The first , 18, rational functions , psi[n], discussed in Gert Almkvist's Beautiful article "Invariants, Mostly Old ones" Pacific Journal of Mathematics v. 86 (1980), No. 1, 1-13. By Shalosh B. Ekhad In a beautiful article of Gert Almkvist (1934-2018) , "Invariants, Mostly Ol\ d ones" , Pacific Journal of Mathematics v. 86 (1980), No. 1, 1-13, he investigated certaion rational functions, that he called, psi[n](t), that turned out to have been studied by Faa Di Bruno, Cayley, Sylvester, and Fabian Franklin Here we present fully rigorous, computer-generated explicit expressions for \ this quantities, as well as explicit expressions as sum of quasi-polynomials for the coefficients. These generating functions turned out to enumerate covariantes of binary qua\ dratic forms of degree n. and have been computed by James Joseph Sylvester and his student, Fabian Fra\ nklin (of Franklin's involution fame) up to n=10 and n=12 Here we will do it up to n=, 18 For , psi[2](t), we have the following theorem. Theorem Number, 1, : Let f(t) be the constant term of the rational function in z and t 2 (1 + z) ---------------------------------- 2 / t \ 2 z (1 - t) (-t z + 1) |1 - ----| | 2 | \ z / We have 1 f(t) = ----------------- 2 (-1 + t) (t + 1) and in Maple notation 1/(-1+t)^2/(t+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16] Furthermore, a(n) is a quasi-polynomial given as sum of, 2, quasi-polynomials 2 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), are defined as followed Q[1](n), is the polynomial n/2 + 3/4 and in Maple notation 1/2*n+3/4 This is the leading term in particular, a(n) , is asymptotic to n/2 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 [-1/4, 1/4] and in Maple format [-1/4, 1/4] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 5000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000001 ------------------------------------ --------------------------------------------------------- For , psi[3](t), we have the following theorem. Theorem Number, 2, : Let f(t) be the constant term of the rational function in z and t 2 (1 + z) ----------------------------------------------- 3 / t \ 2 z (-t z + 1) (-t z + 1) (1 - t/z) |1 - ----| | 3 | \ z / We have 2 t - t + 1 f(t) = - -------------------------- 3 2 (-1 + t) (t + 1) (t + 1) and in Maple notation -(t^2-t+1)/(-1+t)^3/(t+1)/(t^2+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 25, 28, 32, 36, 41, 45, 50, 55, 61, 66, 72, 78, 85, 91, 98, 105, 113, 120, 128] Furthermore, a(n) is a quasi-polynomial given as sum of, 3, quasi-polynomials 3 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), are defined as followed Q[1](n), is the polynomial 2 1/8 n + 1/2 n + 9/16 and in Maple notation 1/8*n^2+1/2*n+9/16 This is the leading term in particular, a(n) , is asymptotic to 2 n ---- 8 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 -3 [--, 3/16] 16 and in Maple format [-3/16, 3/16] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 [0, -1/4, 0, 1/4] and in Maple format [0, -1/4, 0, 1/4] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 1250000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000005000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000001 ------------------------------------ --------------------------------------------------------- For , psi[4](t), we have the following theorem. Theorem Number, 3, : Let f(t) be the constant term of the rational function in z and t 2 (1 + z) --------------------------------------------------------- 2 4 / t \ / t \ 2 z (1 - t) (-t z + 1) (-t z + 1) |1 - ----| |1 - ----| | 2 | | 4 | \ z / \ z / We have 2 t - t + 1 f(t) = ------------------------------ 2 4 (t + 1) (t + t + 1) (-1 + t) and in Maple notation (t^2-t+1)/(t+1)/(t^2+t+1)/(-1+t)^4 For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 3, 5, 8, 12, 18, 24, 33, 43, 55, 69, 86, 104, 126, 150, 177, 207, 241, 277, 318, 362, 410, 462, 519, 579, 645, 715, 790, 870, 956] Furthermore, a(n) is a quasi-polynomial given as sum of, 3, quasi-polynomials 3 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), are defined as followed Q[1](n), is the polynomial 3 2 85 1/36 n + 5/24 n + 7/12 n + --- 144 and in Maple notation 1/36*n^3+5/24*n^2+7/12*n+85/144 This is the leading term in particular, a(n) , is asymptotic to 3 n ---- 36 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 -3 [--, 3/16] 16 and in Maple format [-3/16, 3/16] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 [-2/9, 0, 2/9] and in Maple format [-2/9, 0, 2/9] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 2777777777777777777777777777777777777777777777777777777777777777777777777777777\ 7777777777777777777798611111111111111111111111111111111111111111111111111111111\ 1111111111111111111111111111111111111111169444444444444444444444444444444444444\ 44444444444444444444444444444444444444444444444444444444444445 ------------------------------------ --------------------------------------------------------- For , psi[5](t), we have the following theorem. Theorem Number, 4, : Let f(t) be the constant term of the rational function in z and t 2 (1 + z) ---------------------------------------------------------------------- 3 5 / t \ / t \ 2 z (-t z + 1) (-t z + 1) (-t z + 1) (1 - t/z) |1 - ----| |1 - ----| | 3 | | 5 | \ z / \ z / We have 14 13 12 11 10 9 8 7 6 5 4 f(t) = - (t - t + 2 t + t + 2 t + 3 t + t + 5 t + t + 3 t + 2 t 3 2 / 5 3 2 2 2 + t + 2 t - t + 1) / ((-1 + t) (t + 1) (t + 1) (t + t + 1) / 2 4 (t - t + 1) (t + 1)) and in Maple notation -(t^14-t^13+2*t^12+t^11+2*t^10+3*t^9+t^8+5*t^7+t^6+3*t^5+2*t^4+t^3+2*t^2-t+1)/( -1+t)^5/(t+1)^3/(t^2+1)^2/(t^2+t+1)/(t^2-t+1)/(t^4+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 3, 6, 12, 20, 32, 49, 73, 102, 141, 190, 252, 325, 414, 521, 649, 795, 967, 1165, 1394, 1651, 1944, 2275, 2649, 3061, 3523, 4035, 4604, 5225, 5910 ] Furthermore, a(n) is a quasi-polynomial given as sum of, 6, quasi-polynomials 6 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), are defined as followed Q[1](n), is the polynomial 23 4 23 3 1205 2 377 11693 ---- n + --- n + ---- n + --- n + ----- 4608 384 4608 768 27648 and in Maple notation 23/4608*n^4+23/384*n^3+1205/4608*n^2+377/768*n+11693/27648 This is the leading term in particular, a(n) , is asymptotic to 4 23 n ----- 4608 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 2 1505 2 1505 [-1/1536 n - 1/256 n - ----, 1/1536 n + 1/256 n + ----] 9216 9216 and in Maple format [-1/1536*n^2-1/256*n-1505/9216, 1/1536*n^2+1/256*n+1505/9216] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 -1 -1 [--, --, 1/27] 54 54 and in Maple format [-1/54, -1/54, 1/27] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 3 n -1 3 n [1/64, - --- - 9/64, --, --- + 9/64] 64 64 64 and in Maple format [1/64, -3/64*n-9/64, -1/64, 3/64*n+9/64] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 -1 -1 [1/18, --, -1/9, --, 1/18, 1/9] 18 18 and in Maple format [1/18, -1/18, -1/9, -1/18, 1/18, 1/9] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 [-1/8, 1/8, 0, -1/8, 1/8, -1/8, 0, 1/8] and in Maple format [-1/8, 1/8, 0, -1/8, 1/8, -1/8, 0, 1/8] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 4991319444444444444444444444444444444444444444444444444444444444444444444444444\ 4444444444444444444504340277777777777777777777777777777777777777777777777777777\ 7777777777777777777777777777777777777778039930555555555555555555555555555555555\ 5555555555555555555555555555555555555555555555555555555555556097222222222222222\ 2222222222222222222222222222222222222222222222222222222222222222222222222222222\ 223 ------------------------------------ --------------------------------------------------------- For , psi[6](t), we have the following theorem. Theorem Number, 5, : Let f(t) be the constant term of the rational function in z and t 2 / / 2 4 6 / t \ (1 + z) / |2 z (1 - t) (-t z + 1) (-t z + 1) (-t z + 1) |1 - ----| / | | 2 | \ \ z / / t \ / t \\ |1 - ----| |1 - ----|| | 4 | | 6 || \ z / \ z // We have 10 8 7 6 5 4 3 2 t + t + 3 t + 4 t + 4 t + 4 t + 3 t + t + 1 f(t) = --------------------------------------------------------------- 2 4 3 2 3 6 2 (t + 1) (t + t + t + t + 1) (t + 1) (-1 + t) (t + t + 1) and in Maple notation (t^10+t^8+3*t^7+4*t^6+4*t^5+4*t^4+3*t^3+t^2+1)/(t^2+1)/(t^4+t^3+t^2+t+1)/(t+1)^ 3/(-1+t)^6/(t^2+t+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 4, 8, 18, 32, 58, 94, 151, 227, 338, 480, 676, 920, 1242, 1636, 2137, 2739, 3486, 4370, 5444, 6698, 8196, 9926, 11963, 14293, 17002, 20076, 23612, 27594, 32134] Furthermore, a(n) is a quasi-polynomial given as sum of, 5, quasi-polynomials 5 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), are defined as followed Q[1](n), is the polynomial 11 5 77 4 49 3 343 2 203 84427 ----- n + ---- n + --- n + ---- n + --- n + ------ 14400 5760 540 1152 384 172800 and in Maple notation 11/14400*n^5+77/5760*n^4+49/540*n^3+343/1152*n^2+203/384*n+84427/172800 This is the leading term in particular, a(n) , is asymptotic to 5 11 n ----- 14400 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 2 55 2 55 [-1/128 n - 7/128 n - ---, 1/128 n + 7/128 n + ---] 256 256 and in Maple format [-1/128*n^2-7/128*n-55/256, 1/128*n^2+7/128*n+55/256] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 -2 [0, --, 2/27] 27 and in Maple format [0, -2/27, 2/27] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 -1 -1 [--, --, 1/16, 1/16] 16 16 and in Maple format [-1/16, -1/16, 1/16, 1/16] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 -2 -4 [--, 2/25, --, 0, 4/25] 25 25 and in Maple format [-2/25, 2/25, -4/25, 0, 4/25] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 7638888888888888888888888888888888888888888888888888888888888888888888888888888\ 8888888888888888889022569444444444444444444444444444444444444444444444444444444\ 4444444444444444444444444444444444444445351851851851851851851851851851851851851\ 8518518518518518518518518518518518518518518518518518518518521574074074074074074\ 0740740740740740740740740740740740740740740740740740740740740740740740740740740\ 7465740740740740740740740740740740740740740740740740740740740740740740740740740\ 74074074074074074074075 ------------------------------------ --------------------------------------------------------- For , psi[7](t), we have the following theorem. Theorem Number, 6, : Let f(t) be the constant term of the rational function in z and t 2 / / 3 5 7 (1 + z) / |2 z (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (1 - t/z) / | \ / t \ / t \ / t \\ |1 - ----| |1 - ----| |1 - ----|| | 3 | | 5 | | 7 || \ z / \ z / \ z // We have 34 33 32 31 30 29 28 27 26 f(t) = - (t - t + 3 t + 3 t + 7 t + 12 t + 16 t + 28 t + 33 t 25 24 23 22 21 20 19 18 + 46 t + 56 t + 73 t + 83 t + 90 t + 106 t + 109 t + 121 t 17 16 15 14 13 12 11 + 110 t + 121 t + 109 t + 106 t + 90 t + 83 t + 73 t 10 9 8 7 6 5 4 3 2 + 56 t + 46 t + 33 t + 28 t + 16 t + 12 t + 7 t + 3 t + 3 t - t / 7 5 2 3 2 2 2 2 + 1) / ((-1 + t) (t + 1) (t + 1) (t + t + 1) (t - t + 1) / 4 4 3 2 4 3 2 4 2 (t + 1) (t + t + t + t + 1) (t - t + t - t + 1) (t - t + 1)) and in Maple notation -(t^34-t^33+3*t^32+3*t^31+7*t^30+12*t^29+16*t^28+28*t^27+33*t^26+46*t^25+56*t^ 24+73*t^23+83*t^22+90*t^21+106*t^20+109*t^19+121*t^18+110*t^17+121*t^16+109*t^ 15+106*t^14+90*t^13+83*t^12+73*t^11+56*t^10+46*t^9+33*t^8+28*t^7+16*t^6+12*t^5+ 7*t^4+3*t^3+3*t^2-t+1)/(-1+t)^7/(t+1)^5/(t^2+1)^3/(t^2+t+1)^2/(t^2-t+1)^2/(t^4+ 1)/(t^4+t^3+t^2+t+1)/(t^4-t^3+t^2-t+1)/(t^4-t^2+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 4, 10, 24, 49, 94, 169, 289, 468, 734, 1117, 1656, 2385, 3370, 4672, 6375, 8550, 11322, 14800, 19138, 24460, 30982, 38882, 48417, 59779, 73316, 89291, 108108, 130053, 155646] Furthermore, a(n) is a quasi-polynomial given as sum of, 9, quasi-polynomials 9 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), are defined as followed Q[1](n), is the polynomial 841 6 841 5 78263 4 24439 3 589391 2 37151 ------- n + ------ n + ------- n + ------ n + ------- n + ----- n 8294400 345600 3317760 207360 1843200 82944 16740419 + -------- 49766400 and in Maple notation 841/8294400*n^6+841/345600*n^5+78263/3317760*n^4+24439/207360*n^3+589391/ 1843200*n^2+37151/82944*n+16740419/49766400 This is the leading term in particular, a(n) , is asymptotic to 6 841 n ------- 8294400 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 11 4 11 3 73 2 391 848341 [- ------- n - ----- n - ----- n - ------ n - -------, 1105920 69120 73728 138240 5529600 11 4 11 3 73 2 391 848341 ------- n + ----- n + ----- n + ------ n + -------] 1105920 69120 73728 138240 5529600 and in Maple format [-11/1105920*n^4-11/69120*n^3-73/73728*n^2-391/138240*n-848341/5529600, 11/ 1105920*n^4+11/69120*n^3+73/73728*n^2+391/138240*n+848341/5529600] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 5 n -1 5 n [- --- - 7/243, ---, --- + 8/243] 648 243 648 and in Maple format [-5/648*n-7/243, -1/243, 5/648*n+8/243] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 n 11 2 11 143 n 11 2 11 143 [- --- - 1/192, - ---- n - --- n - ----, --- + 1/192, ---- n + --- n + ----] 768 1536 192 1152 768 1536 192 1152 and in Maple format [-1/768*n-1/192, -11/1536*n^2-11/192*n-143/1152, 1/768*n+1/192, 11/1536*n^2+11/ 192*n+143/1152] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 -1 -1 -1 [--, 1/25, --, --, 1/25] 25 50 50 and in Maple format [-1/25, 1/25, -1/50, -1/50, 1/25] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 5 n 5 n 5 n 5 n [--- + 5/54, 0, - --- - 5/54, - --- - 5/54, 0, --- + 5/54] 216 216 216 216 and in Maple format [5/216*n+5/54, 0, -5/216*n-5/54, -5/216*n-5/54, 0, 5/216*n+5/54] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 [0, 0, 0, -1/8, 0, 0, 0, 1/8] and in Maple format [0, 0, 0, -1/8, 0, 0, 0, 1/8] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 10 -1 -3 -1 -1 -3 [--, 1/25, 3/50, --, --, 1/25, --, --, 3/50, 1/25] 25 50 25 25 50 and in Maple format [-1/25, 1/25, 3/50, -3/50, -1/25, 1/25, -1/25, -3/50, 3/50, 1/25] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 12 -1 -1 -1 -1 [--, 1/9, --, 1/18, 0, --, 1/12, -1/9, 1/12, --, 0, 1/18] 12 12 18 18 and in Maple format [-1/12, 1/9, -1/12, 1/18, 0, -1/18, 1/12, -1/9, 1/12, -1/18, 0, 1/18] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 1013937114197530864197530864197530864197530864197530864197530864197530864197530\ 8641975308641975308666309799382716049382716049382716049382716049382716049382716\ 0493827160493827160493827160493827160494063151041666666666666666666666666666666\ 6666666666666666666666666666666666666666666666666666666666667846836419753086419\ 7530864197530864197530864197530864197530864197530864197530864197530864197530864\ 2008100308641975308641975308641975308641975308641975308641975308641975308641975\ 3086419753086419753086467469135802469135802469135802469135802469135802469135802\ 46913580246913580246913580246913580246913581 ------------------------------------ --------------------------------------------------------- For , psi[8](t), we have the following theorem. Theorem Number, 7, : Let f(t) be the constant term of the rational function in z and t 2 / / 2 4 6 8 (1 + z) / |2 z (1 - t) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ / t \ / t \ / t \ / t \\ |1 - ----| |1 - ----| |1 - ----| |1 - ----|| | 2 | | 4 | | 6 | | 8 || \ z / \ z / \ z / \ z // We have 18 16 15 14 13 12 11 10 f(t) = (t + 2 t + 6 t + 12 t + 19 t + 25 t + 31 t + 36 t 9 8 7 6 5 4 3 2 / + 38 t + 36 t + 31 t + 25 t + 19 t + 12 t + 6 t + 2 t + 1) / ( / 3 2 2 8 2 4 3 2 (t + 1) (t + t + 1) (-1 + t) (t + 1) (t + t + t + t + 1) 6 5 4 3 2 (t + t + t + t + t + t + 1)) and in Maple notation (t^18+2*t^16+6*t^15+12*t^14+19*t^13+25*t^12+31*t^11+36*t^10+38*t^9+36*t^8+31*t^ 7+25*t^6+19*t^5+12*t^4+6*t^3+2*t^2+1)/(t+1)^3/(t^2+t+1)^2/(-1+t)^8/(t^2+1)/(t^4 +t^3+t^2+t+1)/(t^6+t^5+t^4+t^3+t^2+t+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 5, 13, 33, 73, 151, 289, 526, 910, 1514, 2430, 3788, 5744, 8512, 12346, 17575, 24591, 33885, 46029, 61731, 81805, 107233, 139143, 178870, 227930, 288100, 361384, 450096, 556834, 684572] Furthermore, a(n) is a quasi-polynomial given as sum of, 6, quasi-polynomials 6 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), are defined as followed Q[1](n), is the polynomial 151 7 151 6 5939 5 133 4 129329 3 13349 2 -------- n + ------ n + ------- n + ---- n + ------ n + ----- n 12700800 403200 1209600 3840 907200 38400 2217407 7357871 + ------- n + -------- 4354560 16934400 and in Maple notation 151/12700800*n^7+151/403200*n^6+5939/1209600*n^5+133/3840*n^4+129329/907200*n^3 +13349/38400*n^2+2217407/4354560*n+7357871/16934400 This is the leading term in particular, a(n) , is asymptotic to 7 151 n -------- 12700800 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 2 91 2 91 [-1/512 n - 9/512 n - ---, 1/512 n + 9/512 n + ---] 512 512 and in Maple format [-1/512*n^2-9/512*n-91/512, 1/512*n^2+9/512*n+91/512] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 2 n 2 n 4 n [- --- - 8/81, - --- + 2/81, --- + 2/27] 243 243 243 and in Maple format [-2/243*n-8/81, -2/243*n+2/81, 4/243*n+2/27] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 -1 -1 [1/32, --, --, 1/32] 32 32 and in Maple format [1/32, -1/32, -1/32, 1/32] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 -4 [--, 0, 0, 0, 4/25] 25 and in Maple format [-4/25, 0, 0, 0, 4/25] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 -2 -4 -6 [--, 4/49, --, 2/49, --, 0, 6/49] 49 49 49 and in Maple format [-2/49, 4/49, -4/49, 2/49, -6/49, 0, 6/49] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 1188901486520534139581758629377676996724615772234819853867472915091962711010330\ 0579491055681531872045512881078357268833459309649785840262030738221214411690602\ 1667926429831191735953640715545477450239845990016376921138825900730662635424540\ 1864449483497102544721592340639959687578735197782816830435881518467340639959687\ 5787351977828168304358780549256739732930209120685311161501637692113882590073066\ 2777982961703199798437893675988914084152179390274628369866465104560342655580750\ 8188460569412950365331667295603426555807508188460569412950365331317712270093222\ 4741748551272360796170319979843789367598891926977828168304358780549256739732930\ 2091206853111615016376921138825900730662635424540186444948349711 ------------------------------------ --------------------------------------------------------- For , psi[9](t), we have the following theorem. Theorem Number, 8, : Let f(t) be the constant term of the rational function in z and t 2 / / 3 5 7 9 (1 + z) / |2 z (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ / t \ / t \ / t \ / t \\ (1 - t/z) |1 - ----| |1 - ----| |1 - ----| |1 - ----|| | 3 | | 5 | | 7 | | 9 || \ z / \ z / \ z / \ z // We have 62 61 60 59 58 57 56 55 f(t) = - (t - t + 4 t + 6 t + 17 t + 32 t + 61 t + 111 t 54 53 52 51 50 49 48 + 178 t + 279 t + 422 t + 614 t + 863 t + 1160 t + 1560 t 47 46 45 44 43 42 + 2008 t + 2565 t + 3137 t + 3876 t + 4590 t + 5453 t 41 40 39 38 37 36 + 6219 t + 7181 t + 7974 t + 8906 t + 9581 t + 10432 t 35 34 33 32 31 30 + 10960 t + 11579 t + 11819 t + 12194 t + 12161 t + 12194 t 29 28 27 26 25 24 + 11819 t + 11579 t + 10960 t + 10432 t + 9581 t + 8906 t 23 22 21 20 19 18 + 7974 t + 7181 t + 6219 t + 5453 t + 4590 t + 3876 t 17 16 15 14 13 12 11 + 3137 t + 2565 t + 2008 t + 1560 t + 1160 t + 863 t + 614 t 10 9 8 7 6 5 4 3 2 + 422 t + 279 t + 178 t + 111 t + 61 t + 32 t + 17 t + 6 t + 4 t / 9 7 2 4 2 2 2 2 - t + 1) / ((-1 + t) (t + 1) (t + 1) (t + t + 1) (t - t + 1) / 4 2 4 3 2 4 3 2 4 2 (t + 1) (t + t + t + t + 1) (t - t + t - t + 1) (t - t + 1) 8 6 5 4 3 2 6 5 4 3 2 (t + 1) (t + t + t + t + t + t + 1) (t - t + t - t + t - t + 1)) and in Maple notation -(t^62-t^61+4*t^60+6*t^59+17*t^58+32*t^57+61*t^56+111*t^55+178*t^54+279*t^53+ 422*t^52+614*t^51+863*t^50+1160*t^49+1560*t^48+2008*t^47+2565*t^46+3137*t^45+ 3876*t^44+4590*t^43+5453*t^42+6219*t^41+7181*t^40+7974*t^39+8906*t^38+9581*t^37 +10432*t^36+10960*t^35+11579*t^34+11819*t^33+12194*t^32+12161*t^31+12194*t^30+ 11819*t^29+11579*t^28+10960*t^27+10432*t^26+9581*t^25+8906*t^24+7974*t^23+7181* t^22+6219*t^21+5453*t^20+4590*t^19+3876*t^18+3137*t^17+2565*t^16+2008*t^15+1560 *t^14+1160*t^13+863*t^12+614*t^11+422*t^10+279*t^9+178*t^8+111*t^7+61*t^6+32*t^ 5+17*t^4+6*t^3+4*t^2-t+1)/(-1+t)^9/(t+1)^7/(t^2+1)^4/(t^2+t+1)^2/(t^2-t+1)^2/(t ^4+1)^2/(t^4+t^3+t^2+t+1)/(t^4-t^3+t^2-t+1)/(t^4-t^2+1)/(t^8+1)/(t^6+t^5+t^4+t^ 3+t^2+t+1)/(t^6-t^5+t^4-t^3+t^2-t+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 5, 15, 43, 102, 227, 468, 910, 1667, 2934, 4968, 8150, 12954, 20094, 30441, 45207, 65809, 94257, 132856, 184717, 253305, 343363, 460085, 610358, 801543, 1043534, 1346749, 1724882, 2192069, 2767118] Furthermore, a(n) is a quasi-polynomial given as sum of, 12, quasi-polynomials 12 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), are defined as followed Q[1](n), is the polynomial 259723 8 259723 7 1515757 6 264925 5 4084755761 4 ------------ n + ---------- n + ---------- n + -------- n + ----------- n 208089907200 5202247680 1783627776 33030144 89181388800 240391087 3 479110865 2 369233665 5546529445271 + ---------- n + ---------- n + --------- n + -------------- 1486356480 1387266048 891813888 18728091648000 and in Maple notation 259723/208089907200*n^8+259723/5202247680*n^7+1515757/1783627776*n^6+264925/ 33030144*n^5+4084755761/89181388800*n^4+240391087/1486356480*n^3+479110865/ 1387266048*n^2+369233665/891813888*n+5546529445271/18728091648000 This is the leading term in particular, a(n) , is asymptotic to 8 259723 n ------------ 208089907200 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 289 6 289 5 216577 4 72077 3 1067039 2 [- ---------- n - -------- n - ---------- n - --------- n - --------- n 2972712960 99090432 5945425920 297271296 330301440 7444501 222446739719 289 6 289 5 216577 4 - --------- n - -------------, ---------- n + -------- n + ---------- n 297271296 1248539443200 2972712960 99090432 5945425920 72077 3 1067039 2 7444501 222446739719 + --------- n + --------- n + --------- n + -------------] 297271296 330301440 297271296 1248539443200 and in Maple format [-289/2972712960*n^6-289/99090432*n^5-216577/5945425920*n^4-72077/297271296*n^3 -1067039/330301440*n^2-7444501/297271296*n-222446739719/1248539443200, 289/ 2972712960*n^6+289/99090432*n^5+216577/5945425920*n^4+72077/297271296*n^3+ 1067039/330301440*n^2+7444501/297271296*n+222446739719/1248539443200] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 -1 5 n 37 5 n 19 [----, - ---- - ----, ---- + ----] 2916 1944 2916 1944 1458 and in Maple format [-1/2916, -5/1944*n-37/2916, 5/1944*n+19/1458] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 13 2 65 6815 235 3 1175 2 1445 13975 [----- n + ----- n - ------, - ------ n - ----- n - ----- n - ------, 98304 49152 294912 294912 98304 24576 147456 13 2 65 6815 235 3 1175 2 1445 13975 - ----- n - ----- n + ------, ------ n + ----- n + ----- n + ------] 98304 49152 294912 294912 98304 24576 147456 and in Maple format [13/98304*n^2+65/49152*n-6815/294912, -235/294912*n^3-1175/98304*n^2-1445/24576 *n-13975/147456, -13/98304*n^2-65/49152*n+6815/294912, 235/294912*n^3+1175/ 98304*n^2+1445/24576*n+13975/147456] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 -3 -3 -3 -3 [---, ---, ---, ---, 6/125] 250 250 250 250 and in Maple format [-3/250, -3/250, -3/250, -3/250, 6/125] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 5 n 35 5 n 10 -5 5 n 35 5 n 10 [5/972, - --- - ---, - --- - ---, ---, --- + ---, --- + ---] 648 972 648 243 972 648 972 648 243 and in Maple format [5/972, -5/648*n-35/972, -5/648*n-10/243, -5/972, 5/648*n+35/972, 5/648*n+10/ 243] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 -1 -1 -1 -1 [--, 0, --, 3/98, --, --, 3/98] 98 98 49 49 and in Maple format [-1/98, 0, -1/98, 3/98, -1/49, -1/49, 3/98] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 7 n 35 7 n 7 n 23 7 n 35 7 n -3 [--- + ---, --- + 3/64, 3/256, - --- - ---, - --- - ---, - --- - 3/64, ---, 512 512 512 512 256 512 512 512 256 7 n 23 --- + ---] 512 256 and in Maple format [7/512*n+35/512, 7/512*n+3/64, 3/256, -7/512*n-23/256, -7/512*n-35/512, -7/512* n-3/64, -3/256, 7/512*n+23/256] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 10 -1 -1 -2 -1 -1 [1/50, --, 1/50, --, --, --, 1/50, --, 1/50, 2/25] 50 50 25 50 50 and in Maple format [1/50, -1/50, 1/50, -1/50, -2/25, -1/50, 1/50, -1/50, 1/50, 2/25] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 -1 -1 -1 -1 -1 [--, 1/18, --, 0, 1/36, --, 1/18, --, 1/36, 0, --, 1/18] 18 36 18 18 36 and in Maple format [-1/18, 1/18, -1/36, 0, 1/36, -1/18, 1/18, -1/18, 1/36, 0, -1/36, 1/18] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 14 -1 -1 -3 -1 -4 -1 -3 [--, 4/49, --, 1/98, 3/49, --, --, 1/98, --, 1/98, --, --, 3/49, 1/98] 98 98 49 98 49 98 49 and in Maple format [-1/98, 4/49, -1/98, 1/98, 3/49, -3/49, -1/98, 1/98, -4/49, 1/98, -1/98, -3/49, 3/49, 1/98] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 16 -1 -1 -1 -1 -1 -1 -1 [--, 1/16, --, 1/16, --, 1/16, 0, --, 1/16, --, 1/16, --, 1/16, --, 0, 1/16] 16 16 16 16 16 16 16 and in Maple format [-1/16, 1/16, -1/16, 1/16, -1/16, 1/16, 0, -1/16, 1/16, -1/16, 1/16, -1/16, 1/ 16, -1/16, 0, 1/16] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 1248128770370271951373141849332325522801713277903754094230284706475182665658856\ 1350466112370874275686105548899970867976820357772738725119677500629881582262534\ 6434870244394053917863441672965482489292863015639168234809355841101872847904593\ 9363399680859998320315780633240950701268161585621903082220550561623137994588267\ 4057277231880406483581086755689930293104896279499454102628705803308977912152515\ 3729577117968618774670361971949273536575123876711178298479885781473083060384647\ 6862349878222894095911596636793220374569580918787267993617199966406315612664819\ 0140253632317124380616444108507600571092638190163874191651969429747207524985302\ 7630805408583186360964138741916519694297472075249853027630805413698559670781893\ 0041152263374485596707818930041152263374485596707818930041152263374485596707818\ 93005 ------------------------------------ --------------------------------------------------------- For , psi[10](t), we have the following theorem. Theorem Number, 9, : Let f(t) be the constant term of the rational function in z and t 2 / / 2 4 6 8 (1 + z) / |2 z (1 - t) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 10 / t \ / t \ / t \ / t \ / t \\ (-t z + 1) |1 - ----| |1 - ----| |1 - ----| |1 - ----| |1 - ---|| | 2 | | 4 | | 6 | | 8 | | 10|| \ z / \ z / \ z / \ z / \ z // We have 36 34 33 32 31 30 29 28 f(t) = (t + 3 t + 11 t + 27 t + 58 t + 112 t + 193 t + 318 t 27 26 25 24 23 22 21 + 485 t + 699 t + 951 t + 1245 t + 1541 t + 1842 t + 2108 t 20 19 18 17 16 15 + 2321 t + 2451 t + 2506 t + 2451 t + 2321 t + 2108 t 14 13 12 11 10 9 8 + 1842 t + 1541 t + 1245 t + 951 t + 699 t + 485 t + 318 t 7 6 5 4 3 2 / 5 + 193 t + 112 t + 58 t + 27 t + 11 t + 3 t + 1) / ((t + 1) / 10 2 2 2 3 2 4 3 2 (-1 + t) (t + 1) (t + t + 1) (t - t + 1) (t + t + t + t + 1) 4 6 5 4 3 2 6 3 (t + 1) (t + t + t + t + t + t + 1) (t + t + 1)) and in Maple notation (t^36+3*t^34+11*t^33+27*t^32+58*t^31+112*t^30+193*t^29+318*t^28+485*t^27+699*t^ 26+951*t^25+1245*t^24+1541*t^23+1842*t^22+2108*t^21+2321*t^20+2451*t^19+2506*t^ 18+2451*t^17+2321*t^16+2108*t^15+1842*t^14+1541*t^13+1245*t^12+951*t^11+699*t^ 10+485*t^9+318*t^8+193*t^7+112*t^6+58*t^5+27*t^4+11*t^3+3*t^2+1)/(t+1)^5/(-1+t) ^10/(t^2+1)^2/(t^2+t+1)^3/(t^2-t+1)/(t^4+t^3+t^2+t+1)/(t^4+1)/(t^6+t^5+t^4+t^3+ t^2+t+1)/(t^6+t^3+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 6, 18, 55, 141, 338, 734, 1514, 2934, 5448, 9686, 16660, 27718, 44916, 70922, 109583, 165821, 246448, 360002, 517971, 734517, 1028172, 1421530, 1943488, 2628824, 3521260, 4672836, 6147894, 8022362, 10388788] Furthermore, a(n) is a quasi-polynomial given as sum of, 9, quasi-polynomials 9 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), are defined as followed Q[1](n), is the polynomial 15619 9 171809 8 2762419 7 79981 6 91001801 5 ------------ n + ----------- n + ----------- n + -------- n + ---------- n 131681894400 29262643200 21946982400 52254720 7838208000 237638071 4 2627273611 3 1558151551 2 3292681451 + ---------- n + ----------- n + ---------- n + ---------- n 4180377600 14631321600 4389396480 7524679680 335369173931 + ------------ 877879296000 and in Maple notation 15619/131681894400*n^9+171809/29262643200*n^8+2762419/21946982400*n^7+79981/ 52254720*n^6+91001801/7838208000*n^5+237638071/4180377600*n^4+2627273611/ 14631321600*n^3+1558151551/4389396480*n^2+3292681451/7524679680*n+335369173931/ 877879296000 This is the leading term in particular, a(n) , is asymptotic to 9 15619 n ------------ 131681894400 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 23 4 253 3 1007 2 4565 330809 [- ------ n - ------ n - ----- n - ----- n - -------, 294912 147456 73728 98304 1769472 23 4 253 3 1007 2 4565 330809 ------ n + ------ n + ----- n + ----- n + -------] 294912 147456 73728 98304 1769472 and in Maple format [-23/294912*n^4-253/147456*n^3-1007/73728*n^2-4565/98304*n-330809/1769472, 23/ 294912*n^4+253/147456*n^3+1007/73728*n^2+4565/98304*n+330809/1769472] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 13 2 209 3067 11 n 121 13 2 220 1103 [- ---- n - ---- n - -----, - ---- - -----, ---- n + ---- n + ----] 4374 6561 26244 6561 13122 4374 6561 8748 and in Maple format [-13/4374*n^2-209/6561*n-3067/26244, -11/6561*n-121/13122, 13/4374*n^2+220/6561 *n+1103/8748] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 3 n 15 3 n 3 n 15 3 n [--- + ---, - --- - 9/256, - --- - ---, --- + 9/256] 512 512 512 512 512 512 and in Maple format [3/512*n+15/512, -3/512*n-9/256, -3/512*n-15/512, 3/512*n+9/256] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 -4 -8 [4/125, 0, ---, ---, 8/125] 125 125 and in Maple format [4/125, 0, -4/125, -8/125, 8/125] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 -1 -1 -1 [---, --, ---, 1/108, 1/54, 1/108] 108 54 108 and in Maple format [-1/108, -1/54, -1/108, 1/108, 1/54, 1/108] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 -4 -6 [--, 4/49, --, 0, 0, 0, 6/49] 49 49 and in Maple format [-4/49, 4/49, -6/49, 0, 0, 0, 6/49] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 -1 -1 [0, 1/16, --, 0, 0, --, 1/16, 0] 16 16 and in Maple format [0, 1/16, -1/16, 0, 0, -1/16, 1/16, 0] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -2 -2 -2 [0, 2/27, --, 2/27, --, 0, --, 0, 2/27] 27 27 27 and in Maple format [0, 2/27, -2/27, 2/27, -2/27, 0, -2/27, 0, 2/27] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 1186115985889097294153143653437598175987358820986099057821573988534599939655789\ 1560831008214900043295028758334752510972381636696745456298660296308738401624938\ 9559207313469512176155327242922774962752705850653375776465135665605977187399879\ 9336835785983346242017611800092693684698387814202041127379179134297144497945497\ 3576078808295151622606061171610848271636043534926544920666025898242241569696008\ 3776582011869962891420857323267670122461421697165377353501986071063084584542550\ 4449607918155831148194277397200020840526425476454870928709846993209721016893268\ 5099645711050766900267194211932601115434760840893418981675889377241523038113279\ 1479646270945506689186877311509880586894108352074254484601339429205330508975423\ 7314495985865768346661938666641782456009381347417796565356824028193810674704266\ 7090156274164901442973165489332449943854999704499998444736833919667546945618668\ 1348350954465005023500027 ------------------------------------ --------------------------------------------------------- For , psi[11](t), we have the following theorem. Theorem Number, 10, : Let f(t) be the constant term of the rational function in z and t 2 / / 3 5 7 9 (1 + z) / |2 z (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 11 / t \ / t \ / t \ / t \ (-t z + 1) (1 - t/z) |1 - ----| |1 - ----| |1 - ----| |1 - ----| | 3 | | 5 | | 7 | | 9 | \ z / \ z / \ z / \ z / / t \\ |1 - ---|| | 11|| \ z // We have 98 97 96 95 94 93 92 91 f(t) = - (t - t + 5 t + 10 t + 32 t + 74 t + 163 t + 338 t 90 89 88 87 86 85 + 637 t + 1143 t + 1973 t + 3250 t + 5180 t + 7915 t 84 83 82 81 80 79 + 11918 t + 17279 t + 24715 t + 34181 t + 46976 t + 62630 t 78 77 76 75 74 + 83012 t + 107026 t + 137688 t + 172650 t + 216084 t 73 72 71 70 69 + 264009 t + 322697 t + 385521 t + 460814 t + 538920 t 68 67 66 65 64 + 631503 t + 724577 t + 833058 t + 938425 t + 1060328 t 63 62 61 60 59 + 1174455 t + 1304785 t + 1421587 t + 1554667 t + 1667902 t 58 57 56 55 54 + 1795958 t + 1897576 t + 2013423 t + 2096662 t + 2192256 t 53 52 51 50 49 + 2249942 t + 2319522 t + 2347366 t + 2385695 t + 2380240 t 48 47 46 45 44 + 2385695 t + 2347366 t + 2319522 t + 2249942 t + 2192256 t 43 42 41 40 39 + 2096662 t + 2013423 t + 1897576 t + 1795958 t + 1667902 t 38 37 36 35 34 + 1554667 t + 1421587 t + 1304785 t + 1174455 t + 1060328 t 33 32 31 30 29 + 938425 t + 833058 t + 724577 t + 631503 t + 538920 t 28 27 26 25 24 + 460814 t + 385521 t + 322697 t + 264009 t + 216084 t 23 22 21 20 19 18 + 172650 t + 137688 t + 107026 t + 83012 t + 62630 t + 46976 t 17 16 15 14 13 12 + 34181 t + 24715 t + 17279 t + 11918 t + 7915 t + 5180 t 11 10 9 8 7 6 5 4 + 3250 t + 1973 t + 1143 t + 637 t + 338 t + 163 t + 74 t + 32 t 3 2 / 8 11 9 + 10 t + 5 t - t + 1) / ((t + 1) (-1 + t) (t + 1) / 8 6 4 2 6 5 4 3 2 (t - t + t - t + 1) (t + t + t + t + t + t + 1) 6 5 4 3 2 2 5 2 3 2 3 (t - t + t - t + t - t + 1) (t + 1) (t + t + 1) (t - t + 1) 4 2 4 3 2 2 4 3 2 2 4 2 (t + 1) (t + t + t + t + 1) (t - t + t - t + 1) (t - t + 1) 6 3 6 3 (t + t + 1) (t - t + 1)) and in Maple notation -(t^98-t^97+5*t^96+10*t^95+32*t^94+74*t^93+163*t^92+338*t^91+637*t^90+1143*t^89 +1973*t^88+3250*t^87+5180*t^86+7915*t^85+11918*t^84+17279*t^83+24715*t^82+34181 *t^81+46976*t^80+62630*t^79+83012*t^78+107026*t^77+137688*t^76+172650*t^75+ 216084*t^74+264009*t^73+322697*t^72+385521*t^71+460814*t^70+538920*t^69+631503* t^68+724577*t^67+833058*t^66+938425*t^65+1060328*t^64+1174455*t^63+1304785*t^62 +1421587*t^61+1554667*t^60+1667902*t^59+1795958*t^58+1897576*t^57+2013423*t^56+ 2096662*t^55+2192256*t^54+2249942*t^53+2319522*t^52+2347366*t^51+2385695*t^50+ 2380240*t^49+2385695*t^48+2347366*t^47+2319522*t^46+2249942*t^45+2192256*t^44+ 2096662*t^43+2013423*t^42+1897576*t^41+1795958*t^40+1667902*t^39+1554667*t^38+ 1421587*t^37+1304785*t^36+1174455*t^35+1060328*t^34+938425*t^33+833058*t^32+ 724577*t^31+631503*t^30+538920*t^29+460814*t^28+385521*t^27+322697*t^26+264009* t^25+216084*t^24+172650*t^23+137688*t^22+107026*t^21+83012*t^20+62630*t^19+ 46976*t^18+34181*t^17+24715*t^16+17279*t^15+11918*t^14+7915*t^13+5180*t^12+3250 *t^11+1973*t^10+1143*t^9+637*t^8+338*t^7+163*t^6+74*t^5+32*t^4+10*t^3+5*t^2-t+1 )/(t^8+1)/(-1+t)^11/(t+1)^9/(t^8-t^6+t^4-t^2+1)/(t^6+t^5+t^4+t^3+t^2+t+1)/(t^6- t^5+t^4-t^3+t^2-t+1)/(t^2+1)^5/(t^2+t+1)^3/(t^2-t+1)^3/(t^4+1)^2/(t^4+t^3+t^2+t +1)^2/(t^4-t^3+t^2-t+1)^2/(t^4-t^2+1)/(t^6+t^3+1)/(t^6-t^3+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 6, 21, 69, 190, 480, 1117, 2430, 4968, 9686, 18084, 32540, 56556, 95514, 157004, 252117, 395867, 609734, 921921, 1371535, 2008475, 2900900, 4133811, 5820904, 8100945, 11157558, 15210856, 20547836, 27506731, 36525052] Furthermore, a(n) is a quasi-polynomial given as sum of, 15, quasi-polynomials 15 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), Q[13](n), Q[14](n), Q[15](n), are defined as followed Q[1](n), is the polynomial 34706647 10 34706647 9 1045223387 8 2318806541 7 ---------------- n + -------------- n + -------------- n + ------------- n 3371056496640000 56184274944000 64210599936000 9364045824000 385831909199 6 9887240489 5 5048781668113 4 + --------------- n + ------------ n + -------------- n 160526499840000 637009920000 74912366592000 605410284601 3 13642253109517 2 4284915913651 + ------------- n + -------------- n + -------------- n 3121348608000 38526359961600 11466178560000 1725797485777489 + ---------------- 6742112993280000 and in Maple notation 34706647/3371056496640000*n^10+34706647/56184274944000*n^9+1045223387/ 64210599936000*n^8+2318806541/9364045824000*n^7+385831909199/160526499840000*n^ 6+9887240489/637009920000*n^5+5048781668113/74912366592000*n^4+605410284601/ 3121348608000*n^3+13642253109517/38526359961600*n^2+4284915913651/ 11466178560000*n+1725797485777489/6742112993280000 This is the leading term in particular, a(n) , is asymptotic to 10 34706647 n ---------------- 3371056496640000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 14219 8 14219 7 56747 6 2315791 5 [- -------------- n - ------------ n - ----------- n - ------------ n 21403533312000 445906944000 85614133248 297271296000 606916409 4 117217697 3 19724631889 2 - -------------- n - ------------ n - -------------- n 10701766656000 445906944000 12842119987200 56492382773 384403782108133 14219 8 14219 7 - ------------- n - ----------------, -------------- n + ------------ n 5350883328000 2247370997760000 21403533312000 445906944000 56747 6 2315791 5 606916409 4 117217697 3 + ----------- n + ------------ n + -------------- n + ------------ n 85614133248 297271296000 10701766656000 445906944000 19724631889 2 56492382773 384403782108133 + -------------- n + ------------- n + ----------------] 12842119987200 5350883328000 2247370997760000 and in Maple format [-14219/21403533312000*n^8-14219/445906944000*n^7-56747/85614133248*n^6-2315791 /297271296000*n^5-606916409/10701766656000*n^4-117217697/445906944000*n^3-\ 19724631889/12842119987200*n^2-56492382773/5350883328000*n-384403782108133/ 2247370997760000, 14219/21403533312000*n^8+14219/445906944000*n^7+56747/ 85614133248*n^6+2315791/297271296000*n^5+606916409/10701766656000*n^4+117217697 /445906944000*n^3+19724631889/12842119987200*n^2+56492382773/5350883328000*n+ 384403782108133/2247370997760000] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 23 2 19 1091 23 2 31 755 [- ------ n - ---- n - ------, - ------ n - ----- n - ------, 104976 6561 104976 104976 13122 104976 23 2 23 923 ----- n + ---- n + -----] 52488 4374 52488 and in Maple format [-23/104976*n^2-19/6561*n-1091/104976, -23/104976*n^2-31/13122*n-755/104976, 23 /52488*n^2+23/4374*n+923/52488] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 19 3 57 2 2189 821 [- ------- n - ------ n - ------- n - ------, 1966080 327680 1966080 327680 1723 4 1723 3 18095 2 28447 4487603 - -------- n - ------ n - ------- n - ------ n - --------, 23592960 983040 1179648 491520 44236800 19 3 57 2 2189 821 ------- n + ------ n + ------- n + ------, 1966080 327680 1966080 327680 1723 4 1723 3 18095 2 28447 4487603 -------- n + ------ n + ------- n + ------ n + --------] 23592960 983040 1179648 491520 44236800 and in Maple format [-19/1966080*n^3-57/327680*n^2-2189/1966080*n-821/327680, -1723/23592960*n^4-\ 1723/983040*n^3-18095/1179648*n^2-28447/491520*n-4487603/44236800, 19/1966080*n ^3+57/327680*n^2+2189/1966080*n+821/327680, 1723/23592960*n^4+1723/983040*n^3+ 18095/1179648*n^2+28447/491520*n+4487603/44236800] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 9 n 9 n 27 n 59 -13 27 n 103 [---- + 9/1250, - ---- - 9/625, - ---- - ----, ----, ---- + ----] 5000 5000 5000 2500 1250 5000 2500 and in Maple format [9/5000*n+9/1250, -9/5000*n-9/625, -27/5000*n-59/2500, -13/1250, 27/5000*n+103/ 2500] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 23 2 983 23 2 11 839 [----- n + 2/243 n + -----, - ----- n - ---- n - -----, 34992 34992 34992 1458 34992 23 2 23 911 23 2 983 - ----- n - ---- n - -----, - ----- n - 2/243 n - -----, 17496 1458 17496 34992 34992 23 2 11 839 23 2 23 911 ----- n + ---- n + -----, ----- n + ---- n + -----] 34992 1458 34992 17496 1458 17496 and in Maple format [23/34992*n^2+2/243*n+983/34992, -23/34992*n^2-11/1458*n-839/34992, -23/17496*n ^2-23/1458*n-911/17496, -23/34992*n^2-2/243*n-983/34992, 23/34992*n^2+11/1458*n +839/34992, 23/17496*n^2+23/1458*n+911/17496] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 -2 -1 -1 -1 -1 [--, 2/49, --, --, --, --, 2/49] 49 98 98 98 98 and in Maple format [-2/49, 2/49, -1/98, -1/98, -1/98, -1/98, 2/49] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 n n 3 n n -3 [---- + 7/1024, 3/256, - ---- - 5/1024, - --- - 9/128, - ---- - 7/1024, ---, 1024 1024 256 1024 256 n 3 n ---- + 5/1024, --- + 9/128] 1024 256 and in Maple format [1/1024*n+7/1024, 3/256, -1/1024*n-5/1024, -3/256*n-9/128, -1/1024*n-7/1024, -3 /256, 1/1024*n+5/1024, 3/256*n+9/128] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -1 -1 -1 [0, 1/54, --, 1/54, 0, 1/54, --, --, 1/54] 27 54 54 and in Maple format [0, 1/54, -1/27, 1/54, 0, 1/54, -1/54, -1/54, 1/54] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 10 9 n 43 9 n 49 9 n 97 -1 9 n 173 9 n 43 [---- + ---, ---- + ----, ---- + ----, ----, - ---- - ----, - ---- - ---, 1000 625 1000 1250 1000 2500 1250 1000 2500 1000 625 9 n 49 9 n 97 9 n 173 - ---- - ----, - ---- - ----, 1/1250, ---- + ----] 1000 1250 1000 2500 1000 2500 and in Maple format [9/1000*n+43/625, 9/1000*n+49/1250, 9/1000*n+97/2500, -1/1250, -9/1000*n-173/ 2500, -9/1000*n-43/625, -9/1000*n-49/1250, -9/1000*n-97/2500, 1/1250, 9/1000*n+ 173/2500] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 -1 -1 -1 [0, 1/54, 0, --, 0, --, 0, --, 0, 1/54, 0, 1/27] 54 27 54 and in Maple format [0, 1/54, 0, -1/54, 0, -1/27, 0, -1/54, 0, 1/54, 0, 1/27] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 14 -2 -3 -3 -2 -2 -3 -3 [--, 2/49, 3/98, --, 3/98, --, --, 2/49, --, --, 3/98, --, 3/98, 2/49] 49 98 98 49 49 98 98 and in Maple format [-2/49, 2/49, 3/98, -3/98, 3/98, -3/98, -2/49, 2/49, -2/49, -3/98, 3/98, -3/98, 3/98, 2/49] Q[13](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 16 -1 -1 -1 -1 [--, 1/8, --, 1/16, 0, 0, 0, --, 1/16, -1/8, 1/16, --, 0, 0, 0, 1/16] 16 16 16 16 and in Maple format [-1/16, 1/8, -1/16, 1/16, 0, 0, 0, -1/16, 1/16, -1/8, 1/16, -1/16, 0, 0, 0, 1/ 16] Q[14](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 18 -1 -1 -1 -1 -1 -1 [0, 1/18, --, 1/18, 0, --, 1/18, --, 1/54, 0, --, 1/27, --, 0, 1/54, --, 1/18, 27 54 18 18 18 18 -1 --] 54 and in Maple format [0, 1/18, -1/27, 1/18, 0, -1/54, 1/18, -1/18, 1/54, 0, -1/18, 1/27, -1/18, 0, 1 /54, -1/18, 1/18, -1/54] Q[15](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 20 -1 -1 -1 -1 -1 -3 -1 -3 [--, 3/50, --, 1/25, --, 3/50, --, 1/25, 0, --, 1/20, --, 1/20, --, 1/20, --, 20 20 20 20 25 50 25 50 -1 1/20, --, 0, 1/25] 25 and in Maple format [-1/20, 3/50, -1/20, 1/25, -1/20, 3/50, -1/20, 1/25, 0, -1/25, 1/20, -3/50, 1/ 20, -1/25, 1/20, -3/50, 1/20, -1/25, 0, 1/25] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 1029548067040490571800279325264628027708568566887202983616857808509777939525147\ 0501323528954334362979322162535846689641791787588377829412514891644815219183237\ 9957012526089539610997576899987246842039769145365740481783348341622886008541505\ 3081131858321746622746034856558078192410937854794409762075865619090023700327324\ 5740911819689009577310576722687245908880241625685482240449906528683718572019571\ 6721022656987990598045345252870238173001253542111860747957161831353483322913070\ 1205951058978689784113896403801128790565151529290271206790901088373163623016650\ 6474560560392424002065389484554681497656285934296261916474980481447265691827714\ 1675142850921804478535813044925331815396483238928859152257153628905907842103462\ 3268603280361069896637209982301105170005816684241140443374423386181175716743030\ 0884913116457717060303774001598810534730581761739904009157389521880997483584197\ 2820220909580110050421636415584203503074755279340817259688511893097431016302268\ 506854044772916025120610658529529781734367273 ------------------------------------ --------------------------------------------------------- For , psi[12](t), we have the following theorem. Theorem Number, 11, : Let f(t) be the constant term of the rational function in z and t 2 / / 2 4 6 8 (1 + z) / |2 z (1 - t) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 10 12 / t \ / t \ / t \ / t \ (-t z + 1) (-t z + 1) |1 - ----| |1 - ----| |1 - ----| |1 - ----| | 2 | | 4 | | 6 | | 8 | \ z / \ z / \ z / \ z / / t \ / t \\ |1 - ---| |1 - ---|| | 10| | 12|| \ z / \ z // We have 50 48 47 46 45 44 43 42 f(t) = (t + 4 t + 17 t + 49 t + 124 t + 285 t + 590 t + 1126 t 41 40 39 38 37 36 + 2014 t + 3393 t + 5408 t + 8227 t + 11978 t + 16755 t 35 34 33 32 31 30 + 22586 t + 29422 t + 37090 t + 45349 t + 53840 t + 62144 t 29 28 27 26 25 24 + 69792 t + 76340 t + 81349 t + 84505 t + 85578 t + 84505 t 23 22 21 20 19 18 + 81349 t + 76340 t + 69792 t + 62144 t + 53840 t + 45349 t 17 16 15 14 13 12 + 37090 t + 29422 t + 22586 t + 16755 t + 11978 t + 8227 t 11 10 9 8 7 6 5 + 5408 t + 3393 t + 2014 t + 1126 t + 590 t + 285 t + 124 t 4 3 2 / 12 2 3 5 + 49 t + 17 t + 4 t + 1) / ((-1 + t) (t + t + 1) (t + 1) / 6 5 4 3 2 4 3 2 2 2 2 (t + t + t + t + t + t + 1) (t + t + t + t + 1) (t + 1) 10 9 8 7 6 5 4 3 2 2 (t + t + t + t + t + t + t + t + t + t + 1) (t - t + 1) 6 3 4 (t + t + 1) (t + 1)) and in Maple notation (t^50+4*t^48+17*t^47+49*t^46+124*t^45+285*t^44+590*t^43+1126*t^42+2014*t^41+ 3393*t^40+5408*t^39+8227*t^38+11978*t^37+16755*t^36+22586*t^35+29422*t^34+37090 *t^33+45349*t^32+53840*t^31+62144*t^30+69792*t^29+76340*t^28+81349*t^27+84505*t ^26+85578*t^25+84505*t^24+81349*t^23+76340*t^22+69792*t^21+62144*t^20+53840*t^ 19+45349*t^18+37090*t^17+29422*t^16+22586*t^15+16755*t^14+11978*t^13+8227*t^12+ 5408*t^11+3393*t^10+2014*t^9+1126*t^8+590*t^7+285*t^6+124*t^5+49*t^4+17*t^3+4*t ^2+1)/(-1+t)^12/(t^2+t+1)^3/(t+1)^5/(t^6+t^5+t^4+t^3+t^2+t+1)/(t^4+t^3+t^2+t+1) ^2/(t^2+1)^2/(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^2-t+1)/(t^6+t^3+1)/( t^4+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 7, 25, 86, 252, 676, 1656, 3788, 8150, 16660, 32540, 61108, 110780, 194668, 332578, 553981, 901723, 1437269, 2247151, 3451798, 5216252, 7764392, 11396054, 16509188, 23626234, 33427622, 46791278, 64841876, 89008530, 121095602] Furthermore, a(n) is a quasi-polynomial given as sum of, 10, quasi-polynomials 10 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), are defined as followed Q[1](n), is the polynomial 655177 11 8517301 10 4509557 9 61004437 8 --------------- n + --------------- n + ------------- n + ------------- n 796675461120000 144850083840000 2414168064000 1755758592000 337767547 7 135254587 6 118750282837 5 45503350373 4 + ------------ n + ----------- n + ------------- n + ------------ n 804722688000 39191040000 6035420160000 585252864000 674840112907 3 395130194693 2 1523477894647 2306847774201139 + ------------- n + ------------- n + ------------- n + ---------------- 3218890752000 1053455155200 3448811520000 6373403688960000 and in Maple notation 655177/796675461120000*n^11+8517301/144850083840000*n^10+4509557/2414168064000* n^9+61004437/1755758592000*n^8+337767547/804722688000*n^7+135254587/39191040000 *n^6+118750282837/6035420160000*n^5+45503350373/585252864000*n^4+674840112907/ 3218890752000*n^3+395130194693/1053455155200*n^2+1523477894647/3448811520000*n+ 2306847774201139/6373403688960000 This is the leading term in particular, a(n) , is asymptotic to 11 655177 n --------------- 796675461120000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 23 4 299 3 6875 2 1079 1404695 [- ------- n - ------ n - ------- n - ----- n - -------, 1179648 589824 1179648 32768 7077888 23 4 299 3 6875 2 1079 1404695 ------- n + ------ n + ------- n + ----- n + -------] 1179648 589824 1179648 32768 7077888 and in Maple format [-23/1179648*n^4-299/589824*n^3-6875/1179648*n^2-1079/32768*n-1404695/7077888, 23/1179648*n^4+299/589824*n^3+6875/1179648*n^2+1079/32768*n+1404695/7077888] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 5 n 65 13 2 82 1387 13 2 29 1517 [- ---- - -----, - ----- n - ---- n - -----, ----- n + ---- n + -----] 6561 13122 13122 6561 26244 13122 2187 26244 and in Maple format [-5/6561*n-65/13122, -13/13122*n^2-82/6561*n-1387/26244, 13/13122*n^2+29/2187*n +1517/26244] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 3 n 3 n 21 3 n 3 n 21 [- ---- - 9/512, - ---- - ----, ---- + 9/512, ---- + ----] 1024 1024 1024 1024 1024 1024 and in Maple format [-3/1024*n-9/512, -3/1024*n-21/1024, 3/1024*n+9/512, 3/1024*n+21/1024] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 4 n 26 4 n 66 4 n 14 4 n 26 [0, --- + ---, - --- - ---, - --- + ---, --- + ---] 625 625 625 625 625 625 625 625 and in Maple format [0, 4/625*n+26/625, -4/625*n-66/625, -4/625*n+14/625, 4/625*n+26/625] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 -1 -1 -1 [1/54, 1/108, ---, --, ---, 1/108] 108 54 108 and in Maple format [1/54, 1/108, -1/108, -1/54, -1/108, 1/108] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 -6 [--, 0, 0, 0, 0, 0, 6/49] 49 and in Maple format [-6/49, 0, 0, 0, 0, 0, 6/49] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 -1 -1 -1 -1 [--, 1/32, --, --, 1/32, --, 1/32, 1/32] 32 32 32 32 and in Maple format [-1/32, 1/32, -1/32, -1/32, 1/32, -1/32, 1/32, 1/32] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -2 -2 -2 [--, 2/27, --, 2/27, --, 0, 0, 0, 2/27] 27 27 27 and in Maple format [-2/27, 2/27, -2/27, 2/27, -2/27, 0, 0, 0, 2/27] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 11 -2 -4 -6 -8 -10 10 [---, 8/121, ---, 6/121, ---, 4/121, ---, 2/121, ---, 0, ---] 121 121 121 121 121 121 and in Maple format [-2/121, 8/121, -4/121, 6/121, -6/121, 4/121, -8/121, 2/121, -10/121, 0, 10/121 ] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 8223888295478870541667826687333929063393015079188987585118203471044046106843391\ 8628991046285685803135549644629676930195341824864565498417243380099454066639144\ 9353343426348610464905692312030829480476720746445576174746181693966419528922869\ 2039872628830995567140081062372762442240289495914529317340845382900510919655323\ 3483381524640677010940492315084800788397603105528924565854713895973048343311828\ 8837332418275049781917399896128985969186920498982922156456441101836858343725961\ 7046907945309954823420563402596295597070542289931953766062044614767611935053546\ 9396032701040450492644389282730365515894713420608398517658110945482010630853557\ 4742618727440490040030417348572444505318215969704398970606022367775869275665936\ 0528742175901600838803553784046542317078365392191483795353177000336425273627988\ 6294970494870713158084223233066159686864085346261606215643947459457052744423809\ 5726524992732036716858860366252698670794977780173654258417232069094609846039486\ 3580155654336617406469365210212839949358575249325107045164765222485280205337925\ 3956454533655110855688056265256842457419657996858574059151259729 ------------------------------------ --------------------------------------------------------- For , psi[13](t), we have the following theorem. Theorem Number, 12, : Let f(t) be the constant term of the rational function in z and t 2 / / 3 5 7 9 (1 + z) / |2 z (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 11 13 / t \ / t \ / t \ (-t z + 1) (-t z + 1) (1 - t/z) |1 - ----| |1 - ----| |1 - ----| | 3 | | 5 | | 7 | \ z / \ z / \ z / / t \ / t \ / t \\ |1 - ----| |1 - ---| |1 - ---|| | 9 | | 11| | 13|| \ z / \ z / \ z // We have 142 141 140 139 138 137 136 f(t) = - (t - t + 6 t + 15 t + 54 t + 146 t + 367 t 135 134 133 132 131 130 + 856 t + 1832 t + 3692 t + 7119 t + 13044 t + 23063 t 129 128 127 126 125 + 39090 t + 64715 t + 103384 t + 161966 t + 246071 t 124 123 122 121 120 + 368602 t + 537454 t + 774602 t + 1089888 t + 1520199 t 119 118 117 116 115 + 2074893 t + 2811903 t + 3735750 t + 4936965 t + 6405528 t 114 113 112 111 + 8276196 t + 10511120 t + 13309479 t + 16582517 t 110 109 108 107 + 20614251 t + 25236133 t + 30850246 t + 37165105 t 106 105 104 103 + 44733003 t + 53090558 t + 62990057 t + 73729232 t 102 101 100 99 + 86306486 t + 99711441 t + 115255899 t + 131533708 t 98 97 96 95 + 150228851 t + 169457708 t + 191357334 t + 213470586 t 94 93 92 91 + 238450991 t + 263190673 t + 290944549 t + 317868327 t 90 89 88 87 + 347873047 t + 376328269 t + 407872001 t + 437036713 t 86 85 84 83 + 469215173 t + 498101569 t + 529881906 t + 557414441 t 82 81 80 79 + 587659913 t + 612703335 t + 640269158 t + 661732364 t 78 77 76 75 + 685504885 t + 702377539 t + 721381446 t + 732839945 t 74 73 72 71 + 746270269 t + 751700367 t + 759014750 t + 758094326 t 70 69 68 67 + 759014750 t + 751700367 t + 746270269 t + 732839945 t 66 65 64 63 + 721381446 t + 702377539 t + 685504885 t + 661732364 t 62 61 60 59 + 640269158 t + 612703335 t + 587659913 t + 557414441 t 58 57 56 55 + 529881906 t + 498101569 t + 469215173 t + 437036713 t 54 53 52 51 + 407872001 t + 376328269 t + 347873047 t + 317868327 t 50 49 48 47 + 290944549 t + 263190673 t + 238450991 t + 213470586 t 46 45 44 43 + 191357334 t + 169457708 t + 150228851 t + 131533708 t 42 41 40 39 + 115255899 t + 99711441 t + 86306486 t + 73729232 t 38 37 36 35 34 + 62990057 t + 53090558 t + 44733003 t + 37165105 t + 30850246 t 33 32 31 30 29 + 25236133 t + 20614251 t + 16582517 t + 13309479 t + 10511120 t 28 27 26 25 24 + 8276196 t + 6405528 t + 4936965 t + 3735750 t + 2811903 t 23 22 21 20 19 + 2074893 t + 1520199 t + 1089888 t + 774602 t + 537454 t 18 17 16 15 14 + 368602 t + 246071 t + 161966 t + 103384 t + 64715 t 13 12 11 10 9 8 + 39090 t + 23063 t + 13044 t + 7119 t + 3692 t + 1832 t 7 6 5 4 3 2 / 13 + 856 t + 367 t + 146 t + 54 t + 15 t + 6 t - t + 1) / ((-1 + t) / 10 9 8 7 6 5 4 3 2 11 (t + t + t + t + t + t + t + t + t + t + 1) (t + 1) 10 9 8 7 6 5 4 3 2 8 4 3 (t - t + t - t + t - t + t - t + t - t + 1) (t + 1) (t + 1) 2 6 2 4 2 4 4 3 2 2 (t + 1) (t + t + 1) (t - t + 1) (t + t + t + t + 1) 4 3 2 2 4 2 2 6 5 4 3 2 (t - t + t - t + 1) (t - t + 1) (t + t + t + t + t + t + 1) 6 5 4 3 2 8 6 4 2 6 3 (t - t + t - t + t - t + 1) (t - t + t - t + 1) (t + t + 1) 6 3 8 4 (t - t + 1) (t - t + 1)) and in Maple notation -(t^142-t^141+6*t^140+15*t^139+54*t^138+146*t^137+367*t^136+856*t^135+1832*t^ 134+3692*t^133+7119*t^132+13044*t^131+23063*t^130+39090*t^129+64715*t^128+ 103384*t^127+161966*t^126+246071*t^125+368602*t^124+537454*t^123+774602*t^122+ 1089888*t^121+1520199*t^120+2074893*t^119+2811903*t^118+3735750*t^117+4936965*t ^116+6405528*t^115+8276196*t^114+10511120*t^113+13309479*t^112+16582517*t^111+ 20614251*t^110+25236133*t^109+30850246*t^108+37165105*t^107+44733003*t^106+ 53090558*t^105+62990057*t^104+73729232*t^103+86306486*t^102+99711441*t^101+ 115255899*t^100+131533708*t^99+150228851*t^98+169457708*t^97+191357334*t^96+ 213470586*t^95+238450991*t^94+263190673*t^93+290944549*t^92+317868327*t^91+ 347873047*t^90+376328269*t^89+407872001*t^88+437036713*t^87+469215173*t^86+ 498101569*t^85+529881906*t^84+557414441*t^83+587659913*t^82+612703335*t^81+ 640269158*t^80+661732364*t^79+685504885*t^78+702377539*t^77+721381446*t^76+ 732839945*t^75+746270269*t^74+751700367*t^73+759014750*t^72+758094326*t^71+ 759014750*t^70+751700367*t^69+746270269*t^68+732839945*t^67+721381446*t^66+ 702377539*t^65+685504885*t^64+661732364*t^63+640269158*t^62+612703335*t^61+ 587659913*t^60+557414441*t^59+529881906*t^58+498101569*t^57+469215173*t^56+ 437036713*t^55+407872001*t^54+376328269*t^53+347873047*t^52+317868327*t^51+ 290944549*t^50+263190673*t^49+238450991*t^48+213470586*t^47+191357334*t^46+ 169457708*t^45+150228851*t^44+131533708*t^43+115255899*t^42+99711441*t^41+ 86306486*t^40+73729232*t^39+62990057*t^38+53090558*t^37+44733003*t^36+37165105* t^35+30850246*t^34+25236133*t^33+20614251*t^32+16582517*t^31+13309479*t^30+ 10511120*t^29+8276196*t^28+6405528*t^27+4936965*t^26+3735750*t^25+2811903*t^24+ 2074893*t^23+1520199*t^22+1089888*t^21+774602*t^20+537454*t^19+368602*t^18+ 246071*t^17+161966*t^16+103384*t^15+64715*t^14+39090*t^13+23063*t^12+13044*t^11 +7119*t^10+3692*t^9+1832*t^8+856*t^7+367*t^6+146*t^5+54*t^4+15*t^3+6*t^2-t+1)/( -1+t)^13/(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t+1)^11/(t^10-t^9+t^8-t^7+ t^6-t^5+t^4-t^3+t^2-t+1)/(t^8+1)/(t^4+1)^3/(t^2+1)^6/(t^2+t+1)^4/(t^2-t+1)^4/(t ^4+t^3+t^2+t+1)^2/(t^4-t^3+t^2-t+1)^2/(t^4-t^2+1)^2/(t^6+t^5+t^4+t^3+t^2+t+1)/( t^6-t^5+t^4-t^3+t^2-t+1)/(t^8-t^6+t^4-t^2+1)/(t^6+t^3+1)/(t^6-t^3+1)/(t^8-t^4+1 ) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 7, 28, 104, 325, 920, 2385, 5744, 12954, 27718, 56556, 110780, 208960, 381676, 676572, 1168261, 1967888, 3243531, 5236679, 8300708, 12927670, 19819846, 29929658, 44585180, 65545873, 95220378, 136735590, 194299052, 273274429, 380777798] Furthermore, a(n) is a quasi-polynomial given as sum of, 18, quasi-polynomials 18 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), Q[13](n), Q[14](n), Q[15](n), Q[16](n), Q[17](n), Q[18](n), are defined as followed Q[1](n), is the polynomial 1588223323 12 1588223323 11 195864868861 10 -------------------- n + ------------------ n + ------------------- n 26105461509980160000 310779303690240000 1017095902986240000 5677646339 9 6121382313119 8 519543829819 7 + ---------------- n + ----------------- n + --------------- n 1320903770112000 96866276474880000 807218970624000 16545679137850717 6 25043104448651 5 2185954508593259 4 + ------------------- n + ---------------- n + ----------------- n 3559835660451840000 1046394961920000 25113479086080000 3163385451112807 3 4543914539309773 2 8276233381674013 + ----------------- n + ----------------- n + ----------------- n 14529941471232000 12786348494684160 24216569118720000 3735590085437632039577 + ----------------------- 16446440751287500800000 and in Maple notation 1588223323/26105461509980160000*n^12+1588223323/310779303690240000*n^11+ 195864868861/1017095902986240000*n^10+5677646339/1320903770112000*n^9+ 6121382313119/96866276474880000*n^8+519543829819/807218970624000*n^7+ 16545679137850717/3559835660451840000*n^6+25043104448651/1046394961920000*n^5+ 2185954508593259/25113479086080000*n^4+3163385451112807/14529941471232000*n^3+ 4543914539309773/12786348494684160*n^2+8276233381674013/24216569118720000*n+ 3735590085437632039577/16446440751287500800000 This is the leading term in particular, a(n) , is asymptotic to 12 1588223323 n -------------------- 26105461509980160000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 162263 10 1135841 9 21977191 8 [- ----------------- n - ---------------- n - ---------------- n 48433138237440000 4843313823744000 3013617490329600 107033029 7 264629843671 6 50227191697 5 - --------------- n - ------------------ n - ---------------- n 807218970624000 169515983831040000 4036094853120000 29341123057 4 1231149112627 3 1326224537681 2 - --------------- n - ---------------- n - ---------------- n 430516784332800 4843313823744000 1506808745164800 109512602534371 5216075321438620739 162263 10 - ----------------- n - --------------------, ----------------- n 24216569118720000 31326553811976192000 48433138237440000 1135841 9 21977191 8 107033029 7 + ---------------- n + ---------------- n + --------------- n 4843313823744000 3013617490329600 807218970624000 264629843671 6 50227191697 5 29341123057 4 + ------------------ n + ---------------- n + --------------- n 169515983831040000 4036094853120000 430516784332800 1231149112627 3 1326224537681 2 109512602534371 + ---------------- n + ---------------- n + ----------------- n 4843313823744000 1506808745164800 24216569118720000 5216075321438620739 + --------------------] 31326553811976192000 and in Maple format [-162263/48433138237440000*n^10-1135841/4843313823744000*n^9-21977191/ 3013617490329600*n^8-107033029/807218970624000*n^7-264629843671/ 169515983831040000*n^6-50227191697/4036094853120000*n^5-29341123057/ 430516784332800*n^4-1231149112627/4843313823744000*n^3-1326224537681/ 1506808745164800*n^2-109512602534371/24216569118720000*n-5216075321438620739/ 31326553811976192000, 162263/48433138237440000*n^10+1135841/4843313823744000*n^ 9+21977191/3013617490329600*n^8+107033029/807218970624000*n^7+264629843671/ 169515983831040000*n^6+50227191697/4036094853120000*n^5+29341123057/ 430516784332800*n^4+1231149112627/4843313823744000*n^3+1326224537681/ 1506808745164800*n^2+109512602534371/24216569118720000*n+5216075321438620739/ 31326553811976192000] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 277 3 5675 2 37763 243881 [- ------- n - ------- n - ------- n - --------, 5038848 5038848 5038848 15116544 71 2 497 2603 - ------- n - ------ n - ------, 1259712 629856 944784 277 3 5959 2 13913 285529 ------- n + ------- n + ------- n + --------] 5038848 5038848 1679616 15116544 and in Maple format [-277/5038848*n^3-5675/5038848*n^2-37763/5038848*n-243881/15116544, -71/1259712 *n^2-497/629856*n-2603/944784, 277/5038848*n^3+5959/5038848*n^2+13913/1679616*n +285529/15116544] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 145 4 1015 3 4037 2 265643 12794389 [--------- n + -------- n + -------- n + --------- n + ----------, 226492416 56623104 20971520 283115520 1132462080 31927 5 223489 4 454811 3 539 2 23676751 - ---------- n - ---------- n - --------- n - ----- n - --------- n 5662310400 1132462080 169869312 30720 424673280 24710749 - ---------, 353894400 145 4 1015 3 4037 2 265643 12794389 - --------- n - -------- n - -------- n - --------- n - ----------, 226492416 56623104 20971520 283115520 1132462080 31927 5 223489 4 454811 3 539 2 23676751 ---------- n + ---------- n + --------- n + ----- n + --------- n 5662310400 1132462080 169869312 30720 424673280 24710749 + ---------] 353894400 and in Maple format [145/226492416*n^4+1015/56623104*n^3+4037/20971520*n^2+265643/283115520*n+ 12794389/1132462080, -31927/5662310400*n^5-223489/1132462080*n^4-454811/ 169869312*n^3-539/30720*n^2-23676751/424673280*n-24710749/353894400, -145/ 226492416*n^4-1015/56623104*n^3-4037/20971520*n^2-265643/283115520*n-12794389/ 1132462080, 31927/5662310400*n^5+223489/1132462080*n^4+454811/169869312*n^3+539 /30720*n^2+23676751/424673280*n+24710749/353894400] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 23 n 199 n 23 -33 n 47 23 n 407 [- ---- - ----, - ---- - -----, -----, ---- + -----, ---- + -----] 5000 6250 2500 12500 12500 2500 12500 5000 12500 and in Maple format [-23/5000*n-199/6250, -1/2500*n-23/12500, -33/12500, 1/2500*n+47/12500, 23/5000 *n+407/12500] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 277 3 1901 2 38231 251741 19 2 133 455 [------- n + ------ n + ------- n + -------, - ------ n - ----- n - -----, 1679616 559872 1679616 5038848 139968 69984 78732 277 3 659 2 41423 280861 - ------- n - ------ n - ------- n - -------, 1679616 186624 1679616 5038848 277 3 1901 2 38231 251741 19 2 133 455 - ------- n - ------ n - ------- n - -------, ------ n + ----- n + -----, 1679616 559872 1679616 5038848 139968 69984 78732 277 3 659 2 41423 280861 ------- n + ------ n + ------- n + -------] 1679616 186624 1679616 5038848 and in Maple format [277/1679616*n^3+1901/559872*n^2+38231/1679616*n+251741/5038848, -19/139968*n^2 -133/69984*n-455/78732, -277/1679616*n^3-659/186624*n^2-41423/1679616*n-280861/ 5038848, -277/1679616*n^3-1901/559872*n^2-38231/1679616*n-251741/5038848, 19/ 139968*n^2+133/69984*n+455/78732, 277/1679616*n^3+659/186624*n^2+41423/1679616* n+280861/5038848] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 -5 -5 -5 -5 -5 -5 15 [---, ---, ---, ---, ---, ---, ---] 686 686 686 686 686 686 343 and in Maple format [-5/686, -5/686, -5/686, -5/686, -5/686, -5/686, 15/343] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 47 2 329 6965 47 2 11 1121 5 n 35 [- ----- n - ----- n - ------, ----- n + ---- n + -----, - ---- - ----, 49152 24576 147456 49152 1024 36864 6144 6144 47 2 197 1243 47 2 329 6965 - ----- n - ----- n - -----, ----- n + ----- n + ------, 49152 12288 18432 49152 24576 147456 47 2 11 1121 5 n 35 47 2 197 1243 - ----- n - ---- n - -----, ---- + ----, ----- n + ----- n + -----] 49152 1024 36864 6144 6144 49152 12288 18432 and in Maple format [-47/49152*n^2-329/24576*n-6965/147456, 47/49152*n^2+11/1024*n+1121/36864, -5/ 6144*n-35/6144, -47/49152*n^2-197/12288*n-1243/18432, 47/49152*n^2+329/24576*n+ 6965/147456, -47/49152*n^2-11/1024*n-1121/36864, 5/6144*n+35/6144, 47/49152*n^2 +197/12288*n+1243/18432] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -1 -1 -1 -1 -1 -1 [--, 2/81, --, 2/81, --, --, --, --, 2/81] 81 81 81 81 81 81 and in Maple format [-1/81, 2/81, -1/81, 2/81, -1/81, -1/81, -1/81, -1/81, 2/81] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 10 41 n 71 n 19 n 41 n 29 41 n 71 [---- + ----, ---- + 3/500, ----, - ---- + 1/2500, - ---- - ---, - ---- - ----, 5000 1250 2500 2500 2500 5000 500 5000 1250 n -19 n 41 n 29 - ---- - 3/500, ----, ---- - 1/2500, ---- + ---] 2500 2500 2500 5000 500 and in Maple format [41/5000*n+71/1250, 1/2500*n+3/500, 19/2500, -1/2500*n+1/2500, -41/5000*n-29/ 500, -41/5000*n-71/1250, -1/2500*n-3/500, -19/2500, 1/2500*n-1/2500, 41/5000*n+ 29/500] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 11 -3 -1 -3 -5 -5 [0, 5/242, ---, ---, ---, 5/242, 0, 2/121, ---, ---, 2/121] 242 121 242 242 242 and in Maple format [0, 5/242, -3/242, -1/121, -3/242, 5/242, 0, 2/121, -5/242, -5/242, 2/121] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 11 n 11 n 77 11 n 73 11 n -1 11 n 19 [---- + 3/64, ---- + ----, ---- + ----, ---- + 5/324, ---, - ---- - ---, 1728 1296 1296 1728 1728 2592 216 2592 432 11 n 11 n 77 11 n 73 11 n - ---- - 3/64, - ---- - ----, - ---- - ----, - ---- - 5/324, 1/216, 1728 1296 1296 1728 1728 2592 11 n 19 ---- + ---] 2592 432 and in Maple format [11/1728*n+3/64, 11/1296*n+77/1296, 11/1728*n+73/1728, 11/2592*n+5/324, -1/216, -11/2592*n-19/432, -11/1728*n-3/64, -11/1296*n-77/1296, -11/1728*n-73/1728, -11 /2592*n-5/324, 1/216, 11/2592*n+19/432] Q[13](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 14 -1 -1 -1 -3 -1 -1 -1 [1/98, --, 1/98, --, 1/98, --, --, --, 1/98, --, 1/98, --, 1/98, 3/49] 98 98 98 49 98 98 98 and in Maple format [1/98, -1/98, 1/98, -1/98, 1/98, -1/98, -3/49, -1/98, 1/98, -1/98, 1/98, -1/98, 1/98, 3/49] Q[14](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 16 -1 -1 -1 [--, 1/16, 0, 0, 0, 0, 0, --, 1/16, --, 0, 0, 0, 0, 0, 1/16] 16 16 16 and in Maple format [-1/16, 1/16, 0, 0, 0, 0, 0, -1/16, 1/16, -1/16, 0, 0, 0, 0, 0, 1/16] Q[15](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 18 -1 -1 -1 -1 -2 -1 -1 [--, 2/27, --, 0, 1/27, --, 1/27, --, 0, 1/27, --, 1/27, 0, --, 1/27, --, 1/27, 27 27 27 27 27 27 27 0] and in Maple format [-1/27, 2/27, -1/27, 0, 1/27, -1/27, 1/27, -1/27, 0, 1/27, -2/27, 1/27, 0, -1/ 27, 1/27, -1/27, 1/27, 0] Q[16](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 20 -1 -1 -1 -1 -3 -3 -1 [--, 3/25, --, 3/25, --, 1/25, 0, 0, 0, --, 1/20, --, 1/10, --, 1/20, --, 0, 0, 20 10 20 25 25 25 25 0, 1/25] and in Maple format [-1/20, 3/25, -1/10, 3/25, -1/20, 1/25, 0, 0, 0, -1/25, 1/20, -3/25, 1/10, -3/ 25, 1/20, -1/25, 0, 0, 0, 1/25] Q[17](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 22 -7 -7 -2 15 -15 -2 -7 [2/121, 7/242, ---, 9/121, ---, 7/242, 2/121, ---, ---, ---, 2/121, ---, ---, 242 242 121 242 242 121 242 -9 -7 -2 -15 15 -2 7/242, ---, 7/242, ---, ---, 2/121, ---, ---, ---] 121 242 121 242 242 121 and in Maple format [2/121, 7/242, -7/242, 9/121, -7/242, 7/242, 2/121, -2/121, 15/242, -15/242, 2/ 121, -2/121, -7/242, 7/242, -9/121, 7/242, -7/242, -2/121, 2/121, -15/242, 15/ 242, -2/121] Q[18](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 24 -1 -1 -1 -1 -1 -1 -1 -1 [--, 1/36, --, 1/18, --, 1/18, --, 1/36, --, 1/36, 0, --, 1/36, --, 1/24, --, 36 24 18 24 36 36 36 18 -1 -1 -1 1/18, --, 1/24, --, 1/36, --, 0, 1/36] 18 36 36 and in Maple format [-1/36, 1/36, -1/24, 1/18, -1/18, 1/18, -1/24, 1/36, -1/36, 1/36, 0, -1/36, 1/ 36, -1/36, 1/24, -1/18, 1/18, -1/18, 1/24, -1/36, 1/36, -1/36, 0, 1/36] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 6083873761024372867155594812454413758459506949699884337263419086945354916067331\ 2241908254948712434125408356662798875827887630770961432069870956087620082061854\ 2814042787320538376067209373820458209339633561870922273403401163030080066404510\ 1363612756198222857763591378399067280830236667266578370563978476681280424768506\ 2257671896832461918952931369869712733981524176240296882343450315944214364038716\ 1091260455453154099046974630967113107743396521833448418523151746033117062636240\ 3035661190400826622588079065836434654244100670445244937967578838367857305024390\ 3396238030054145351742130820391380133426006382390926335525002662370609923742662\ 0605855636942519881254671841314279808570154712352706402552328200886439720341732\ 1860906739676803632052772487098106608911304901186076252436503981940712562628369\ 8166705942773588981172978108826896527536080027484370740565848612808511620262283\ 8647993191666253857836772200487794534870786808149856624016209286905668713161631\ 7656614808023070542444172928914524767231734752132108025031973511477546537137152\ 2833229850492423310551684235613475098270138620734526885988593009904116752251984\ 6464763002272132373855068170355078234578706671771357428635678506520969956025983\ 67461 ------------------------------------ --------------------------------------------------------- For , psi[14](t), we have the following theorem. Theorem Number, 13, : Let f(t) be the constant term of the rational function in z and t 2 / / 2 4 6 8 (1 + z) / |2 z (1 - t) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 10 12 14 / t \ / t \ / t \ (-t z + 1) (-t z + 1) (-t z + 1) |1 - ----| |1 - ----| |1 - ----| | 2 | | 4 | | 6 | \ z / \ z / \ z / / t \ / t \ / t \ / t \\ |1 - ----| |1 - ---| |1 - ---| |1 - ---|| | 8 | | 10| | 12| | 14|| \ z / \ z / \ z / \ z // We have 78 76 75 74 73 72 71 70 f(t) = (t + 5 t + 23 t + 80 t + 234 t + 617 t + 1466 t + 3230 t 69 68 67 66 65 64 + 6623 t + 12823 t + 23526 t + 41265 t + 69375 t + 112446 t 63 62 61 60 59 + 176099 t + 267440 t + 394590 t + 567091 t + 794896 t 58 57 56 55 54 + 1088804 t + 1458912 t + 1914914 t + 2464162 t + 3112079 t 53 52 51 50 49 + 3859790 t + 4705097 t + 5639973 t + 6652162 t + 7723070 t 48 47 46 45 44 + 8830269 t + 9945630 t + 11039080 t + 12077027 t + 13026793 t 43 42 41 40 39 + 13855414 t + 14534372 t + 15038019 t + 15348416 t + 15452946 t 38 37 36 35 34 + 15348416 t + 15038019 t + 14534372 t + 13855414 t + 13026793 t 33 32 31 30 29 + 12077027 t + 11039080 t + 9945630 t + 8830269 t + 7723070 t 28 27 26 25 24 + 6652162 t + 5639973 t + 4705097 t + 3859790 t + 3112079 t 23 22 21 20 19 + 2464162 t + 1914914 t + 1458912 t + 1088804 t + 794896 t 18 17 16 15 14 + 567091 t + 394590 t + 267440 t + 176099 t + 112446 t 13 12 11 10 9 8 + 69375 t + 41265 t + 23526 t + 12823 t + 6623 t + 3230 t 7 6 5 4 3 2 / 2 2 + 1466 t + 617 t + 234 t + 80 t + 23 t + 5 t + 1) / ((t - t + 1) / 2 4 7 2 3 14 6 3 4 (t + t + 1) (t + 1) (t + 1) (-1 + t) (t + t + 1) (t + 1) 4 2 10 9 8 7 6 5 4 3 2 (t - t + 1) (t + t + t + t + t + t + t + t + t + t + 1) 6 5 4 3 2 4 3 2 2 (t + t + t + t + t + t + 1) (t + t + t + t + 1) 12 11 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + t + t + 1) 4 3 2 (t - t + t - t + 1)) and in Maple notation (t^78+5*t^76+23*t^75+80*t^74+234*t^73+617*t^72+1466*t^71+3230*t^70+6623*t^69+ 12823*t^68+23526*t^67+41265*t^66+69375*t^65+112446*t^64+176099*t^63+267440*t^62 +394590*t^61+567091*t^60+794896*t^59+1088804*t^58+1458912*t^57+1914914*t^56+ 2464162*t^55+3112079*t^54+3859790*t^53+4705097*t^52+5639973*t^51+6652162*t^50+ 7723070*t^49+8830269*t^48+9945630*t^47+11039080*t^46+12077027*t^45+13026793*t^ 44+13855414*t^43+14534372*t^42+15038019*t^41+15348416*t^40+15452946*t^39+ 15348416*t^38+15038019*t^37+14534372*t^36+13855414*t^35+13026793*t^34+12077027* t^33+11039080*t^32+9945630*t^31+8830269*t^30+7723070*t^29+6652162*t^28+5639973* t^27+4705097*t^26+3859790*t^25+3112079*t^24+2464162*t^23+1914914*t^22+1458912*t ^21+1088804*t^20+794896*t^19+567091*t^18+394590*t^17+267440*t^16+176099*t^15+ 112446*t^14+69375*t^13+41265*t^12+23526*t^11+12823*t^10+6623*t^9+3230*t^8+1466* t^7+617*t^6+234*t^5+80*t^4+23*t^3+5*t^2+1)/(t^2-t+1)^2/(t^2+t+1)^4/(t+1)^7/(t^2 +1)^3/(-1+t)^14/(t^6+t^3+1)/(t^4+1)/(t^4-t^2+1)/(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t ^3+t^2+t+1)/(t^6+t^5+t^4+t^3+t^2+t+1)/(t^4+t^3+t^2+t+1)^2/(t^12+t^11+t^10+t^9+t ^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^4-t^3+t^2-t+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 8, 32, 126, 414, 1242, 3370, 8512, 20094, 44916, 95514, 194668, 381676, 723354, 1328980, 2374753, 4136477, 7040196, 11728606, 19159798, 30734578, 48479188, 75277670, 115195490, 173885716, 259140928, 381577586, 555546058, 800247348, 1141190188] Furthermore, a(n) is a quasi-polynomial given as sum of, 13, quasi-polynomials 13 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), Q[13](n), are defined as followed Q[1](n), is the polynomial 27085381 13 27085381 12 27145847 11 ------------------- n + ----------------- n + ---------------- n 6462631340605440000 66283398365184000 1506440871936000 2098109 10 4424080709 9 142313909 8 + ------------- n + --------------- n + ------------- n 4414478745600 529608119040000 1379524608000 1817892949133 7 554103740797 6 473555366540713 5 + ---------------- n + -------------- n + ----------------- n 1977203644416000 92704053657600 16738231910400000 1415057532029 4 68106992670098687 3 105593177409169 2 + -------------- n + ------------------ n + --------------- n 14714929152000 298275292643328000 291355597209600 5245185888906677 30031950115859018923 + ----------------- n + -------------------- 13834661068800000 90476838768476160000 and in Maple notation 27085381/6462631340605440000*n^13+27085381/66283398365184000*n^12+27145847/ 1506440871936000*n^11+2098109/4414478745600*n^10+4424080709/529608119040000*n^9 +142313909/1379524608000*n^8+1817892949133/1977203644416000*n^7+554103740797/ 92704053657600*n^6+473555366540713/16738231910400000*n^5+1415057532029/ 14714929152000*n^4+68106992670098687/298275292643328000*n^3+105593177409169/ 291355597209600*n^2+5245185888906677/13834661068800000*n+30031950115859018923/ 90476838768476160000 This is the leading term in particular, a(n) , is asymptotic to 13 27085381 n ------------------- 6462631340605440000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 841 6 841 5 54883 4 42595 3 2459587 2 [- ---------- n - -------- n - --------- n - -------- n - --------- n 2123366400 47185920 169869312 14155776 157286400 365159 54550099 841 6 841 5 54883 4 - ------- n - ---------, ---------- n + -------- n + --------- n 7864320 254803968 2123366400 47185920 169869312 42595 3 2459587 2 365159 54550099 + -------- n + --------- n + ------- n + ---------] 14155776 157286400 7864320 254803968 and in Maple format [-841/2123366400*n^6-841/47185920*n^5-54883/169869312*n^4-42595/14155776*n^3-\ 2459587/157286400*n^2-365159/7864320*n-54550099/254803968, 841/2123366400*n^6+ 841/47185920*n^5+54883/169869312*n^4+42595/14155776*n^3+2459587/157286400*n^2+ 365159/7864320*n+54550099/254803968] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 31 3 10 2 10915 11741 [- ------ n - ---- n - ------ n - ------, 472392 6561 944784 236196 31 3 25 2 9565 2011 - ------ n - ----- n - ------ n - ------, 472392 17496 944784 944784 31 3 155 2 1280 16325 ------ n + ----- n + ----- n + ------] 236196 52488 59049 314928 and in Maple format [-31/472392*n^3-10/6561*n^2-10915/944784*n-11741/236196, -31/472392*n^3-25/ 17496*n^2-9565/944784*n-2011/944784, 31/236196*n^3+155/52488*n^2+1280/59049*n+ 16325/314928] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 11 2 77 2849 11 2 11 209 [- ----- n - ----- n - -----, - ----- n - ---- n - ----, 24576 12288 73728 24576 1536 4608 11 2 77 2849 11 2 11 209 ----- n + ----- n + -----, ----- n + ---- n + ----] 24576 12288 73728 24576 1536 4608 and in Maple format [-11/24576*n^2-77/12288*n-2849/73728, -11/24576*n^2-11/1536*n-209/4608, 11/ 24576*n^2+77/12288*n+2849/73728, 11/24576*n^2+11/1536*n+209/4608] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 7 n 227 7 n 291 7 n 501 7 n 17 28 n 42 [- ---- - ----, - ---- + ----, - ---- - ----, - ---- + ----, ---- + ---] 3125 6250 3125 6250 3125 6250 3125 6250 3125 625 and in Maple format [-7/3125*n-227/6250, -7/3125*n+291/6250, -7/3125*n-501/6250, -7/3125*n+17/6250, 28/3125*n+42/625] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 5 n 31 5 n 101 -23 5 n 31 5 n 101 23 [---- + ---, ---- + ----, ----, - ---- - ---, - ---- - ----, ----] 1296 972 1296 3888 3888 1296 972 1296 3888 3888 and in Maple format [5/1296*n+31/972, 5/1296*n+101/3888, -23/3888, -5/1296*n-31/972, -5/1296*n-101/ 3888, 23/3888] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 12 -6 -12 -18 18 [---, 6/343, 0, ---, ---, ---, ---] 343 343 343 343 343 and in Maple format [12/343, 6/343, 0, -6/343, -12/343, -18/343, 18/343] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 -1 -1 [--, 0, 0, --, 1/32, 0, 0, 1/32] 32 32 and in Maple format [-1/32, 0, 0, -1/32, 1/32, 0, 0, 1/32] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -4 -2 -2 -2 [--, 4/81, --, 2/81, --, 0, 2/81, --, 2/27] 81 27 81 81 and in Maple format [-4/81, 4/81, -2/27, 2/81, -2/81, 0, 2/81, -2/81, 2/27] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 10 -1 -1 -1 -1 [1/50, 1/50, --, --, 0, --, --, 1/50, 1/50, 0] 50 50 50 50 and in Maple format [1/50, 1/50, -1/50, -1/50, 0, -1/50, -1/50, 1/50, 1/50, 0] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 11 -4 12 -20 20 -12 -10 10 [---, ---, ---, ---, ---, 4/121, ---, 0, 0, 0, ---] 121 121 121 121 121 121 121 and in Maple format [-4/121, 12/121, -20/121, 20/121, -12/121, 4/121, -10/121, 0, 0, 0, 10/121] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 -1 -1 -1 -1 -1 -1 [1/36, 1/36, --, 1/18, --, 1/36, --, --, 1/36, --, 1/18, --] 36 18 36 36 18 36 and in Maple format [1/36, 1/36, -1/36, 1/18, -1/18, 1/36, -1/36, -1/36, 1/36, -1/18, 1/18, -1/36] Q[13](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 13 -2 10 -4 -6 -8 -10 -12 12 [---, ---, ---, 8/169, ---, 6/169, ---, 4/169, ---, 2/169, ---, 0, ---] 169 169 169 169 169 169 169 169 and in Maple format [-2/169, 10/169, -4/169, 8/169, -6/169, 6/169, -8/169, 4/169, -10/169, 2/169, -\ 12/169, 0, 12/169] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 4191076292688939729827511456033825296935300786045377530709756782723596431150879\ 4460684819982586688169430280857502871385764898288040440812223213635244676905348\ 1956491155796651398443296531490915981614788429463646690585561042950018857513283\ 5262280576911076726761727631913687284927841353582964509761097020909002356872928\ 7307691071280020054154013473157977487667002541702601587846658276913892354692989\ 0322905027532675833163639824285991104997430176594861932386776084459288315331677\ 5478792702538321346700163665738108812431323482229594864102560338651636471329352\ 7506292336341117007181623863524496042421705208848708969234897190278408446222733\ 9664749730025756501077509026382936456141061973533767204855468346691537985175883\ 9242875227793046926033607521170842949332276322485146909138881631735958805453423\ 7320199022730981146385880132765555135986324759665208490666134465929941210425309\ 2515597532952638491175195231739107325584116315757135581376922373113457487963379\ 2241205730453434868687784822535172453070894979066234347900514707774989462078909\ 3094500144175271451809663092142452223223238738522331407335227121013043728187572\ 5201293747320847139834084041486802681451001319779742350052045582199666145637216\ 8287892017253129101630720855444136604807536201571652044492288055635675420684976\ 4176362067333856012875733 ------------------------------------ --------------------------------------------------------- For , psi[15](t), we have the following theorem. Theorem Number, 14, : Let f(t) be the constant term of the rational function in z and t 2 / / 3 5 7 9 (1 + z) / |2 z (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 11 13 15 / t \ / t \ (-t z + 1) (-t z + 1) (-t z + 1) (1 - t/z) |1 - ----| |1 - ----| | 3 | | 5 | \ z / \ z / / t \ / t \ / t \ / t \ / t \\ |1 - ----| |1 - ----| |1 - ---| |1 - ---| |1 - ---|| | 7 | | 9 | | 11| | 13| | 15|| \ z / \ z / \ z / \ z / \ z // We have 194 193 192 191 190 189 188 f(t) = - (t - t + 7 t + 21 t + 85 t + 258 t + 736 t 187 186 185 184 183 + 1901 t + 4544 t + 10144 t + 21549 t + 43402 t 182 181 180 179 178 + 84024 t + 155821 t + 280647 t + 487742 t + 827691 t 177 176 175 174 173 + 1362682 t + 2201633 t + 3465115 t + 5369185 t + 8130213 t 172 171 170 169 + 12155242 t + 17803927 t + 25800797 t + 36704288 t 168 167 166 165 + 51756039 t + 71760588 t + 98759488 t + 133827024 t 164 163 162 161 + 180224915 t + 239249951 t + 315953723 t + 411697344 t 160 159 158 157 + 534131070 t + 684313410 t + 873552750 t + 1101956171 t 156 155 154 153 + 1385939549 t + 1723561880 t + 2138190435 t + 2624188805 t 152 151 150 149 + 3214315609 t + 3896801345 t + 4716816092 t + 5653120763 t 148 147 146 145 + 6767167464 t + 8023713855 t + 9505153670 t + 11156447014 t 144 143 142 + 13086713143 t + 15213741691 t + 17680136424 t 141 140 139 + 20367623720 t + 23460408483 t + 26793457989 t 138 137 136 + 30601760521 t + 34661295188 t + 39268478025 t 135 134 133 + 44126347227 t + 49604353924 t + 55317477082 t 132 131 130 + 61721148931 t + 68325939523 t + 75686781729 t 129 128 127 + 83193143729 t + 91513881235 t + 99900368243 t 126 125 124 + 109149645372 t + 118358899101 t + 128467741372 t 123 122 121 + 138404055200 t + 149263187973 t + 159791000351 t 120 119 118 + 171250802765 t + 182196092817 t + 194067870888 t 117 116 115 + 205220808947 t + 217281202976 t + 228402032159 t 114 113 112 + 240398714151 t + 251224679033 t + 262885105530 t 111 110 109 + 273140332079 t + 284181151368 t + 293586767079 t 108 107 106 + 303725587113 t + 312011926326 t + 320978562341 t 105 104 103 + 327896335469 t + 335445266956 t + 340777574677 t 102 101 100 + 346698430553 t + 350270302483 t + 354398267351 t 99 98 97 + 356085783482 t + 358308683983 t + 358044408872 t 96 95 94 + 358308683983 t + 356085783482 t + 354398267351 t 93 92 91 + 350270302483 t + 346698430553 t + 340777574677 t 90 89 88 + 335445266956 t + 327896335469 t + 320978562341 t 87 86 85 + 312011926326 t + 303725587113 t + 293586767079 t 84 83 82 + 284181151368 t + 273140332079 t + 262885105530 t 81 80 79 + 251224679033 t + 240398714151 t + 228402032159 t 78 77 76 + 217281202976 t + 205220808947 t + 194067870888 t 75 74 73 + 182196092817 t + 171250802765 t + 159791000351 t 72 71 70 + 149263187973 t + 138404055200 t + 128467741372 t 69 68 67 66 + 118358899101 t + 109149645372 t + 99900368243 t + 91513881235 t 65 64 63 62 + 83193143729 t + 75686781729 t + 68325939523 t + 61721148931 t 61 60 59 58 + 55317477082 t + 49604353924 t + 44126347227 t + 39268478025 t 57 56 55 54 + 34661295188 t + 30601760521 t + 26793457989 t + 23460408483 t 53 52 51 50 + 20367623720 t + 17680136424 t + 15213741691 t + 13086713143 t 49 48 47 46 + 11156447014 t + 9505153670 t + 8023713855 t + 6767167464 t 45 44 43 42 + 5653120763 t + 4716816092 t + 3896801345 t + 3214315609 t 41 40 39 38 + 2624188805 t + 2138190435 t + 1723561880 t + 1385939549 t 37 36 35 34 + 1101956171 t + 873552750 t + 684313410 t + 534131070 t 33 32 31 30 + 411697344 t + 315953723 t + 239249951 t + 180224915 t 29 28 27 26 + 133827024 t + 98759488 t + 71760588 t + 51756039 t 25 24 23 22 21 + 36704288 t + 25800797 t + 17803927 t + 12155242 t + 8130213 t 20 19 18 17 16 + 5369185 t + 3465115 t + 2201633 t + 1362682 t + 827691 t 15 14 13 12 11 10 + 487742 t + 280647 t + 155821 t + 84024 t + 43402 t + 21549 t 9 8 7 6 5 4 3 2 + 10144 t + 4544 t + 1901 t + 736 t + 258 t + 85 t + 21 t + 7 t / 8 6 4 2 15 13 6 3 - t + 1) / ((t - t + t - t + 1) (-1 + t) (t + 1) (t + t + 1) / 6 3 (t - t + 1) 12 11 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + t + t + 1) 12 11 10 9 8 7 6 5 4 3 2 (t - t + t - t + t - t + t - t + t - t + t - t + 1) 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + 1) 10 9 8 7 6 5 4 3 2 8 4 (t - t + t - t + t - t + t - t + t - t + 1) (t - t + 1) 12 10 8 6 4 2 6 5 4 3 2 2 (t - t + t - t + t - t + 1) (t - t + t - t + t - t + 1) 6 5 4 3 2 2 2 7 2 4 2 4 (t + t + t + t + t + t + 1) (t + 1) (t + t + 1) (t - t + 1) 4 3 4 3 2 2 4 3 2 2 4 2 2 (t + 1) (t + t + t + t + 1) (t - t + t - t + 1) (t - t + 1) 8 (t + 1)) and in Maple notation -(t^194-t^193+7*t^192+21*t^191+85*t^190+258*t^189+736*t^188+1901*t^187+4544*t^ 186+10144*t^185+21549*t^184+43402*t^183+84024*t^182+155821*t^181+280647*t^180+ 487742*t^179+827691*t^178+1362682*t^177+2201633*t^176+3465115*t^175+5369185*t^ 174+8130213*t^173+12155242*t^172+17803927*t^171+25800797*t^170+36704288*t^169+ 51756039*t^168+71760588*t^167+98759488*t^166+133827024*t^165+180224915*t^164+ 239249951*t^163+315953723*t^162+411697344*t^161+534131070*t^160+684313410*t^159 +873552750*t^158+1101956171*t^157+1385939549*t^156+1723561880*t^155+2138190435* t^154+2624188805*t^153+3214315609*t^152+3896801345*t^151+4716816092*t^150+ 5653120763*t^149+6767167464*t^148+8023713855*t^147+9505153670*t^146+11156447014 *t^145+13086713143*t^144+15213741691*t^143+17680136424*t^142+20367623720*t^141+ 23460408483*t^140+26793457989*t^139+30601760521*t^138+34661295188*t^137+ 39268478025*t^136+44126347227*t^135+49604353924*t^134+55317477082*t^133+ 61721148931*t^132+68325939523*t^131+75686781729*t^130+83193143729*t^129+ 91513881235*t^128+99900368243*t^127+109149645372*t^126+118358899101*t^125+ 128467741372*t^124+138404055200*t^123+149263187973*t^122+159791000351*t^121+ 171250802765*t^120+182196092817*t^119+194067870888*t^118+205220808947*t^117+ 217281202976*t^116+228402032159*t^115+240398714151*t^114+251224679033*t^113+ 262885105530*t^112+273140332079*t^111+284181151368*t^110+293586767079*t^109+ 303725587113*t^108+312011926326*t^107+320978562341*t^106+327896335469*t^105+ 335445266956*t^104+340777574677*t^103+346698430553*t^102+350270302483*t^101+ 354398267351*t^100+356085783482*t^99+358308683983*t^98+358044408872*t^97+ 358308683983*t^96+356085783482*t^95+354398267351*t^94+350270302483*t^93+ 346698430553*t^92+340777574677*t^91+335445266956*t^90+327896335469*t^89+ 320978562341*t^88+312011926326*t^87+303725587113*t^86+293586767079*t^85+ 284181151368*t^84+273140332079*t^83+262885105530*t^82+251224679033*t^81+ 240398714151*t^80+228402032159*t^79+217281202976*t^78+205220808947*t^77+ 194067870888*t^76+182196092817*t^75+171250802765*t^74+159791000351*t^73+ 149263187973*t^72+138404055200*t^71+128467741372*t^70+118358899101*t^69+ 109149645372*t^68+99900368243*t^67+91513881235*t^66+83193143729*t^65+ 75686781729*t^64+68325939523*t^63+61721148931*t^62+55317477082*t^61+49604353924 *t^60+44126347227*t^59+39268478025*t^58+34661295188*t^57+30601760521*t^56+ 26793457989*t^55+23460408483*t^54+20367623720*t^53+17680136424*t^52+15213741691 *t^51+13086713143*t^50+11156447014*t^49+9505153670*t^48+8023713855*t^47+ 6767167464*t^46+5653120763*t^45+4716816092*t^44+3896801345*t^43+3214315609*t^42 +2624188805*t^41+2138190435*t^40+1723561880*t^39+1385939549*t^38+1101956171*t^ 37+873552750*t^36+684313410*t^35+534131070*t^34+411697344*t^33+315953723*t^32+ 239249951*t^31+180224915*t^30+133827024*t^29+98759488*t^28+71760588*t^27+ 51756039*t^26+36704288*t^25+25800797*t^24+17803927*t^23+12155242*t^22+8130213*t ^21+5369185*t^20+3465115*t^19+2201633*t^18+1362682*t^17+827691*t^16+487742*t^15 +280647*t^14+155821*t^13+84024*t^12+43402*t^11+21549*t^10+10144*t^9+4544*t^8+ 1901*t^7+736*t^6+258*t^5+85*t^4+21*t^3+7*t^2-t+1)/(t^8-t^6+t^4-t^2+1)/(-1+t)^15 /(t+1)^13/(t^6+t^3+1)/(t^6-t^3+1)/(t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t ^2+t+1)/(t^12-t^11+t^10-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1)/(t^10+t^9+t^8+t^7+ t^6+t^5+t^4+t^3+t^2+t+1)/(t^10-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1)/(t^8-t^4+1) /(t^12-t^10+t^8-t^6+t^4-t^2+1)/(t^6-t^5+t^4-t^3+t^2-t+1)^2/(t^6+t^5+t^4+t^3+t^2 +t+1)^2/(t^2+1)^7/(t^2+t+1)^4/(t^2-t+1)^4/(t^4+1)^3/(t^4+t^3+t^2+t+1)^2/(t^4-t^ 3+t^2-t+1)^2/(t^4-t^2+1)^2/(t^8+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 8, 36, 150, 521, 1636, 4672, 12346, 30441, 70922, 157004, 332578, 676572, 1328980, 2527074, 4669367, 8398764, 14749992, 25325765, 42611990, 70327440, 114067100, 181955403, 285893124, 442717786, 676537634, 1020693863, 1521967986, 2243761893, 3273480400] Furthermore, a(n) is a quasi-polynomial given as sum of, 21, quasi-polynomials 21 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), Q[13](n), Q[14](n), Q[15](n), Q[16](n), Q[17](n), Q[18](n), Q[19](n), Q[20](n), Q[21](n), are defined as followed Q[1](n), is the polynomial 467168310097 14 467168310097 13 ------------------------- n + ----------------------- n 1729434614113165639680000 15441380483153264640000 75423902448211 12 169182054982981 11 + ----------------------- n + ---------------------- n 48869423946682859520000 3563395496112291840000 16978546764566813 10 2161762893418201 9 + ----------------------- n + --------------------- n 17277069072059596800000 149513097738977280000 3766214918472728401 8 94092178406811997 7 + ----------------------- n + -------------------- n 24187896700883435520000 75587177190260736000 4221207577178045719 6 356008093671311663 5 + --------------------- n + -------------------- n 569573705672294400000 10907240575795200000 49031922402236155303 4 38732291703039319631 3 + --------------------- n + --------------------- n 469253727883100160000 164464407512875008000 3000281260069776294721 2 2465811238591860669313 + ---------------------- n + ---------------------- n 8470700150758362316800 7558717719026073600000 4971613507605681994165439 + -------------------------- 22235587895740701081600000 and in Maple notation 467168310097/1729434614113165639680000*n^14+467168310097/ 15441380483153264640000*n^13+75423902448211/48869423946682859520000*n^12+ 169182054982981/3563395496112291840000*n^11+16978546764566813/ 17277069072059596800000*n^10+2161762893418201/149513097738977280000*n^9+ 3766214918472728401/24187896700883435520000*n^8+94092178406811997/ 75587177190260736000*n^7+4221207577178045719/569573705672294400000*n^6+ 356008093671311663/10907240575795200000*n^5+49031922402236155303/ 469253727883100160000*n^4+38732291703039319631/164464407512875008000*n^3+ 3000281260069776294721/8470700150758362316800*n^2+2465811238591860669313/ 7558717719026073600000*n+4971613507605681994165439/22235587895740701081600000 This is the leading term in particular, a(n) , is asymptotic to 14 467168310097 n ------------------------- 1729434614113165639680000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 134478763 12 134478763 11 [- ----------------------- n - --------------------- n 10366241443235758080000 107981681700372480000 560473886197 10 181781689589 9 - ----------------------- n - --------------------- n 10366241443235758080000 129578018040446976000 83692104400921 8 351653871773 7 - ---------------------- n - ------------------- n 3455413814411919360000 1199796463337472000 2402763880142149 6 385826256727039 5 - --------------------- n - -------------------- n 942385585748705280000 23995929266749440000 820642108692421 4 36482196116001373 3 - ------------------- n - -------------------- n 9844483801743360000 64789009020223488000 284595355134052561 2 72950625694618213 - -------------------- n - ------------------- n 59235665389918617600 2666214362972160000 33486362557596245729674951 134478763 12 - ---------------------------, ----------------------- n 155649115270184907571200000 10366241443235758080000 134478763 11 560473886197 10 + --------------------- n + ----------------------- n 107981681700372480000 10366241443235758080000 181781689589 9 83692104400921 8 + --------------------- n + ---------------------- n 129578018040446976000 3455413814411919360000 351653871773 7 2402763880142149 6 + ------------------- n + --------------------- n 1199796463337472000 942385585748705280000 385826256727039 5 820642108692421 4 + -------------------- n + ------------------- n 23995929266749440000 9844483801743360000 36482196116001373 3 284595355134052561 2 + -------------------- n + -------------------- n 64789009020223488000 59235665389918617600 72950625694618213 33486362557596245729674951 + ------------------- n + ---------------------------] 2666214362972160000 155649115270184907571200000 and in Maple format [-134478763/10366241443235758080000*n^12-134478763/107981681700372480000*n^11-\ 560473886197/10366241443235758080000*n^10-181781689589/129578018040446976000*n^ 9-83692104400921/3455413814411919360000*n^8-351653871773/1199796463337472000*n^ 7-2402763880142149/942385585748705280000*n^6-385826256727039/ 23995929266749440000*n^5-820642108692421/9844483801743360000*n^4-\ 36482196116001373/64789009020223488000*n^3-284595355134052561/ 59235665389918617600*n^2-72950625694618213/2666214362972160000*n-\ 33486362557596245729674951/155649115270184907571200000, 134478763/ 10366241443235758080000*n^12+134478763/107981681700372480000*n^11+560473886197/ 10366241443235758080000*n^10+181781689589/129578018040446976000*n^9+ 83692104400921/3455413814411919360000*n^8+351653871773/1199796463337472000*n^7+ 2402763880142149/942385585748705280000*n^6+385826256727039/23995929266749440000 *n^5+820642108692421/9844483801743360000*n^4+36482196116001373/ 64789009020223488000*n^3+284595355134052561/59235665389918617600*n^2+ 72950625694618213/2666214362972160000*n+33486362557596245729674951/ 155649115270184907571200000] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 2 44405 [-5/279936 n - 5/17496 n - -------, 2519424 277 3 2171 2 5651 7289 - -------- n - ------- n - ------- n - --------, 15116544 5038848 1679616 15116544 277 3 2261 2 6131 273719 -------- n + ------- n + ------- n + --------] 15116544 5038848 1679616 15116544 and in Maple format [-5/279936*n^2-5/17496*n-44405/2519424, -277/15116544*n^3-2171/5038848*n^2-5651 /1679616*n-7289/15116544, 277/15116544*n^3+2261/5038848*n^2+6131/1679616*n+ 273719/15116544] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 1177 5 1177 4 1127929 3 124883 2 [- ----------- n - --------- n - ----------- n - --------- n 31708938240 792723456 47563407360 660602880 91851197 3315409 358703 6 358703 5 + ----------- n + ---------, - ------------ n - ----------- n 95126814720 264241152 951268147200 19818086400 1192417 4 2607671 3 17983807 2 39738829 5093017067 - ---------- n - --------- n - --------- n - --------- n - -----------, 3397386240 743178240 943718400 743178240 78033715200 1177 5 1177 4 1127929 3 124883 2 ----------- n + --------- n + ----------- n + --------- n 31708938240 792723456 47563407360 660602880 91851197 3315409 358703 6 358703 5 - ----------- n - ---------, ------------ n + ----------- n 95126814720 264241152 951268147200 19818086400 1192417 4 2607671 3 17983807 2 39738829 5093017067 + ---------- n + --------- n + --------- n + --------- n + -----------] 3397386240 743178240 943718400 743178240 78033715200 and in Maple format [-1177/31708938240*n^5-1177/792723456*n^4-1127929/47563407360*n^3-124883/ 660602880*n^2+91851197/95126814720*n+3315409/264241152, -358703/951268147200*n^ 6-358703/19818086400*n^5-1192417/3397386240*n^4-2607671/743178240*n^3-17983807/ 943718400*n^2-39738829/743178240*n-5093017067/78033715200, 1177/31708938240*n^5 +1177/792723456*n^4+1127929/47563407360*n^3+124883/660602880*n^2-91851197/ 95126814720*n-3315409/264241152, 358703/951268147200*n^6+358703/19818086400*n^5 +1192417/3397386240*n^4+2607671/743178240*n^3+17983807/943718400*n^2+39738829/ 743178240*n+5093017067/78033715200] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 29 n 51 -3 29 n 13 6 n 43 6 n 53 [----- + ----, ----, - ----- - ----, - ---- - ----, ---- + ----] 25000 6250 3125 25000 1250 3125 3125 3125 3125 and in Maple format [29/25000*n+51/6250, -3/3125, -29/25000*n-13/1250, -6/3125*n-43/3125, 6/3125*n+ 53/3125] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 163 2 163 35713 [------- n + ------ n + -------, 2519424 157464 7558272 277 3 6485 2 50431 386279 - ------- n - ------- n - ------- n - --------, 5038848 5038848 5038848 15116544 277 3 6811 2 229 457705 - ------- n - ------- n - ----- n - --------, 5038848 5038848 20736 15116544 163 2 163 35713 - ------- n - ------ n - -------, 2519424 157464 7558272 277 3 6485 2 50431 386279 ------- n + ------- n + ------- n + --------, 5038848 5038848 5038848 15116544 277 3 6811 2 229 457705 ------- n + ------- n + ----- n + --------] 5038848 5038848 20736 15116544 and in Maple format [163/2519424*n^2+163/157464*n+35713/7558272, -277/5038848*n^3-6485/5038848*n^2-\ 50431/5038848*n-386279/15116544, -277/5038848*n^3-6811/5038848*n^2-229/20736*n-\ 457705/15116544, -163/2519424*n^2-163/157464*n-35713/7558272, 277/5038848*n^3+ 6485/5038848*n^2+50431/5038848*n+386279/15116544, 277/5038848*n^3+6811/5038848* n^2+229/20736*n+457705/15116544] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 39 n 73 13 n 13 13 n 13 39 n 83 65 n 85 [----- + ----, ----- + ----, - ----- - ----, - ----- - ----, - ----- - ----, 19208 4802 19208 9604 19208 1372 19208 4802 19208 4802 -41 65 n 25 ----, ----- + ---] 4802 19208 686 and in Maple format [39/19208*n+73/4802, 13/19208*n+13/9604, -13/19208*n-13/1372, -39/19208*n-83/ 4802, -65/19208*n-85/4802, -41/4802, 65/19208*n+25/686] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 2 13 485 n 2 19 449 [-1/24576 n - ----- n + -----, ---- + 1/192, 1/24576 n + ----- n - -----, 24576 18432 1536 24576 18432 43 2 43 79 2 13 485 n - ----- n - ---- n - ----, 1/24576 n + ----- n - -----, - ---- - 1/192, 49152 3072 2304 24576 18432 1536 2 19 449 43 2 43 79 -1/24576 n - ----- n + -----, ----- n + ---- n + ----] 24576 18432 49152 3072 2304 and in Maple format [-1/24576*n^2-13/24576*n+485/18432, 1/1536*n+1/192, 1/24576*n^2+19/24576*n-449/ 18432, -43/49152*n^2-43/3072*n-79/2304, 1/24576*n^2+13/24576*n-485/18432, -1/ 1536*n-1/192, -1/24576*n^2-19/24576*n+449/18432, 43/49152*n^2+43/3072*n+79/2304 ] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -2 -1 -1 -1 -1 [--, 2/81, --, 1/81, --, --, 1/81, --, 2/81] 81 81 81 81 81 and in Maple format [-2/81, 2/81, -1/81, 1/81, -1/81, -1/81, 1/81, -1/81, 2/81] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 10 7 n 57 7 n 83 2 n 74 2 n 86 7 n 57 [- ---- - ----, 1/3125, ---- + ----, - --- - ----, - --- - ----, ---- + ----, 5000 6250 5000 6250 625 3125 625 3125 5000 6250 -1 7 n 83 2 n 74 2 n 86 ----, - ---- - ----, --- + ----, --- + ----] 3125 5000 6250 625 3125 625 3125 and in Maple format [-7/5000*n-57/6250, 1/3125, 7/5000*n+83/6250, -2/625*n-74/3125, -2/625*n-86/ 3125, 7/5000*n+57/6250, -1/3125, -7/5000*n-83/6250, 2/625*n+74/3125, 2/625*n+86 /3125] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 11 -1 -1 -3 -3 -3 -3 [---, 0, 2/121, 0, ---, 3/121, ---, ---, ---, ---, 3/121] 121 121 242 242 242 242 and in Maple format [-1/121, 0, 2/121, 0, -1/121, 3/121, -3/242, -3/242, -3/242, -3/242, 3/121] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 11 n 11 11 n 67 11 n 13 11 n 59 11 n 65 [---- + ---, ---- + ----, ---- + ---, 1/972, - ---- - ----, - ---- - ----, 2592 324 2592 1944 5184 576 5184 5184 2592 1944 11 n 11 11 n 67 11 n 13 -1 11 n 59 11 n 65 - ---- - ---, - ---- - ----, - ---- - ---, ---, ---- + ----, ---- + ----] 2592 324 2592 1944 5184 576 972 5184 5184 2592 1944 and in Maple format [11/2592*n+11/324, 11/2592*n+67/1944, 11/5184*n+13/576, 1/972, -11/5184*n-59/ 5184, -11/2592*n-65/1944, -11/2592*n-11/324, -11/2592*n-67/1944, -11/5184*n-13/ 576, -1/972, 11/5184*n+59/5184, 11/2592*n+65/1944] Q[13](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 13 -1 -5 -1 -7 -7 [0, 7/338, ---, 1/169, ---, 1/169, ---, 7/338, 0, 2/169, ---, ---, 2/169] 338 169 338 338 338 and in Maple format [0, 7/338, -1/338, 1/169, -5/169, 1/169, -1/338, 7/338, 0, 2/169, -7/338, -7/ 338, 2/169] Q[14](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 14 13 n 138 13 n 379 13 n 349 13 n 44 13 n 47 -3 [---- + ----, ---- + ----, ---- + ----, ---- + ----, ---- + ----, ----, 2744 2401 2744 9604 2744 9604 2744 2401 2744 2401 4802 13 n 135 13 n 138 13 n 379 13 n 349 13 n 44 - ---- - ----, - ---- - ----, - ---- - ----, - ---- - ----, - ---- - ----, 2744 2401 2744 2401 2744 9604 2744 9604 2744 2401 13 n 47 13 n 135 - ---- - ----, 3/4802, ---- + ----] 2744 2401 2744 2401 and in Maple format [13/2744*n+138/2401, 13/2744*n+379/9604, 13/2744*n+349/9604, 13/2744*n+44/2401, 13/2744*n+47/2401, -3/4802, -13/2744*n-135/2401, -13/2744*n-138/2401, -13/2744* n-379/9604, -13/2744*n-349/9604, -13/2744*n-44/2401, -13/2744*n-47/2401, 3/4802 , 13/2744*n+135/2401] Q[15](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 16 -1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 1/16] 16 and in Maple format [0, 0, 0, 0, 0, 0, 0, -1/16, 0, 0, 0, 0, 0, 0, 0, 1/16] Q[16](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 18 -2 -1 -1 -1 -2 -2 -1 -1 -1 [--, 2/81, 1/81, --, 1/81, --, 1/81, --, --, 2/81, --, --, 1/81, --, 1/81, --, 81 81 81 81 81 81 81 81 81 1/81, 2/81] and in Maple format [-2/81, 2/81, 1/81, -1/81, 1/81, -1/81, 1/81, -1/81, -2/81, 2/81, -2/81, -1/81, 1/81, -1/81, 1/81, -1/81, 1/81, 2/81] Q[17](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 20 -1 -1 -1 -2 -1 [--, 2/25, --, 1/25, 0, 0, 0, 0, 0, --, 1/20, --, 1/20, --, 0, 0, 0, 0, 0, 1/25 20 20 25 25 25 ] and in Maple format [-1/20, 2/25, -1/20, 1/25, 0, 0, 0, 0, 0, -1/25, 1/20, -2/25, 1/20, -1/25, 0, 0 , 0, 0, 0, 1/25] Q[18](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 22 10 -10 10 -1 13 -13 13 -13 -1 -10 10 [1/121, ---, ---, ---, 1/121, ---, ---, ---, ---, ---, 1/121, ---, ---, ---, 121 121 121 121 242 242 242 242 121 121 121 -10 -1 -13 13 -13 13 -1 ---, ---, 1/121, ---, ---, ---, ---, ---] 121 121 242 242 242 242 121 and in Maple format [1/121, 10/121, -10/121, 10/121, 1/121, -1/121, 13/242, -13/242, 13/242, -13/ 242, 1/121, -1/121, -10/121, 10/121, -10/121, -1/121, 1/121, -13/242, 13/242, -\ 13/242, 13/242, -1/121] Q[19](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 24 -5 -7 -7 -5 -1 -1 -1 [--, 1/12, --, 1/9, --, 1/12, --, 1/18, --, 0, 1/36, --, 5/72, --, 7/72, -1/9, 72 72 72 72 36 18 12 -1 -1 -1 7/72, --, 5/72, --, 1/36, 0, --, 1/18] 12 18 36 and in Maple format [-5/72, 1/12, -7/72, 1/9, -7/72, 1/12, -5/72, 1/18, -1/36, 0, 1/36, -1/18, 5/72 , -1/12, 7/72, -1/9, 7/72, -1/12, 5/72, -1/18, 1/36, 0, -1/36, 1/18] Q[20](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 26 -5 -9 -5 -4 21 -21 [4/169, 5/338, ---, 9/169, ---, 9/169, ---, 5/338, 4/169, ---, ---, ---, 4/169, 338 169 338 169 338 338 -4 -5 -9 -9 -5 -4 -21 21 -4 ---, ---, 5/338, ---, 9/169, ---, 5/338, ---, ---, 4/169, ---, ---, ---] 169 338 169 169 338 169 338 338 169 and in Maple format [4/169, 5/338, -5/338, 9/169, -9/169, 9/169, -5/338, 5/338, 4/169, -4/169, 21/ 338, -21/338, 4/169, -4/169, -5/338, 5/338, -9/169, 9/169, -9/169, 5/338, -5/ 338, -4/169, 4/169, -21/338, 21/338, -4/169] Q[21](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 28 -1 -1 -1 -1 -1 -1 -3 -2 [--, 2/49, --, 3/98, --, 2/49, --, 3/98, --, 2/49, --, 3/98, 0, --, 1/28, --, 28 28 28 28 28 28 98 49 -3 -2 -3 -2 -3 1/28, --, 1/28, --, 1/28, --, 1/28, --, 1/28, --, 0, 3/98] 98 49 98 49 98 and in Maple format [-1/28, 2/49, -1/28, 3/98, -1/28, 2/49, -1/28, 3/98, -1/28, 2/49, -1/28, 3/98, 0, -3/98, 1/28, -2/49, 1/28, -3/98, 1/28, -2/49, 1/28, -3/98, 1/28, -2/49, 1/28 , -3/98, 0, 3/98] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 2701277667768657361204903796070607758961397428414580238988299224023858841464180\ 6114698721378176130495783585784861303216527396138622235109102571026722293277187\ 9499964550045064978187621060830827464645813503469121733330526768024671973508147\ 0860692152637052543655650057194352414636549004835552512375292357095926551329632\ 0398415332772801773757799153229940447071831499764176626054798086661224596420994\ 4559621991068136065249357474657900229174880822422366663331413240481617677986811\ 8716927961302839340035139309680267089459013390535418240363556797913292738419460\ 7495088543113052966738935315304496360011960610746312674721603194011919633041323\ 7019410443422113843459245908200889279831054162095310974706388287510678377859260\ 1130117923114685877132690977841935615720298046174627531178653413827047337793472\ 1079231577046186680468614551545422080223076097390187879637401588813184971568733\ 8829572704988742158402326806029646462897115139878115823253521981545728465922853\ 9932813683194765397465136068124860078126815077224913335705075313011538483667205\ 9848927772583300169288828595704077938592674997056148224903324359074172541989935\ 4872074173339707913380557096032759102274527547070481492982556955024303656933216\ 7488445024226140979891740949128360761174258437239197322502127399272936378161062\ 1209438636424367768879666923867087630304807480160518451324441802893858569396230\ 320638536555948744461965980982410555655447621 ------------------------------------ --------------------------------------------------------- For , psi[16](t), we have the following theorem. Theorem Number, 15, : Let f(t) be the constant term of the rational function in z and t 2 / / 2 4 6 8 (1 + z) / |2 z (1 - t) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 10 12 14 16 / t \ / t \ (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) |1 - ----| |1 - ----| | 2 | | 4 | \ z / \ z / / t \ / t \ / t \ / t \ / t \ / t \\ |1 - ----| |1 - ----| |1 - ---| |1 - ---| |1 - ---| |1 - ---|| | 6 | | 8 | | 10| | 12| | 14| | 16|| \ z / \ z / \ z / \ z / \ z / \ z // We have 98 96 95 94 93 92 91 f(t) = (t + 6 t + 31 t + 120 t + 401 t + 1185 t + 3166 t 90 89 88 87 86 85 + 7793 t + 17838 t + 38355 t + 78078 t + 151412 t + 281048 t 84 83 82 81 80 + 501614 t + 863802 t + 1439651 t + 2328324 t + 3662383 t 79 78 77 76 75 + 5613867 t + 8400606 t + 12290246 t + 17603620 t + 24714716 t 74 73 72 71 + 34047543 t + 46068401 t + 61274954 t + 80178037 t 70 69 68 67 + 103281200 t + 131054107 t + 163903310 t + 202139763 t 66 65 64 63 + 245948195 t + 295353664 t + 350196212 t + 410107889 t 62 61 60 59 + 474499204 t + 542553704 t + 613236818 t + 685310641 t 58 57 56 55 + 757367236 t + 827867884 t + 895194845 t + 957708542 t 54 53 52 51 + 1013813491 t + 1062018794 t + 1101004852 t + 1129678048 t 50 49 48 47 + 1147220258 t + 1153124382 t + 1147220258 t + 1129678048 t 46 45 44 43 + 1101004852 t + 1062018794 t + 1013813491 t + 957708542 t 42 41 40 39 + 895194845 t + 827867884 t + 757367236 t + 685310641 t 38 37 36 35 + 613236818 t + 542553704 t + 474499204 t + 410107889 t 34 33 32 31 + 350196212 t + 295353664 t + 245948195 t + 202139763 t 30 29 28 27 + 163903310 t + 131054107 t + 103281200 t + 80178037 t 26 25 24 23 22 + 61274954 t + 46068401 t + 34047543 t + 24714716 t + 17603620 t 21 20 19 18 17 + 12290246 t + 8400606 t + 5613867 t + 3662383 t + 2328324 t 16 15 14 13 12 + 1439651 t + 863802 t + 501614 t + 281048 t + 151412 t 11 10 9 8 7 6 5 + 78078 t + 38355 t + 17838 t + 7793 t + 3166 t + 1185 t + 401 t 4 3 2 / 2 5 6 3 + 120 t + 31 t + 6 t + 1) / ((t + t + 1) (t + t + 1) / 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + 1) 12 11 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + t + t + 1) 6 5 4 3 2 2 7 16 2 3 (t + t + t + t + t + t + 1) (t + 1) (-1 + t) (t + 1) 2 2 4 2 4 4 3 2 3 (t - t + 1) (t - t + 1) (t + 1) (t + t + t + t + 1) 4 3 2 8 7 5 4 3 (t - t + t - t + 1) (t - t + t - t + t - t + 1)) and in Maple notation (t^98+6*t^96+31*t^95+120*t^94+401*t^93+1185*t^92+3166*t^91+7793*t^90+17838*t^89 +38355*t^88+78078*t^87+151412*t^86+281048*t^85+501614*t^84+863802*t^83+1439651* t^82+2328324*t^81+3662383*t^80+5613867*t^79+8400606*t^78+12290246*t^77+17603620 *t^76+24714716*t^75+34047543*t^74+46068401*t^73+61274954*t^72+80178037*t^71+ 103281200*t^70+131054107*t^69+163903310*t^68+202139763*t^67+245948195*t^66+ 295353664*t^65+350196212*t^64+410107889*t^63+474499204*t^62+542553704*t^61+ 613236818*t^60+685310641*t^59+757367236*t^58+827867884*t^57+895194845*t^56+ 957708542*t^55+1013813491*t^54+1062018794*t^53+1101004852*t^52+1129678048*t^51+ 1147220258*t^50+1153124382*t^49+1147220258*t^48+1129678048*t^47+1101004852*t^46 +1062018794*t^45+1013813491*t^44+957708542*t^43+895194845*t^42+827867884*t^41+ 757367236*t^40+685310641*t^39+613236818*t^38+542553704*t^37+474499204*t^36+ 410107889*t^35+350196212*t^34+295353664*t^33+245948195*t^32+202139763*t^31+ 163903310*t^30+131054107*t^29+103281200*t^28+80178037*t^27+61274954*t^26+ 46068401*t^25+34047543*t^24+24714716*t^23+17603620*t^22+12290246*t^21+8400606*t ^20+5613867*t^19+3662383*t^18+2328324*t^17+1439651*t^16+863802*t^15+501614*t^14 +281048*t^13+151412*t^12+78078*t^11+38355*t^10+17838*t^9+7793*t^8+3166*t^7+1185 *t^6+401*t^5+120*t^4+31*t^3+6*t^2+1)/(t^2+t+1)^5/(t^6+t^3+1)/(t^10+t^9+t^8+t^7+ t^6+t^5+t^4+t^3+t^2+t+1)/(t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/( t^6+t^5+t^4+t^3+t^2+t+1)^2/(t+1)^7/(-1+t)^16/(t^2+1)^3/(t^2-t+1)^2/(t^4-t^2+1)/ (t^4+1)/(t^4+t^3+t^2+t+1)^3/(t^4-t^3+t^2-t+1)/(t^8-t^7+t^5-t^4+t^3-t+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 9, 41, 177, 649, 2137, 6375, 17575, 45207, 109583, 252117, 553981, 1168261, 2374753, 4669367, 8908546, 16535154, 29927526, 52925886, 91617530, 155484150, 259062316, 424306230, 683919048, 1085984406, 1700352360, 2627337510, 4009478902, 6047272980, 9020077206] Furthermore, a(n) is a quasi-polynomial given as sum of, 14, quasi-polynomials 14 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), Q[13](n), Q[14](n), are defined as followed Q[1](n), is the polynomial 2330931341 15 39625832797 14 ------------------------ n + ----------------------- n 142501021060349952000000 19000136141379993600000 992281114729 13 270845663603 12 + ---------------------- n + -------------------- n 8142915489162854400000 62637811455098880000 108665507237459 11 512045186010973 10 + ---------------------- n + --------------------- n 1043963524251648000000 284717324795904000000 1965325657605229 9 368124179842669 8 + -------------------- n + ------------------- n 85415197438771200000 1660851061309440000 22398597351724703 7 3956429327218451 6 + -------------------- n + ------------------ n 13840425510912000000 442451165184000000 2864215387505719333 5 14148085947879274871 4 + -------------------- n + --------------------- n 77330631426048000000 125275622910197760000 481295185711394862331 3 4283492722695300227723 2 + ---------------------- n + ----------------------- n 1954299717399085056000 11632736413089792000000 184799090256800308193 8978713891052398660577 + --------------------- n + ----------------------- 498255318392832000000 28500204212069990400000 and in Maple notation 2330931341/142501021060349952000000*n^15+39625832797/19000136141379993600000*n^ 14+992281114729/8142915489162854400000*n^13+270845663603/62637811455098880000*n ^12+108665507237459/1043963524251648000000*n^11+512045186010973/ 284717324795904000000*n^10+1965325657605229/85415197438771200000*n^9+ 368124179842669/1660851061309440000*n^8+22398597351724703/13840425510912000000* n^7+3956429327218451/442451165184000000*n^6+2864215387505719333/ 77330631426048000000*n^5+14148085947879274871/125275622910197760000*n^4+ 481295185711394862331/1954299717399085056000*n^3+4283492722695300227723/ 11632736413089792000000*n^2+184799090256800308193/498255318392832000000*n+ 8978713891052398660577/28500204212069990400000 This is the leading term in particular, a(n) , is asymptotic to 15 2330931341 n ------------------------ 142501021060349952000000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 841 6 14297 5 17737 4 1898747 3 2095981 2 [- ---------- n - ---------- n - --------- n - ---------- n - --------- n 8493465600 2831155200 169869312 1698693120 314572800 4681511 53243899 841 6 14297 5 17737 4 - --------- n - ---------, ---------- n + ---------- n + --------- n 212336640 254803968 8493465600 2831155200 169869312 1898747 3 2095981 2 4681511 53243899 + ---------- n + --------- n + --------- n + ---------] 1698693120 314572800 212336640 254803968 and in Maple format [-841/8493465600*n^6-14297/2831155200*n^5-17737/169869312*n^4-1898747/ 1698693120*n^3-2095981/314572800*n^2-4681511/212336640*n-53243899/254803968, 841/8493465600*n^6+14297/2831155200*n^5+17737/169869312*n^4+1898747/1698693120* n^3+2095981/314572800*n^2+4681511/212336640*n+53243899/254803968] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 377 4 18883 3 76087 2 257159 52401547 [- -------- n - -------- n - -------- n - ------- n - ---------, 28343520 42515280 14171760 9447840 850305600 43 3 731 2 7837 47107 - ------- n - ------- n - ------- n - -------, 2657205 1771470 1771470 2657205 377 4 19571 3 16387 2 896869 22491929 -------- n + -------- n + ------- n + -------- n + ---------] 28343520 42515280 2834352 28343520 283435200 and in Maple format [-377/28343520*n^4-18883/42515280*n^3-76087/14171760*n^2-257159/9447840*n-\ 52401547/850305600, -43/2657205*n^3-731/1771470*n^2-7837/1771470*n-47107/ 2657205, 377/28343520*n^4+19571/42515280*n^3+16387/2834352*n^2+896869/28343520* n+22491929/283435200] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 11 2 11 3503 11 2 33 127 [----- n + ---- n + ------, - ----- n - ---- n - ----, 49152 3072 147456 49152 8192 4608 11 2 11 3503 11 2 33 127 - ----- n - ---- n - ------, ----- n + ---- n + ----] 49152 3072 147456 49152 8192 4608 and in Maple format [11/49152*n^2+11/3072*n+3503/147456, -11/49152*n^2-33/8192*n-127/4608, -11/ 49152*n^2-11/3072*n-3503/147456, 11/49152*n^2+33/8192*n+127/4608] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 37 2 392 41969 37 2 79 26159 [- ----- n - ----- n - ------, ----- n + ----- n + ------, 93750 46875 562500 93750 15625 562500 37 2 177 8722 41 n 697 37 2 727 2744 - ----- n - ----- n - ------, - ----- - -----, ----- n + ----- n + -----] 46875 15625 140625 46875 93750 46875 46875 28125 and in Maple format [-37/93750*n^2-392/46875*n-41969/562500, 37/93750*n^2+79/15625*n+26159/562500, -37/46875*n^2-177/15625*n-8722/140625, -41/46875*n-697/93750, 37/46875*n^2+727/ 46875*n+2744/28125] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 5 n 271 5 n 239 5 n 271 -1 5 n 239 [---- + -----, 1/486, - ---- - -----, - ---- - -----, ---, ---- + -----] 2592 15552 2592 15552 2592 15552 486 2592 15552 and in Maple format [5/2592*n+271/15552, 1/486, -5/2592*n-239/15552, -5/2592*n-271/15552, -1/486, 5 /2592*n+239/15552] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 74 12 n 102 -74 6 n 64 12 n 254 12 n 50 [----, ---- + ----, ----, ---- + ----, - ---- - ----, - ---- + ----, 2401 2401 2401 2401 2401 2401 2401 2401 2401 2401 6 n 38 ---- + ----] 2401 2401 and in Maple format [74/2401, 12/2401*n+102/2401, -74/2401, 6/2401*n+64/2401, -12/2401*n-254/2401, -12/2401*n+50/2401, 6/2401*n+38/2401] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 -1 -1 -1 -1 [1/64, 1/64, 1/64, --, --, --, --, 1/64] 64 64 64 64 and in Maple format [1/64, 1/64, 1/64, -1/64, -1/64, -1/64, -1/64, 1/64] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -4 -2 -2 [--, 0, --, 2/81, 0, --, 2/81, 0, 4/81] 81 81 81 and in Maple format [-4/81, 0, -2/81, 2/81, 0, -2/81, 2/81, 0, 4/81] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 10 -1 -1 -1 [---, 1/100, 0, --, 0, 1/100, ---, 0, 1/50, 0] 100 50 100 and in Maple format [-1/100, 1/100, 0, -1/50, 0, 1/100, -1/100, 0, 1/50, 0] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 11 -6 12 -12 -10 10 [---, ---, ---, 6/121, ---, 0, 0, 0, 0, 0, ---] 121 121 121 121 121 and in Maple format [-6/121, 12/121, -12/121, 6/121, -10/121, 0, 0, 0, 0, 0, 10/121] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 -1 -5 -1 -1 -5 -1 [--, 1/18, --, 5/72, --, 1/72, 1/72, --, 5/72, --, 1/18, --] 72 72 18 18 72 72 and in Maple format [-1/72, 1/18, -5/72, 5/72, -1/18, 1/72, 1/72, -1/18, 5/72, -5/72, 1/18, -1/72] Q[13](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 13 -4 16 -20 30 -30 20 -16 -12 12 [---, ---, ---, ---, ---, ---, ---, 4/169, ---, 0, 0, 0, ---] 169 169 169 169 169 169 169 169 169 and in Maple format [-4/169, 16/169, -20/169, 30/169, -30/169, 20/169, -16/169, 4/169, -12/169, 0, 0, 0, 12/169] Q[14](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 15 -2 -2 -8 -4 -2 -2 -8 [2/225, 2/45, ---, 4/75, --, 8/225, ---, 2/45, --, 2/225, --, ---, ---, 0, 225 45 225 75 45 225 225 8/225] and in Maple format [2/225, 2/45, -2/225, 4/75, -2/45, 8/225, -8/225, 2/45, -4/75, 2/225, -2/45, -2 /225, -8/225, 0, 8/225] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 1635729571378185415824816695930005104958690763036624491749405344126615004768389\ 0106285480725870703548265471588487419373489404600168053251150241192725482291320\ 5260090168493511431161918992972877608701028120800579530998832799464909998759005\ 7750573867250225801111940811383609843793035646673456471508576487033294476670535\ 9671147060382841320982215451320216889247176510435974921608887380587258275210215\ 6135683721811849207041636195180659046782440914569404638601854855424514553928449\ 4463174173641907962956628955280549540532826699502888854815168192728735822158734\ 1757505011568492389342824705731723967563398384252540952684226248181099054009331\ 7515010980488214616942264888299777589335001285403353663264870391145511462277035\ 1348532755582335415431947389093550769260889863370702127121074464128075203298217\ 0954259620588707163826332474167732680183880409778992386513713555770700568530117\ 6264285715119473783449804886024102039701829956672561673336938102572678682885374\ 5427534709254899964200720812582936608774226961503334004645476879165736129417426\ 1241140572514178964071072259420754591567166707728186858015139333418569383324334\ 6876864842568357388638256902104365862246461760834125910558622275900494676431881\ 3069158156503318997154059296279692748290555036789878542762903636962447809142533\ 6682259719760008737020127278178491235378718724027998062671722215906101513799444\ 7805985158744761721323909554304161899855691383256211225238941061259498286370127\ 49688869867579559460775095525065819953670341086810079942660320839 ------------------------------------ --------------------------------------------------------- For , psi[17](t), we have the following theorem. Theorem Number, 16, : Let f(t) be the constant term of the rational function in z and t 2 / / 3 5 7 9 (1 + z) / |2 z (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 11 13 15 17 / t \ (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (1 - t/z) |1 - ----| | 3 | \ z / / t \ / t \ / t \ / t \ / t \ / t \ / t \\ |1 - ----| |1 - ----| |1 - ----| |1 - ---| |1 - ---| |1 - ---| |1 - ---|| | 5 | | 7 | | 9 | | 11| | 13| | 15| | 17|| \ z / \ z / \ z / \ z / \ z / \ z / \ z // We have 254 253 252 251 250 249 248 f(t) = - (t - t + 8 t + 28 t + 125 t + 427 t + 1347 t 247 246 245 244 243 + 3840 t + 10095 t + 24724 t + 57304 t + 125713 t 242 241 240 239 238 + 264098 t + 530839 t + 1032045 t + 1934619 t + 3529278 t 237 236 235 234 + 6243353 t + 10802695 t + 18203969 t + 30112588 t 233 232 231 230 + 48675709 t + 77467356 t + 120800285 t + 185891183 t 229 228 227 226 + 280886262 t + 419644752 t + 616713182 t + 897513480 t 225 224 223 222 + 1286732274 t + 1829226628 t + 2564916974 t + 3570184087 t 221 220 219 218 + 4906705619 t + 6700608511 t + 9042955299 t + 12136208028 t 217 216 215 + 16108814873 t + 21277832837 t + 27815749485 t 214 213 212 + 36207832755 t + 46673294914 t + 59939943259 t 211 210 209 + 76267899428 t + 96728305390 t + 121601793102 t 208 207 206 + 152438330028 t + 189494865183 t + 234978902728 t 205 204 203 + 289044316648 t + 354790092041 t + 432136549654 t 202 201 200 + 525375285488 t + 633993110504 t + 763859162370 t 199 198 197 + 913733218608 t + 1091546464637 t + 1294918419434 t 196 195 194 + 1534448037511 t + 1806049148200 t + 2123734755624 t 193 192 191 + 2480962521587 t + 2896072631712 t + 3359092062362 t 190 189 188 + 3893789727583 t + 4485527417350 t + 5164822019639 t 187 186 185 + 5910834852458 t + 6762388641394 t + 7690575616866 t 184 183 182 + 8744349763642 t + 9884497158475 t + 11172211639160 t 181 180 179 + 12555347513575 t + 14109753753728 t + 15767318790038 t 178 177 176 + 17621272662901 t + 19584098076332 t + 21769454311526 t 175 174 173 + 24066577857293 t + 26612912654297 t + 29270243119655 t 172 171 170 + 32203453182941 t + 35242342067638 t + 38583147916740 t 169 168 167 + 42018898582082 t + 45781327369296 t + 49621726058364 t 166 165 164 + 53811618672390 t + 58055531597778 t + 62669169687081 t 163 162 161 + 67305307752876 t + 72328211795922 t + 77334108954707 t 160 159 158 + 82740110592104 t + 88081420329346 t + 93832050784069 t 157 156 155 + 99462192114931 t + 105506480760965 t + 111366724048769 t 154 153 152 + 117641421984311 t + 123661414127611 t + 130091710857232 t 151 150 149 + 136190500020373 t + 142691205834949 t + 148778719971641 t 148 147 146 + 155255943682569 t + 161234942493065 t + 167588186368218 t 145 144 143 + 173356590020049 t + 179481248961241 t + 184934821472250 t 142 141 140 + 190724958961843 t + 195760203421117 t + 201111555004700 t 139 138 137 + 205628766128684 t + 210441805931306 t + 214348106347950 t 136 135 134 + 218531107366999 t + 221743399249260 t + 225215307199710 t 133 132 131 + 227662958057852 t + 230356012479774 t + 231983216787385 t 130 129 128 + 233845149228825 t + 234612803074640 t + 235608573536788 t 127 126 125 + 235495637641341 t + 235608573536788 t + 234612803074640 t 124 123 122 + 233845149228825 t + 231983216787385 t + 230356012479774 t 121 120 119 + 227662958057852 t + 225215307199710 t + 221743399249260 t 118 117 116 + 218531107366999 t + 214348106347950 t + 210441805931306 t 115 114 113 + 205628766128684 t + 201111555004700 t + 195760203421117 t 112 111 110 + 190724958961843 t + 184934821472250 t + 179481248961241 t 109 108 107 + 173356590020049 t + 167588186368218 t + 161234942493065 t 106 105 104 + 155255943682569 t + 148778719971641 t + 142691205834949 t 103 102 101 + 136190500020373 t + 130091710857232 t + 123661414127611 t 100 99 98 + 117641421984311 t + 111366724048769 t + 105506480760965 t 97 96 95 + 99462192114931 t + 93832050784069 t + 88081420329346 t 94 93 92 + 82740110592104 t + 77334108954707 t + 72328211795922 t 91 90 89 + 67305307752876 t + 62669169687081 t + 58055531597778 t 88 87 86 + 53811618672390 t + 49621726058364 t + 45781327369296 t 85 84 83 + 42018898582082 t + 38583147916740 t + 35242342067638 t 82 81 80 + 32203453182941 t + 29270243119655 t + 26612912654297 t 79 78 77 + 24066577857293 t + 21769454311526 t + 19584098076332 t 76 75 74 + 17621272662901 t + 15767318790038 t + 14109753753728 t 73 72 71 + 12555347513575 t + 11172211639160 t + 9884497158475 t 70 69 68 + 8744349763642 t + 7690575616866 t + 6762388641394 t 67 66 65 + 5910834852458 t + 5164822019639 t + 4485527417350 t 64 63 62 + 3893789727583 t + 3359092062362 t + 2896072631712 t 61 60 59 + 2480962521587 t + 2123734755624 t + 1806049148200 t 58 57 56 + 1534448037511 t + 1294918419434 t + 1091546464637 t 55 54 53 + 913733218608 t + 763859162370 t + 633993110504 t 52 51 50 + 525375285488 t + 432136549654 t + 354790092041 t 49 48 47 + 289044316648 t + 234978902728 t + 189494865183 t 46 45 44 43 + 152438330028 t + 121601793102 t + 96728305390 t + 76267899428 t 42 41 40 39 + 59939943259 t + 46673294914 t + 36207832755 t + 27815749485 t 38 37 36 35 + 21277832837 t + 16108814873 t + 12136208028 t + 9042955299 t 34 33 32 31 + 6700608511 t + 4906705619 t + 3570184087 t + 2564916974 t 30 29 28 27 + 1829226628 t + 1286732274 t + 897513480 t + 616713182 t 26 25 24 23 + 419644752 t + 280886262 t + 185891183 t + 120800285 t 22 21 20 19 18 + 77467356 t + 48675709 t + 30112588 t + 18203969 t + 10802695 t 17 16 15 14 13 + 6243353 t + 3529278 t + 1934619 t + 1032045 t + 530839 t 12 11 10 9 8 7 + 264098 t + 125713 t + 57304 t + 24724 t + 10095 t + 3840 t 6 5 4 3 2 / 16 + 1347 t + 427 t + 125 t + 28 t + 8 t - t + 1) / ((t + 1) / 8 4 10 9 8 7 6 5 4 3 2 (t - t + 1) (t + t + t + t + t + t + t + t + t + t + 1) 12 10 8 6 4 2 (t - t + t - t + t - t + 1) 10 9 8 7 6 5 4 3 2 (t - t + t - t + t - t + t - t + t - t + 1) 6 5 4 3 2 2 8 7 5 4 3 (t + t + t + t + t + t + 1) (t - t + t - t + t - t + 1) 6 5 4 3 2 2 8 7 5 4 3 (t - t + t - t + t - t + 1) (t + t - t - t - t + t + 1) 4 3 2 3 (t + t + t + t + 1) 12 11 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + t + t + 1) 4 3 2 3 8 2 (t - t + t - t + 1) (t + 1) 12 11 10 9 8 7 6 5 4 3 2 (t - t + t - t + t - t + t - t + t - t + t - t + 1) 4 2 2 4 4 2 8 6 3 17 (t - t + 1) (t + 1) (t + 1) (t - t + 1) (-1 + t) 8 6 4 2 2 5 2 5 6 3 15 (t - t + t - t + 1) (t + t + 1) (t - t + 1) (t + t + 1) (t + 1) ) and in Maple notation -(t^254-t^253+8*t^252+28*t^251+125*t^250+427*t^249+1347*t^248+3840*t^247+10095* t^246+24724*t^245+57304*t^244+125713*t^243+264098*t^242+530839*t^241+1032045*t^ 240+1934619*t^239+3529278*t^238+6243353*t^237+10802695*t^236+18203969*t^235+ 30112588*t^234+48675709*t^233+77467356*t^232+120800285*t^231+185891183*t^230+ 280886262*t^229+419644752*t^228+616713182*t^227+897513480*t^226+1286732274*t^ 225+1829226628*t^224+2564916974*t^223+3570184087*t^222+4906705619*t^221+ 6700608511*t^220+9042955299*t^219+12136208028*t^218+16108814873*t^217+ 21277832837*t^216+27815749485*t^215+36207832755*t^214+46673294914*t^213+ 59939943259*t^212+76267899428*t^211+96728305390*t^210+121601793102*t^209+ 152438330028*t^208+189494865183*t^207+234978902728*t^206+289044316648*t^205+ 354790092041*t^204+432136549654*t^203+525375285488*t^202+633993110504*t^201+ 763859162370*t^200+913733218608*t^199+1091546464637*t^198+1294918419434*t^197+ 1534448037511*t^196+1806049148200*t^195+2123734755624*t^194+2480962521587*t^193 +2896072631712*t^192+3359092062362*t^191+3893789727583*t^190+4485527417350*t^ 189+5164822019639*t^188+5910834852458*t^187+6762388641394*t^186+7690575616866*t ^185+8744349763642*t^184+9884497158475*t^183+11172211639160*t^182+ 12555347513575*t^181+14109753753728*t^180+15767318790038*t^179+17621272662901*t ^178+19584098076332*t^177+21769454311526*t^176+24066577857293*t^175+ 26612912654297*t^174+29270243119655*t^173+32203453182941*t^172+35242342067638*t ^171+38583147916740*t^170+42018898582082*t^169+45781327369296*t^168+ 49621726058364*t^167+53811618672390*t^166+58055531597778*t^165+62669169687081*t ^164+67305307752876*t^163+72328211795922*t^162+77334108954707*t^161+ 82740110592104*t^160+88081420329346*t^159+93832050784069*t^158+99462192114931*t ^157+105506480760965*t^156+111366724048769*t^155+117641421984311*t^154+ 123661414127611*t^153+130091710857232*t^152+136190500020373*t^151+ 142691205834949*t^150+148778719971641*t^149+155255943682569*t^148+ 161234942493065*t^147+167588186368218*t^146+173356590020049*t^145+ 179481248961241*t^144+184934821472250*t^143+190724958961843*t^142+ 195760203421117*t^141+201111555004700*t^140+205628766128684*t^139+ 210441805931306*t^138+214348106347950*t^137+218531107366999*t^136+ 221743399249260*t^135+225215307199710*t^134+227662958057852*t^133+ 230356012479774*t^132+231983216787385*t^131+233845149228825*t^130+ 234612803074640*t^129+235608573536788*t^128+235495637641341*t^127+ 235608573536788*t^126+234612803074640*t^125+233845149228825*t^124+ 231983216787385*t^123+230356012479774*t^122+227662958057852*t^121+ 225215307199710*t^120+221743399249260*t^119+218531107366999*t^118+ 214348106347950*t^117+210441805931306*t^116+205628766128684*t^115+ 201111555004700*t^114+195760203421117*t^113+190724958961843*t^112+ 184934821472250*t^111+179481248961241*t^110+173356590020049*t^109+ 167588186368218*t^108+161234942493065*t^107+155255943682569*t^106+ 148778719971641*t^105+142691205834949*t^104+136190500020373*t^103+ 130091710857232*t^102+123661414127611*t^101+117641421984311*t^100+ 111366724048769*t^99+105506480760965*t^98+99462192114931*t^97+93832050784069*t^ 96+88081420329346*t^95+82740110592104*t^94+77334108954707*t^93+72328211795922*t ^92+67305307752876*t^91+62669169687081*t^90+58055531597778*t^89+53811618672390* t^88+49621726058364*t^87+45781327369296*t^86+42018898582082*t^85+38583147916740 *t^84+35242342067638*t^83+32203453182941*t^82+29270243119655*t^81+ 26612912654297*t^80+24066577857293*t^79+21769454311526*t^78+19584098076332*t^77 +17621272662901*t^76+15767318790038*t^75+14109753753728*t^74+12555347513575*t^ 73+11172211639160*t^72+9884497158475*t^71+8744349763642*t^70+7690575616866*t^69 +6762388641394*t^68+5910834852458*t^67+5164822019639*t^66+4485527417350*t^65+ 3893789727583*t^64+3359092062362*t^63+2896072631712*t^62+2480962521587*t^61+ 2123734755624*t^60+1806049148200*t^59+1534448037511*t^58+1294918419434*t^57+ 1091546464637*t^56+913733218608*t^55+763859162370*t^54+633993110504*t^53+ 525375285488*t^52+432136549654*t^51+354790092041*t^50+289044316648*t^49+ 234978902728*t^48+189494865183*t^47+152438330028*t^46+121601793102*t^45+ 96728305390*t^44+76267899428*t^43+59939943259*t^42+46673294914*t^41+36207832755 *t^40+27815749485*t^39+21277832837*t^38+16108814873*t^37+12136208028*t^36+ 9042955299*t^35+6700608511*t^34+4906705619*t^33+3570184087*t^32+2564916974*t^31 +1829226628*t^30+1286732274*t^29+897513480*t^28+616713182*t^27+419644752*t^26+ 280886262*t^25+185891183*t^24+120800285*t^23+77467356*t^22+48675709*t^21+ 30112588*t^20+18203969*t^19+10802695*t^18+6243353*t^17+3529278*t^16+1934619*t^ 15+1032045*t^14+530839*t^13+264098*t^12+125713*t^11+57304*t^10+24724*t^9+10095* t^8+3840*t^7+1347*t^6+427*t^5+125*t^4+28*t^3+8*t^2-t+1)/(t^16+1)/(t^8-t^4+1)/(t ^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^12-t^10+t^8-t^6+t^4-t^2+1)/(t^10-t^ 9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1)/(t^6+t^5+t^4+t^3+t^2+t+1)^2/(t^8-t^7+t^5-t^4 +t^3-t+1)/(t^6-t^5+t^4-t^3+t^2-t+1)^2/(t^8+t^7-t^5-t^4-t^3+t+1)/(t^4+t^3+t^2+t+ 1)^3/(t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^4-t^3+t^2-t+1)^3/( t^8+1)^2/(t^12-t^11+t^10-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1)/(t^4-t^2+1)^2/(t^ 4+1)^4/(t^2+1)^8/(t^6-t^3+1)/(-1+t)^17/(t^8-t^6+t^4-t^2+1)/(t^2+t+1)^5/(t^2-t+1 )^5/(t^6+t^3+1)/(t+1)^15 For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 9, 45, 207, 795, 2739, 8550, 24591, 65809, 165821, 395867, 901723, 1967888, 4136477, 8398764, 16535154, 31630390, 58965214, 107287595, 190973410, 332943467, 569585274, 957037970, 1581788460, 2573496325, 4126777398, 6526134528, 10188889502, 15711616171, 23951790860] Furthermore, a(n) is a quasi-polynomial given as sum of, 24, quasi-polynomials 24 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), Q[13](n), Q[14](n), Q[15](n), Q[16](n), Q[17](n), Q[18](n), Q[19](n), Q[20](n), Q[21](n), Q[22](n), Q[23](n), Q[24](n), are defined as followed Q[1](n), is the polynomial 4465908195054673 16 4465908195054673 15 ------------------------------- n + ----------------------------- n 4781540821100080360587264000000 33205144590972780281856000000 711099854146560173 14 228781769080655489 13 + ----------------------------- n + --------------------------- n 79692347018334672676454400000 632478944589957719654400000 12533221274975478821 12 2325820972627441937 11 + ---------------------------- n + -------------------------- n 1251057253035081203712000000 11583863454028529664000000 5803940568424517281459 10 1366957567864019844619 9 + ---------------------------- n + -------------------------- n 1929077792267380457472000000 39806367142025311027200000 671622017086239000933499 8 315116563154870699385407 7 + ---------------------------- n + --------------------------- n 2229156559953417417523200000 154802538885653987328000000 2197303960092350654703923 6 159435383533581760172333 5 + --------------------------- n + ------------------------- n 208509542172513533952000000 3861287818009509888000000 34379942586871224670515166529 4 56629669781588683043608783 3 + ------------------------------ n + --------------------------- n 286892449266004821635235840000 227692420052384779075584000 8499901565407448835725413421 2 12598219713162956527411691 + ----------------------------- n + -------------------------- n 24395616434184083472384000000 42218874241541996544000000 92334573618104101409952301117 + ------------------------------ 455384840104769558151168000000 and in Maple notation 4465908195054673/4781540821100080360587264000000*n^16+4465908195054673/ 33205144590972780281856000000*n^15+711099854146560173/ 79692347018334672676454400000*n^14+228781769080655489/ 632478944589957719654400000*n^13+12533221274975478821/ 1251057253035081203712000000*n^12+2325820972627441937/ 11583863454028529664000000*n^11+5803940568424517281459/ 1929077792267380457472000000*n^10+1366957567864019844619/ 39806367142025311027200000*n^9+671622017086239000933499/ 2229156559953417417523200000*n^8+315116563154870699385407/ 154802538885653987328000000*n^7+2197303960092350654703923/ 208509542172513533952000000*n^6+159435383533581760172333/ 3861287818009509888000000*n^5+34379942586871224670515166529/ 286892449266004821635235840000*n^4+56629669781588683043608783/ 227692420052384779075584000*n^3+8499901565407448835725413421/ 24395616434184083472384000000*n^2+12598219713162956527411691/ 42218874241541996544000000*n+92334573618104101409952301117/ 455384840104769558151168000000 This is the leading term in particular, a(n) , is asymptotic to 16 4465908195054673 n ------------------------------- 4781540821100080360587264000000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 22161558721 14 22161558721 13 [- --------------------------- n - ------------------------- n 557289139988354354380800000 4422929682447256780800000 27637701211997 12 44844020727133 11 - -------------------------- n - ------------------------- n 95535281140860746465280000 4422929682447256780800000 287644336140068357 10 54423984324455591 9 - ---------------------------- n - -------------------------- n 1194191014260759330816000000 13268789047341770342400000 23011814424862360961 8 15096439826515433101 7 - --------------------------- n - -------------------------- n 445831311990683483504640000 30960507777130797465600000 461203331728786246321 6 3519779183586471019 5 - --------------------------- n - ------------------------ n 132687890473417703424000000 189014088993472512000000 7391247766213922222797 4 16164609837677551711 3 - -------------------------- n - ----------------------- n 95535281140860746465280000 48250141990333710336000 399498712579001263043197 2 7059510986114799556758247 - --------------------------- n - --------------------------- n 170598716322965618688000000 464407616656961961984000000 438297468824106417175116591667 22161558721 14 - -------------------------------, --------------------------- n 1912616328440032144234905600000 557289139988354354380800000 22161558721 13 27637701211997 12 + ------------------------- n + -------------------------- n 4422929682447256780800000 95535281140860746465280000 44844020727133 11 287644336140068357 10 + ------------------------- n + ---------------------------- n 4422929682447256780800000 1194191014260759330816000000 54423984324455591 9 23011814424862360961 8 + -------------------------- n + --------------------------- n 13268789047341770342400000 445831311990683483504640000 15096439826515433101 7 461203331728786246321 6 + -------------------------- n + --------------------------- n 30960507777130797465600000 132687890473417703424000000 3519779183586471019 5 7391247766213922222797 4 + ------------------------ n + -------------------------- n 189014088993472512000000 95535281140860746465280000 16164609837677551711 3 399498712579001263043197 2 + ----------------------- n + --------------------------- n 48250141990333710336000 170598716322965618688000000 7059510986114799556758247 438297468824106417175116591667 + --------------------------- n + -------------------------------] 464407616656961961984000000 1912616328440032144234905600000 and in Maple format [-22161558721/557289139988354354380800000*n^14-22161558721/ 4422929682447256780800000*n^13-27637701211997/95535281140860746465280000*n^12-\ 44844020727133/4422929682447256780800000*n^11-287644336140068357/ 1194191014260759330816000000*n^10-54423984324455591/13268789047341770342400000* n^9-23011814424862360961/445831311990683483504640000*n^8-15096439826515433101/ 30960507777130797465600000*n^7-461203331728786246321/ 132687890473417703424000000*n^6-3519779183586471019/189014088993472512000000*n^ 5-7391247766213922222797/95535281140860746465280000*n^4-16164609837677551711/ 48250141990333710336000*n^3-399498712579001263043197/ 170598716322965618688000000*n^2-7059510986114799556758247/ 464407616656961961984000000*n-438297468824106417175116591667/ 1912616328440032144234905600000, 22161558721/557289139988354354380800000*n^14+ 22161558721/4422929682447256780800000*n^13+27637701211997/ 95535281140860746465280000*n^12+44844020727133/4422929682447256780800000*n^11+ 287644336140068357/1194191014260759330816000000*n^10+54423984324455591/ 13268789047341770342400000*n^9+23011814424862360961/445831311990683483504640000 *n^8+15096439826515433101/30960507777130797465600000*n^7+461203331728786246321/ 132687890473417703424000000*n^6+3519779183586471019/189014088993472512000000*n^ 5+7391247766213922222797/95535281140860746465280000*n^4+16164609837677551711/ 48250141990333710336000*n^3+399498712579001263043197/ 170598716322965618688000000*n^2+7059510986114799556758247/ 464407616656961961984000000*n+438297468824106417175116591667/ 1912616328440032144234905600000] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 2761 4 2843 3 46115 2 2036657 16584899 [- ---------- n - -------- n - -------- n - --------- n - ----------, 2720977920 75582720 90699264 680244480 1511654400 2761 4 893 3 204007 2 1645981 4517873 - ---------- n - -------- n - --------- n - --------- n - ---------, 2720977920 25194240 453496320 680244480 503884800 2761 4 2761 3 217291 2 68197 15069259 ---------- n + -------- n + --------- n + -------- n + ---------] 1360488960 37791360 226748160 12597120 755827200 and in Maple format [-2761/2720977920*n^4-2843/75582720*n^3-46115/90699264*n^2-2036657/680244480*n-\ 16584899/1511654400, -2761/2720977920*n^4-893/25194240*n^3-204007/453496320*n^2 -1645981/680244480*n-4517873/503884800, 2761/1360488960*n^4+2761/37791360*n^3+ 217291/226748160*n^2+68197/12597120*n+15069259/755827200] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 13471 6 13471 5 113368133 4 36987563 3 [------------- n + ------------ n + -------------- n + ------------- n 6957847019520 128849018880 48704929136640 1352914698240 325240913 2 38822115 5899298448871 1403657 7 + ------------- n + ----------- n - ---------------, - -------------- n 1803886264320 60129542144 730573937049600 63136019251200 4210971 6 124199443 5 155709121 4 21485714989 3 - ------------- n - ------------- n - ------------ n - ------------- n 3006477107200 3382286745600 300647710720 5073430118400 14721517 2 3314570947 257931980897 13471 6 - --------- n - ----------- n - -------------, - ------------- n 734003200 63417876480 3946001203200 6957847019520 13471 5 113368133 4 36987563 3 - ------------ n - -------------- n - ------------- n 128849018880 48704929136640 1352914698240 325240913 2 38822115 5899298448871 1403657 7 - ------------- n - ----------- n + ---------------, -------------- n 1803886264320 60129542144 730573937049600 63136019251200 4210971 6 124199443 5 155709121 4 21485714989 3 + ------------- n + ------------- n + ------------ n + ------------- n 3006477107200 3382286745600 300647710720 5073430118400 14721517 2 3314570947 257931980897 + --------- n + ----------- n + -------------] 734003200 63417876480 3946001203200 and in Maple format [13471/6957847019520*n^6+13471/128849018880*n^5+113368133/48704929136640*n^4+ 36987563/1352914698240*n^3+325240913/1803886264320*n^2+38822115/60129542144*n-\ 5899298448871/730573937049600, -1403657/63136019251200*n^7-4210971/ 3006477107200*n^6-124199443/3382286745600*n^5-155709121/300647710720*n^4-\ 21485714989/5073430118400*n^3-14721517/734003200*n^2-3314570947/63417876480*n-\ 257931980897/3946001203200, -13471/6957847019520*n^6-13471/128849018880*n^5-\ 113368133/48704929136640*n^4-36987563/1352914698240*n^3-325240913/1803886264320 *n^2-38822115/60129542144*n+5899298448871/730573937049600, 1403657/ 63136019251200*n^7+4210971/3006477107200*n^6+124199443/3382286745600*n^5+ 155709121/300647710720*n^4+21485714989/5073430118400*n^3+14721517/734003200*n^2 +3314570947/63417876480*n+257931980897/3946001203200] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 71 2 213 3517 71 2 191 2399 [- ------ n - ----- n - ------, ------ n + ----- n + ------, 375000 62500 225000 375000 75000 281250 71 2 737 24341 71 2 541 13757 - ------ n - ------ n - -------, - ------ n - ------ n - -------, 750000 375000 2250000 750000 375000 2250000 71 2 1601 13519 ------ n + ------ n + ------] 375000 375000 562500 and in Maple format [-71/375000*n^2-213/62500*n-3517/225000, 71/375000*n^2+191/75000*n+2399/281250, -71/750000*n^2-737/375000*n-24341/2250000, -71/750000*n^2-541/375000*n-13757/ 2250000, 71/375000*n^2+1601/375000*n+13519/562500] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 2761 4 25259 3 673441 2 1945117 105224471 [--------- n + --------- n + --------- n + --------- n + ----------, 906992640 226748160 453496320 226748160 4534963200 2761 4 24439 3 629161 2 345445 89915111 - --------- n - --------- n - --------- n - -------- n - ----------, 906992640 226748160 453496320 45349632 4534963200 2761 4 2761 3 651301 2 204019 97569791 - --------- n - -------- n - --------- n - -------- n - ----------, 453496320 12597120 226748160 12597120 2267481600 2761 4 25259 3 673441 2 1945117 105224471 - --------- n - --------- n - --------- n - --------- n - ----------, 906992640 226748160 453496320 226748160 4534963200 2761 4 24439 3 629161 2 345445 89915111 --------- n + --------- n + --------- n + -------- n + ----------, 906992640 226748160 453496320 45349632 4534963200 2761 4 2761 3 651301 2 204019 97569791 --------- n + -------- n + --------- n + -------- n + ----------] 453496320 12597120 226748160 12597120 2267481600 and in Maple format [2761/906992640*n^4+25259/226748160*n^3+673441/453496320*n^2+1945117/226748160* n+105224471/4534963200, -2761/906992640*n^4-24439/226748160*n^3-629161/ 453496320*n^2-345445/45349632*n-89915111/4534963200, -2761/453496320*n^4-2761/ 12597120*n^3-651301/226748160*n^2-204019/12597120*n-97569791/2267481600, -2761/ 906992640*n^4-25259/226748160*n^3-673441/453496320*n^2-1945117/226748160*n-\ 105224471/4534963200, 2761/906992640*n^4+24439/226748160*n^3+629161/453496320*n ^2+345445/45349632*n+89915111/4534963200, 2761/453496320*n^4+2761/12597120*n^3+ 651301/226748160*n^2+204019/12597120*n+97569791/2267481600] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 9 n 15 9 n 59 n 223 -27 [- ----- - ----, ----- + 3/1372, - ----- - ----, 1/2401, ----, 1/2401, 19208 2401 19208 19208 9604 4802 59 n 11 ----- + ---] 19208 343 and in Maple format [-9/19208*n-15/2401, 9/19208*n+3/1372, -59/19208*n-223/9604, 1/2401, -27/4802, 1/2401, 59/19208*n+11/343] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 2003 3 6009 2 156631 145407 [-------- n + ------- n + -------- n + -------, 37748736 4194304 12582912 4194304 2003 3 971 2 28693 7031 -------- n + ------ n + ------- n + ------, 37748736 786432 3145728 147456 91 2 819 232471 ------- n + ------- n - -------, 2097152 1048576 6291456 2003 3 10259 2 6389 61141 - -------- n - ------- n - ------ n - -------, 37748736 6291456 393216 2359296 2003 3 6009 2 156631 145407 - -------- n - ------- n - -------- n - -------, 37748736 4194304 12582912 4194304 2003 3 971 2 28693 7031 - -------- n - ------ n - ------- n - ------, 37748736 786432 3145728 147456 91 2 819 232471 - ------- n - ------- n + -------, 2097152 1048576 6291456 2003 3 10259 2 6389 61141 -------- n + ------- n + ------ n + -------] 37748736 6291456 393216 2359296 and in Maple format [2003/37748736*n^3+6009/4194304*n^2+156631/12582912*n+145407/4194304, 2003/ 37748736*n^3+971/786432*n^2+28693/3145728*n+7031/147456, 91/2097152*n^2+819/ 1048576*n-232471/6291456, -2003/37748736*n^3-10259/6291456*n^2-6389/393216*n-\ 61141/2359296, -2003/37748736*n^3-6009/4194304*n^2-156631/12582912*n-145407/ 4194304, -2003/37748736*n^3-971/786432*n^2-28693/3145728*n-7031/147456, -91/ 2097152*n^2-819/1048576*n+232471/6291456, 2003/37748736*n^3+10259/6291456*n^2+ 6389/393216*n+61141/2359296] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -1 -1 [0, 0, --, 0, 0, --, 0, 0, 2/81] 81 81 and in Maple format [0, 0, -1/81, 0, 0, -1/81, 0, 0, 2/81] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 10 71 2 213 17633 71 2 809 1391 [- ------ n - ----- n - -------, ------ n + ------ n + ------, 375000 62500 1125000 375000 375000 281250 71 2 17273 71 2 653 3757 ------ n + 1/200 n + ------, - ------ n - ------ n - ------, 250000 750000 250000 125000 150000 71 2 1747 3089 71 2 213 17633 - ------ n - ------ n - ------, ------ n + ----- n + -------, 375000 375000 112500 375000 62500 1125000 71 2 809 1391 71 2 17273 - ------ n - ------ n - ------, - ------ n - 1/200 n - ------, 375000 375000 281250 250000 750000 71 2 653 3757 71 2 1747 3089 ------ n + ------ n + ------, ------ n + ------ n + ------] 250000 125000 150000 375000 375000 112500 and in Maple format [-71/375000*n^2-213/62500*n-17633/1125000, 71/375000*n^2+809/375000*n+1391/ 281250, 71/250000*n^2+1/200*n+17273/750000, -71/250000*n^2-653/125000*n-3757/ 150000, -71/375000*n^2-1747/375000*n-3089/112500, 71/375000*n^2+213/62500*n+ 17633/1125000, -71/375000*n^2-809/375000*n-1391/281250, -71/250000*n^2-1/200*n-\ 17273/750000, 71/250000*n^2+653/125000*n+3757/150000, 71/375000*n^2+1747/375000 *n+3089/112500] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 11 -5 -5 -1 -1 -1 -1 -1 -1 [---, 4/121, ---, 7/242, ---, ---, ---, ---, ---, ---, 7/242] 242 242 121 121 121 121 121 121 and in Maple format [-5/242, 4/121, -5/242, 7/242, -1/121, -1/121, -1/121, -1/121, -1/121, -1/121, 7/242] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 n 49 n 13 -5 n n 59 n [---- + -----, --- + ---, ----, - --- - 5/864, - ---- - -----, - --- - 1/48, 2592 15552 864 864 7776 864 2592 15552 432 n 49 n 13 n n 59 n - ---- - -----, - --- - ---, 5/7776, --- + 5/864, ---- + -----, --- + 1/48] 2592 15552 864 864 864 2592 15552 432 and in Maple format [1/2592*n+49/15552, 1/864*n+13/864, -5/7776, -1/864*n-5/864, -1/2592*n-59/15552 , -1/432*n-1/48, -1/2592*n-49/15552, -1/864*n-13/864, 5/7776, 1/864*n+5/864, 1/ 2592*n+59/15552, 1/432*n+1/48] Q[13](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 13 -1 -1 -1 -5 -5 -5 -5 [---, 2/169, 3/338, ---, 3/338, 2/169, ---, 3/169, ---, ---, ---, ---, 3/169] 169 169 169 338 338 338 338 and in Maple format [-1/169, 2/169, 3/338, -1/169, 3/338, 2/169, -1/169, 3/169, -5/338, -5/338, -5/ 338, -5/338, 3/169] Q[14](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 14 87 n 107 87 n 355 89 n 53 n 17 25 n [----- + ----, ----- + ----, ----- + ----, ---- + ----, ----, - ---- - 1/2401, 19208 2401 19208 9604 19208 1372 2401 2401 4802 2401 89 n 215 87 n 107 87 n 355 89 n 53 - ----- - ----, - ----- - ----, - ----- - ----, - ----- - ----, 19208 4802 19208 2401 19208 9604 19208 1372 n 17 -25 n 89 n 215 - ---- - ----, ----, ---- + 1/2401, ----- + ----] 2401 2401 4802 2401 19208 4802 and in Maple format [87/19208*n+107/2401, 87/19208*n+355/9604, 89/19208*n+53/1372, 1/2401*n+17/2401 , 25/4802, -1/2401*n-1/2401, -89/19208*n-215/4802, -87/19208*n-107/2401, -87/ 19208*n-355/9604, -89/19208*n-53/1372, -1/2401*n-17/2401, -25/4802, 1/2401*n+1/ 2401, 89/19208*n+215/4802] Q[15](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 15 -4 -2 -4 [1/225, 7/450, 1/450, 7/450, ---, ---, ---, 7/450, 1/450, 7/450, 1/225, 1/450, 225 225 225 -4 -4 ---, ---, 1/450] 225 225 and in Maple format [1/225, 7/450, 1/450, 7/450, -4/225, -2/225, -4/225, 7/450, 1/450, 7/450, 1/225 , 1/450, -4/225, -4/225, 1/450] Q[16](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 16 15 n 197 15 n 83 15 n 135 15 n 13 15 n 73 15 n 21 [---- + ----, ---- + ----, ---- + ----, ---- + ---, ---- + ----, ---- + ----, 4096 4096 4096 2048 4096 4096 4096 512 4096 4096 4096 2048 15 n 57 15 n 197 15 n 83 15 n 135 1/1024, - ---- - ----, - ---- - ----, - ---- - ----, - ---- - ----, 4096 1024 4096 4096 4096 2048 4096 4096 15 n 13 15 n 73 15 n 21 -1 15 n 57 - ---- - ---, - ---- - ----, - ---- - ----, ----, ---- + ----] 4096 512 4096 4096 4096 2048 1024 4096 1024 and in Maple format [15/4096*n+197/4096, 15/4096*n+83/2048, 15/4096*n+135/4096, 15/4096*n+13/512, 15/4096*n+73/4096, 15/4096*n+21/2048, 1/1024, -15/4096*n-57/1024, -15/4096*n-\ 197/4096, -15/4096*n-83/2048, -15/4096*n-135/4096, -15/4096*n-13/512, -15/4096* n-73/4096, -15/4096*n-21/2048, -1/1024, 15/4096*n+57/1024] Q[17](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 18 -1 -2 -1 [0, 0, 1/81, 0, 0, --, 0, 0, --, 0, 0, --, 0, 0, 1/81, 0, 0, 2/81] 81 81 81 and in Maple format [0, 0, 1/81, 0, 0, -1/81, 0, 0, -2/81, 0, 0, -1/81, 0, 0, 1/81, 0, 0, 2/81] Q[18](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 20 -1 -1 -1 -1 -1 -1 [--, 1/25, ---, 0, 1/100, 0, ---, 0, 1/100, --, 1/25, --, 1/100, 0, ---, 0, 25 100 100 25 25 100 -1 1/100, 0, ---, 1/25] 100 and in Maple format [-1/25, 1/25, -1/100, 0, 1/100, 0, -1/100, 0, 1/100, -1/25, 1/25, -1/25, 1/100, 0, -1/100, 0, 1/100, 0, -1/100, 1/25] Q[19](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 22 -5 -5 -3 -3 -3 -5 -8 [---, 8/121, ---, 5/242, 3/121, ---, 3/121, ---, 3/121, ---, ---, 5/242, ---, 242 242 121 121 121 242 121 -5 -3 -3 -3 5/242, ---, ---, 3/121, ---, 3/121, ---, 3/121, 5/242] 242 121 121 121 and in Maple format [-5/242, 8/121, -5/242, 5/242, 3/121, -3/121, 3/121, -3/121, 3/121, -3/121, -5/ 242, 5/242, -8/121, 5/242, -5/242, -3/121, 3/121, -3/121, 3/121, -3/121, 3/121, 5/242] Q[20](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 24 -1 -1 -1 -1 -1 -1 -1 -1 [--, 1/12, --, 1/12, --, 1/18, --, 1/36, 0, --, 1/24, --, 1/18, --, 1/12, --, 18 12 18 24 36 18 12 12 -1 -1 -1 1/18, --, 1/24, --, 0, 1/36, --, 1/18] 18 36 24 and in Maple format [-1/18, 1/12, -1/12, 1/12, -1/18, 1/18, -1/24, 1/36, 0, -1/36, 1/24, -1/18, 1/ 18, -1/12, 1/12, -1/12, 1/18, -1/18, 1/24, -1/36, 0, 1/36, -1/24, 1/18] Q[21](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 26 -29 21 -29 -5 23 -23 23 -23 -5 [5/169, 8/169, ---, ---, ---, 8/169, 5/169, ---, ---, ---, ---, ---, 5/169, ---, 338 169 338 169 338 338 338 338 169 -8 29 -21 29 -8 -5 -23 23 -23 23 -5 ---, ---, ---, ---, ---, ---, 5/169, ---, ---, ---, ---, ---] 169 338 169 338 169 169 338 338 338 338 169 and in Maple format [5/169, 8/169, -29/338, 21/169, -29/338, 8/169, 5/169, -5/169, 23/338, -23/338, 23/338, -23/338, 5/169, -5/169, -8/169, 29/338, -21/169, 29/338, -8/169, -5/169 , 5/169, -23/338, 23/338, -23/338, 23/338, -5/169] Q[22](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 28 -1 -3 -3 -1 -3 -4 [--, 4/49, --, 8/49, -1/7, 8/49, --, 4/49, --, 3/98, 0, 0, 0, --, 1/28, --, 28 28 28 28 98 49 -8 -8 -4 -3 3/28, --, 1/7, --, 3/28, --, 1/28, --, 0, 0, 0, 3/98] 49 49 49 98 and in Maple format [-1/28, 4/49, -3/28, 8/49, -1/7, 8/49, -3/28, 4/49, -1/28, 3/98, 0, 0, 0, -3/98 , 1/28, -4/49, 3/28, -8/49, 1/7, -8/49, 3/28, -4/49, 1/28, -3/98, 0, 0, 0, 3/98 ] Q[23](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 30 -1 -8 14 -8 -1 -17 [7/225, 1/450, ---, 7/150, ---, ---, ---, 7/150, ---, 1/450, 7/225, ---, 4/75, 150 225 225 225 150 450 -4 17 -7 -1 -7 -14 -7 -1 -7 17 --, ---, ---, ---, 1/150, ---, 8/225, ---, 8/225, ---, 1/150, ---, ---, ---, 75 450 225 450 150 225 150 450 225 450 -4 -17 --, 4/75, ---] 75 450 and in Maple format [7/225, 1/450, -1/150, 7/150, -8/225, 14/225, -8/225, 7/150, -1/150, 1/450, 7/ 225, -17/450, 4/75, -4/75, 17/450, -7/225, -1/450, 1/150, -7/150, 8/225, -14/ 225, 8/225, -7/150, 1/150, -1/450, -7/225, 17/450, -4/75, 4/75, -17/450] Q[24](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 32 -1 -1 -1 -1 -1 -1 -1 -1 [--, 1/32, --, 1/32, --, 1/32, --, 1/32, --, 1/32, --, 1/32, --, 1/32, 0, --, 32 32 32 32 32 32 32 32 -1 -1 -1 -1 -1 -1 -1 1/32, --, 1/32, --, 1/32, --, 1/32, --, 1/32, --, 1/32, --, 1/32, --, 0, 32 32 32 32 32 32 32 1/32] and in Maple format [-1/32, 1/32, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32, 0, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32, -1/32, 1/32 , -1/32, 1/32, -1/32, 1/32, -1/32, 0, 1/32] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 9339893482342392009548533886376066304021996758910626251973227149042502904384019\ 1415931248326829442996283867540290185857352067664748337973885013155072497358261\ 0767954344496375728925144056233832490406636604784509529581578683501084503573390\ 9461701472856211818123031650941239991334626379290818429100102931493902351144895\ 1126567695603116370973940712756777263611905661127369373181087540714892699681586\ 5354491142191548237275023522658779543822859642189570866942225089597346317045717\ 5211321058941333389447063482941926904682861392440519705073281355252698952437713\ 9048704337553577020100437241138332457008615858670726529095462270141484333031436\ 4598562008091855123451826110369142921861653578733606331260050269306767195333432\ 2476063071379330017759088146950281963952488946885053512060729756897930376369554\ 9968395001377580966700699917647133875180619590287770287128025070909006346341744\ 4845108420528217780347045936257422693257682223213835692665844573457918321060222\ 3589501742928660387939763178937975795938763656398758668214976806004607906702941\ 6889931639447256799720825010359104237981297620374000303099962244133323896908574\ 3949122155427360206851820418627848988931263453811204991973165546691713930263241\ 1034105296575039518845510124260118121671010126840829529122990351139857892987165\ 0828714889275492486080031936274434263700217044001880466093207565824181913825074\ 3789037269997529603403833728667340083074063474909168346395714731534678451739111\ 4161592122406300011720863411416398125707446365833399741835626703766960759637966\ 3930265116542795639824699060863361275583078583415219428348260827055512955263102\ 93465 ------------------------------------ --------------------------------------------------------- For , psi[18](t), we have the following theorem. Theorem Number, 17, : Let f(t) be the constant term of the rational function in z and t 2 / / 2 4 6 8 (1 + z) / |2 z (1 - t) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) / | \ 10 12 14 16 18 / t \ (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) (-t z + 1) |1 - ----| | 2 | \ z / / t \ / t \ / t \ / t \ / t \ / t \ / t \ |1 - ----| |1 - ----| |1 - ----| |1 - ---| |1 - ---| |1 - ---| |1 - ---| | 4 | | 6 | | 8 | | 10| | 12| | 14| | 16| \ z / \ z / \ z / \ z / \ z / \ z / \ z / / t \\ |1 - ---|| | 18|| \ z // We have 136 134 133 132 131 130 129 f(t) = (t + 7 t + 39 t + 173 t + 638 t + 2098 t + 6203 t 128 127 126 125 124 + 16849 t + 42421 t + 100195 t + 223455 t + 474052 t 123 122 121 120 119 + 961142 t + 1871590 t + 3512649 t + 6376122 t + 11223889 t 118 117 116 115 + 19208925 t + 32028449 t + 52129758 t + 82958496 t 114 113 112 111 + 129276372 t + 197526639 t + 296280779 t + 436725806 t 110 109 108 107 + 633233448 t + 903943381 t + 1271410119 t + 1763211992 t 106 105 104 103 + 2412583748 t + 3258929017 t + 4348295502 t + 5733622950 t 102 101 100 99 + 7474877667 t + 9638839413 t + 12298701360 t + 15533205195 t 98 97 96 95 + 19425542338 t + 24061703317 t + 29528595059 t + 35911576529 t 94 93 92 91 + 43291839604 t + 51743251101 t + 61329207368 t + 72099077437 t 90 89 88 87 + 84084909475 t + 97297927517 t + 111725609168 t + 127328794331 t 86 85 84 + 144039719788 t + 161760322947 t + 180361781993 t 83 82 81 + 199684506588 t + 219539589139 t + 239710769151 t 80 79 78 + 259957946653 t + 280021121136 t + 299625795329 t 77 76 75 + 318488553852 t + 336323857563 t + 352850610860 t 74 73 72 + 367799564701 t + 380919998145 t + 391986778079 t 71 70 69 + 400806186738 t + 407221681758 t + 411117971453 t 68 67 66 + 412424671638 t + 411117971453 t + 407221681758 t 65 64 63 + 400806186738 t + 391986778079 t + 380919998145 t 62 61 60 + 367799564701 t + 352850610860 t + 336323857563 t 59 58 57 + 318488553852 t + 299625795329 t + 280021121136 t 56 55 54 + 259957946653 t + 239710769151 t + 219539589139 t 53 52 51 + 199684506588 t + 180361781993 t + 161760322947 t 50 49 48 47 + 144039719788 t + 127328794331 t + 111725609168 t + 97297927517 t 46 45 44 43 + 84084909475 t + 72099077437 t + 61329207368 t + 51743251101 t 42 41 40 39 + 43291839604 t + 35911576529 t + 29528595059 t + 24061703317 t 38 37 36 35 + 19425542338 t + 15533205195 t + 12298701360 t + 9638839413 t 34 33 32 31 + 7474877667 t + 5733622950 t + 4348295502 t + 3258929017 t 30 29 28 27 + 2412583748 t + 1763211992 t + 1271410119 t + 903943381 t 26 25 24 23 + 633233448 t + 436725806 t + 296280779 t + 197526639 t 22 21 20 19 + 129276372 t + 82958496 t + 52129758 t + 32028449 t 18 17 16 15 14 + 19208925 t + 11223889 t + 6376122 t + 3512649 t + 1871590 t 13 12 11 10 9 8 + 961142 t + 474052 t + 223455 t + 100195 t + 42421 t + 16849 t 7 6 5 4 3 2 / 9 + 6203 t + 2098 t + 638 t + 173 t + 39 t + 7 t + 1) / ((t + 1) / 4 3 2 3 18 2 5 (t + t + t + t + 1) (-1 + t) (t + t + 1) 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + 1) 12 11 10 9 8 7 6 5 4 3 2 (t + t + t + t + t + t + t + t + t + t + t + t + 1) 4 3 2 2 4 6 5 4 3 2 2 (t - t + t - t + 1) (t + 1) (t + t + t + t + t + t + 1) 4 2 6 3 16 15 14 13 12 11 10 9 8 (t + 1) (t + t + 1) (t + t + t + t + t + t + t + t + t 7 6 5 4 3 2 8 7 5 4 3 + t + t + t + t + t + t + t + 1) (t - t + t - t + t - t + 1) 6 5 4 3 2 4 2 2 2 8 (t - t + t - t + t - t + 1) (t - t + 1) (t - t + 1) (t + 1)) and in Maple notation (t^136+7*t^134+39*t^133+173*t^132+638*t^131+2098*t^130+6203*t^129+16849*t^128+ 42421*t^127+100195*t^126+223455*t^125+474052*t^124+961142*t^123+1871590*t^122+ 3512649*t^121+6376122*t^120+11223889*t^119+19208925*t^118+32028449*t^117+ 52129758*t^116+82958496*t^115+129276372*t^114+197526639*t^113+296280779*t^112+ 436725806*t^111+633233448*t^110+903943381*t^109+1271410119*t^108+1763211992*t^ 107+2412583748*t^106+3258929017*t^105+4348295502*t^104+5733622950*t^103+ 7474877667*t^102+9638839413*t^101+12298701360*t^100+15533205195*t^99+ 19425542338*t^98+24061703317*t^97+29528595059*t^96+35911576529*t^95+43291839604 *t^94+51743251101*t^93+61329207368*t^92+72099077437*t^91+84084909475*t^90+ 97297927517*t^89+111725609168*t^88+127328794331*t^87+144039719788*t^86+ 161760322947*t^85+180361781993*t^84+199684506588*t^83+219539589139*t^82+ 239710769151*t^81+259957946653*t^80+280021121136*t^79+299625795329*t^78+ 318488553852*t^77+336323857563*t^76+352850610860*t^75+367799564701*t^74+ 380919998145*t^73+391986778079*t^72+400806186738*t^71+407221681758*t^70+ 411117971453*t^69+412424671638*t^68+411117971453*t^67+407221681758*t^66+ 400806186738*t^65+391986778079*t^64+380919998145*t^63+367799564701*t^62+ 352850610860*t^61+336323857563*t^60+318488553852*t^59+299625795329*t^58+ 280021121136*t^57+259957946653*t^56+239710769151*t^55+219539589139*t^54+ 199684506588*t^53+180361781993*t^52+161760322947*t^51+144039719788*t^50+ 127328794331*t^49+111725609168*t^48+97297927517*t^47+84084909475*t^46+ 72099077437*t^45+61329207368*t^44+51743251101*t^43+43291839604*t^42+35911576529 *t^41+29528595059*t^40+24061703317*t^39+19425542338*t^38+15533205195*t^37+ 12298701360*t^36+9638839413*t^35+7474877667*t^34+5733622950*t^33+4348295502*t^ 32+3258929017*t^31+2412583748*t^30+1763211992*t^29+1271410119*t^28+903943381*t^ 27+633233448*t^26+436725806*t^25+296280779*t^24+197526639*t^23+129276372*t^22+ 82958496*t^21+52129758*t^20+32028449*t^19+19208925*t^18+11223889*t^17+6376122*t ^16+3512649*t^15+1871590*t^14+961142*t^13+474052*t^12+223455*t^11+100195*t^10+ 42421*t^9+16849*t^8+6203*t^7+2098*t^6+638*t^5+173*t^4+39*t^3+7*t^2+1)/(t+1)^9/( t^4+t^3+t^2+t+1)^3/(-1+t)^18/(t^2+t+1)^5/(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+ t+1)/(t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^4-t^3+t^2-t+1)/(t^ 2+1)^4/(t^6+t^5+t^4+t^3+t^2+t+1)^2/(t^4+1)^2/(t^6+t^3+1)/(t^16+t^15+t^14+t^13+t ^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^8-t^7+t^5-t^4+t^3-t+1)/(t ^6-t^5+t^4-t^3+t^2-t+1)/(t^4-t^2+1)/(t^2-t+1)^2/(t^8+1) For the sake of Sloane, here are the first 31 term, starting at n=0 [1, 1, 10, 50, 241, 967, 3486, 11322, 33885, 94257, 246448, 609734, 1437269, 3243531, 7040196, 14749992, 29927526, 58965214, 113093022, 211591218, 386905330, 692583902, 1215487708, 2094177794, 3546374322, 5909243454, 9698033520, 15690144942, 25044766998, 39470938574, 61461993500] Furthermore, a(n) is a quasi-polynomial given as sum of, 17, quasi-polynomials 17 ----- \ a(n) = ) Q[i](n) / ----- i = 1 where , Q[1](n), Q[2](n), Q[3](n), Q[4](n), Q[5](n), Q[6](n), Q[7](n), Q[8](n), Q[9](n), Q[10](n), Q[11](n), Q[12](n), Q[13](n), Q[14](n), Q[15](n), Q[16](n), Q[17](n), are defined as followed Q[1](n), is the polynomial 12157712239 17 230996532541 16 --------------------------- n + -------------------------- n 240978183820089847971840000 28350374567069393879040000 226664032241531 15 13607602195427 14 + --------------------------- n + ------------------------ n 372098666192785794662400000 486403485219327836160000 93895333703204053 13 1629142986595393 12 + --------------------------- n + ----------------------- n 106313904626510227046400000 80176398662526566400000 65500120938632143 11 5139328241871253 10 + ------------------------ n + ---------------------- n 185863469626766131200000 1093314527216271360000 13593598766755603697 9 10748129027766753083 8 + ------------------------ n + ----------------------- n 278795204440149196800000 27211383788493864960000 15871128078393686532743 7 16724066997417050959 6 + ------------------------- n + ---------------------- n 6360660960560440934400000 1370536729273958400000 598009753926881786119723 5 79459596143100908576059 4 + -------------------------- n + ------------------------ n 13125173410680274944000000 625375909567707217920000 817329899777788361114737 3 53373594253569206854967 2 + ------------------------- n + ------------------------ n 3189417138795306811392000 148899026087549337600000 35994670127500425491533451 413281211023145512085217587 + --------------------------- n + ---------------------------- 104083542990989033472000000 1349477829392503148642304000 and in Maple notation 12157712239/240978183820089847971840000*n^17+230996532541/ 28350374567069393879040000*n^16+226664032241531/372098666192785794662400000*n^ 15+13607602195427/486403485219327836160000*n^14+93895333703204053/ 106313904626510227046400000*n^13+1629142986595393/80176398662526566400000*n^12+ 65500120938632143/185863469626766131200000*n^11+5139328241871253/ 1093314527216271360000*n^10+13593598766755603697/278795204440149196800000*n^9+ 10748129027766753083/27211383788493864960000*n^8+15871128078393686532743/ 6360660960560440934400000*n^7+16724066997417050959/1370536729273958400000*n^6+ 598009753926881786119723/13125173410680274944000000*n^5+79459596143100908576059 /625375909567707217920000*n^4+817329899777788361114737/ 3189417138795306811392000*n^3+53373594253569206854967/148899026087549337600000* n^2+35994670127500425491533451/104083542990989033472000000*n+ 413281211023145512085217587/1349477829392503148642304000 This is the leading term in particular, a(n) , is asymptotic to 17 12157712239 n --------------------------- 240978183820089847971840000 Q[2](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 2 259723 8 4934737 7 68296877 6 [- --------------- n - -------------- n - -------------- n 213084064972800 53271016243200 22830435532800 271280423 5 6586649047 4 7303626049 3 - ------------- n - -------------- n - ------------- n 5073430118400 11415217766400 1902536294400 138520900541 2 704355317291 4767246392080751 - ------------- n - -------------- n - -----------------, 8878502707200 18264348426240 19177565847552000 259723 8 4934737 7 68296877 6 --------------- n + -------------- n + -------------- n 213084064972800 53271016243200 22830435532800 271280423 5 6586649047 4 7303626049 3 + ------------- n + -------------- n + ------------- n 5073430118400 11415217766400 1902536294400 138520900541 2 704355317291 4767246392080751 + ------------- n + -------------- n + -----------------] 8878502707200 18264348426240 19177565847552000 and in Maple format [-259723/213084064972800*n^8-4934737/53271016243200*n^7-68296877/22830435532800 *n^6-271280423/5073430118400*n^5-6586649047/11415217766400*n^4-7303626049/ 1902536294400*n^3-138520900541/8878502707200*n^2-704355317291/18264348426240*n-\ 4767246392080751/19177565847552000, 259723/213084064972800*n^8+4934737/ 53271016243200*n^7+68296877/22830435532800*n^6+271280423/5073430118400*n^5+ 6586649047/11415217766400*n^4+7303626049/1902536294400*n^3+138520900541/ 8878502707200*n^2+704355317291/18264348426240*n+4767246392080751/ 19177565847552000] Q[3](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 3 41 3 779 2 272 1942769 [- ------- n - ------- n - ----- n - ---------, 6377292 4251528 98415 127545840 377 4 21079 3 95821 2 1085861 49354367 - -------- n - --------- n - -------- n - -------- n - ----------, 85030560 127545840 42515280 85030560 2550916800 377 4 21899 3 34537 2 1320869 9801083 -------- n + --------- n + -------- n + -------- n + ---------] 85030560 127545840 14171760 85030560 283435200 and in Maple format [-41/6377292*n^3-779/4251528*n^2-272/98415*n-1942769/127545840, -377/85030560*n ^4-21079/127545840*n^3-95821/42515280*n^2-1085861/85030560*n-49354367/ 2550916800, 377/85030560*n^4+21899/127545840*n^3+34537/14171760*n^2+1320869/ 85030560*n+9801083/283435200] Q[4](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 4 235 3 705 2 18365 50875 [------- n + ------- n + ------- n + -------, 9437184 1048576 3145728 3145728 235 3 1175 2 3805 53875 - ------- n - ------- n - ------ n - -------, 9437184 1572864 524288 2359296 235 3 705 2 18365 50875 - ------- n - ------- n - ------- n - -------, 9437184 1048576 3145728 3145728 235 3 1175 2 3805 53875 ------- n + ------- n + ------ n + -------] 9437184 1572864 524288 2359296 and in Maple format [235/9437184*n^3+705/1048576*n^2+18365/3145728*n+50875/3145728, -235/9437184*n^ 3-1175/1572864*n^2-3805/524288*n-53875/2359296, -235/9437184*n^3-705/1048576*n^ 2-18365/3145728*n-50875/3145728, 235/9437184*n^3+1175/1572864*n^2+3805/524288*n +53875/2359296] Q[5](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 5 2 646 14959 2 n 1253 26 n 247 [-7/9375 n - ----- n - ------, - ----- - -----, - ----- - -----, 46875 281250 15625 46875 46875 46875 2 n 1139 2 228 137 - ----- + -----, 7/9375 n + ----- n + ----] 15625 46875 15625 2250 and in Maple format [-7/9375*n^2-646/46875*n-14959/281250, -2/15625*n-1253/46875, -26/46875*n-247/ 46875, -2/15625*n+1139/46875, 7/9375*n^2+228/15625*n+137/2250] Q[6](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 6 13 5 n 259 5 n 311 -13 5 n 259 5 n 311 [-----, - ---- - -----, - ---- - -----, -----, ---- + -----, ---- + -----] 11664 7776 46656 7776 46656 11664 7776 46656 7776 46656 and in Maple format [13/11664, -5/7776*n-259/46656, -5/7776*n-311/46656, -13/11664, 5/7776*n+259/ 46656, 5/7776*n+311/46656] Q[7](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 7 11 n 321 5 n 473 6 n 167 6 n 281 5 n 283 [0, ---- + ----, - ---- - ----, - ---- + ----, - ---- - ----, - ---- + ----, 2401 4802 2401 4802 2401 2401 2401 2401 2401 4802 11 n 97 ---- + ----] 2401 4802 and in Maple format [0, 11/2401*n+321/4802, -5/2401*n-473/4802, -6/2401*n+167/2401, -6/2401*n-281/ 2401, -5/2401*n+283/4802, 11/2401*n+97/4802] Q[8](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 8 11 7 n 37 7 n 59 -11 -11 7 n 37 7 n 59 11 [----, ---- + ----, ---- + ----, ----, ----, - ---- - ----, - ---- - ----, ---- 1024 2048 1024 2048 2048 1024 1024 2048 1024 2048 2048 1024 ] and in Maple format [11/1024, 7/2048*n+37/1024, 7/2048*n+59/2048, -11/1024, -11/1024, -7/2048*n-37/ 1024, -7/2048*n-59/2048, 11/1024] Q[9](n), is defined by the following list whose i-th entry is the expression \ if n is congruent to i mod, 9 -2 -2 -2 [2/81, 2/81, 0, 0, 0, --, --, --, 2/81] 81 81 81 and in Maple format [2/81, 2/81, 0, 0, 0, -2/81, -2/81, -2/81, 2/81] Q[10](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 10 -3 -1 -1 -3 -1 [---, ---, 1/125, ---, ---, 3/250, 1/125, ---, 1/125, 3/250] 250 125 125 250 125 and in Maple format [-3/250, -1/125, 1/125, -1/125, -3/250, 3/250, 1/125, -1/125, 1/125, 3/250] Q[11](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 11 -8 -10 10 [---, 8/121, ---, 0, 0, 0, 0, 0, 0, 0, ---] 121 121 121 and in Maple format [-8/121, 8/121, -10/121, 0, 0, 0, 0, 0, 0, 0, 10/121] Q[12](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 12 -1 -1 -1 -1 -1 -1 [--, 1/24, --, 1/36, --, --, 1/36, --, 1/24, --, 1/72, 1/72] 36 24 72 72 24 36 and in Maple format [-1/36, 1/24, -1/24, 1/36, -1/72, -1/72, 1/36, -1/24, 1/24, -1/36, 1/72, 1/72] Q[13](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 13 -6 18 -30 30 -18 -12 12 [---, ---, ---, ---, ---, 6/169, ---, 0, 0, 0, 0, 0, ---] 169 169 169 169 169 169 169 and in Maple format [-6/169, 18/169, -30/169, 30/169, -18/169, 6/169, -12/169, 0, 0, 0, 0, 0, 12/ 169] Q[14](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 14 -1 -1 -1 -2 -1 -1 -1 [2/49, --, --, 1/49, --, 1/98, 1/98, --, 1/98, 1/98, --, 1/49, --, --] 98 98 49 49 49 98 98 and in Maple format [2/49, -1/98, -1/98, 1/49, -1/49, 1/98, 1/98, -2/49, 1/98, 1/98, -1/49, 1/49, -\ 1/98, -1/98] Q[15](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 15 -2 -8 -26 26 -2 -8 -16 -2 16 [--, 8/75, --, 2/15, ---, ---, --, 8/75, --, 2/45, ---, 2/75, 0, --, ---] 45 75 225 225 15 75 225 75 225 and in Maple format [-2/45, 8/75, -8/75, 2/15, -26/225, 26/225, -2/15, 8/75, -8/75, 2/45, -16/225, 2/75, 0, -2/75, 16/225] Q[16](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 16 -1 -1 -1 -1 -1 -1 [1/32, 0, 0, 1/32, --, 1/16, --, 1/32, --, 0, 0, --, 1/32, --, 1/16, --] 32 16 32 32 16 32 and in Maple format [1/32, 0, 0, 1/32, -1/32, 1/16, -1/16, 1/32, -1/32, 0, 0, -1/32, 1/32, -1/16, 1 /16, -1/32] Q[17](n), is defined by the following list whose i-th entry is the expression\ if n is congruent to i mod, 17 -2 14 -4 12 -6 10 -8 -10 -12 -14 [---, ---, ---, ---, ---, ---, ---, 8/289, ---, 6/289, ---, 4/289, ---, 2/289, 289 289 289 289 289 289 289 289 289 289 -16 16 ---, 0, ---] 289 289 and in Maple format [-2/289, 14/289, -4/289, 12/289, -6/289, 10/289, -8/289, 8/289, -10/289, 6/289, -12/289, 4/289, -14/289, 2/289, -16/289, 0, 16/289] That makes it very easy to compute a(n) for large n. In particular, a(100000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000), equals 5045150580135809338978142017848393471561374458888881478112494708647628029911779\ 5837336296667516167088786634135346692246958104689256090526229196919954821777251\ 0809706495252733801582254232459973153967134957333709669529739232114201669875509\ 1993856380837982567986684847263004536068287821312427026889016037426099891724994\ 5805266580228782325866456131808535011479060148956650118312875962370112322591422\ 4396978587310916568953473428308482661715625051549930358726229868705109125506790\ 5985085716066445559362798698144479837522172079052136564061444090670675755140181\ 1257713746946498809599350539461100690856444206031191765960041155808925365772403\ 0427401098780082174261382907748545208853410854562124503461769411631019305141006\ 9774342314299440606236289376071036403497545268128559207727165848109181534990449\ 5036429101383091997889307722897728858869992388799643122244550362920996214164324\ 5353590034913602429662850964752581641509197204754580163170648286377567391779452\ 5649074945931820221345938262590573815489287254368021223400830393593412781970313\ 0334092209655855565338510515261980767953967426579090436977750924390973523613130\ 7032955062321362083046658868299182483871797778456655864262037965145075682777274\ 4305733250678167363497579455824938233834091705117792385431546966328558218767783\ 4335584056135593936274443794159625639415755855963347075182818440080518701529017\ 5444557572438209704042184085580481316247426342376149834224826745659805161329708\ 8419889765684141499369357504267913812053229970610618310538075005841007365314025\ 3974186101161548910799604935184357081969614183441123179084147937215419454573590\ 1148205351365694340275990558696701493996059025879216425845534666243277033336363\ 4346414125634614601440577 ------------------------------------ --------------------------------------------------------- This ends this book, that took, 3021.637, seconds to generate.