680625842, 505295149251798755353290235104, 53055975881168582902685805971120, 5570877985181134915272076910293020, 584942170325958591529373327934078622 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 3 F[[3, 2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (459375 x + 67250 x + 9425 x - 2635 x - 528 x + 9)/((1 + x) (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (1 + 35 x) (-1 + 105 x)) and in Maple notation F[[3, 2, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(459375*x^5+67250*x^ 4+9425*x^3-2635*x^2-528*x+9)/(1+x)/(-1+x)/(1+5*x)/(1+3*x)/(-1+15*x)/(-1+5*x)/(1 +35*x)/(-1+105*x) The first 20 term , starting with k=1 are 0, 0, 9, 210, 40739, 3515845, 394221334, 40492497185, 4282881502314, 448606293669870, 47141949967046759, 4948563425598364660, 519646087899927962389, 54561196471764709758395, 5728983122001178492218684, 601541215512999223206442635, 63161898058351997308861540964, 6631996831081227773240545741420, 696359753539924271467371101457109, 73117771102015132127968553093571110 ---------------------------------- Their sum is 6 5 4 3 2 10725 x + 230 x - 17830 x - 2053 x + 1768 x - 121 x + 1 -------------------------------------------------------------- (1 + x) (-1 + x) (1 + 3 x) (-1 + 15 x) (-1 + 5 x) (-1 + 105 x) and in Maple notation (10725*x^6+230*x^5-17830*x^4-2053*x^3+1768*x^2-121*x+1)/(1+x)/(-1+x)/(1+3*x)/(-\ 1+15*x)/(-1+5*x)/(-1+105*x) The first 20 term , starting with k=1 are 1, 91, 8478, 879430, 92174591, 9675915941, 1015935002308, 106672633814040, 11200618432389261, 1176064813653434431, 123486803607496809338, 12966114351396010955450, 1361442006485716412527531, 142951410674837265689582121, 15009898120765468598489928768, 1576039302678987538690258159660, 165484126781272891601856522477401, 17375833312033341618793908836295011, 1824462497763496189982386310651352598, 191568562265167029748286157648500450670 Regarding Lambda=, [3, 1, 1, 1, 1, 1, 1] 11 10 9 F[[3, 1, 1, 1, 1, 1, 1], [9]](x) = (2122176 x - 362224 x - 5020952 x 8 7 6 5 4 3 + 1123268 x + 1667730 x - 351118 x - 144114 x + 23215 x + 4016 x 2 - 362 x - 21 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[9]](x) = (2122176*x^11-362224*x^10-5020952*x^9+1123268 *x^8+1667730*x^7-351118*x^6-144114*x^5+23215*x^4+4016*x^3-362*x^2-21*x+1)/(-1+x )/(1+x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/( -1+28*x) The first 20 term , starting with k=1 are 0, 1, 0, 16, 25, 2661, 30766, 1237636, 27543411, 849694771, 22498481632, 646023245946, 17843936793397, 502854970483321, 14032800230856498, 393558306893213296, 11010474253799432983, 308419494251036602911, 8633958836601791139364, 241775665401191212905286 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 F[[3, 1, 1, 1, 1, 1, 1], [8, 1]](x) = x 7 6 5 4 3 2 (23520 x - 7216 x + 32960 x - 1724 x - 3004 x + 235 x + 23 x - 1)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[8, 1]](x) = x^2*(23520*x^7-7216*x^6+32960*x^5-1724*x^4 -3004*x^3+235*x^2+23*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(1+2*x)/(-1+10*x)/(-1+4 *x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 0, 74, 235, 17847, 255650, 9453340, 223145265, 6725744393, 180655534840, 5155529703006, 142895198438135, 4020495451765339, 112291883579090670, 3148020409039595072, 88089705153225804445, 2467269866085977444685, 69072842583603000396740, 1934188583445814780089538 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 8 7 6 F[[3, 1, 1, 1, 1, 1, 1], [7, 2]](x) = - x (912576 x + 541136 x - 308472 x 5 4 3 2 - 102404 x + 26630 x + 4828 x - 601 x - 40 x + 2)/((1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[7, 2]](x) = -x^2*(912576*x^8+541136*x^7-308472*x^6-\ 102404*x^5+26630*x^4+4828*x^3-601*x^2-40*x+2)/(1+x)/(-1+3*x)/(-1+2*x)/(1+6*x)/( 1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 0, 165, 820, 51793, 885854, 30741671, 760403784, 22511201829, 611463346978, 17366700269107, 482648459704268, 13563018675076145, 379062494605434822, 10623398021522926623, 297318271518042281872, 8326809817671860278141, 233123919930425403714986, 6527842527265180650460619 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 8 7 6 F[[3, 1, 1, 1, 1, 1, 1], [7, 1, 1]](x) = - x (28224 x + 213744 x - 8424 x 5 4 3 2 - 7236 x - 1346 x - 1238 x + 222 x + 20 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[7, 1, 1]](x) = -x^2*(28224*x^8+213744*x^7-8424*x^6-\ 7236*x^5-1346*x^4-1238*x^3+222*x^2+20*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+2*x)/(1+6* x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 1, 162, 966, 52839, 937797, 31684780, 792050932, 23303478297, 634765383483, 18001474302318, 500649880705338, 14063668875584875, 393126161539810609, 11016524194709975376, 308334795642510837984, 8635144613735819566173, 241759064541626805651975, 6769601591822013961625554 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 F[[3, 1, 1, 1, 1, 1, 1], [6, 3]](x) = - x 6 5 4 3 2 (7168 x + 7688 x - 6508 x + 1348 x + 52 x - 29 x + 1)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[6, 3]](x) = -x^2*(7168*x^6+7688*x^5-6508*x^4+1348*x^3+ 52*x^2-29*x+1)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+10*x)/(-1+4*x)/(1+14*x)/(-1+28 *x) The first 20 term , starting with k=1 are 0, 1, 0, 187, 1490, 81485, 1603496, 53180191, 1361052126, 39781170049, 1089268019972, 30831967561715, 858520595889242, 24104219569926133, 673987147337413728, 18884554133202020359, 528585519617331289238, 14802930494842671751337, 414446430617730339735164, 11604997582626307431685723 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 F[[3, 1, 1, 1, 1, 1, 1], [6, 2, 1]](x) = 2 x 7 6 5 4 3 2 (39200 x + 2744 x + 5436 x - 4202 x - 889 x + 219 x + 18 x - 1)/( (-1 + 2 x) (1 + 6 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[6, 2, 1]](x) = 2*x^2*(39200*x^7+2744*x^6+5436*x^5-4202 *x^4-889*x^3+219*x^2+18*x-1)/(-1+2*x)/(1+6*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/ (-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 2, 2, 396, 3730, 173042, 3604762, 115154426, 2995996250, 86779521882, 2386372008922, 67396606150106, 1878711335543770, 52718422089180122, 1474482789691755482, 41308080291783732186, 1156307378974809443290, 32381040772020167756762, 906606763414342178196442, 25385859670144731160136666 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 F[[3, 1, 1, 1, 1, 1, 1], [6, 1, 1, 1]](x) = x 7 6 5 4 3 2 (61152 x + 20944 x - 15412 x + 2824 x - 1089 x + 169 x + 20 x - 1)/( (-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[6, 1, 1, 1]](x) = x^2*(61152*x^7+20944*x^6-15412*x^5+ 2824*x^4-1089*x^3+169*x^2+20*x-1)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(1+2*x)/(-1+10* x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 3, 209, 2352, 90292, 1984381, 60875515, 1608789570, 46162546058, 1274744396079, 35920004231761, 1002360691749508, 28111486437661264, 786464231012372097, 22029981740577394247, 616711523992567601766, 17269692171815348318710, 483526387848898936178035, 13539086561006121404543773 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 4 F[[3, 1, 1, 1, 1, 1, 1], [5, 4]](x) = x 6 5 4 3 2 (65856 x - 17584 x - 20032 x + 4004 x + 2876 x - 40 x - 115)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[5, 4]](x) = x^4*(65856*x^6-17584*x^5-20032*x^4+4004*x^ 3+2876*x^2-40*x-115)/(-1+x)/(1+x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(1+2*x)/(1+4*x)/(-1 +4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 0, 115, 1305, 65874, 1417570, 45763389, 1195746195, 34683282448, 954275556300, 26955837599163, 751456919658445, 21087090080460222, 589790343975877590, 16523204303706558937, 462522674023726012455, 12952413529761352853196, 362642677595576061187840, 10154343590234421781242711 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [5, 3, 1]](x) = 3 x 6 5 4 3 2 (6048 x - 7032 x - 4816 x + 2904 x + 179 x - 117 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[5, 3, 1]](x) = 3*x^3*(6048*x^6-7032*x^5-4816*x^4+2904* x^3+179*x^2-117*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(1+4*x)/(-1+4*x)/(1 +14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 3, 390, 5874, 241377, 5650947, 173910492, 4648866708, 133262984559, 3688001473101, 103871319221274, 2899893782682702, 81316077537697821, 2275183403433443415, 63728470309952558136, 1784070476878502182056, 49958547047402544523563, 1398775285542326927113089, 39166604440519840463447478 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 F[[3, 1, 1, 1, 1, 1, 1], [5, 2, 2]](x) = 2 3 2 x (48 x + 168 x + 10 x - 1) -------------------------------------------------------------- (-1 + x) (-1 + 2 x) (1 + 2 x) (1 + 4 x) (1 + 14 x) (-1 + 28 x) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[5, 2, 2]](x) = x^2*(48*x^3+168*x^2+10*x-1)/(-1+x)/(-1+ 2*x)/(1+2*x)/(1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 1, 285, 4605, 174381, 4252461, 127864045, 3457167085, 98522606061, 2734527715821, 76904242599405, 2148594288899565, 60226783031682541, 1685423924139353581, 47204833884580816365, 1321553852653915416045, 37006048819881862243821, 1036133793718795443381741, 29012244249456368075154925 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 2 F[[3, 1, 1, 1, 1, 1, 1], [5, 2, 1, 1]](x) = - x 7 6 5 4 3 2 (63504 x + 9324 x + 6142 x - 2046 x - 259 x - 109 x - 12 x + 1)/( (1 + x) (-1 + x) (-1 + 3 x) (1 + 6 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[5, 2, 1, 1]](x) = -x^2*(63504*x^7+9324*x^6+6142*x^5-\ 2046*x^4-259*x^3-109*x^2-12*x+1)/(1+x)/(-1+x)/(-1+3*x)/(1+6*x)/(-1+10*x)/(1+4*x )/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 9, 439, 8238, 270850, 6848961, 200265283, 5470713180, 154913535124, 4311515480943, 121069296258637, 3384904432639002, 94846025083694158, 2654708896337330205, 74345386339141156951, 2081479496381329282104, 58284086392715464490152, 1631916992089131903183147, 45694197955135271414807425 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 F[[3, 1, 1, 1, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 2 4 3 2 x (1680 x - 92 x - 132 x + 30 x - 1) - --------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[5, 1, 1, 1, 1]](x) = -x^2*(1680*x^4-92*x^3-132*x^2+30* x-1)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 1, 7, 167, 3755, 97611, 2648667, 73341787, 2045219995, 57183023771, 1600291068827, 44799819742107, 1254311615246235, 35119891943903131, 983348641028824987, 27533678616279113627, 770942167921408024475, 21586372368478976417691, 604418342984060826165147, 16923712770220575958376347 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 8 7 6 F[[3, 1, 1, 1, 1, 1, 1], [4, 4, 1]](x) = - x (301056 x - 480704 x - 29776 x 5 4 3 2 + 68840 x - 2012 x - 2422 x + 568 x - 134 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[4, 4, 1]](x) = -x^3*(301056*x^8-480704*x^7-29776*x^6+ 68840*x^5-2012*x^4-2422*x^3+568*x^2-134*x-1)/(-1+x)/(1+x)/(-1+3*x)/(-1+2*x)/(1+ 6*x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 155, 3050, 118700, 2952565, 89206005, 2417367184, 68937290150, 1913896191779, 53830164827615, 1503988388256718, 42158469364808160, 1179793975005910993, 33043355906164561385, 925087419291688348652, 25904231394870023107130, 725293627832998310314207, 20308570696795977865277715 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [4, 3, 2]](x) = - x 6 5 4 3 2 (23520 x - 19264 x - 19088 x + 2728 x - 20 x + 215 x + 2)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[4, 3, 2]](x) = -x^3*(23520*x^6-19264*x^5-19088*x^4+ 2728*x^3-20*x^2+215*x+2)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(1+2*x)/(-1+10*x)/(-1+4* x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 2, 261, 6601, 226258, 6033328, 176250039, 4861878527, 137459932836, 3833271921094, 107580235103737, 3009065052759493, 84301367797471734, 2359802620104391100, 66083673035161748955, 1850217048505348136299, 51807868515244606648552, 1450595542189121572544146, 40617025048853662577587293 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [4, 3, 1, 1]](x) = - 3 x 5 4 3 2 (11088 x - 7432 x + 1794 x - 387 x + 37 x + 3)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[4, 3, 1, 1]](x) = -3*x^3*(11088*x^5-7432*x^4+1794*x^3-\ 387*x^2+37*x+3)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(-1+10*x)/(-1+4*x)/(1+14*x)/(-1+ 28*x) The first 20 term , starting with k=1 are 0, 0, 9, 336, 9570, 286755, 7929375, 225329706, 6280318116, 176437600185, 4933788663441, 138254719295616, 3869790410789862, 108374720677296255, 3034221901022683107, 84962177297909378166, 2378887266307752356808, 66609613235491103510565, 1865058573651177744668373, 52221790219659325892890956 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [4, 2, 2, 1]](x) = x 6 5 4 3 2 (61152 x + 3520 x - 27664 x + 9180 x - 148 x - 104 x - 9)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[4, 2, 2, 1]](x) = x^3*(61152*x^6+3520*x^5-27664*x^4+ 9180*x^3-148*x^2-104*x-9)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(1+2*x)/(-1+10*x)/(-1+4 *x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 9, 311, 10082, 278566, 8050207, 223600797, 6304746660, 176094257132, 4938603527825, 138187262822323, 3870735091651558, 108361493403810738, 3034407093300027363, 84959584543335243689, 2378923565247938194376, 66609105048071613924184, 1865065688288591926302421, 52221690614654279384941695 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [4, 2, 1, 1, 1]](x) = x ( 7 6 5 4 3 2 249312 x + 63896 x - 43656 x - 45482 x + 2344 x + 3117 x + 125 x - 21 )/((-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[4, 2, 1, 1, 1]](x) = x^3*(249312*x^7+63896*x^6-43656*x ^5-45482*x^4+2344*x^3+3117*x^2+125*x-21)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(1+2*x)/ (-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 21, 274, 10510, 239285, 7289345, 194109174, 5556842844, 153708055495, 4328390180719, 120833062553984, 3388211631946778, 94799724728737665, 2655357098694590493, 74336311521812329354, 2081606543729895904312, 58282307730399753695195, 1631941893358166522554667, 45693849337389098734677684 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (14112 x + 85008 x - 8008 x - 14740 x + 2214 x + 241 x - 14)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = x^3*(14112*x^6+85008*x^5-8008* x^4-14740*x^3+2214*x^2+241*x-14)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(1+2*x)/(-1+10*x )/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 14, 81, 3975, 68600, 2281860, 56748111, 1666349321, 45358053150, 1285999234806, 35762484862181, 1004565672686667, 28080618445965540, 786896372447559752, 22023931823176092891, 616796222460037844013, 17268506395527652831370, 483542988703385345488698, 13538854149124559639441841 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [3, 3, 3]](x) = - x 5 4 3 2 (3472 x - 540 x - 1708 x + 392 x + 3 x + 1)/((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[3, 3, 3]](x) = -x^3*(3472*x^5-540*x^4-1708*x^3+392*x^2 +3*x+1)/(1+x)/(-1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 42, 1735, 52956, 1534225, 43448538, 1221585499, 34254242232, 959619228469, 26874336019014, 752531415855583, 21071379606392388, 590003629097769433, 16520151614141048370, 462564745197864071587, 12951817865530691431824, 362650950234890122102717, 10154227106576771752407006 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [3, 3, 2, 1]](x) = x 6 5 4 3 2 (23520 x - 24640 x + 20708 x + 4632 x - 2063 x - 38 x - 6)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[3, 3, 2, 1]](x) = x^3*(23520*x^6-24640*x^5+20708*x^4+ 4632*x^3-2063*x^2-38*x-6)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(1+2*x)/(-1+10*x)/(-1+4 *x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 6, 176, 7965, 206121, 6321476, 172178622, 4919102355, 136657455467, 3844514666586, 107422788293568, 3011269598340185, 84270502417914813, 2360234745866745936, 66077623211797444514, 1850301746408596397055, 51806682742342243050159, 1450612143023295990521126, 40616792637093972760487460 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 F[[3, 1, 1, 1, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 3 4 3 2 x (896 x - 656 x - 432 x - 22 x - 11) ---------------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 2 x) (1 + 2 x) (1 + 4 x) (1 + 14 x) (-1 + 28 x) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[3, 3, 1, 1, 1]](x) = x^3*(896*x^4-656*x^3-432*x^2-22*x -11)/(1+x)/(-1+x)/(-1+2*x)/(1+2*x)/(1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 11, 132, 6735, 144500, 4670751, 122007732, 3539155295, 97374770100, 2750597418591, 76679266756532, 2151743950697055, 60182687766501300, 1686041257851880031, 47196191212605380532, 1321674850061571474015, 37004354856174677170100, 1036157509210696034239071, 29011912232569759802103732 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 8 7 F[[3, 1, 1, 1, 1, 1, 1], [3, 2, 2, 2]](x) = - x (301056 x + 379456 x 6 5 4 3 2 + 50608 x - 196760 x + 28164 x + 8930 x - 1600 x - 16 x - 3)/( (-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[3, 2, 2, 2]](x) = -x^3*(301056*x^8+379456*x^7+50608*x^ 6-196760*x^5+28164*x^4+8930*x^3-1600*x^2-16*x-3)/(-1+x)/(1+x)/(-1+3*x)/(-1+2*x) /(1+6*x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 3, 79, 4348, 98944, 3238375, 85148553, 2474507010, 68135316538, 1925135913877, 53672736156787, 1506192824997952, 42127604638283892, 1180226096850056259, 33037306106309497981, 925172117053881108574, 25903045622813992448206, 725310228662094730435921, 20308338285066756035047335 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [3, 2, 2, 1, 1]](x) = - 3 x 6 5 4 3 2 (4704 x - 10376 x + 2512 x + 504 x - 157 x - 28 x + 6)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[3, 2, 2, 1, 1]](x) = -3*x^3*(4704*x^6-10376*x^5+2512*x ^4+504*x^3-157*x^2-28*x+6)/(-1+x)/(1+x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(1+4*x)/(-1+4 *x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 18, 150, 9525, 188769, 6396918, 163411164, 4796193735, 131198391663, 3716917868928, 103466417124378, 2905562847403905, 81236708019478557, 2276294592361378698, 63712913570924532792, 1784288271789116649435, 49955497915268610353451, 1398817973412513997352628, 39166006810215349533140406 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 7 6 F[[3, 1, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = - x (172480 x - 261296 x 5 4 3 2 - 47528 x + 89132 x + 9646 x - 4636 x - 348 x + 25)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = -x^3*(172480*x^7-261296*x^6-\ 47528*x^5+89132*x^4+9646*x^3-4636*x^2-348*x+25)/(-1+x)/(1+x)/(-1+2*x)/(1+6*x)/( 1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 25, 102, 7625, 119780, 4342625, 104870892, 3139685665, 84769549260, 2414501547425, 67002853075532, 1884223515788705, 52641253742526540, 1475563133484203425, 41292955557053597772, 1156519124790846307745, 32378076333416761452620, 906648265537863206878625, 25385278640516996714817612 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 9 F[[3, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x (2393664 x 8 7 6 5 4 3 - 680720 x - 1136232 x + 267484 x + 117438 x - 19890 x - 3726 x 2 + 347 x + 21 x - 1)/((-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x*(2393664*x^9-680720*x^8-\ 1136232*x^7+267484*x^6+117438*x^5-19890*x^4-3726*x^3+347*x^2+21*x-1)/(-1+x)/(1+ x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28 *x) The first 20 term , starting with k=1 are 1, 0, 16, 25, 2661, 30766, 1237636, 27543411, 849694771, 22498481632, 646023245946, 17843936793397, 502854970483321, 14032800230856498, 393558306893213296, 11010474253799432983, 308419494251036602911, 8633958836601791139364, 241775665401191212905286, 6769369179909984209654169 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 1]](x) = - x 7 6 5 4 3 2 (9408 x + 43568 x + 4448 x - 13340 x - 116 x + 458 x - 15 x + 4)/( (-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[2, 2, 2, 2, 1]](x) = -x^3*(9408*x^7+43568*x^6+4448*x^5 -13340*x^4-116*x^3+458*x^2-15*x+4)/(-1+x)/(1+x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(1+2* x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 4, 29, 2669, 45732, 1705718, 41691951, 1252970023, 33880804994, 965518301792, 26798390788653, 753661465239137, 21056224700901936, 590222469738232426, 16517154480342249035, 462607371926974273211, 12951227756858989232958, 362659278429750479164820, 10154111178474731964055497 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = x 5 4 3 2 (6272 x - 552 x + 20 x - 1154 x + 285 x - 11)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 2 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = x^3*(6272*x^5-552*x^4+20*x^3-\ 1154*x^2+285*x-11)/(-1+x)/(1+x)/(-1+3*x)/(1+2*x)/(-1+10*x)/(-1+4*x)/(1+14*x)/(-\ 1+28*x) The first 20 term , starting with k=1 are 0, 0, 11, 34, 3625, 51604, 2021807, 47323878, 1443040421, 38633334088, 1105337723083, 30606991718842, 861670257688097, 24060124304744892, 674604481049945639, 18875911461226584526, 528706517024987369053, 14801236531135486677616, 414470146109630930679875, 11604665565739699158634530 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - 3 x 6 5 4 3 2 (8736 x - 53016 x + 6520 x + 5502 x - 650 x - 81 x + 4)/((1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -3*x^3*(8736*x^6-53016*x^5+ 6520*x^4+5502*x^3-650*x^2-81*x+4)/(1+x)/(-1+3*x)/(-1+2*x)/(1+6*x)/(-1+10*x)/(1+ 4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 12, 21, 2580, 28401, 1205406, 26314407, 822104904, 21649064997, 623523182370, 17197923036723, 485010978150348, 13529945593601073, 379525504685350854, 10616915958768217119, 297409019926423066512, 8325539342775639004029, 233141706562045841452458, 6527593514524054478017995 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 F[[3, 1, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (28560 x + 32192 x - 16322 x + 957 x + 169 x - 7)/((-1 + x) (1 + x) (-1 + 3 x) (1 + 6 x) (-1 + 10 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(28560*x^5+32192*x^4 -16322*x^3+957*x^2+169*x-7)/(-1+x)/(1+x)/(-1+3*x)/(1+6*x)/(-1+10*x)/(-1+4*x)/(1 +14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 7, 6, 1006, 8103, 385813, 7668444, 247909732, 6380385801, 185482492459, 5088000670302, 143840314656298, 4007265566141019, 112477091529267745, 3145427560428463680, 88126004657633623504, 2466761675281155969357, 69079957241329173364471, 1934088978318896324138178 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 3 8 F[[3, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x (28224 x 7 6 5 4 3 2 + 213744 x - 8424 x - 7236 x - 1346 x - 1238 x + 222 x + 20 x - 1)/( (-1 + x) (1 + x) (-1 + 3 x) (-1 + 2 x) (1 + 6 x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (1 + 14 x) (-1 + 28 x)) and in Maple notation F[[3, 1, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(28224*x^8+ 213744*x^7-8424*x^6-7236*x^5-1346*x^4-1238*x^3+222*x^2+20*x-1)/(-1+x)/(1+x)/(-1 +3*x)/(-1+2*x)/(1+6*x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(1+14*x)/(-1+28*x) The first 20 term , starting with k=1 are 0, 0, 1, 1, 162, 966, 52839, 937797, 31684780, 792050932, 23303478297, 634765383483, 18001474302318, 500649880705338, 14063668875584875, 393126161539810609, 11016524194709975376, 308334795642510837984, 8635144613735819566173, 241759064541626805651975 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (8992 x - 6888 x - 22380 x + 1422 x + 5680 x - 635 x - 268 x + 38 x - 1) /((1 + x) (-1 + 3 x) (-1 + x) (1 + 2 x) (-1 + 10 x) (1 + 4 x) (-1 + 4 x) (-1 + 28 x)) and in Maple notation (8992*x^8-6888*x^7-22380*x^6+1422*x^5+5680*x^4-635*x^3-268*x^2+38*x-1)/(1+x)/(-\ 1+3*x)/(-1+x)/(1+2*x)/(-1+10*x)/(1+4*x)/(-1+4*x)/(-1+28*x) The first 20 term , starting with k=1 are 1, 12, 197, 4820, 127819, 3514858, 97771581, 2731261776, 76411831343, 2138897825894, 59882803254385, 1676655156167212, 46945710999643587, 1314473574633253410, 36805196755770355109, 1030544875827884675528, 28855250189837469312151, 807946941982110384731806, 22622513742165598576579353, 633430378447303341927449124 Regarding Lambda=, [2, 2, 2, 2, 1] 10 9 8 7 F[[2, 2, 2, 2, 1], [9]](x) = (284844 x - 125000 x - 550143 x + 210142 x 6 5 4 3 2 + 172149 x - 69403 x - 10367 x + 5702 x - 297 x - 38 x + 1)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[9]](x) = (284844*x^10-125000*x^9-550143*x^8+210142*x^7+ 172149*x^6-69403*x^5-10367*x^4+5702*x^3-297*x^2-38*x+1)/(-1+x)/(1+x)/(1+3*x)/(-\ 1+2*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 0, 18, 315, 15954, 625158, 26831267, 1118638535, 47097209772, 1976473425696, 83034367580781, 3487128388700385, 146463801357320630, 6151417920671391914, 258360416917683288135, 10851125410695756950715, 455747436644956438463328, 19141389967535166023862612, 803938411838142761417758529 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [8, 1]](x) = - x 7 6 5 4 3 2 (22344 x + 14708 x - 28122 x - 1045 x + 3522 x - 303 x - 35 x + 1)/( (-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[8, 1]](x) = -x^2*(22344*x^7+14708*x^6-28122*x^5-1045*x^4+ 3522*x^3-303*x^2-35*x+1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+6*x )/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 1, 103, 2600, 125752, 5020985, 214348597, 8953164410, 376720015978, 15812589351779, 664263682560391, 27897184537781780, 1171708205765809804, 49211374230089312333, 2066882903196070496185, 86809009335365772895310, 3645979408461042349937230, 153131120926053381979381847, 6431507278104282517238903179 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [7, 2]](x) = - x 6 5 4 3 2 (12348 x - 7500 x - 1019 x + 1911 x - 173 x - 33 x + 1)/((-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x) ) and in Maple notation F[[2, 2, 2, 2, 1],[7, 2]](x) = -x^2*(12348*x^6-7500*x^5-1019*x^4+1911*x^3-173*x ^2-33*x+1)/(-1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 3, 302, 8986, 420009, 16998530, 722644567, 30227617752, 1271278965797, 53369595677662, 2241860385467187, 94153411118480948, 3954509406420182065, 166088469048398422074, 6975728663916859104287, 292980422387654324570224, 12305180281222592458118013, 516817536238084345374036566, 21706337020024712353253905867 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 8 7 6 5 F[[2, 2, 2, 2, 1], [7, 1, 1]](x) = x (5880 x - 6524 x + 19998 x - 7421 x 4 3 2 - 5815 x + 2633 x - 107 x - 35 x + 1)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[7, 1, 1]](x) = x^2*(5880*x^8-6524*x^7+19998*x^6-7421*x^5-\ 5815*x^4+2633*x^3-107*x^2-35*x+1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x )/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 3, 305, 9377, 434268, 17643444, 749175995, 31350313739, 1318318531730, 55346871076970, 2324883495185505, 97640696935852161, 4100971002687724712, 172239917833796338256, 7234088648689188989735, 303831553848149936584343, 12760927633168940370816414, 535958927391391567428558102, 22510275415261995550264411085 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[2, 2, 2, 2, 1], [6, 3]](x) = 2 4 3 2 x (672 x - 208 x - 236 x - 24 x + 1) - -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x) and in Maple notation F[[2, 2, 2, 2, 1],[6, 3]](x) = -x^2*(672*x^4-208*x^3-236*x^2-24*x+1)/(-1+x)/(1+ 3*x)/(1+2*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 6, 499, 16284, 741393, 30287794, 1283716239, 53751600888, 2259859887805, 94881957736302, 3985492067787099, 167384366870379412, 7030231595168226537, 295268492306130794730, 12401293962060389967079, 520854104410874729865456, 21875875773179075037490389, 918786735042506882373290278, 38589043535818879468399408179 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 8 7 6 5 F[[2, 2, 2, 2, 1], [6, 2, 1]](x) = x (40572 x + 60144 x - 62619 x + 5524 x 4 3 2 + 7307 x - 1914 x + 209 x - 24 x + 1)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[6, 2, 1]](x) = x^2*(40572*x^8+60144*x^7-62619*x^6+5524*x^5+ 7307*x^4-1914*x^3+209*x^2-24*x+1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x )/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 14, 1039, 36000, 1614656, 66342584, 2806826674, 117599433350, 4943191636036, 207557793114804, 8718214656622934, 366153991348104050, 15378621967573425016, 645899961955299643424, 27127828651385917793194, 1139368379866751175436350, 47853477883273343998815596, 2009845988093239726931546444, 84413532661975057296611931454 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [6, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (15288 x + 18200 x - 18902 x - 3522 x + 2582 x - 88 x - 29 x + 1)/( (-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[6, 1, 1, 1]](x) = -x^2*(15288*x^7+18200*x^6-18902*x^5-3522*x ^4+2582*x^3-88*x^2-29*x+1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+6 *x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 7, 534, 19396, 857401, 35429611, 1496279940, 62729193842, 2636234545191, 110699361952465, 4649688221314786, 195282496089022888, 8201926571048412021, 344480051728497451319, 14468174272407849331872, 607663150045180688598334, 25521854673449312565952291, 1071917863083197678534306173, 45020550714318035067182359998 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [5, 4]](x) = x 7 6 5 4 3 2 (53508 x + 43804 x - 41085 x + 2125 x + 2091 x - 143 x - 31 x + 1)/( (-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[5, 4]](x) = x^2*(53508*x^7+43804*x^6-41085*x^5+2125*x^4+2091 *x^3-143*x^2-31*x+1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-\ 1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 7, 421, 14435, 646020, 26538148, 1122736917, 47039813485, 1977276893614, 83023118718254, 3487285871511663, 146461596592941175, 6151448787352548408, 258359984784066613000, 10851131460566068187209, 455747351946770792847905, 19141391153309759908146402, 803938395237298426422685186, 33765413064790038138904268355 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [5, 3, 1]](x) = x 6 5 4 3 2 (6804 x - 11430 x - 3390 x + 2780 x - 311 x + 13 x - 1)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[5, 3, 1]](x) = x^2*(6804*x^6-11430*x^5-3390*x^4+2780*x^3-311 *x^2+13*x-1)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 24, 1528, 56463, 2477569, 102544458, 4327892668, 181475995641, 7626121862767, 320239254268032, 13450858512461338, 564924718259704959, 23726996906953711645, 996531647673191779446, 41854360315787868664888, 1757882697671790261063717, 73831079400480736348105003, 3100905249444397266581229900, 130238021671925340009514132918 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [5, 2, 2]](x) = - x 5 4 3 2 (2684 x - 298 x - 756 x + x + 10 x - 1)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[5, 2, 2]](x) = -x^2*(2684*x^5-298*x^4-756*x^3+x^2+10*x-1)/(-\ 1+x)/(1+x)/(1+3*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 18, 1105, 42140, 1830091, 76027298, 3204863285, 134440282680, 5648787582031, 237216939052478, 9963561392228665, 418463279150133620, 17575545914839027571, 738171693755815268058, 31003228423088219621245, 1302135351774889922606960, 54689688162472791367114711, 2296966855392873436333434038, 96472608590534457544777825025 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [5, 2, 1, 1]](x) = - x 7 6 5 4 3 2 (12348 x - 1596 x - 3599 x - 1705 x - 108 x + 198 x - 4 x + 1)/( (-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[5, 2, 1, 1]](x) = -x^2*(12348*x^7-1596*x^6-3599*x^5-1705*x^4 -108*x^3+198*x^2-4*x+1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(-1+6*x) /(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 31, 1686, 66864, 2874026, 119849807, 5046103315, 211764853828, 8896538444436, 373620893170833, 15692550111697229, 659080491298298342, 27681473239967024206, 1162620579710124453859, 48830082497630236348503, 2050863210807077232806856, 86136258411228009704864936, 3617722803469085304550364885, 151944358442937296007183678337 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[2, 2, 2, 2, 1], [5, 1, 1, 1, 1]](x) = 3 2 x (147 x - 62 x + 13) ------------------------------------------------------ (-1 + 2 x) (1 + 3 x) (-1 + 3 x) (-1 + 6 x) (-1 + 42 x) and in Maple notation F[[2, 2, 2, 2, 1],[5, 1, 1, 1, 1]](x) = x^3*(147*x^2-62*x+13)/(-1+2*x)/(1+3*x)/ (-1+3*x)/(-1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 13, 588, 25140, 1058370, 44468508, 1867778178, 78447307980, 3294790675890, 138381231011628, 5812011838186818, 244104498019675020, 10252388921720937810, 430600334741661728748, 18085214059326083340258, 759578990492753378365260, 31902317600701989130598130, 1339897339229521628124989868, 56275688247640136888828538498 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 8 7 6 F[[2, 2, 2, 2, 1], [4, 4, 1]](x) = - x (24696 x - 29820 x - 16170 x 5 4 3 2 + 11335 x + 2771 x - 1496 x + 51 x + 24 x - 1)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[4, 4, 1]](x) = -x^2*(24696*x^8-29820*x^7-16170*x^6+11335*x^5 +2771*x^4-1496*x^3+51*x^2+24*x-1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x )/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 14, 779, 29530, 1281238, 53220181, 2243410765, 94108237580, 3954151548860, 166051858804843, 6974492983443811, 292924295458754050, 12302882140710693442, 516720185631017087825, 21702259896173456584217, 911494746242493265282840, 38282781713731376281686184, 1607876798775013941054466327, 67530826013374135501749028783 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [4, 3, 2]](x) = x 6 5 4 3 2 (1176 x + 4676 x - 986 x - 565 x + 5 x - 2 x + 1)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[4, 3, 2]](x) = x^2*(1176*x^6+4676*x^5-986*x^4-565*x^3+5*x^2-\ 2*x+1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 31, 1497, 59782, 2551838, 106585787, 4484765447, 188245132624, 7907901116520, 332109340556653, 13948907216515217, 585849693245770226, 24605748847756197122, 1033440587326891755679, 43404516767400035731707, 1822989534834008963396788, 76565562834575033000518844, 3215753605850447512849719665, 135061651910542345474691997317 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [4, 3, 1, 1]](x) = - x 6 5 4 3 2 (8568 x + 6504 x + 1738 x - 150 x - 46 x - 8 x - 1)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[4, 3, 1, 1]](x) = -x^2*(8568*x^6+6504*x^5+1738*x^4-150*x^3-\ 46*x^2-8*x-1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 41, 1868, 77430, 3271539, 137161729, 5764354282, 242053957340, 10166956511417, 427002541038447, 17934241764932736, 753236264661730330, 31635949574932727935, 1328709511758784905245, 55805804679543024397430, 2343843723942786941523000, 98441437421977820342521893, 4134540357493788569640986923, 173650695213949343381326303564 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [4, 2, 2, 1]](x) = x 6 5 4 3 2 (3528 x + 516 x + 1322 x + 283 x - 127 x + 12 x + 1)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[4, 2, 2, 1]](x) = x^2*(3528*x^6+516*x^5+1322*x^4+283*x^3-127 *x^2+12*x+1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 45, 1827, 78086, 3262646, 137287489, 5762598493, 242078558172, 10166612178156, 427007362019183, 17934174272459699, 753237209561388058, 31635936346357648306, 1328709696958916558877, 55805802086741503352745, 2343843760242009524671544, 98441436913788709332337496, 4134540364608436144399542571, 173650695114344277417169634431 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [4, 2, 1, 1, 1]](x) = x 6 5 4 3 2 (19404 x + 2352 x - 6391 x + 136 x + 1201 x - 55 x - 42)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[4, 2, 1, 1, 1]](x) = x^3*(19404*x^6+2352*x^5-6391*x^4+136*x^ 3+1201*x^2-55*x-42)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(-1+6*x)/(1+ 14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 42, 1525, 69100, 2842637, 120288959, 5039953926, 211850940420, 8895333212359, 373637766341521, 15692313886992512, 659083798442907110, 27681426939937466241, 1162621227910518136803, 48830073422824644248458, 2050863337854355200101320, 86136256632566116874219483, 3617722828370351799025506005, 151944358094319565063915728564 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [4, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (15288 x - 2212 x - 3674 x - 1227 x + 503 x - 82 x + 14)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[4, 1, 1, 1, 1, 1]](x) = -x^3*(15288*x^6-2212*x^5-3674*x^4-\ 1227*x^3+503*x^2-82*x+14)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+6* x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 14, 422, 20875, 836413, 35722162, 1492179416, 62786581305, 2635431042491, 110710610674840, 4649530737944950, 195284700851164015, 8201895704358310409, 344480483862078329598, 14468168222537394940724, 607663234743365761529605, 25521853487674716391062967, 1071917879684042004366695236, 45020550481906214423065977938 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[2, 2, 2, 2, 1], [3, 3, 3]](x) = 2 3 2 x (231 x - 229 x + 40 x - 1) - ---------------------------------------------------- (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 6 x) (-1 + 42 x) and in Maple notation F[[2, 2, 2, 2, 1],[3, 3, 3]](x) = -x^2*(231*x^3-229*x^2+40*x-1)/(-1+x)/(1+3*x)/ (-1+2*x)/(-1+6*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 8, 368, 15131, 635444, 26683265, 1120681178, 47068467167, 1976874911792, 83028741615677, 3487207121072918, 146462698920511883, 6151433353685784620, 258360200848914052169, 10851128435619160932338, 455747394295793071762679, 19141390560422039826417128, 803938403537718054776011541, 33765412948584112601419799438 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 F[[2, 2, 2, 2, 1], [3, 3, 2, 1]](x) = x 6 5 4 3 2 (1176 x + 896 x + 4522 x + 524 x - 360 x + 6 x + 1)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[3, 3, 2, 1]](x) = x^2*(1176*x^6+896*x^5+4522*x^4+524*x^3-360 *x^2+6*x+1)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 39, 1396, 61290, 2530997, 106878863, 4480667150, 188302528740, 7907097648943, 332120589418497, 13948749733705304, 585851898010146950, 24605717981075046089, 1033441019460508419891, 43404510717529724517058, 1822989619532194608968520, 76565561648800439116322435, 3215753622451291847844618245, 135061651678130524867226177612 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [3, 3, 1, 1, 1]](x) = x 5 4 3 2 (3024 x - 488 x - 1226 x + 161 x + 140 x + 29)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[3, 3, 1, 1, 1]](x) = x^3*(3024*x^5-488*x^4-1226*x^3+161*x^2+ 140*x+29)/(-1+x)/(1+x)/(1+3*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 29, 952, 44275, 1800210, 76445609, 3199006972, 134522270975, 5647639746070, 237233008755589, 9963336416385792, 418466428811932475, 17575501819573846330, 738172311089527799969, 31003219780416244185412, 1302135472772297578686775, 54689686468509084182040990, 2296966879108365336924378749, 96472608258517570936504773832 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 2 8 7 6 F[[2, 2, 2, 2, 1], [3, 2, 2, 2]](x) = - x (88200 x - 109956 x - 32694 x 5 4 3 2 + 44701 x - 155 x - 4133 x + 493 x + 15 x - 1)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[3, 2, 2, 2]](x) = -x^2*(88200*x^8-109956*x^7-32694*x^6+44701 *x^5-155*x^4-4133*x^3+493*x^2+15*x-1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1 +4*x)/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 1, 23, 679, 31049, 1260426, 53513404, 2239312993, 94165635923, 3953348089936, 166063107701810, 6974335500773367, 292926500223690697, 12302851274031778006, 516720617764642702616, 21702253846303181155501, 911494830940679054031071, 38282780527956782970130236, 1607876815375858278340057822, 67530825780962314903445718595 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [3, 2, 2, 1, 1]](x) = 3 x 5 4 3 2 (1764 x + 1914 x - 3548 x + 365 x + 135 x - 15)/((1 + x) (-1 + x) (1 + 3 x) (-1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[3, 2, 2, 1, 1]](x) = 3*x^3*(1764*x^5+1914*x^4-3548*x^3+365*x ^2+135*x-15)/(1+x)/(-1+x)/(1+3*x)/(-1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 45, 1260, 60330, 2423889, 103297821, 4317352956, 181623581100, 7624055784303, 320268179838387, 13450453556363802, 564930387652620210, 23726917535483097117, 996532758873901178793, 41854344758978420257848, 1757882915467124471497560, 73831076351346065132972331, 3100905292132282694516851839, 130238021074294944142110483894 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [3, 2, 1, 1, 1, 1]](x) = - x ( 7 6 5 4 3 2 118188 x - 38220 x - 52779 x + 24971 x - 2088 x - 1250 x + 409 x - 31 )/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[3, 2, 1, 1, 1, 1]](x) = -x^3*(118188*x^7-38220*x^6-52779*x^5 +24971*x^4-2088*x^3-1250*x^2+409*x-31)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-\ 1+4*x)/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 31, 769, 39710, 1562290, 67074266, 2796576804, 117742907320, 4941182901430, 207585915007526, 8717820948549064, 366159503254852580, 15378544800853768170, 645901042289274199186, 27127813526709871312924, 1139368591612214215640240, 47853474918836854993324510, 2009846029595350547238989246, 84413532080945505709227774384 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [3, 1, 1, 1, 1, 1, 1]](x) = x 7 6 5 4 3 2 (69384 x - 68516 x + 4554 x + 11797 x - 4415 x + 867 x - 150 x + 9)/( (-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[3, 1, 1, 1, 1, 1, 1]](x) = x^3*(69384*x^7-68516*x^6+4554*x^5 +11797*x^4-4415*x^3+867*x^2-150*x+9)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+ 4*x)/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 9, 192, 10845, 413251, 17935848, 745074946, 31407698975, 1317515020377, 55358119764222, 2324726011676200, 97642901697433365, 4100940135995387503, 172240349967368265956, 7234082598818698812654, 303831638546334866339115, 12760926447394343623286629, 535958943992235890970254250, 22510275182850174896985519508 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 8 7 6 F[[2, 2, 2, 2, 1], [2, 2, 2, 2, 1]](x) = - x (223188 x - 62488 x - 131121 x 5 4 3 2 + 46418 x + 11467 x - 5371 x + 280 x + 38 x - 1)/((-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[2, 2, 2, 2, 1]](x) = -x*(223188*x^8-62488*x^7-131121*x^6+ 46418*x^5+11467*x^4-5371*x^3+280*x^2+38*x-1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2 *x)/(-1+4*x)/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 1, 0, 18, 315, 15954, 625158, 26831267, 1118638535, 47097209772, 1976473425696, 83034367580781, 3487128388700385, 146463801357320630, 6151417920671391914, 258360416917683288135, 10851125410695756950715, 455747436644956438463328, 19141389967535166023862612, 803938411838142761417758529, 33765412832378217531438099125 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 F[[2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1]](x) = 3 3 2 x (2352 x + 1396 x + 164 x - 17) -------------------------------------------------------------- (-1 + x) (1 + 3 x) (1 + 2 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x) and in Maple notation F[[2, 2, 2, 2, 1],[2, 2, 2, 1, 1, 1]](x) = x^3*(2352*x^3+1396*x^2+164*x-17)/(-1 +x)/(1+3*x)/(1+2*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 17, 346, 18419, 711512, 30706105, 1277859926, 53833589183, 2258712051844, 94898027439413, 3985267091944226, 167387516532178267, 7030187499903045296, 295269109639843326641, 12401285319388414531246, 520854225408282385945271, 21875874079215367852416668, 918786758757998782964234989, 38589043203801992860126356986 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - x 5 4 3 2 (4788 x + 1248 x - 677 x + 344 x - 178 x + 10)/((-1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -x^3*(4788*x^5+1248*x^4-677*x^3+ 344*x^2-178*x+10)/(-1+x)/(1+3*x)/(-1+2*x)/(1+2*x)/(-1+4*x)/(-1+6*x)/(1+14*x)/(-\ 1+42*x) The first 20 term , starting with k=1 are 0, 0, 10, 182, 10566, 397513, 17311922, 718250967, 30289103512, 1270418066981, 53381647867614, 2241691653235507, 94155773363431988, 3954476334965703729, 166088932048660451386, 6975722181912788048927, 292980513135709708716144, 12305179010749810637656957, 516817554024703265090622038, 21706336771012047374142625227 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = x 6 5 4 3 2 (13944 x + 26008 x - 5250 x - 1670 x + 89 x + 93 x - 4)/((-1 + x) (1 + x) (1 + 3 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(13944*x^6+26008*x^5-5250* x^4-1670*x^3+89*x^2+93*x-4)/(-1+x)/(1+x)/(1+3*x)/(1+2*x)/(-1+4*x)/(-1+3*x)/(-1+ 6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 4, 51, 3227, 116712, 5146220, 212590581, 8977756589, 376375647594, 15817410193046, 664196189527431, 27898129435203911, 1171694977181779596, 49211559430185180032, 2066880310394406275001, 86809045634587783403393, 3645978900271929049059918, 153131128040700947575427978, 6431507178499216516431671691 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 3 F[[2, 2, 2, 2, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 7 6 5 4 3 2 (53508 x + 43804 x - 41085 x + 2125 x + 2091 x - 143 x - 31 x + 1)/( (-1 + x) (1 + x) (1 + 3 x) (-1 + 2 x) (1 + 2 x) (-1 + 4 x) (-1 + 3 x) (-1 + 6 x) (1 + 14 x) (-1 + 42 x)) and in Maple notation F[[2, 2, 2, 2, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(53508*x^7+43804*x^6-\ 41085*x^5+2125*x^4+2091*x^3-143*x^2-31*x+1)/(-1+x)/(1+x)/(1+3*x)/(-1+2*x)/(1+2* x)/(-1+4*x)/(-1+3*x)/(-1+6*x)/(1+14*x)/(-1+42*x) The first 20 term , starting with k=1 are 0, 0, 1, 7, 421, 14435, 646020, 26538148, 1122736917, 47039813485, 1977276893614, 83023118718254, 3487285871511663, 146461596592941175, 6151448787352548408, 258359984784066613000, 10851131460566068187209, 455747351946770792847905, 19141391153309759908146402, 803938395237298426422685186 ---------------------------------- Their sum is 6 5 4 3 2 1992 x - 3995 x + 1854 x + 374 x - 356 x + 50 x - 1 --------------------------------------------------------------- (-1 + 42 x) (-1 + 6 x) (-1 + 3 x) (-1 + 2 x) (1 + 3 x) (-1 + x) and in Maple notation (1992*x^6-3995*x^5+1854*x^4+374*x^3-356*x^2+50*x-1)/(-1+42*x)/(-1+6*x)/(-1+3*x) /(-1+2*x)/(1+3*x)/(-1+x) The first 20 term , starting with k=1 are 1, 18, 548, 22563, 944020, 39633657, 1664513242, 69909011127, 2936175109744, 123319334969301, 5179411950068566, 217535301195545571, 9136482645961623148, 383732271104920963425, 16116755386253811384370, 676903726221743227980495, 28429956501307713880737832, 1194058173054890976101107629, 50150443268305222949561954254, 2106318617268818175630977477499 Regarding Lambda=, [2, 2, 2, 1, 1, 1] 10 9 8 F[[2, 2, 2, 1, 1, 1], [9]](x) = (1327232 x - 472832 x - 2377576 x 7 6 5 4 3 2 + 826332 x + 665088 x - 240690 x - 25410 x + 12244 x - 578 x - 41 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[9]](x) = (1327232*x^10-472832*x^9-2377576*x^8+826332*x^7+ 665088*x^6-240690*x^5-25410*x^4+12244*x^3-578*x^2-41*x+1)/(-1+x)/(1+x)/(-1+2*x) /(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 0, 39, 420, 40375, 1493088, 80212779, 3676768076, 179933089839, 8567662820376, 412629038421139, 19778559189343812, 949923452860368663, 45585272854059418064, 2188314149488137957819, 105034658084281775837628, 5041752009538675771941247, 242002328025357818075455752, 11616147113846079303686066019 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 8 7 6 F[[2, 2, 2, 1, 1, 1], [8, 1]](x) = - x (94208 x - 178368 x - 145296 x 5 4 3 2 + 77296 x + 10236 x - 6510 x + 344 x + 41 x - 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[8, 1]](x) = -x^2*(94208*x^8-178368*x^7-145296*x^6+77296*x ^5+10236*x^4-6510*x^3+344*x^2+41*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-\ 1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 0, 235, 3860, 308959, 12189784, 636548667, 29515163116, 1437428388727, 68581901266208, 3300219319219219, 158244725175526132, 7599062525077964175, 364688684272543924072, 17506383162982249376491, 840279865299769183907708, 40333964063535790974417703, 1936019664456229477950742576, 92929156105684460409421942083 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [7, 2]](x) = - x 7 6 5 4 3 2 (63360 x + 37696 x - 107816 x + 3188 x + 9294 x - 713 x - 76 x + 2)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[7, 2]](x) = -x^2*(63360*x^7+37696*x^6-107816*x^5+3188*x^4 +9294*x^3-713*x^2-76*x+2)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(8*x-1 )/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 0, 673, 14456, 1007409, 41792200, 2134897937, 99879400392, 4845979975441, 231570524014280, 11136106395599121, 534118610365050568, 25645982658737254673, 1230841375846574948040, 59083701839815597822225, 2835951372038079615696584, 136126992180977617557983505, 6534069098205463156810350280, 313635847243343900965123199249 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [7, 1, 1]](x) = - x 7 6 5 4 3 2 (51712 x - 93312 x - 15744 x + 12264 x + 324 x - 6 x - 39 x + 1)/( (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[7, 1, 1]](x) = -x^2*(51712*x^7-93312*x^6-15744*x^5+12264* x^4+324*x^3-6*x^2-39*x+1)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4*x)/(8*x -1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 1, 653, 15397, 1033889, 43530865, 2209960005, 103657202013, 5023876823177, 240178786051369, 11547922448925677, 553913421233397589, 26595581013901752945, 1276433150141356774113, 61271885956384802133269, 2940988630747899885150925, 141168692177742819205989593, 6776072466484188754630390297, 325251973552105811326934856381 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [6, 3]](x) = x 7 6 5 4 3 2 (128 x + 52480 x - 35560 x - 13812 x + 6930 x - 219 x - 79 x + 2)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[6, 3]](x) = x^2*(128*x^7+52480*x^6-35560*x^5-13812*x^4+ 6930*x^3-219*x^2-79*x+2)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4*x )/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 3, 1062, 27639, 1747170, 75132923, 3778357198, 177901336383, 8608297770138, 411816339891363, 19794813158063814, 949598373493422887, 45591774441368461906, 2188184117742076376523, 105037258719202530062910, 5041699996840262596273551, 242003368279326073952584074, 11616126308766714216685343603, 557574770193452585808869362486 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [6, 2, 1]](x) = 2 x 7 6 5 4 3 2 (168000 x + 64032 x - 49932 x - 2550 x + 1750 x - 280 x + 36 x - 1)/( (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[6, 2, 1]](x) = 2*x^2*(168000*x^7+64032*x^6-49932*x^5-2550 *x^4+1750*x^3-280*x^2+36*x-1)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4*x)/ (8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 8, 2118, 62840, 3762822, 165437748, 8242714538, 389601886640, 18821748515742, 901025909888588, 43297597334214258, 2077317545983210440, 99730584311063711462, 4786681201547550325028, 229768434555703907798778, 11028730120836547164326240, 529382140555235754545495982, 25410280851536422498460851068, 1219694718775940300372241348098 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [6, 1, 1, 1]](x) = - x 6 5 4 3 2 (157696 x - 94368 x + 4056 x + 1568 x + 171 x - 34 x + 1)/((-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[6, 1, 1, 1]](x) = -x^2*(157696*x^6-94368*x^5+4056*x^4+ 1568*x^3+171*x^2-34*x+1)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x +1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 6, 1028, 34802, 1975400, 88812234, 4384147328, 208023797218, 10033517743304, 480641907690362, 23090155140783248, 1107940613318671314, 53188886416943990168, 2552911810952068358890, 122542861687017160395488, 5881995465912026308625090, 282337020266371145898060392, 13552152214744351310097539418, 650503801468641640337877747248 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [5, 4]](x) = x 6 5 4 3 2 (32128 x + 8544 x - 7392 x - 632 x - 97 x - 30 x + 1)/((1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[5, 4]](x) = x^2*(32128*x^6+8544*x^5-7392*x^4-632*x^3-97*x ^2-30*x+1)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 4, 862, 25240, 1506118, 66182900, 3297151530, 155841283504, 7528703720350, 360410398702156, 17319039214104178, 830927020647076808, 39892233742535184102, 1914672480764306503972, 91907373823446574419706, 4411492048343954201856352, 211752856222169080269356974, 10164112340615167784365153148, 487877887510380913841627264514 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 F[[2, 2, 2, 1, 1, 1], [5, 3, 1]](x) = 2 3 2 3 x (392 x - 36 x - 28 x + 1) - ------------------------------------------------- (-1 + x) (1 + x) (8 x - 1) (20 x + 1) (-1 + 48 x) and in Maple notation F[[2, 2, 2, 1, 1, 1],[5, 3, 1]](x) = -3*x^2*(392*x^3-36*x^2-28*x+1)/(-1+x)/(1+x )/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 3, 24, 2967, 102528, 5690295, 257525760, 12671569335, 602013990912, 29020980866487, 1390519956013056, 66794692884000183, 3205150498394996736, 153867118642663746999, 7385223796605491085312, 354498700171124621929911, 17015778449286470251315200, 816760548742330479252565431, 39204442676084502261638627328, 1881814521722877222618778004919 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [5, 2, 2]](x) = - x 6 5 4 3 2 (4160 x - 6688 x - 2516 x + 284 x + 499 x + 33 x - 2)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[5, 2, 2]](x) = -x^2*(4160*x^6-6688*x^5-2516*x^4+284*x^3+ 499*x^2+33*x-2)/(-1+x)/(1+x)/(-1+2*x)/(-1+3*x)/(1+2*x)/(1+4*x)/(20*x+1)/(-1+48* x) The first 20 term , starting with k=1 are 0, 2, 21, 2078, 77905, 4171562, 191596501, 9369339798, 446274289305, 21490245421922, 1030150192144381, 49474840971832718, 2374239731714776705, 113974559820769075482, 5470557817428463802661, 262591196315932105240838, 12604289001577436326480105, 605007640507462987716444242, 29040331375723302725515163341, 1393936613407416943720692124158 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 F[[2, 2, 2, 1, 1, 1], [5, 2, 1, 1]](x) = 2 5 4 3 2 x (50112 x - 11744 x - 4780 x + 956 x + 1) ------------------------------------------------------------------------ (-1 + x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x) and in Maple notation F[[2, 2, 2, 1, 1, 1],[5, 2, 1, 1]](x) = x^2*(50112*x^5-11744*x^4-4780*x^3+956*x ^2+1)/(-1+x)/(1+2*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 37, 3053, 125713, 6500577, 303075409, 14729855953, 703413850897, 33836457785873, 1622699789176081, 77918607085071633, 3739512904541360401, 179508225000968053009, 8616162695382002913553, 413580451527712779653393, 19851768830792064753406225, 952886760732381544939917585, 45738527378095886353997566225, 2195450056890001824332903158033 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 F[[2, 2, 2, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 3 2 x (328 x - 152 x + 17) - ------------------------------------------------------ (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (8 x - 1) (-1 + 48 x) and in Maple notation F[[2, 2, 2, 1, 1, 1],[5, 1, 1, 1, 1]](x) = -x^3*(328*x^2-152*x+17)/(-1+3*x)/(-1 +2*x)/(-1+6*x)/(8*x-1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 17, 987, 48845, 2356775, 113225637, 5435650927, 260917911565, 12524113729575, 601157894685557, 28855582453596767, 1385067985982784285, 66483263553699858775, 3191196652394877750277, 153177439329522870101007, 7352517087933830479785005, 352920820221758821728650375, 16940199370651909694799547797, 813129569791351594859438483647 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 8 7 6 F[[2, 2, 2, 1, 1, 1], [4, 4, 1]](x) = x (158208 x + 176640 x - 223232 x 5 4 3 2 + 12656 x + 16620 x - 2876 x + 187 x - 24 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[4, 4, 1]](x) = x^2*(158208*x^8+176640*x^7-223232*x^6+ 12656*x^5+16620*x^4-2876*x^3+187*x^2-24*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+ 3*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 17, 1463, 54665, 2920987, 134125789, 6558602023, 312392538209, 15043176084083, 721105169355461, 34632388960290463, 1661967814455840073, 79782191892641419819, 3829390472345238116333, 183813837422317378306583, 8823002301113541196302257, 423505348355298868141117795, 20328231963006910699732710805, 975755629385196654269785447183 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [4, 3, 2]](x) = - 2 x 7 6 5 4 3 2 (55296 x + 99616 x - 24952 x - 8284 x + 2282 x - 74 x - 20 x + 1)/( (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[4, 3, 2]](x) = -2*x^2*(55296*x^7+99616*x^6-24952*x^5-8284 *x^4+2282*x^3-74*x^2-20*x+1)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4*x)/( 8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 2, 40, 2690, 113592, 5751990, 270019764, 13081577322, 625495607728, 30072125406878, 1442494746762828, 69259088742658674, 3324049404379182024, 159562108211298582246, 7658826455654309591332, 367626764621235890884346, 17646022806662120463485280, 847010332621633647754264494, 40656471207790997845168708476, 1951511113134832944089128039938 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [4, 3, 1, 1]](x) = x 6 5 4 3 2 (18432 x - 7072 x - 3736 x - 3052 x + 339 x + 23 x + 1)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[4, 3, 1, 1]](x) = x^2*(18432*x^6-7072*x^5-3736*x^4-3052*x ^3+339*x^2+23*x+1)/(-1+x)/(1+x)/(1+2*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+ 48*x) The first 20 term , starting with k=1 are 0, 1, 59, 3228, 149615, 7316840, 348673967, 16788541048, 804816940271, 38651960793144, 1854879832552175, 89042522976920888, 4273875324268359407, 205149331468243972408, 9847101595032246726383, 472662202891302547931448, 22687759212353742667575023, 1089012972722881701090591032, 52272612080110865709061566191, 2509085592057155203382663151928 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 2 F[[2, 2, 2, 1, 1, 1], [4, 2, 2, 1]](x) = - x 7 6 5 4 3 2 (36864 x + 129344 x - 5104 x - 19928 x + 2064 x - 154 x + 28 x + 1)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[4, 2, 2, 1]](x) = -x^2*(36864*x^7+129344*x^6-5104*x^5-\ 19928*x^4+2064*x^3-154*x^2+28*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+4* x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 66, 3047, 153382, 7240815, 350197094, 16758067567, 805426453158, 38639770360559, 1855123641901222, 89037646787143407, 4273972848075087014, 205147380992064679663, 9847140604556011510950, 472661422700826536406767, 22687774816163265761270950, 1089012660646691227763404527, 52272618321634675221417843878, 2509085467226679012952285572847 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [4, 2, 1, 1, 1]](x) = x 5 4 3 2 (26496 x + 4736 x + 3656 x - 572 x + 298 x - 69)/((-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[4, 2, 1, 1, 1]](x) = x^3*(26496*x^5+4736*x^4+3656*x^3-572 *x^2+298*x-69)/(-1+x)/(-1+2*x)/(1+2*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48 *x) The first 20 term , starting with k=1 are 0, 0, 69, 2393, 139025, 6234001, 308408401, 14623190673, 705547178769, 33793791141137, 1623553122422033, 77901540418754833, 3739854237873295633, 179501398334306980113, 8616299228715313877265, 413577720861046202470673, 19851823444125397728825617, 952885668465714879704928529, 45738549223429219681604276497, 2195449619983335157689143070993 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = - 2 x 6 5 4 3 2 (47104 x - 5024 x - 15384 x + 10024 x - 2412 x + 250 x - 13)/((-1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -2*x^3*(47104*x^6-5024*x^5-15384* x^4+10024*x^3-2412*x^2+250*x-13)/(-1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4 *x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 26, 592, 43650, 1797784, 92367114, 4313038912, 209446008546, 10005073342408, 481210796404602, 23078777363703952, 1108168168871431122, 53184335305844061592, 2553002833174245843690, 122541041242572894898912, 5882031874800914481693378, 282336292088593370983557736, 13552166778299906854199877978, 650503510197530529272579474992 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 F[[2, 2, 2, 1, 1, 1], [3, 3, 3]](x) = 2 5 4 3 2 x (2160 x - 5408 x + 3206 x - 667 x + 52 x - 1) - ------------------------------------------------------------------------- (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (-1 + 6 x) (8 x - 1) (-1 + 48 x) and in Maple notation F[[2, 2, 2, 1, 1, 1],[3, 3, 3]](x) = -x^2*(2160*x^5-5408*x^4+3206*x^3-667*x^2+ 52*x-1)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(-1+6*x)/(8*x-1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 1, 14, 624, 29546, 1416028, 67951394, 3261521348, 156551810282, 7514476852380, 360694806386354, 17313350032751092, 831040796098480058, 39889958168435160332, 1914717991727518248194, 91906463600043675952356, 4411510252778969255054474, 211752492133204848820028284, 10164119622392343391500101714, 487877741874820544283031869140 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [3, 3, 2, 1]](x) = x 6 5 4 3 2 (110592 x + 183232 x - 73744 x - 23360 x + 6062 x + 281 x - 63)/( (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[3, 3, 2, 1]](x) = x^3*(110592*x^6+183232*x^5-73744*x^4-\ 23360*x^3+6062*x^2+281*x-63)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4*x)/( 8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 63, 2239, 122495, 5574143, 273575547, 13010465251, 626917833591, 30043680947671, 1443063635709971, 69247710964647083, 3324276959935669647, 159557557100183739679, 7658917477876546726635, 367624944176791386774835, 17646059215551009590983463, 847009604443855869021998567, 40656485771346553404542012739, 1951510821863721832962745730107 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [3, 3, 1, 1, 1]](x) = - x 5 4 3 2 (4160 x + 224 x - 1508 x + 1412 x - 5 x - 53)/((-1 + x) (1 + x) (-1 + 2 x) (-1 + 3 x) (1 + 2 x) (1 + 4 x) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -x^3*(4160*x^5+224*x^4-1508*x^3+1412 *x^2-5*x-53)/(-1+x)/(1+x)/(-1+2*x)/(-1+3*x)/(1+2*x)/(1+4*x)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 53, 1436, 90625, 3917480, 196676293, 9267750676, 448306042505, 21449610472160, 1030962890673133, 49458587003112716, 2374564811081718385, 113968058233460031640, 5470687849174525367573, 262588595681011351015556, 12604341014275849502082265, 605006600253494731839315920, 29040352180802667812515623613, 1393936197305829642102848547196 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [3, 2, 2, 2]](x) = 2 x ( 7 6 5 4 3 2 189696 x - 38400 x - 102880 x + 14456 x + 6650 x - 183 x - 268 x + 19 )/((-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[3, 2, 2, 2]](x) = 2*x^3*(189696*x^7-38400*x^6-102880*x^5+ 14456*x^4+6650*x^3-183*x^2-268*x+19)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1 +6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 38, 1022, 63538, 2743266, 137681110, 6487491822, 313814756762, 15014731654202, 721674058185982, 34621011182745702, 1662195370010463106, 79777640781534036978, 3829481494567445423654, 183812016977872993514462, 8823038710002429846574570, 423504620177521091317777194, 20328246526562466251470488526, 975755338114085543173945330902 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 F[[2, 2, 2, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 3 2 3 x (8 x - 364 x + 27) - ------------------------------------------------- (-1 + x) (1 + x) (8 x - 1) (20 x + 1) (-1 + 48 x) and in Maple notation F[[2, 2, 2, 1, 1, 1],[3, 2, 2, 1, 1]](x) = -3*x^3*(8*x^2-364*x+27)/(-1+x)/(1+x) /(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 81, 1824, 125385, 5233152, 266668617, 12488712192, 605671133769, 28947838009344, 1391982813155913, 66765435741143040, 3205735641252139593, 153855415785520889856, 7385457853748348228169, 354494019028267479072768, 17015872072143613108458057, 816758676285187622109708288, 39204480125227359404495770185, 1881813772740020079761635147776 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (216960 x - 15168 x - 96648 x + 25364 x + 5618 x - 1509 x + 63)/( (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = -x^3*(216960*x^6-15168*x^5-96648* x^4+25364*x^3+5618*x^2-1509*x+63)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4 *x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 63, 1011, 85055, 3318415, 174326523, 8064937251, 393157440375, 18750637412055, 902448132081683, 43269152889886891, 2077886434871633295, 99719206533287800095, 4786908757103098424043, 229763883444592826522931, 11028821143058769267243815, 529380320110791310578306535, 25410317260425311385440865603, 1219693990598162522602099213371 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = 2 x 6 5 4 3 2 (84736 x + 30528 x + 480 x - 5396 x - 1086 x + 334 x - 11)/((1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = 2*x^3*(84736*x^6+30528*x^5+480 *x^4-5396*x^3-1086*x^2+334*x-11)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(-1+6*x)/(1+4* x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 22, 212, 24270, 856168, 47086186, 2138849804, 105079420566, 4995432393296, 240747674881890, 11536544671380916, 554140976788020622, 26591029902794370104, 1276524172363564081434, 61270065511940417341148, 2941025039636788535423238, 141167963999965042382648992, 6776087030039744306368168018, 325251682280994700231094740100 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [2, 2, 2, 2, 1]](x) = - x 5 4 3 2 (23168 x - 2784 x - 7536 x + 2188 x + 503 x - 27)/((1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[2, 2, 2, 2, 1]](x) = -x^3*(23168*x^5-2784*x^4-7536*x^3+ 2188*x^2+503*x-27)/(1+x)/(-1+2*x)/(1+2*x)/(-1+3*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1 +48*x) The first 20 term , starting with k=1 are 0, 0, 27, 415, 34143, 1328287, 69738683, 3226039523, 157263509367, 7500259261399, 360979287649299, 17307661436093611, 831154576203564431, 39887682631420345631, 1914763502986543639275, 91905553379002070326579, 4411528457232843329354535, 211752128044391301537156583, 10164126904170723343738457411, 487877596239269802715245216827 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 8 7 F[[2, 2, 2, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = - x (884608 x - 292096 x 6 5 4 3 2 - 498200 x + 178148 x + 24296 x - 11106 x + 540 x + 41 x - 1)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = -x*(884608*x^8-292096*x^7-498200* x^6+178148*x^5+24296*x^4-11106*x^3+540*x^2+41*x-1)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x )/(-1+3*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 1, 0, 39, 420, 40375, 1493088, 80212779, 3676768076, 179933089839, 8567662820376, 412629038421139, 19778559189343812, 949923452860368663, 45585272854059418064, 2188314149488137957819, 105034658084281775837628, 5041752009538675771941247, 242002328025357818075455752, 11616147113846079303686066019, 557574354091865284191025785524 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = x 5 4 3 2 (60480 x + 60928 x + 24356 x - 1964 x - 710 x + 25)/((-1 + x) (1 + x) (1 + 2 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = x^3*(60480*x^5+60928*x^4+24356 *x^3-1964*x^2-710*x+25)/(-1+x)/(1+x)/(1+2*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/ (-1+48*x) The first 20 term , starting with k=1 are 0, 0, 25, 190, 24001, 816842, 45602065, 2058706074, 101403215377, 4815503763034, 232180047911185, 11123915919058778, 534362419890258193, 25641106468255470426, 1230938899656121127185, 59081751363625032147802, 2835990381561889497420049, 136126211990501425650115418, 6534084702014986972060782865, 313635535167153424751740429146 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 7 F[[2, 2, 2, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x (315392 x 6 5 4 3 2 - 238272 x + 44400 x + 44472 x - 18010 x - 293 x + 371 x - 10)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(315392*x^7-238272*x^6 +44400*x^5+44472*x^4-18010*x^3-293*x^2+371*x-10)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/ (-1+3*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 10, 39, 7682, 232703, 13713814, 606071531, 30124690538, 1425237897831, 68825710848158, 3295343128509443, 158342248985981554, 7597112048883757439, 364727693796368359142, 17505602972505999238875, 840295469109293232033530, 40333651987345313829467927, 1936025905980039005577985966, 92929031275208269917960325427 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 3 F[[2, 2, 2, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x 7 6 5 4 3 2 (128 x + 52480 x - 35560 x - 13812 x + 6930 x - 219 x - 79 x + 2)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 2 x) (-1 + 3 x) (-1 + 6 x) (1 + 4 x) (8 x - 1) (20 x + 1) (-1 + 48 x)) and in Maple notation F[[2, 2, 2, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x^3*(128*x^7+52480*x^6-\ 35560*x^5-13812*x^4+6930*x^3-219*x^2-79*x+2)/(-1+x)/(1+x)/(-1+2*x)/(1+2*x)/(-1+ 3*x)/(-1+6*x)/(1+4*x)/(8*x-1)/(20*x+1)/(-1+48*x) The first 20 term , starting with k=1 are 0, 0, 2, 3, 1062, 27639, 1747170, 75132923, 3778357198, 177901336383, 8608297770138, 411816339891363, 19794813158063814, 949598373493422887, 45591774441368461906, 2188184117742076376523, 105037258719202530062910, 5041699996840262596273551, 242003368279326073952584074, 11616126308766714216685343603 ---------------------------------- Their sum is 8 7 6 5 4 3 2 (11488 x - 5544 x - 21596 x + 14140 x + 3298 x - 3898 x + 859 x - 64 x + 1)/((-1 + 48 x) (8 x - 1) (-1 + 6 x) (-1 + 3 x) (1 + 2 x) (-1 + 2 x) (1 + x) (-1 + x)) and in Maple notation (11488*x^8-5544*x^7-21596*x^6+14140*x^5+3298*x^4-3898*x^3+859*x^2-64*x+1)/(-1+ 48*x)/(8*x-1)/(-1+6*x)/(-1+3*x)/(1+2*x)/(-1+2*x)/(1+x)/(-1+x) The first 20 term , starting with k=1 are 1, 23, 835, 38603, 1841751, 88320979, 4238767359, 203455870011, 9765842906871, 468760153778915, 22500484965416943, 1080023259191868619, 51841116289099174311, 2488373580666317561331, 119441931862338323891487, 5733212729315311871912027, 275194211006520938308445271, 13209322128308101125009524227, 634047462158749672719560621391, 30434278183619671140463235993835 Regarding Lambda=, [2, 2, 1, 1, 1, 1, 1] 10 9 8 F[[2, 2, 1, 1, 1, 1, 1], [9]](x) = (966735 x - 482787 x - 2335953 x 7 6 5 4 3 2 + 1240507 x + 315542 x - 184146 x - 4632 x + 5779 x - 253 x - 25 x + 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[9]](x) = (966735*x^10-482787*x^9-2335953*x^8+1240507*x ^7+315542*x^6-184146*x^5-4632*x^4+5779*x^3-253*x^2-25*x+1)/(-1+x)/(1+x)/(-1+2*x )/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 0, 16, 0, 2616, 14757, 1059443, 17417331, 626518201, 14477653674, 426637930170, 10975955844282, 304436835281606, 8097952181541651, 220467454152300397, 5925236749577988453, 160391773988776248831, 4324418729895027753888, 116851662456365429597324 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [8, 1]](x) = 2 5 4 3 2 x (37800 x - 5184 x - 2949 x + 265 x + 21 x - 1) - ------------------------------------------------------------------------- (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[8, 1]](x) = -x^2*(37800*x^5-5184*x^4-2949*x^3+265*x^2+ 21*x-1)/(-1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 0, 74, 40, 17187, 142712, 7878988, 146278128, 4891621253, 117486050704, 3386908075662, 88189872821336, 2429667784241959, 64870195675029576, 1762433664205012496, 47421418705340090464, 1282840744112659024905, 34599746403379143386528, 934747305115940967527890 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[2, 2, 1, 1, 1, 1, 1], [7, 2]](x) = - x 7 6 5 4 3 2 (2835 x - 290952 x + 125520 x + 10042 x - 7573 x + 336 x + 50 x - 2)/ ((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[7, 2]](x) = -x^2*(2835*x^7-290952*x^6+125520*x^5+10042 *x^4-7573*x^3+336*x^2+50*x-2)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-\ 1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 2, 0, 172, 265, 49059, 549077, 25053803, 512060700, 16194084676, 400893573454, 11362110801054, 298644562456955, 8184836017605353, 219164197888889331, 5944785587153389525, 160098541456988185030, 4328817217712741566290, 116785685139894935283908, 3154598919444838159868816 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[2, 2, 1, 1, 1, 1, 1], [7, 1, 1]](x) = x 7 6 5 4 3 2 (113400 x - 23652 x - 8841 x + 727 x - 1926 x + 142 x + 23 x - 1)/( (-1 + x) (1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[7, 1, 1]](x) = x^2*(113400*x^7-23652*x^6-8841*x^5+727* x^4-1926*x^3+142*x^2+23*x-1)/(-1+x)/(1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+3*x)/(-1 +7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 0, 158, 356, 48847, 594412, 25580052, 536938584, 16705005293, 417079219304, 11762947686106, 310006264499932, 8483477779263579, 227349013998127476, 6163949647204820720, 166043326071671529200, 4488915752400543425305, 121114502310032831546128, 3271384604252662465182294 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[2, 2, 1, 1, 1, 1, 1], [6, 3]](x) = x 6 5 4 3 2 (11340 x - 16875 x - 307 x - 816 x - 24 x + 27 x - 1)/((1 + x) (-1 + 2 x) (1 + 3 x) (-1 + x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[6, 3]](x) = x^2*(11340*x^6-16875*x^5-307*x^4-816*x^3-\ 24*x^2+27*x-1)/(1+x)/(-1+2*x)/(1+3*x)/(-1+x)/(-1+9*x)/(-1+7*x)/(1+15*x)/(-1+27* x) The first 20 term , starting with k=1 are 0, 1, 0, 209, 690, 76309, 1064748, 42611193, 933812550, 28390464557, 718266902496, 20112120825457, 532198638890490, 14531402931624645, 389913878439649044, 10564154789591439401, 284684713766143758510, 7694696909499739305373, 207633651123689152389192, 5607955875747250102267425 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[2, 2, 1, 1, 1, 1, 1], [6, 2, 1]](x) = - x 6 5 4 3 2 (85050 x + 17955 x - 7683 x - 3103 x + 251 x + 44 x - 2)/((1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x) ) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[6, 2, 1]](x) = -x^2*(85050*x^6+17955*x^5-7683*x^4-3103 *x^3+251*x^2+44*x-2)/(1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/ (-1+27*x) The first 20 term , starting with k=1 are 0, 2, 0, 393, 1985, 156107, 2470195, 90953368, 2075877510, 61603024512, 1578718312490, 43882704735443, 1165874860285135, 31762113921971617, 853316841901590885, 23103388036258482618, 622833348953640792860, 16830866706400603509422, 454217856240917248769380, 12267114836954850681650893 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[2, 2, 1, 1, 1, 1, 1], [6, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (95445 x - 40149 x - 37492 x + 95 x + 2229 x - 139 x - 22 x + 1)/( (-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[6, 1, 1, 1]](x) = -x^2*(95445*x^7-40149*x^6-37492*x^5+ 95*x^4+2229*x^3-139*x^2-22*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1+9*x)/ (-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 0, 181, 1367, 77951, 1397242, 47342793, 1125184893, 32590886971, 846017333744, 23344338265205, 622703795853139, 16926305928587511, 455305297239708006, 12318768366155687617, 332223421600666025705, 8975778231376875343571, 242259788263515961411228, 6542307315636881765981229 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 2 F[[2, 2, 1, 1, 1, 1, 1], [5, 4]](x) = - x 6 5 4 3 2 (23625 x + 3768 x - 7649 x - 70 x + 281 x + 14 x - 1)/((-1 + x) (1 + x) (1 + 5 x) (1 + 3 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[5, 4]](x) = -x^2*(23625*x^6+3768*x^5-7649*x^4-70*x^3+ 281*x^2+14*x-1)/(-1+x)/(1+x)/(1+5*x)/(1+3*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27* x) The first 20 term , starting with k=1 are 0, 1, 0, 145, 730, 61295, 979118, 36281211, 829466652, 24632581909, 631408997956, 17552352012317, 466343302278854, 12704784771997683, 341326185003262914, 9241350203790259063, 249133294234777304536, 6732346272368089611017, 181687138791734788682792, 4906845901341923173991049 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [5, 3, 1]](x) = 2 3 2 x (846 x - 33 x + 16 x - 1) -------------------------------------------------------------- (1 + x) (1 + 5 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[5, 3, 1]](x) = x^2*(846*x^3-33*x^2+16*x-1)/(1+x)/(1+5* x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 0, 424, 3834, 214165, 4065678, 135268840, 3267263196, 93976792129, 2450844447876, 67469996840896, 1802226331715238, 48952027617755653, 1317325724440909794, 35633478037814854192, 961118641674579107160, 25964982214630048295137, 700832835810980318360232, 18925811822194057389750328 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [5, 2, 2]](x) = 2 2 x (135 x + 12 x - 1) - ------------------------------------------- (1 + 3 x) (-1 + 3 x) (1 + 15 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[5, 2, 2]](x) = -x^2*(135*x^2+12*x-1)/(1+3*x)/(-1+3*x)/ (1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 1, 0, 279, 3240, 150741, 3120120, 98481339, 2445415920, 69229841481, 1821151466640, 49891902572799, 1336269174154200, 36241450624393821, 976086423018799560, 26390824579038130659, 712004896271013994080, 19232342709762008191761, 519150095506904249342880, 14018899943536458885890919 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [5, 2, 1, 1]](x) = x 6 5 4 3 2 (229635 x + 107973 x - 108486 x + 4893 x + 3958 x - 306 x - 3)/( (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[5, 2, 1, 1]](x) = x^3*(229635*x^6+107973*x^5-108486*x^ 4+4893*x^3+3958*x^2-306*x-3)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+3*x)/( -1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 3, 378, 5948, 226365, 5102566, 152551791, 3892723677, 108447627960, 2877431723969, 78445612756104, 2106660714253186, 57049962996306735, 1537793059141229712, 41558713960384269117, 1121510409828519006875, 30289400903910103506690, 817684497981896531112595, 22079421079903390758975630 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 3 3 2 x (693 x - 348 x + 75 x - 4) ------------------------------------------------------------- (1 + x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[5, 1, 1, 1, 1]](x) = x^3*(693*x^3-348*x^2+75*x-4)/(1+x )/(-1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 4, 109, 2870, 75980, 2031414, 54620349, 1472360740, 39729854560, 1072474334624, 28954603823189, 781753650833610, 21107156934967740, 569891472139437634, 15387053579240408629, 415450299144653021480, 11217156735605131927520, 302863219692704974908444, 8177306821506827244216669 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [4, 4, 1]](x) = x ( 7 6 5 4 3 2 195615 x + 2322 x - 153078 x + 27533 x + 9791 x - 2464 x + 152 x + 1) /((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[4, 4, 1]](x) = x^3*(195615*x^7+2322*x^6-153078*x^5+ 27533*x^4+9791*x^3-2464*x^2+152*x+1)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1+ 9*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 1, 177, 2215, 104320, 2175229, 68836355, 1710907759, 48452257170, 1274727708397, 34923601883533, 935381780152183, 25368954639965300, 683259944356502605, 18473572194610685511, 498403382043037235887, 13462639486641228104710, 363405063150200921810653, 9813229927035503888967089 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [4, 3, 2]](x) = 3 4 3 2 x (14175 x - 3249 x - 1177 x + 233 x + 2) ------------------------------------------------------------------------- (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[4, 3, 2]](x) = x^3*(14175*x^4-3249*x^3-1177*x^2+233*x+ 2)/(-1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 2, 275, 5276, 191846, 4576334, 134035769, 3474469784, 96097753292, 2561416324346, 69666554646623, 1873462654548212, 50697329051747378, 1367127768421432838, 36938018890271279237, 996943581782053085360, 26923226130485040951704, 726840913790572986289010, 19625997995641971876996011 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [4, 3, 1, 1]](x) = x 6 5 4 3 2 (76545 x - 32994 x + 9900 x + 5776 x + 857 x - 174 x - 6)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[4, 3, 1, 1]](x) = x^3*(76545*x^6-32994*x^5+9900*x^4+ 5776*x^3+857*x^2-174*x-6)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+7* x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 6, 306, 7795, 234570, 6101948, 169431858, 4514396661, 122881704210, 3303677368750, 89418038297610, 2411065776012647, 65147629218540570, 1758257939652585312, 47483927547215277762, 1281901975352507867353, 34613817757622238475650, 934536143552615344514834, 25233030187634328797591514 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [4, 2, 2, 1]](x) = - x 6 5 4 3 2 (76545 x - 33669 x - 16605 x - 6578 x + 211 x + 55 x + 9)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[4, 2, 2, 1]](x) = -x^3*(76545*x^6-33669*x^5-16605*x^4-\ 6578*x^3+211*x^2+55*x+9)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+7*x )/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 9, 253, 8657, 221240, 6304141, 166387035, 4560130833, 122195375950, 3313973890733, 89263582418417, 2413382654624929, 65112875836702980, 1758779241395010285, 47476108015999814599, 1282019268346149606545, 34612058362590520382330, 934562534478726705081997, 25232634323739480055826781 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 7 6 F[[2, 2, 1, 1, 1, 1, 1], [4, 2, 1, 1, 1]](x) = - x (229635 x + 329913 x 5 4 3 2 - 276264 x + 31515 x + 5467 x - 965 x + 202 x - 15)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[4, 2, 1, 1, 1]](x) = -x^3*(229635*x^7+329913*x^6-\ 276264*x^5+31515*x^4+5467*x^3-965*x^2+202*x-15)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/( 1+3*x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 15, 173, 9100, 178915, 5814578, 141872136, 4052910429, 106044883460, 2913472558381, 77905002052999, 2114769865291538, 56928325779601185, 1539617617143478524, 41531345591604166562, 1121920935353916524827, 30283243021060768456090, 817776866224478095427807, 22078035556265460570017625 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (27405 x - 91791 x - 6515 x + 11874 x - 677 x - 211 x + 11)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = -x^3*(27405*x^6-91791*x^5-6515 *x^4+11874*x^3-677*x^2-211*x+11)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1+9*x) /(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 11, 31, 3525, 46060, 1873123, 40217233, 1232004281, 30988918390, 870045244695, 22983927709435, 628109913588157, 16845214365747040, 456521669665859627, 12300522784847284837, 332497105294867926753, 8971672976090981880010, 242321367092168645057119, 6541383633210270231970639 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [3, 3, 3]](x) = 3 4 3 2 x (378 x + 96 x - 59 x + 1) ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[3, 3, 3]](x) = x^3*(378*x^4+96*x^3-59*x^2+1)/(1+x)/(-1 +2*x)/(-1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 1, 48, 1530, 43602, 1199093, 32580954, 881599930, 23820894264, 643326734985, 17371308571980, 469038876392450, 12664172719429686, 341933779006695877, 9232222129927656726, 249270088774366767090, 6730293221074097821068, 181717924405881984284969, 4906384026026445928260792 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [3, 3, 2, 1]](x) = 3 4 3 2 x (28350 x + 2781 x - 1213 x + 43 x - 9) - ------------------------------------------------------------------------- (-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[3, 3, 2, 1]](x) = -x^3*(28350*x^4+2781*x^3-1213*x^2+43 *x-9)/(-1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 9, 146, 7330, 160476, 5049611, 126923230, 3581224068, 94496110232, 2585442607693, 69306152228874, 1878868731593126, 50616237692357428, 1368344139830331855, 36919773314049139478, 997217265450823671304, 26919120875326304063664, 726902492618589887057297, 19625074313218539257373442 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 3 2 2 x (270 x + 39 x - 7) - --------------------------------------------------- (1 + x) (1 + 3 x) (-1 + 3 x) (1 + 15 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -2*x^3*(270*x^2+39*x-7)/(1+x)/(1+ 3*x)/(-1+3*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 14, 76, 6260, 105520, 3798194, 88310956, 2597969480, 66941544640, 1855475899574, 49377036137836, 1343992170501500, 36125605679715760, 977824097187376154, 26364759466514264716, 712395872958857634320, 19226478059444396634880, 519238065261668293555934, 14017580397214998610115596 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [3, 2, 2, 2]](x) = x ( 7 6 5 4 3 2 399735 x - 370062 x + 35016 x + 19987 x - 5829 x + 1140 x - 122 x + 7 )/((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[3, 2, 2, 2]](x) = x^3*(399735*x^7-370062*x^6+35016*x^5 +19987*x^4-5829*x^3+1140*x^2-122*x+7)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x)/(1+3*x)/(-1 +9*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 7, 53, 4243, 73080, 2647855, 61727071, 1817645767, 46850695490, 1298753584843, 34563201500289, 940787847024571, 25287863331437980, 684476315511088471, 18455326619660111507, 498677065705449993055, 13458534231514280360550, 363466641978058876859539, 9812306244612865997941525 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 F[[2, 2, 1, 1, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 3 3 x (93 x - 7) ------------------------------------------------------ (1 + 5 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[3, 2, 2, 1, 1]](x) = 3*x^3*(93*x-7)/(1+5*x)/(-1+3*x)/( -1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 21, 78, 9180, 133194, 5284149, 116972244, 3541809792, 89858104908, 2512627197597, 66543243388050, 1816127694543084, 48743506870162182, 1320453537180690765, 35586560839088745096, 961822399693617716256, 25954425844153734295416, 700991181369078702672453, 18923436638817813253484982 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = x 5 4 3 2 (170100 x + 18360 x - 29589 x + 329 x + 569 x - 25)/((-1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = x^3*(170100*x^5+18360*x^4-\ 29589*x^3+329*x^2+569*x-25)/(-1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/( 1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 25, 31, 7315, 76705, 3658270, 73147606, 2342885290, 57598306510, 1638787072615, 42981683432281, 1179390129191365, 31559385142027015, 856357772331187060, 23057774086166390056, 623517558173250973540, 16820603568265342710220, 454371803312151593585605, 12264805630890308668116931 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x 6 5 4 3 2 (14175 x - 122337 x + 9726 x + 19954 x - 1129 x - 273 x + 12)/( (-1 + x) (1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^3*(14175*x^6-122337*x^5+ 9726*x^4+19954*x^3-1129*x^2-273*x+12)/(-1+x)/(1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1 +3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 12, 3, 2540, 16826, 1070944, 18451237, 643774248, 15102955332, 441107537156, 11402535095831, 315412392407476, 8402386165560478, 228565386678592248, 6145704064624852185, 166317009772231259024, 4484810497082860817864, 121176081138844460911420, 3270460921825256202574699 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [2, 2, 2, 2, 1]](x) = x 5 4 3 2 (4725 x + 17403 x - 896 x - 1156 x - 117 x + 9)/((-1 + x) (1 + x) (1 + 5 x) (1 + 3 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[2, 2, 2, 2, 1]](x) = x^3*(4725*x^5+17403*x^4-896*x^3-\ 1156*x^2-117*x+9)/(-1+x)/(1+x)/(1+5*x)/(1+3*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+ 27*x) The first 20 term , starting with k=1 are 0, 0, 9, 9, 2804, 29864, 1452577, 29168125, 936222576, 23030933928, 655435296065, 17191949550281, 471749379456628, 12623693412209152, 342542556413357673, 9223104627564532077, 249406977903558652160, 6728241017209320437936, 181748717619751786306201, 4905922218918490263803113 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = - 2 x 5 4 3 2 (2835 x - 5094 x + 5287 x + 479 x - 186 x + 7)/((1 + x) (-1 + 2 x) (1 + 3 x) (-1 + x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = -2*x^3*(2835*x^5-5094*x^4+5287 *x^3+479*x^2-186*x+7)/(1+x)/(-1+2*x)/(1+3*x)/(-1+x)/(-1+9*x)/(-1+7*x)/(1+15*x)/ (-1+27*x) The first 20 term , starting with k=1 are 0, 0, 14, 6, 3710, 31088, 1742822, 32440810, 1086366110, 26102167716, 752591335430, 19597254390494, 539921635237790, 14415557986946584, 391651552608225638, 10538089677067573458, 285075690453987398750, 7688832259182127748492, 207721620878453196602246, 5606636329425789826492102 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 8 F[[2, 2, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = - x (1329615 x 7 6 5 4 3 2 - 721278 x - 277134 x + 147329 x + 5842 x - 5404 x + 238 x + 25 x - 1)/((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = -x*(1329615*x^8-721278*x^7-\ 277134*x^6+147329*x^5+5842*x^4-5404*x^3+238*x^2+25*x-1)/(-1+x)/(1+x)/(-1+2*x)/( 1+5*x)/(1+3*x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 0, 16, 0, 2616, 14757, 1059443, 17417331, 626518201, 14477653674, 426637930170, 10975955844282, 304436835281606, 8097952181541651, 220467454152300397, 5925236749577988453, 160391773988776248831, 4324418729895027753888, 116851662456365429597324, 3153609259701755840979444 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x 5 4 3 2 (37800 x - 47331 x - 12084 x + 1514 x + 140 x - 7)/((-1 + x) (1 + x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(37800*x^5-47331*x^ 4-12084*x^3+1514*x^2+140*x-7)/(-1+x)/(1+x)/(1+5*x)/(1+3*x)/(-1+9*x)/(-1+7*x)/(1 +15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 7, 0, 1006, 3336, 347509, 4821144, 192077404, 4204967472, 127784200291, 3232444058448, 90506792123722, 2394914198953848, 65391498434707153, 1754614127903286312, 47538711724413144760, 1281081348953784356064, 34626137330126286831295, 934351441217913311375136 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 3 F[[2, 2, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (2835 x - 290952 x + 125520 x + 10042 x - 7573 x + 336 x + 50 x - 2)/ ((-1 + x) (1 + x) (-1 + 2 x) (1 + 5 x) (1 + 3 x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (1 + 15 x) (-1 + 27 x)) and in Maple notation F[[2, 2, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(2835*x^7-290952 *x^6+125520*x^5+10042*x^4-7573*x^3+336*x^2+50*x-2)/(-1+x)/(1+x)/(-1+2*x)/(1+5*x )/(1+3*x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(1+15*x)/(-1+27*x) The first 20 term , starting with k=1 are 0, 0, 2, 0, 172, 265, 49059, 549077, 25053803, 512060700, 16194084676, 400893573454, 11362110801054, 298644562456955, 8184836017605353, 219164197888889331, 5944785587153389525, 160098541456988185030, 4328817217712741566290, 116785685139894935283908 ---------------------------------- Their sum is 7 6 5 4 3 2 3276 x - 2028 x - 8370 x + 10132 x - 4059 x + 679 x - 47 x + 1 ------------------------------------------------------------------------ (1 + x) (-1 + 2 x) (-1 + x) (-1 + 9 x) (-1 + 3 x) (-1 + 7 x) (-1 + 27 x) and in Maple notation (3276*x^7-2028*x^6-8370*x^5+10132*x^4-4059*x^3+679*x^2-47*x+1)/(1+x)/(-1+2*x)/( -1+x)/(-1+9*x)/(-1+3*x)/(-1+7*x)/(-1+27*x) The first 20 term , starting with k=1 are 1, 12, 188, 4203, 106599, 2822286, 75736918, 2040973341, 55072766297, 1486675590840, 40137726593328, 1083696601105119, 29259614362658395, 790007872979101074, 21330197347055203418, 575915192829570992337, 15549708996816589899693, 419842132099816363391388, 11335737469870991100959188, 306064910818628534542071795 Regarding Lambda=, [2, 1, 1, 1, 1, 1, 1, 1] 11 10 9 F[[2, 1, 1, 1, 1, 1, 1, 1], [9]](x) = (50856 x - 8090 x - 152543 x 8 7 6 5 4 3 2 + 24121 x + 63662 x - 10012 x - 8689 x + 1349 x + 443 x - 67 x - 7 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[9]](x) = (50856*x^11-8090*x^10-152543*x^9+24121*x^8 +63662*x^7-10012*x^6-8689*x^5+1349*x^4+443*x^3-67*x^2-7*x+1)/(1+x)/(-1+x)/(1+2* x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 4, 0, 41, 0, 715, 28, 17721, 6160, 579151, 716716, 23632701, 63343280, 1140267587, 4861620764, 61704992481, 345236069360, 3587497622423 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 2 F[[2, 1, 1, 1, 1, 1, 1, 1], [8, 1]](x) = - x 8 7 6 5 4 3 2 (24 x - 20366 x + 3249 x + 5294 x - 827 x - 373 x + 57 x + 7 x - 1)/ ((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[8, 1]](x) = -x^2*(24*x^8-20366*x^7+3249*x^6+5294*x^ 5-827*x^4-373*x^3+57*x^2+7*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/ (1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 11, 0, 162, 1, 3425, 470, 98208, 69707, 3610399, 6898320, 161309214, 562025893, 8293469533, 41262479050, 467302279580, 2857439568959, 27828087318427 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 2 9 8 7 F[[2, 1, 1, 1, 1, 1, 1, 1], [7, 2]](x) = - x (25416 x - 4674 x - 22311 x 6 5 4 3 2 + 3485 x + 5047 x - 785 x - 359 x + 55 x + 7 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[7, 2]](x) = -x^2*(25416*x^9-4674*x^8-22311*x^7+3485 *x^6+5047*x^5-785*x^4-359*x^3+55*x^2+7*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2 *x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 13, 0, 256, 6, 6790, 2232, 231376, 288882, 9686722, 26338884, 474213676, 2041946478, 25854614734, 145480838256, 1508613935656, 9894679957194, 91640527613626 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 2 9 8 7 F[[2, 1, 1, 1, 1, 1, 1, 1], [7, 1, 1]](x) = - x (25440 x - 3440 x - 22482 x 6 5 4 3 2 + 3589 x + 5032 x - 785 x - 359 x + 55 x + 7 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[7, 1, 1]](x) = -x^2*(25440*x^9-3440*x^8-22482*x^7+ 3589*x^6+5032*x^5-785*x^4-359*x^3+55*x^2+7*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/( -1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 1, 0, 13, 0, 256, 21, 6791, 3430, 231846, 351967, 9756429, 29389360, 481111996, 2194305113, 26416640627, 153631131290, 1549876414706, 10359691543859, 94497967182585 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 4 F[[2, 1, 1, 1, 1, 1, 1, 1], [6, 3]](x) = 2 x 5 4 3 2 (1056 x + 227 x - 493 x + 26 x + 27 x - 3)/((-1 + x) (1 + 2 x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[6, 3]](x) = 2*x^4*(1056*x^5+227*x^4-493*x^3+26*x^2+ 27*x-3)/(-1+x)/(1+2*x)/(1+x)/(-1+3*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 6, 0, 200, 14, 7154, 4788, 297660, 581658, 14225222, 50706656, 756413840, 3814392582, 43289104410, 266530977804, 2595532752740, 17909045526386, 160031189543918 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 4 F[[2, 1, 1, 1, 1, 1, 1, 1], [6, 2, 1]](x) = x 6 5 4 3 2 (11320 x + 1442 x - 6453 x - 1025 x + 560 x + 48 x - 12)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[6, 2, 1]](x) = x^4*(11320*x^6+1442*x^5-6453*x^4-\ 1025*x^3+560*x^2+48*x-12)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x )/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 12, 0, 400, 105, 14357, 16170, 604350, 1544235, 29395927, 121861740, 1591226000, 8754510765, 92367269097, 597998929710, 5592700290850, 39715195291695, 346974396383867 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 4 F[[2, 1, 1, 1, 1, 1, 1, 1], [6, 1, 1, 1]](x) = x 7 6 5 4 3 2 (16896 x - 1576 x - 10790 x + 1383 x + 1393 x - 215 x - 41 x + 6)/( (1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[6, 1, 1, 1]](x) = x^4*(16896*x^7-1576*x^6-10790*x^5 +1383*x^4+1393*x^3-215*x^2-41*x+6)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4* x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 6, 1, 200, 161, 7182, 13685, 303820, 1020657, 14941938, 72101029, 819757120, 4918863313, 48150725174, 327663286133, 2940768822100, 21487380464529, 183488236683290 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 4 F[[2, 1, 1, 1, 1, 1, 1, 1], [5, 4]](x) = - x 6 5 4 3 2 (1200 x - 412 x - 410 x - 84 x + x - 2 x + 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[5, 4]](x) = -x^4*(1200*x^6-412*x^5-410*x^4-84*x^3+x ^2-2*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+4*x)/(1+6* x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 1, 0, 79, 14, 4067, 4620, 205423, 544698, 10936123, 46406360, 617260007, 3434332902, 36486904819, 237361253220, 2225907854431, 15837629110226, 138515657408555 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[2, 1, 1, 1, 1, 1, 1, 1], [5, 3, 1]](x) = 4 3 2 3 x (108 x - 21 x + 5 x - 1) - -------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 4 x) (1 + 6 x) (8 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[5, 3, 1]](x) = -3*x^4*(108*x^3-21*x^2+5*x-1)/(-1+x) /(1+x)/(-1+3*x)/(1+4*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 0, 240, 189, 12915, 28350, 687270, 2613303, 38334417, 199999800, 2241488340, 14047850817, 135705944559, 945134312850, 8404417140450, 62165977652331, 527747849868141 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[2, 1, 1, 1, 1, 1, 1, 1], [5, 2, 2]](x) = 4 3 2 2 x (26 x + 21 x + x + 1) - ------------------------------------------------------------------- (-1 + x) (1 + 2 x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 6 x) (8 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[5, 2, 2]](x) = -2*x^4*(26*x^3+21*x^2+x+1)/(-1+x)/(1 +2*x)/(1+x)/(-1+2*x)/(1+3*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 2, 0, 160, 190, 8702, 24900, 473420, 2125530, 27029002, 155927200, 1609732280, 10702533270, 98676594902, 711061731900, 6158639456740, 46448124059410, 388511297090402 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 4 F[[2, 1, 1, 1, 1, 1, 1, 1], [5, 2, 1, 1]](x) = - x 7 6 5 4 3 2 (18936 x - 8718 x - 4853 x + 1727 x - 42 x - 8 x + 17 x - 3)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[5, 2, 1, 1]](x) = -x^4*(18936*x^7-8718*x^6-4853*x^5 +1727*x^4-42*x^3-8*x^2+17*x-3)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/( 1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 3, 4, 240, 644, 13062, 53760, 714360, 3867028, 41170866, 263319056, 2476683300, 17458438452, 153073125390, 1140496417792, 9609321496560, 73866259430516, 608483901756234 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[2, 1, 1, 1, 1, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 4 2 x (15 x - 9 x + 1) - ----------------------------------------------------------- (1 + x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[5, 1, 1, 1, 1]](x) = -x^4*(15*x^2-9*x+1)/(1+x)/(-1+ 2*x)/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 1, 10, 80, 595, 4361, 32200, 241410, 1839365, 14211571, 111000890, 873849340, 6917208935, 54958556381, 437728133580, 3491944518870, 27885437563305, 222830044234791 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 6 F[[2, 1, 1, 1, 1, 1, 1, 1], [4, 4, 1]](x) = x 5 4 3 2 (8544 x - 2176 x + 754 x + 159 x - 280 x + 55)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[4, 4, 1]](x) = x^6*(8544*x^5-2176*x^4+754*x^3+159*x ^2-280*x+55)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-\ 1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 55, 105, 4634, 15582, 297045, 1414875, 18115768, 106810704, 1107741635, 7424301885, 68615048502, 495904893666, 4299888271825, 32465237835135, 271683923044436 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 6 F[[2, 1, 1, 1, 1, 1, 1, 1], [4, 3, 2]](x) = x 5 4 3 2 (7680 x - 9784 x + 1046 x + 31 x - 224 x + 75)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[4, 3, 2]](x) = x^6*(7680*x^5-9784*x^4+1046*x^3+31*x ^2-224*x+75)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-\ 1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 75, 301, 7238, 38430, 509985, 3191727, 33082236, 228788560, 2101554455, 15437975553, 133153942194, 1013911333890, 8454284521485, 65744679788179, 538171161518312 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 F[[2, 1, 1, 1, 1, 1, 1, 1], [4, 3, 1, 1]](x) = x 6 5 4 3 2 (9216 x + 6456 x - 1694 x - 769 x - 165 x + 55 x + 5)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[4, 3, 1, 1]](x) = x^5*(9216*x^6+6456*x^5-1694*x^4-\ 769*x^3-165*x^2+55*x+5)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/ (-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 90, 805, 8736, 66633, 623070, 4709045, 40988772, 314635321, 2635883250, 20540346645, 168535648008, 1327172701769, 10766566350630, 85341793883605, 688028078672844 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 F[[2, 1, 1, 1, 1, 1, 1, 1], [4, 2, 2, 1]](x) = 3 x 6 5 4 3 2 (3072 x + 1832 x - 114 x - 497 x + 6 x + 11 x + 2)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[4, 2, 2, 1]](x) = 3*x^5*(3072*x^6+1832*x^5-114*x^4-\ 497*x^3+6*x^2+11*x+2)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-\ 1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 6, 75, 951, 7566, 75060, 566145, 5078337, 38655012, 329131374, 2546867895, 21082791483, 165246975738, 1347042531048, 10646790833325, 86062689252789, 683693689056144 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 F[[2, 1, 1, 1, 1, 1, 1, 1], [4, 2, 1, 1, 1]](x) = 3 x 6 5 4 3 2 (2328 x - 3182 x + 513 x + 998 x - 225 x - 50 x + 10)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[4, 2, 1, 1, 1]](x) = 3*x^5*(2328*x^6-3182*x^5+513*x ^4+998*x^3-225*x^2-50*x+10)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3 *x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 30, 60, 1785, 6069, 96012, 460530, 5388735, 32053263, 317951634, 2149247100, 19421417925, 141302400057, 1211090364696, 9185884827270, 76406355233355, 593245472262051 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 F[[2, 1, 1, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = x 8 7 6 5 4 3 2 (1536 x - 2024 x + 114 x - 1087 x + 104 x + 220 x - 33 x - 7 x + 1)/ ((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = x^3*(1536*x^8-2024*x^7+114* x^6-1087*x^5+104*x^4+220*x^3-33*x^2-7*x+1)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2* x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 0, 35, 15, 1176, 1526, 45753, 119245, 2095786, 8623252, 109476731, 597672075, 6242627756, 40243326578, 374966316669, 2657517898505, 23184611543086, 173313964293904 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[2, 1, 1, 1, 1, 1, 1, 1], [3, 3, 3]](x) = 6 x (19 x - 5) - ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[3, 3, 3]](x) = -x^6*(19*x-5)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 5, 91, 1092, 10962, 100275, 870177, 7319114, 60404344, 492717225, 3989968983, 32163929616, 258543640446, 2074553057855, 16627608724909, 133177428145398 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 F[[2, 1, 1, 1, 1, 1, 1, 1], [3, 3, 2, 1]](x) = - x 6 5 4 3 2 (7680 x - 7048 x + 3746 x + 651 x - 339 x + 19 x - 5)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[3, 3, 2, 1]](x) = -x^5*(7680*x^6-7048*x^5+3746*x^4+ 651*x^3-339*x^2+19*x-5)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/ (-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 5, 16, 791, 3724, 61845, 360192, 4127387, 27322108, 263928665, 1888414528, 16725954063, 125389698252, 1060641723965, 8173324203424, 67432748357219, 528033539341756 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[2, 1, 1, 1, 1, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 5 2 10 x (2 x + 3 x + 2) - ------------------------------------------------------------------- (-1 + x) (1 + 2 x) (1 + x) (-1 + 2 x) (1 + 3 x) (1 + 6 x) (8 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[3, 3, 1, 1, 1]](x) = -10*x^5*(2*x^2+3*x+2)/(-1+x)/( 1+2*x)/(1+x)/(-1+2*x)/(1+3*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 20, 10, 1190, 2380, 63840, 236670, 3555530, 18420160, 207667460, 1299027730, 12567552270, 87484100340, 778223851880, 5755645257190, 48866153739410, 374002925474920 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 F[[2, 1, 1, 1, 1, 1, 1, 1], [3, 2, 2, 2]](x) = - x 6 5 4 3 2 (1056 x - 7504 x + 1058 x + 899 x - 232 x + 23 x - 4)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[3, 2, 2, 2]](x) = -x^5*(1056*x^6-7504*x^5+1058*x^4+ 899*x^3-232*x^2+23*x-4)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/ (-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 4, 5, 539, 1414, 37590, 153615, 2322793, 12473648, 141457316, 896640745, 8703927687, 60884807802, 542497488082, 4019484495395, 34151064138821, 261555318269476 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 F[[2, 1, 1, 1, 1, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 5 2 3 x (12 x + 47 x - 10) - -------------------------------------------------------------------- (-1 + x) (1 + x) (-1 + 3 x) (1 + 4 x) (-1 + 4 x) (1 + 6 x) (8 x - 1) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[3, 2, 2, 1, 1]](x) = -3*x^5*(12*x^2+47*x-10)/(-1+x) /(1+x)/(-1+3*x)/(1+4*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 30, 9, 1779, 2556, 93780, 281583, 5099853, 23205402, 291612750, 1688455197, 17379471447, 115662533688, 1065609526440, 7680706865451, 66511675276161, 501659920229814 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 F[[2, 1, 1, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = - x 7 6 5 4 3 2 (520 x + 1190 x + 363 x + 818 x + 145 x - 90 x - 8 x + 2)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = -x^3*(520*x^7+1190*x^6+363* x^5+818*x^4+145*x^3-90*x^2-8*x+2)/(1+x)/(-1+x)/(1+2*x)/(-1+2*x)/(1+4*x)/(1+3*x) /(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 2, 0, 70, 5, 2352, 1435, 91310, 163485, 4148452, 13937495, 213936450, 1041505465, 12041852552, 72687351555, 715901287990, 4885996555445, 43952570600652, 321561549217615 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 8 7 F[[2, 1, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = - x (480 x - 33568 x 6 5 4 3 2 + 5562 x + 11275 x - 1766 x - 966 x + 149 x + 21 x - 3)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = -x^3*(480*x^8-33568*x^7+ 5562*x^6+11275*x^5-1766*x^4-966*x^3+149*x^2+21*x-3)/(1+x)/(-1+x)/(1+2*x)/(-1+3* x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 3, 0, 55, 1, 1274, 294, 38545, 35987, 1469600, 3275448, 67391415, 256589333, 3527618006, 18471651562, 201082719165, 1266036664439, 12059261742892, 84314386826236 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 5 F[[2, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 1]](x) = 2 x 5 4 3 2 (168 x - 242 x + 113 x + 103 x + 10 x - 5)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[2, 2, 2, 2, 1]](x) = 2*x^5*(168*x^5-242*x^4+113*x^3 +103*x^2+10*x-5)/(1+x)/(-1+x)/(1+2*x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+4*x )/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 0, 0, 10, 0, 574, 322, 28770, 53340, 1487398, 5154534, 81611530, 403923520, 4722903822, 28720878186, 284097002290, 1944931437540, 17525746019446, 128377890124078 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 F[[2, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = x 5 4 3 2 (192 x - 10 x + 132 x - 15 x - 8 x + 1)/((1 + 2 x) (1 + x) (-1 + 3 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = x^3*(192*x^5-10*x^4+132*x^3 -15*x^2-8*x+1)/(1+2*x)/(1+x)/(-1+3*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 0, 35, 0, 1175, 328, 45283, 56160, 2026079, 5572776, 102578411, 445313440, 5680601863, 32093033544, 333703837619, 2192506311840, 20327171974127, 145522527537832 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 8 7 F[[2, 1, 1, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = x (504 x + 33450 x 6 5 4 3 2 - 5011 x - 11354 x + 1773 x + 965 x - 149 x - 21 x + 3)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = x^3*(504*x^8+33450*x^7-\ 5011*x^6-11354*x^5+1773*x^4+965*x^3-149*x^2-21*x+3)/(1+x)/(-1+x)/(1+2*x)/(-1+3* x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 3, 0, 55, 0, 1274, 147, 38517, 27090, 1463440, 2836449, 66674699, 235194960, 3464274726, 17367180831, 196221098401, 1204904356110, 11714025673532, 80736051888093 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 9 F[[2, 1, 1, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = - x (87384 x 8 7 6 5 4 3 2 - 13798 x - 48289 x + 7615 x + 7598 x - 1182 x - 422 x + 64 x + 7 x - 1)/((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = -x*(87384*x^9-13798*x ^8-48289*x^7+7615*x^6+7598*x^5-1182*x^4-422*x^3+64*x^2+7*x-1)/(1+x)/(-1+x)/(1+2 *x)/(-1+3*x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 0, 4, 0, 41, 0, 715, 28, 17721, 6160, 579151, 716716, 23632701, 63343280, 1140267587, 4861620764, 61704992481, 345236069360, 3587497622423, 23457047139372 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 3 F[[2, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = - x 8 7 6 5 4 3 2 (24 x - 20366 x + 3249 x + 5294 x - 827 x - 373 x + 57 x + 7 x - 1)/ ((1 + x) (-1 + x) (1 + 2 x) (-1 + 3 x) (-1 + 2 x) (1 + 4 x) (1 + 3 x) (-1 + 5 x) (-1 + 4 x) (1 + 6 x) (8 x - 1)) and in Maple notation F[[2, 1, 1, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x^3*(24*x^8-20366 *x^7+3249*x^6+5294*x^5-827*x^4-373*x^3+57*x^2+7*x-1)/(1+x)/(-1+x)/(1+2*x)/(-1+3 *x)/(-1+2*x)/(1+4*x)/(1+3*x)/(-1+5*x)/(-1+4*x)/(1+6*x)/(8*x-1) The first 20 term , starting with k=1 are 0, 0, 1, 0, 11, 0, 162, 1, 3425, 470, 98208, 69707, 3610399, 6898320, 161309214, 562025893, 8293469533, 41262479050, 467302279580, 2857439568959 ---------------------------------- Their sum is 7 6 5 4 3 2 214 x - 423 x - 362 x + 942 x - 591 x + 164 x - 21 x + 1 ---------------------------------------------------------------------- (1 + x) (-1 + 3 x) (-1 + 2 x) (-1 + x) (-1 + 5 x) (-1 + 4 x) (8 x - 1) and in Maple notation (214*x^7-423*x^6-362*x^5+942*x^4-591*x^3+164*x^2-21*x+1)/(1+x)/(-1+3*x)/(-1+2*x )/(-1+x)/(-1+5*x)/(-1+4*x)/(8*x-1) The first 20 term , starting with k=1 are 1, 4, 15, 75, 427, 2735, 18979, 139011, 1053691, 8157911, 63979763, 505699627, 4015966475, 31983667967, 255164915267, 2037859370323, 16285819892379, 130202250411303, 1041200289923091, 8327528965317499 Regarding Lambda=, [1, 1, 1, 1, 1, 1, 1, 1, 1] 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [9]](x) = - ---------------- (-1 + x) (1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[9]](x) = -1/(-1+x)/(1+x) The first 20 term , starting with k=1 are 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [8, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[8, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [7, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[7, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [7, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[7, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [6, 3]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[6, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [6, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[6, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [6, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[6, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [5, 4]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[5, 4]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [5, 3, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[5, 3, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [5, 2, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[5, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [5, 2, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[5, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [5, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[5, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[4, 4, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 3, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[4, 3, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 3, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[4, 3, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 2, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[4, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[4, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[4, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [3, 3, 3]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[3, 3, 3]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [3, 3, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[3, 3, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [3, 3, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[3, 3, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [3, 2, 2, 2]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[3, 2, 2, 2]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [3, 2, 2, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[3, 2, 2, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [3, 2, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[3, 2, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[3, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[2, 2, 2, 2, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[2, 2, 2, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[2, 2, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1]](x) = 0 and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1, 1, 1]](x) = 0 The first 20 term , starting with k=1 are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 F[[1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = x - ---------------- (-1 + x) (1 + x) and in Maple notation F[[1, 1, 1, 1, 1, 1, 1, 1, 1],[1, 1, 1, 1, 1, 1, 1, 1, 1]](x) = -x/(-1+x)/(1+x) The first 20 term , starting with k=1 are 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ---------------------------------- Their sum is 1 - ------ -1 + x and in Maple notation -1/(-1+x) The first 20 term , starting with k=1 are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 This ends this book that took, 40.710, seconds to generate