The Sorting Probabilities of The entries in the first row vs. those not rel\ ated to it in lower rows in a random Standard Young tableau of shape, [n, n], and its Limiting behavior as n goes to infinity for i from 2 to, 60 By Shalosh B. Ekhad --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 2], vs. those in the, 2, -th row from j=1 to j=, 1, are as follws -n + 2 [--------] -1 + 2 n and in Maple notation [(-n+2)/(-1+2*n)] The limits, as n goes to infinity are [-1/2] and in Maple notation [-1/2] and in floating point [-.5000000000] The cut off is at j=, 1 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 3], vs. those in the, 2, -th row from j=1 to j=, 2, are as follws 2 3 -3 n + 9 n - 3 [--------, ---------------] -1 + 2 n 2 4 n - 8 n + 3 and in Maple notation [3/(-1+2*n), (-3*n^2+9*n-3)/(4*n^2-8*n+3)] The limits, as n goes to infinity are [0, -3/4] and in Maple notation [0, -3/4] and in floating point [0., -.7500000000] The cut off is at j=, 1 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 4], vs. those in the, 2, -th row from j=1 to j=, 3, are as follws 2 2 3 2 3 n + 9 n - 24 -3 n + 21 n - 6 -27 n + 144 n - 189 n + 60 [---------------, ----------------, ----------------------------] 2 2 3 2 8 n - 16 n + 6 8 n - 16 n + 6 32 n - 144 n + 184 n - 60 and in Maple notation [(3*n^2+9*n-24)/(8*n^2-16*n+6), (-3*n^2+21*n-6)/(8*n^2-16*n+6), (-27*n^3+144*n^ 2-189*n+60)/(32*n^3-144*n^2+184*n-60)] The limits, as n goes to infinity are -27 [3/8, -3/8, ---] 32 and in Maple notation [3/8, -3/8, -27/32] and in floating point [.3750000000, -.3750000000, -.8437500000] The cut off is at j=, 2 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 5], vs. those in the, 2, -th row from j=1 to j=, 4, are as follws 2 3 2 5 (n + 3) (n - 2) 45 n - 135 n + 30 -9 n + 72 n - 99 n + 30 [-----------------, -------------------------, -------------------------, 2 3 2 3 2 8 n - 16 n + 6 16 n - 72 n + 92 n - 30 16 n - 72 n + 92 n - 30 4 3 2 -57 n + 498 n - 1383 n + 1422 n - 420 ----------------------------------------] 4 3 2 64 n - 512 n + 1376 n - 1408 n + 420 and in Maple notation [5*(n+3)*(n-2)/(8*n^2-16*n+6), (45*n^2-135*n+30)/(16*n^3-72*n^2+92*n-30), (-9*n ^3+72*n^2-99*n+30)/(16*n^3-72*n^2+92*n-30), (-57*n^4+498*n^3-1383*n^2+1422*n-\ 420)/(64*n^4-512*n^3+1376*n^2-1408*n+420)] The limits, as n goes to infinity are -9 -57 [5/8, 0, --, ---] 16 64 and in Maple notation [5/8, 0, -9/16, -57/64] and in floating point [.6250000000, 0., -.5625000000, -.8906250000] The cut off is at j=, 2 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 6], vs. those in the, 2, -th row from j=1 to j=, 5, are as follws 3 2 3 2 25 n - 60 n - 145 n + 360 5 n + 30 n - 155 n + 30 [---------------------------, -------------------------, 3 2 3 2 32 n - 144 n + 184 n - 60 16 n - 72 n + 92 n - 30 4 3 2 -15 n + 330 n - 1425 n + 1590 n - 420 ----------------------------------------, 4 3 2 64 n - 512 n + 1376 n - 1408 n + 420 4 3 2 -43 n + 470 n - 1397 n + 1450 n - 420 ----------------------------------------, 4 3 2 64 n - 512 n + 1376 n - 1408 n + 420 5 4 3 2 -235 n + 3095 n - 14615 n + 30505 n - 27150 n + 7560 --------------------------------------------------------] 5 4 3 2 256 n - 3200 n + 14720 n - 30400 n + 27024 n - 7560 and in Maple notation [(25*n^3-60*n^2-145*n+360)/(32*n^3-144*n^2+184*n-60), (5*n^3+30*n^2-155*n+30)/( 16*n^3-72*n^2+92*n-30), (-15*n^4+330*n^3-1425*n^2+1590*n-420)/(64*n^4-512*n^3+ 1376*n^2-1408*n+420), (-43*n^4+470*n^3-1397*n^2+1450*n-420)/(64*n^4-512*n^3+ 1376*n^2-1408*n+420), (-235*n^5+3095*n^4-14615*n^3+30505*n^2-27150*n+7560)/(256 *n^5-3200*n^4+14720*n^3-30400*n^2+27024*n-7560)] The limits, as n goes to infinity are 25 -15 -43 -235 [--, 5/16, ---, ---, ----] 32 64 64 256 and in Maple notation [25/32, 5/16, -15/64, -43/64, -235/256] and in floating point [.7812500000, .3125000000, -.2343750000, -.6718750000, -.9179687500] The cut off is at j=, 3 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 7], vs. those in the, 2, -th row from j=1 to j=, 6, are as follws 3 2 4 3 2 7 n - 21 n - 28 n + 105 35 n - 70 n - 875 n + 2590 n - 420 [-------------------------, ---------------------------------------, 3 2 4 3 2 8 n - 36 n + 46 n - 15 64 n - 512 n + 1376 n - 1408 n + 420 4 3 2 5 n + 230 n - 1445 n + 1690 n - 420 ---------------------------------------, 4 3 2 64 n - 512 n + 1376 n - 1408 n + 420 5 4 3 2 -50 n + 1075 n - 6700 n + 15725 n - 14250 n + 3780 ------------------------------------------------------, 5 4 3 2 128 n - 1600 n + 7360 n - 15200 n + 13512 n - 3780 5 4 3 2 -95 n + 1435 n - 7195 n + 15365 n - 13710 n + 3780 ------------------------------------------------------, 5 4 3 2 128 n - 1600 n + 7360 n - 15200 n + 13512 n - 3780 6 5 4 3 2 -479 n + 8919 n - 63815 n + 222225 n - 389306 n + 313176 n - 83160 -----------------------------------------------------------------------] 6 5 4 3 2 512 n - 9216 n + 64640 n - 222720 n + 388448 n - 312384 n + 83160 and in Maple notation [(7*n^3-21*n^2-28*n+105)/(8*n^3-36*n^2+46*n-15), (35*n^4-70*n^3-875*n^2+2590*n-\ 420)/(64*n^4-512*n^3+1376*n^2-1408*n+420), (5*n^4+230*n^3-1445*n^2+1690*n-420)/ (64*n^4-512*n^3+1376*n^2-1408*n+420), (-50*n^5+1075*n^4-6700*n^3+15725*n^2-\ 14250*n+3780)/(128*n^5-1600*n^4+7360*n^3-15200*n^2+13512*n-3780), (-95*n^5+1435 *n^4-7195*n^3+15365*n^2-13710*n+3780)/(128*n^5-1600*n^4+7360*n^3-15200*n^2+ 13512*n-3780), (-479*n^6+8919*n^5-63815*n^4+222225*n^3-389306*n^2+313176*n-\ 83160)/(512*n^6-9216*n^5+64640*n^4-222720*n^3+388448*n^2-312384*n+83160)] The limits, as n goes to infinity are 35 -25 -95 -479 [7/8, --, 5/64, ---, ---, ----] 64 64 128 512 and in Maple notation [7/8, 35/64, 5/64, -25/64, -95/128, -479/512] and in floating point [.8750000000, .5468750000, .7812500000e-1, -.3906250000, -.7421875000, -.935546\ 8750] The cut off is at j=, 4 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 8], vs. those in the, 2, -th row from j=1 to j=, 7, are as follws 4 3 2 119 n - 826 n + 1141 n + 2926 n - 6720 [-----------------------------------------, 4 3 2 128 n - 1024 n + 2752 n - 2816 n + 840 4 3 2 91 n - 350 n - 1351 n + 5810 n - 840 -----------------------------------------, 4 3 2 128 n - 1024 n + 2752 n - 2816 n + 840 5 4 3 2 175 n + 175 n - 15925 n + 67025 n - 68250 n + 15120 --------------------------------------------------------, 5 4 3 2 512 n - 6400 n + 29440 n - 60800 n + 54048 n - 15120 5 4 3 2 -25 n + 1550 n - 12575 n + 32050 n - 29400 n + 7560 -------------------------------------------------------, - 5 5 4 3 2 256 n - 3200 n + 14720 n - 30400 n + 27024 n - 7560 6 5 4 3 2 (205 n - 5175 n + 44815 n - 173325 n + 317860 n - 256956 n + 66528)/( 32 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), ( 6 5 4 3 2 -1619 n + 33003 n - 247835 n + 884445 n - 1564946 n + 1259832 n - 332640)/(32 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) 7 6 5 4 3 (-1 + 2 n)), - 7 (1109 n - 27814 n + 280382 n - 1460860 n + 4193861 n 2 - 6494686 n + 4838568 n - 1235520)/(64 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [(119*n^4-826*n^3+1141*n^2+2926*n-6720)/(128*n^4-1024*n^3+2752*n^2-2816*n+840), (91*n^4-350*n^3-1351*n^2+5810*n-840)/(128*n^4-1024*n^3+2752*n^2-2816*n+840), ( 175*n^5+175*n^4-15925*n^3+67025*n^2-68250*n+15120)/(512*n^5-6400*n^4+29440*n^3-\ 60800*n^2+54048*n-15120), (-25*n^5+1550*n^4-12575*n^3+32050*n^2-29400*n+7560)/( 256*n^5-3200*n^4+14720*n^3-30400*n^2+27024*n-7560), -5/32*(205*n^6-5175*n^5+ 44815*n^4-173325*n^3+317860*n^2-256956*n+66528)/(2*n-9)/(2*n-11)/(2*n-7)/(2*n-5 )/(2*n-3)/(-1+2*n), 1/32*(-1619*n^6+33003*n^5-247835*n^4+884445*n^3-1564946*n^2 +1259832*n-332640)/(2*n-9)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -7/64*( 1109*n^7-27814*n^6+280382*n^5-1460860*n^4+4193861*n^3-6494686*n^2+4838568*n-\ 1235520)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n)] The limits, as n goes to infinity are 119 91 175 -25 -1025 -1619 -7763 [---, ---, ---, ---, -----, -----, -----] 128 128 512 256 2048 2048 8192 and in Maple notation [119/128, 91/128, 175/512, -25/256, -1025/2048, -1619/2048, -7763/8192] and in floating point [.9296875000, .7109375000, .3417968750, -.9765625000e-1, -.5004882812, -.790527\ 3438, -.9476318359] The cut off is at j=, 4 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 9], vs. those in the, 2, -th row from j=1 to j=, 8, are as follws 4 3 2 123 n - 894 n + 1497 n + 2514 n - 7560 [-----------------------------------------, 4 3 2 128 n - 1024 n + 2752 n - 2816 n + 840 5 4 3 2 210 n - 1995 n + 2940 n + 20895 n - 59850 n + 7560 -------------------------------------------------------, 5 4 3 2 256 n - 3200 n + 14720 n - 30400 n + 27024 n - 7560 5 4 3 2 140 n - 595 n - 5810 n + 34195 n - 36330 n + 7560 -------------------------------------------------------, 5 4 3 2 256 n - 3200 n + 14720 n - 30400 n + 27024 n - 7560 6 5 4 3 2 35 (5 n + 75 n - 1885 n + 11025 n - 24040 n + 20004 n - 4752) ------------------------------------------------------------------, 16 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n) 6 5 4 3 2 5 (95 n - 3855 n + 40415 n - 170025 n + 322370 n - 261576 n + 66528) - ------------------------------------------------------------------------- 32 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n) 7 6 5 4 3 2 , - 35 (68 n - 2095 n + 24263 n - 137935 n + 415997 n - 658306 n + 491064 n - 123552)/(32 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) 7 6 5 (2 n - 5) (2 n - 3) (-1 + 2 n)), - 21 (161 n - 4302 n + 45062 n 4 3 2 - 240140 n + 697809 n - 1086118 n + 809288 n - 205920)/(32 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 3 ( 8 7 6 5 4 3 5223 n - 169996 n + 2295286 n - 16687440 n + 70803887 n - 177121364 n 2 + 249778164 n - 175029840 n + 43243200)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [(123*n^4-894*n^3+1497*n^2+2514*n-7560)/(128*n^4-1024*n^3+2752*n^2-2816*n+840), (210*n^5-1995*n^4+2940*n^3+20895*n^2-59850*n+7560)/(256*n^5-3200*n^4+14720*n^3-\ 30400*n^2+27024*n-7560), (140*n^5-595*n^4-5810*n^3+34195*n^2-36330*n+7560)/(256 *n^5-3200*n^4+14720*n^3-30400*n^2+27024*n-7560), 35/16*(5*n^6+75*n^5-1885*n^4+ 11025*n^3-24040*n^2+20004*n-4752)/(2*n-9)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+ 2*n), -5/32*(95*n^6-3855*n^5+40415*n^4-170025*n^3+322370*n^2-261576*n+66528)/(2 *n-9)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -35/32*(68*n^7-2095*n^6+24263* n^5-137935*n^4+415997*n^3-658306*n^2+491064*n-123552)/(2*n-9)/(2*n-13)/(2*n-11) /(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -21/32*(161*n^7-4302*n^6+45062*n^5-240140*n^ 4+697809*n^3-1086118*n^2+809288*n-205920)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2* n-5)/(2*n-3)/(-1+2*n), -3/64*(5223*n^8-169996*n^7+2295286*n^6-16687440*n^5+ 70803887*n^4-177121364*n^3+249778164*n^2-175029840*n+43243200)/(2*n-15)/(2*n-13 )/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n)] The limits, as n goes to infinity are 123 105 35 175 -475 -595 -3381 -15669 [---, ---, --, ----, ----, ----, -----, ------] 128 128 64 1024 2048 1024 4096 16384 and in Maple notation [123/128, 105/128, 35/64, 175/1024, -475/2048, -595/1024, -3381/4096, -15669/ 16384] and in floating point [.9609375000, .8203125000, .5468750000, .1708984375, -.2319335938, -.5810546875 , -.8254394531, -.9563598633] The cut off is at j=, 5 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 10], vs. those in the, 2, -th row from j=1 to j=, 9, are as follws 5 4 3 2 501 n - 6015 n + 24105 n - 24225 n - 69966 n + 151200 [---------------------------------------------------------, 5 4 3 2 512 n - 6400 n + 29440 n - 60800 n + 54048 n - 15120 5 4 3 2 228 n - 2355 n + 5190 n + 17475 n - 65898 n + 7560 -------------------------------------------------------, 5 4 3 2 256 n - 3200 n + 14720 n - 30400 n + 27024 n - 7560 6 5 4 3 2 357 n - 4347 n + 6090 n + 108045 n - 460047 n + 440622 n - 83160 ----------------------------------------------------------------------, 6 5 4 3 2 512 n - 9216 n + 64640 n - 222720 n + 388448 n - 312384 n + 83160 6 5 4 3 2 203 n - 651 n - 23170 n + 181965 n - 430633 n + 363006 n - 83160 ----------------------------------------------------------------------, 35 6 5 4 3 2 512 n - 9216 n + 64640 n - 222720 n + 388448 n - 312384 n + 83160 7 6 5 4 3 2 (7 n + 1330 n - 29078 n + 225820 n - 803537 n + 1369522 n - 1030176 n + 247104)/(64 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) 7 6 5 4 (2 n - 3) (-1 + 2 n)), - 7 (197 n - 8044 n + 107444 n - 658930 n 3 2 + 2066543 n - 3324706 n + 2482776 n - 617760)/(32 (2 n - 9) (2 n - 13) 8 (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 21 (501 n 7 6 5 4 3 - 18892 n + 283282 n - 2212320 n + 9828269 n - 25244708 n 2 + 36014028 n - 25239600 n + 6177600)/(64 (2 n - 15) (2 n - 13) (2 n - 11) 8 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 3 (4651 n 7 6 5 4 3 - 158556 n + 2207198 n - 16367120 n + 70295379 n - 177041284 n 2 + 250375332 n - 175441680 n + 43243200)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 3 ( 9 8 7 6 5 21035 n - 862857 n + 15031650 n - 145243938 n + 852076995 n 4 3 2 - 3117998793 n + 7021430080 n - 9189936012 n + 6122237040 n - 1470268800)/(128 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [(501*n^5-6015*n^4+24105*n^3-24225*n^2-69966*n+151200)/(512*n^5-6400*n^4+29440* n^3-60800*n^2+54048*n-15120), (228*n^5-2355*n^4+5190*n^3+17475*n^2-65898*n+7560 )/(256*n^5-3200*n^4+14720*n^3-30400*n^2+27024*n-7560), (357*n^6-4347*n^5+6090*n ^4+108045*n^3-460047*n^2+440622*n-83160)/(512*n^6-9216*n^5+64640*n^4-222720*n^3 +388448*n^2-312384*n+83160), (203*n^6-651*n^5-23170*n^4+181965*n^3-430633*n^2+ 363006*n-83160)/(512*n^6-9216*n^5+64640*n^4-222720*n^3+388448*n^2-312384*n+ 83160), 35/64*(7*n^7+1330*n^6-29078*n^5+225820*n^4-803537*n^3+1369522*n^2-\ 1030176*n+247104)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -\ 7/32*(197*n^7-8044*n^6+107444*n^5-658930*n^4+2066543*n^3-3324706*n^2+2482776*n-\ 617760)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -21/64*(501 *n^8-18892*n^7+283282*n^6-2212320*n^5+9828269*n^4-25244708*n^3+36014028*n^2-\ 25239600*n+6177600)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/ (-1+2*n), -3/64*(4651*n^8-158556*n^7+2207198*n^6-16367120*n^5+70295379*n^4-\ 177041284*n^3+250375332*n^2-175441680*n+43243200)/(2*n-15)/(2*n-13)/(2*n-11)/(2 *n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -3/128*(21035*n^9-862857*n^8+15031650*n ^7-145243938*n^6+852076995*n^5-3117998793*n^4+7021430080*n^3-9189936012*n^2+ 6122237040*n-1470268800)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2 *n-5)/(2*n-3)/(-1+2*n)] The limits, as n goes to infinity are 501 57 357 203 245 -1379 -10521 -13953 -63105 [---, --, ---, ---, ----, -----, ------, ------, ------] 512 64 512 512 8192 4096 16384 16384 65536 and in Maple notation [501/512, 57/64, 357/512, 203/512, 245/8192, -1379/4096, -10521/16384, -13953/ 16384, -63105/65536] and in floating point [.9785156250, .8906250000, .6972656250, .3964843750, .2990722656e-1, -.33666992\ 19, -.6421508789, -.8516235352, -.9629058838] The cut off is at j=, 6 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 11], vs. those in the, 2, -th row from j=1 to j=, 10, are as follws 5 4 3 2 253 n - 3080 n + 12815 n - 15400 n - 31548 n + 83160 [--------------------------------------------------------, 5 4 3 2 256 n - 3200 n + 14720 n - 30400 n + 27024 n - 7560 6 5 4 3 2 33 (29 n - 477 n + 2615 n - 3135 n - 17764 n + 48972 n - 5040) -------------------------------------------------------------------, 16 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n) 6 5 4 3 2 411 n - 5643 n + 16350 n + 82125 n - 470361 n + 467838 n - 83160 ----------------------------------------------------------------------, 21 6 5 4 3 2 512 n - 9216 n + 64640 n - 222720 n + 388448 n - 312384 n + 83160 7 6 5 4 3 2 (28 n - 413 n + 259 n + 24220 n - 146993 n + 308833 n - 241374 n + 51480)/(8 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) 7 6 5 4 3 (-1 + 2 n)), 7 (77 n + 161 n - 25291 n + 252245 n - 986062 n 2 + 1743434 n - 1317204 n + 308880)/(16 (2 n - 9) (2 n - 13) (2 n - 11) 8 7 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 7 (203 n - 22876 n 6 5 4 3 2 + 524846 n - 5222560 n + 26847107 n - 74935924 n + 111006084 n - 77965200 n + 18532800)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 8 7 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 7 (983 n - 44716 n 6 5 4 3 2 + 747926 n - 6236560 n + 28804127 n - 75584884 n + 108825204 n - 76280400 n + 18532800)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 9 8 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 3 (7534 n - 344907 n 7 6 5 4 3 + 6519336 n - 66816078 n + 408349326 n - 1534710723 n + 3510070604 n 2 - 4625149092 n + 3080451600 n - 735134400)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 9 8 7 6 5 - 3 (9523 n - 404577 n + 7223442 n - 70992978 n + 421255947 n 4 3 2 - 1552671393 n + 3510778688 n - 4602951852 n + 3066130800 n - 735134400 )/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 10 9 8 (2 n - 5) (2 n - 3) (-1 + 2 n)), - 3 (42291 n - 2135545 n + 46635930 n 7 6 5 4 - 577685730 n + 4471729563 n - 22447059705 n + 73311541320 n 3 2 - 151785000620 n + 186818434896 n - 119241878400 n + 27935107200)/(128 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [(253*n^5-3080*n^4+12815*n^3-15400*n^2-31548*n+83160)/(256*n^5-3200*n^4+14720*n ^3-30400*n^2+27024*n-7560), 33/16*(29*n^6-477*n^5+2615*n^4-3135*n^3-17764*n^2+ 48972*n-5040)/(2*n-9)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), (411*n^6-5643* n^5+16350*n^4+82125*n^3-470361*n^2+467838*n-83160)/(512*n^6-9216*n^5+64640*n^4-\ 222720*n^3+388448*n^2-312384*n+83160), 21/8*(28*n^7-413*n^6+259*n^5+24220*n^4-\ 146993*n^3+308833*n^2-241374*n+51480)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5) /(2*n-3)/(-1+2*n), 7/16*(77*n^7+161*n^6-25291*n^5+252245*n^4-986062*n^3+1743434 *n^2-1317204*n+308880)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2* n), -7/64*(203*n^8-22876*n^7+524846*n^6-5222560*n^5+26847107*n^4-74935924*n^3+ 111006084*n^2-77965200*n+18532800)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/( 2*n-5)/(2*n-3)/(-1+2*n), -7/64*(983*n^8-44716*n^7+747926*n^6-6236560*n^5+ 28804127*n^4-75584884*n^3+108825204*n^2-76280400*n+18532800)/(2*n-15)/(2*n-13)/ (2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -3/64*(7534*n^9-344907*n^8+ 6519336*n^7-66816078*n^6+408349326*n^5-1534710723*n^4+3510070604*n^3-4625149092 *n^2+3080451600*n-735134400)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7 )/(2*n-5)/(2*n-3)/(-1+2*n), -3/64*(9523*n^9-404577*n^8+7223442*n^7-70992978*n^6 +421255947*n^5-1552671393*n^4+3510778688*n^3-4602951852*n^2+3066130800*n-\ 735134400)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/ (-1+2*n), -3/128*(42291*n^10-2135545*n^9+46635930*n^8-577685730*n^7+4471729563* n^6-22447059705*n^5+73311541320*n^4-151785000620*n^3+186818434896*n^2-\ 119241878400*n+27935107200)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9 )/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n)] The limits, as n goes to infinity are 253 957 411 147 539 -1421 -6881 -11301 -28569 -126873 [---, ----, ---, ---, ----, -----, -----, ------, ------, -------] 256 1024 512 256 2048 16384 16384 16384 32768 131072 and in Maple notation [253/256, 957/1024, 411/512, 147/256, 539/2048, -1421/16384, -6881/16384, -\ 11301/16384, -28569/32768, -126873/131072] and in floating point [.9882812500, .9345703125, .8027343750, .5742187500, .2631835938, -.8673095703e\ -1, -.4199829102, -.6897583008, -.8718566895, -.9679641724] The cut off is at j=, 6 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 12], vs. those in the, 2, -th row from j=1 to j=, 11, are as follws 6 5 4 3 2 [11 (185 n - 3291 n + 22235 n - 66825 n + 53180 n + 175956 n - 362880)/(32 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 6 5 4 3 2 11 (179 n - 3027 n + 17675 n - 28725 n - 93454 n + 319032 n - 30240) -------------------------------------------------------------------------, 32 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n) 7 6 5 4 3 2 231 (31 n - 662 n + 4762 n - 7700 n - 57881 n + 250282 n - 229152 n + 37440)/(64 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) 7 6 5 4 3 2 (-1 + 2 n)), 63 (23 n - 401 n + 1451 n + 11055 n - 93038 n + 211106 n - 167156 n + 34320)/(16 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) 8 7 6 5 (2 n - 5) (2 n - 3) (-1 + 2 n)), 63 (119 n - 1988 n - 4242 n + 288680 n 4 3 2 - 2322089 n + 8010828 n - 13049148 n + 9274320 n - 2059200)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 8 7 6 5 4 (-1 + 2 n)), 7 (343 n + 7588 n - 368690 n + 4512760 n - 25477193 n 3 2 + 74481652 n - 112532700 n + 79144560 n - 18532800)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 9 8 7 6 5 - 21 (287 n - 23226 n + 595098 n - 7390404 n + 51224943 n 4 3 2 - 208686114 n + 500362072 n - 673474656 n + 448588800 n - 105019200)/( 64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 9 8 7 (2 n - 5) (2 n - 3) (-1 + 2 n)), - 3 (5324 n - 278607 n + 5736996 n 6 5 4 3 - 62175078 n + 394008636 n - 1514754423 n + 3509283844 n 2 - 4649812692 n + 3096363600 n - 735134400)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 10 9 8 7 6 - 3 (63587 n - 3494275 n + 81430680 n - 1059256350 n + 8498596071 n 5 4 3 2 - 43750878675 n + 145288514470 n - 303751985900 n + 375428163192 n - 239541904800 n + 55870214400)/(256 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 10 9 8 7 6 (-232751 n + 12078445 n - 269108130 n + 3382344330 n - 26454293943 n 5 4 3 2 + 133741047405 n - 438801862120 n + 910889301020 n - 1122364807056 n + 716297788800 n - 167610643200)/(256 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 11 10 9 8 - 11 (92653 n - 5649596 n + 151149765 n - 2334403380 n 7 6 5 4 + 23029578579 n - 151579372308 n + 674573001815 n - 2011072326220 n 3 2 + 3884765336868 n - 4539433976496 n + 2792349141120 n - 639967910400)/( 512 (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [11/32*(185*n^6-3291*n^5+22235*n^4-66825*n^3+53180*n^2+175956*n-362880)/(2*n-9) /(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 11/32*(179*n^6-3027*n^5+17675*n^4-\ 28725*n^3-93454*n^2+319032*n-30240)/(2*n-9)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-\ 1+2*n), 231/64*(31*n^7-662*n^6+4762*n^5-7700*n^4-57881*n^3+250282*n^2-229152*n+ 37440)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 63/16*(23*n^ 7-401*n^6+1451*n^5+11055*n^4-93038*n^3+211106*n^2-167156*n+34320)/(2*n-9)/(2*n-\ 13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 63/64*(119*n^8-1988*n^7-4242*n^6 +288680*n^5-2322089*n^4+8010828*n^3-13049148*n^2+9274320*n-2059200)/(2*n-15)/(2 *n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 7/64*(343*n^8+7588*n^ 7-368690*n^6+4512760*n^5-25477193*n^4+74481652*n^3-112532700*n^2+79144560*n-\ 18532800)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -21/64*(287*n^9-23226*n^8+595098*n^7-7390404*n^6+51224943*n^5-208686114*n^4+ 500362072*n^3-673474656*n^2+448588800*n-105019200)/(2*n-17)/(2*n-15)/(2*n-13)/( 2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -3/64*(5324*n^9-278607*n^8+ 5736996*n^7-62175078*n^6+394008636*n^5-1514754423*n^4+3509283844*n^3-4649812692 *n^2+3096363600*n-735134400)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7 )/(2*n-5)/(2*n-3)/(-1+2*n), -3/256*(63587*n^10-3494275*n^9+81430680*n^8-\ 1059256350*n^7+8498596071*n^6-43750878675*n^5+145288514470*n^4-303751985900*n^3 +375428163192*n^2-239541904800*n+55870214400)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-\ 13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 1/256*(-232751*n^10+ 12078445*n^9-269108130*n^8+3382344330*n^7-26454293943*n^6+133741047405*n^5-\ 438801862120*n^4+910889301020*n^3-1122364807056*n^2+716297788800*n-167610643200 )/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/ (-1+2*n), -11/512*(92653*n^11-5649596*n^10+151149765*n^9-2334403380*n^8+ 23029578579*n^7-151579372308*n^6+674573001815*n^5-2011072326220*n^4+ 3884765336868*n^3-4539433976496*n^2+2792349141120*n-639967910400)/(2*n-21)/(2*n -19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2* n)] The limits, as n goes to infinity are 2035 1969 7161 1449 7497 2401 -6027 -3993 -190761 -232751 -1019183 [----, ----, ----, ----, -----, -----, -----, -----, -------, -------, -------- 2048 2048 8192 2048 16384 16384 32768 8192 262144 262144 1048576 ] and in Maple notation [2035/2048, 1969/2048, 7161/8192, 1449/2048, 7497/16384, 2401/16384, -6027/ 32768, -3993/8192, -190761/262144, -232751/262144, -1019183/1048576] and in floating point [.9936523438, .9614257812, .8741455078, .7075195312, .4575805664, .1465454102, -.1839294434, -.4874267578, -.7276954651, -.8878746033, -.9719686508] The cut off is at j=, 7 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 13], vs. those in the, 2, -th row from j=1 to j=, 12, are as follws 6 5 4 3 2 [13 (157 n - 2805 n + 19165 n - 59475 n + 56278 n + 137880 n - 332640)/(32 (2 n - 9) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 1001 7 6 5 4 3 2 (4 n - 95 n + 871 n - 3575 n + 3661 n + 18790 n - 49896 n + 4320)/(32 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 7 6 5 4 3 2 77 (49 n - 1088 n + 8488 n - 19910 n - 68369 n + 383878 n - 363528 n + 56160)/(32 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) 8 7 6 5 4 3 (-1 + 2 n)), 693 (19 n - 508 n + 4638 n - 9920 n - 95989 n + 635148 n 2 - 1306428 n + 972720 n - 187200)/(64 (2 n - 15) (2 n - 13) (2 n - 11) 8 7 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 63 (159 n - 3388 n 6 5 4 3 2 + 13918 n + 183480 n - 2070529 n + 7879828 n - 13318908 n + 9511920 n - 2059200)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 9 8 7 (2 n - 5) (2 n - 3) (-1 + 2 n)), 63 (182 n - 3087 n - 33432 n 6 5 4 3 2 + 1228962 n - 12407682 n + 61667697 n - 165022508 n + 233631228 n - 156126960 n + 35006400)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) 9 8 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 21 (70 n + 10731 n 7 6 5 4 3 - 425880 n + 6265854 n - 47408970 n + 202875939 n - 499635220 n 2 + 680421876 n - 453301200 n + 105019200)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 10 9 8 7 6 - 21 (829 n - 65675 n + 1921935 n - 29187450 n + 260855007 n 5 4 3 2 - 1446584475 n + 5045131865 n - 10858730800 n + 13595546364 n - 8669271600 n + 1995364800)/(64 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 10 9 8 7 6 - 3 (47437 n - 2880575 n + 71886030 n - 980573550 n + 8129940021 n 5 4 3 2 - 42793797375 n + 144159161120 n - 303893007700 n + 376935733392 n - 240437260800 n + 55870214400)/(256 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 11 10 9 8 - 11 (36152 n - 2356771 n + 66468705 n - 1069526490 n 7 6 5 4 + 10885974666 n - 73328698443 n + 331751318545 n - 999933065960 n 3 2 + 1944043187532 n - 2277879216336 n + 1400687164800 n - 319983955200)/( 256 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) 11 (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), - 11 (42935 n 10 9 8 7 - 2675572 n + 72776895 n - 1138102620 n + 11332248585 n 6 5 4 3 - 75090860796 n + 335745555925 n - 1004013311780 n + 1942878247980 n 2 - 2271967913232 n + 1397405278080 n - 319983955200)/(256 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) 12 11 (2 n - 17) (2 n - 19) (2 n - 21)), (-2045149 n + 148186782 n 10 9 8 7 - 4767102989 n + 89784670470 n - 1098958422507 n + 9171024094386 n 6 5 4 - 53225415090287 n + 214984489807890 n - 594335195946844 n 3 2 + 1082685879201432 n - 1209802473193824 n + 720487014186240 n - 161911881331200)/(512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [13/32*(157*n^6-2805*n^5+19165*n^4-59475*n^3+56278*n^2+137880*n-332640)/(2*n-9) /(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 1001/32*(4*n^7-95*n^6+871*n^5-3575* n^4+3661*n^3+18790*n^2-49896*n+4320)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/ (2*n-3)/(-1+2*n), 77/32*(49*n^7-1088*n^6+8488*n^5-19910*n^4-68369*n^3+383878*n^ 2-363528*n+56160)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 693/64*(19*n^8-508*n^7+4638*n^6-9920*n^5-95989*n^4+635148*n^3-1306428*n^2+ 972720*n-187200)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1 +2*n), 63/64*(159*n^8-3388*n^7+13918*n^6+183480*n^5-2070529*n^4+7879828*n^3-\ 13318908*n^2+9511920*n-2059200)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n -5)/(2*n-3)/(-1+2*n), 63/64*(182*n^9-3087*n^8-33432*n^7+1228962*n^6-12407682*n^ 5+61667697*n^4-165022508*n^3+233631228*n^2-156126960*n+35006400)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 21/64*(70*n^9+ 10731*n^8-425880*n^7+6265854*n^6-47408970*n^5+202875939*n^4-499635220*n^3+ 680421876*n^2-453301200*n+105019200)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9 )/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -21/64*(829*n^10-65675*n^9+1921935*n^8-\ 29187450*n^7+260855007*n^6-1446584475*n^5+5045131865*n^4-10858730800*n^3+ 13595546364*n^2-8669271600*n+1995364800)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2 *n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -3/256*(47437*n^10-2880575*n^9 +71886030*n^8-980573550*n^7+8129940021*n^6-42793797375*n^5+144159161120*n^4-\ 303893007700*n^3+376935733392*n^2-240437260800*n+55870214400)/(2*n-19)/(2*n-17) /(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -11/256*( 36152*n^11-2356771*n^10+66468705*n^9-1069526490*n^8+10885974666*n^7-73328698443 *n^6+331751318545*n^5-999933065960*n^4+1944043187532*n^3-2277879216336*n^2+ 1400687164800*n-319983955200)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11) /(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), -11/256*(42935*n^11-2675572*n^10 +72776895*n^9-1138102620*n^8+11332248585*n^7-75090860796*n^6+335745555925*n^5-\ 1004013311780*n^4+1942878247980*n^3-2271967913232*n^2+1397405278080*n-\ 319983955200)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-\ 15)/(2*n-17)/(2*n-19)/(2*n-21), 1/512*(-2045149*n^12+148186782*n^11-4767102989* n^10+89784670470*n^9-1098958422507*n^8+9171024094386*n^7-53225415090287*n^6+ 214984489807890*n^5-594335195946844*n^4+1082685879201432*n^3-1209802473193824*n ^2+720487014186240*n-161911881331200)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n)] The limits, as n goes to infinity are 2041 1001 3773 13167 10017 5733 735 -17409 -142311 -49709 -472285 [----, ----, ----, -----, -----, -----, -----, ------, -------, ------, -------, 2048 1024 4096 16384 16384 16384 16384 65536 262144 65536 524288 -2045149 --------] 2097152 and in Maple notation [2041/2048, 1001/1024, 3773/4096, 13167/16384, 10017/16384, 5733/16384, 735/ 16384, -17409/65536, -142311/262144, -49709/65536, -472285/524288, -2045149/ 2097152] and in floating point [.9965820312, .9775390625, .9211425781, .8036499023, .6113891602, .3499145508, .4486083984e-1, -.2656402588, -.5428733826, -.7584991455, -.9008121490, -.97520\ 30373] The cut off is at j=, 8 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 14], vs. those in the, 2, -th row from j=1 to j=, 13, are as follws 7 6 5 4 3 2 [13 (629 n - 15358 n + 150878 n - 751660 n + 1872101 n - 1228822 n - 4684728 n + 9313920)/(64 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) 7 6 5 (2 n - 5) (2 n - 3) (-1 + 2 n)), 13 (311 n - 7462 n + 69902 n 4 3 2 - 301840 n + 397859 n + 1307222 n - 4127112 n + 332640)/(32 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 143 ( 8 7 6 5 4 3 109 n - 3308 n + 39298 n - 213920 n + 338821 n + 1636348 n 2 - 7151988 n + 6322320 n - 907200)/(64 (2 n - 15) (2 n - 13) (2 n - 11) 8 7 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 11 (1297 n - 36404 n 6 5 4 3 2 + 366394 n - 1210160 n - 4018007 n + 38227924 n - 84408564 n + 63657360 n - 11793600)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 9 8 7 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 99 (241 n - 7848 n + 88134 n 6 5 4 3 2 - 229152 n - 3157791 n + 29385888 n - 100938904 n + 159548712 n - 108276480 n + 22276800)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) 9 8 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 9 (1886 n - 47313 n 7 6 5 4 3 + 184584 n + 5286918 n - 73757586 n + 408694503 n - 1149284804 n 2 + 1661739492 n - 1112276880 n + 245044800)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 10 9 8 7 6 5 63 (262 n - 3410 n - 159375 n + 4971360 n - 60848424 n + 404382510 n 4 3 2 - 1575478075 n + 3616900340 n - 4664666388 n + 2975029200 n - 665121600 )/(64 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 10 9 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 3 (958 n - 251390 n 8 7 6 5 + 9839175 n - 171647160 n + 1660785084 n - 9668204910 n 4 3 2 + 34736829275 n - 76041406540 n + 95916737508 n - 61145370000 n + 13967553600)/(64 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) 11 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 33 (10643 n 10 9 8 7 6 - 898264 n + 30221595 n - 552002160 n + 6163275069 n - 44401670712 n 5 4 3 2 + 210655604905 n - 655446028640 n + 1298695318188 n - 1534407976224 n + 942745003200 n - 213322636800)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 11 10 9 8 (2 n - 21)), - 33 (9359 n - 659082 n + 19652985 n - 329296080 n 7 6 5 4 + 3451565397 n - 23743628706 n + 108998758015 n - 331691876820 n 3 2 + 648476673444 n - 761638827312 n + 468198057600 n - 106661318400)/(256 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) 12 (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), - 11 (149453 n 11 10 9 8 - 11429226 n + 384029581 n - 7488283890 n + 94187492139 n 7 6 5 - 802659958758 n + 4731928239343 n - 19328826992310 n 4 3 2 + 53839294294508 n - 98514142839816 n + 110285734140576 n - 65657628691200 n + 14719261939200)/(512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) 12 11 10 (2 n - 3) (-1 + 2 n)), (-1911427 n + 140965794 n - 4597944659 n 9 8 7 6 + 87534129210 n - 1080287755701 n + 9071497216782 n - 52889120975537 n 5 4 3 + 214316873362350 n - 593754297415972 n + 1082970029893224 n 2 - 1210738505798304 n + 720972264579840 n - 161911881331200)/(512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 13 ( 13 12 11 10 630991 n - 53597327 n + 2041233197 n - 46044134455 n 9 8 7 + 684427403583 n - 7055876904681 n + 51686431494071 n 6 5 4 - 270978554510125 n + 1009957217368726 n - 2619221258941892 n 3 2 + 4536320298938232 n - 4874468515809120 n + 2820767412153600 n - 622738005120000)/(1024 (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [13/64*(629*n^7-15358*n^6+150878*n^5-751660*n^4+1872101*n^3-1228822*n^2-4684728 *n+9313920)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 13/32*( 311*n^7-7462*n^6+69902*n^5-301840*n^4+397859*n^3+1307222*n^2-4127112*n+332640)/ (2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 143/64*(109*n^8-\ 3308*n^7+39298*n^6-213920*n^5+338821*n^4+1636348*n^3-7151988*n^2+6322320*n-\ 907200)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 11 /64*(1297*n^8-36404*n^7+366394*n^6-1210160*n^5-4018007*n^4+38227924*n^3-\ 84408564*n^2+63657360*n-11793600)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2 *n-5)/(2*n-3)/(-1+2*n), 99/64*(241*n^9-7848*n^8+88134*n^7-229152*n^6-3157791*n^ 5+29385888*n^4-100938904*n^3+159548712*n^2-108276480*n+22276800)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 9/64*(1886*n^9-\ 47313*n^8+184584*n^7+5286918*n^6-73757586*n^5+408694503*n^4-1149284804*n^3+ 1661739492*n^2-1112276880*n+245044800)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n -9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 63/64*(262*n^10-3410*n^9-159375*n^8+ 4971360*n^7-60848424*n^6+404382510*n^5-1575478075*n^4+3616900340*n^3-4664666388 *n^2+2975029200*n-665121600)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-\ 9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -3/64*(958*n^10-251390*n^9+9839175*n^8-\ 171647160*n^7+1660785084*n^6-9668204910*n^5+34736829275*n^4-76041406540*n^3+ 95916737508*n^2-61145370000*n+13967553600)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/ (2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -33/512*(10643*n^11-898264*n ^10+30221595*n^9-552002160*n^8+6163275069*n^7-44401670712*n^6+210655604905*n^5-\ 655446028640*n^4+1298695318188*n^3-1534407976224*n^2+942745003200*n-\ 213322636800)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-\ 15)/(2*n-17)/(2*n-19)/(2*n-21), -33/256*(9359*n^11-659082*n^10+19652985*n^9-\ 329296080*n^8+3451565397*n^7-23743628706*n^6+108998758015*n^5-331691876820*n^4+ 648476673444*n^3-761638827312*n^2+468198057600*n-106661318400)/(-1+2*n)/(2*n-3) /(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), -11/512*(149453*n^12-11429226*n^11+384029581*n^10-7488283890*n^9+94187492139*n^ 8-802659958758*n^7+4731928239343*n^6-19328826992310*n^5+53839294294508*n^4-\ 98514142839816*n^3+110285734140576*n^2-65657628691200*n+14719261939200)/(2*n-23 )/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5) /(2*n-3)/(-1+2*n), 1/512*(-1911427*n^12+140965794*n^11-4597944659*n^10+ 87534129210*n^9-1080287755701*n^8+9071497216782*n^7-52889120975537*n^6+ 214316873362350*n^5-593754297415972*n^4+1082970029893224*n^3-1210738505798304*n ^2+720972264579840*n-161911881331200)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -13/1024*( 630991*n^13-53597327*n^12+2041233197*n^11-46044134455*n^10+684427403583*n^9-\ 7055876904681*n^8+51686431494071*n^7-270978554510125*n^6+1009957217368726*n^5-\ 2619221258941892*n^4+4536320298938232*n^3-4874468515809120*n^2+2820767412153600 *n-622738005120000)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-\ 13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n)] The limits, as n goes to infinity are 8177 4043 15587 14267 23859 8487 8253 -1437 -351219 -308847 [----, ----, -----, -----, -----, -----, -----, -----, -------, -------, 8192 4096 16384 16384 32768 16384 32768 32768 1048576 524288 -1643983 -1911427 -8202883 --------, --------, --------] 2097152 2097152 8388608 and in Maple notation [8177/8192, 4043/4096, 15587/16384, 14267/16384, 23859/32768, 8487/16384, 8253/ 32768, -1437/32768, -351219/1048576, -308847/524288, -1643983/2097152, -1911427 /2097152, -8202883/8388608] and in floating point [.9981689453, .9870605469, .9513549805, .8707885742, .7281188965, .5180053711, .2518615723, -.4385375977e-1, -.3349485397, -.5890789032, -.7839121819, -.91143\ 94188, -.9778598547] The cut off is at j=, 8 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 15], vs. those in the, 2, -th row from j=1 to j=, 14, are as follws 7 6 5 4 3 2 [3 (341 n - 8337 n + 82187 n - 413315 n + 1058904 n - 825748 n - 2388832 n + 5405400)/(8 (2 n - 9) (2 n - 13) (2 n - 11) (2 n - 7) 8 7 6 (2 n - 5) (2 n - 3) (-1 + 2 n)), 39 (417 n - 13204 n + 171514 n 5 4 3 2 - 1150800 n + 3907673 n - 3516716 n - 17385204 n + 44597520 n - 3326400)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 8 7 6 (2 n - 5) (2 n - 3) (-1 + 2 n)), 13 (1223 n - 37708 n + 461174 n 5 4 3 2 - 2667280 n + 5411567 n + 14608748 n - 80385324 n + 73252080 n - 9979200)/(64 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 9 8 7 (2 n - 5) (2 n - 3) (-1 + 2 n)), 429 (70 n - 2631 n + 39240 n 6 5 4 3 2 - 275814 n + 657510 n + 2729601 n - 20255620 n + 41094444 n - 29430000 n + 5140800)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) 9 8 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 33 (808 n - 27879 n 7 6 5 4 3 + 351612 n - 1554966 n - 5137608 n + 78663249 n - 298488172 n 2 + 489012396 n - 333581040 n + 66830400)/(64 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 10 9 8 7 6 5 99 (431 n - 16705 n + 223635 n - 618150 n - 14452767 n + 173894175 n 4 3 2 - 871043135 n + 2280062200 n - 3124552164 n + 2002038480 n - 423259200) /(64 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 10 9 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 27 (1042 n - 29490 n 8 7 6 5 4 + 60945 n + 7116600 n - 116448444 n + 865147710 n - 3565996195 n 3 2 + 8433667500 n - 11020305348 n + 7030417680 n - 1551950400)/(64 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 11 10 9 (2 n - 5) (2 n - 3) (-1 + 2 n)), 297 (72 n + 166 n - 132880 n 8 7 6 5 4 + 4052865 n - 58839564 n + 498478008 n - 2629430400 n + 8763233285 n 3 2 - 18089234108 n + 21773243676 n - 13365271920 n + 2962814400)/(64 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) 11 10 (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), - 33 (965 n - 136387 n 9 8 7 6 + 5647680 n - 115616640 n + 1385449065 n - 10452444471 n 5 4 3 2 + 51122099890 n - 162194584910 n + 325039643520 n - 385939361592 n + 237019517280 n - 53330659200)/(128 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 12 11 10 9 (2 n - 21)), - 11 (75163 n - 6971826 n + 269548691 n - 5839045890 n 8 7 6 5 + 79535795469 n - 719819179758 n + 4437283475873 n - 18715435263510 n 4 3 2 + 53271103870468 n - 98744750886216 n + 111163335579936 n - 66124906848000 n + 14719261939200)/(512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) 12 11 10 (2 n - 3) (-1 + 2 n)), - 11 (119737 n - 9735414 n + 342516329 n 9 8 7 6 - 6914616510 n + 89273565231 n - 775744662042 n + 4638827090147 n 5 4 3 - 19139962942650 n + 53670176497132 n - 98590314100344 n 2 + 110552908557024 n - 65797812138240 n + 14719261939200)/(512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 13 12 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 13 (259778 n - 23065591 n 11 10 9 8 + 911020276 n - 21168067415 n + 322243825464 n - 3385336141473 n 7 6 5 + 25163989278868 n - 133389510614525 n + 501120311883158 n 4 3 2 - 1306582491883636 n + 2270089312634856 n - 2442506622176160 n + 1413012145708800 n - 311369002560000)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) 13 (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), - 13 (296923 n 12 11 10 9 - 25591451 n + 986090321 n - 22451055715 n + 336182226699 n 8 7 6 - 3485590645053 n + 25644453279203 n - 134882770073425 n 5 4 3 + 503877958763278 n - 1308762610978196 n + 2268724273109976 n 2 - 2438731703604960 n + 1411125060844800 n - 311369002560000)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), ( 14 13 12 11 -16442911 n + 1618425683 n - 72018966929 n + 1916626789987 n 10 9 8 - 33999501258723 n + 424097213448909 n - 3823144318918067 n 7 6 5 + 25199326059308401 n - 121407312972291346 n + 422655867161636108 n 4 3 2 - 1037335654412324904 n + 1718951148770349312 n - 1784196056987637120 n + 1006240516641561600 n - 218581039797120000)/(1024 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n))] and in Maple notation [3/8*(341*n^7-8337*n^6+82187*n^5-413315*n^4+1058904*n^3-825748*n^2-2388832*n+ 5405400)/(2*n-9)/(2*n-13)/(2*n-11)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 39/64*(417 *n^8-13204*n^7+171514*n^6-1150800*n^5+3907673*n^4-3516716*n^3-17385204*n^2+ 44597520*n-3326400)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/ (-1+2*n), 13/64*(1223*n^8-37708*n^7+461174*n^6-2667280*n^5+5411567*n^4+14608748 *n^3-80385324*n^2+73252080*n-9979200)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7 )/(2*n-5)/(2*n-3)/(-1+2*n), 429/64*(70*n^9-2631*n^8+39240*n^7-275814*n^6+657510 *n^5+2729601*n^4-20255620*n^3+41094444*n^2-29430000*n+5140800)/(2*n-17)/(2*n-15 )/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 33/64*(808*n^9-\ 27879*n^8+351612*n^7-1554966*n^6-5137608*n^5+78663249*n^4-298488172*n^3+ 489012396*n^2-333581040*n+66830400)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9) /(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 99/64*(431*n^10-16705*n^9+223635*n^8-618150* n^7-14452767*n^6+173894175*n^5-871043135*n^4+2280062200*n^3-3124552164*n^2+ 2002038480*n-423259200)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2 *n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 27/64*(1042*n^10-29490*n^9+60945*n^8+7116600*n^ 7-116448444*n^6+865147710*n^5-3565996195*n^4+8433667500*n^3-11020305348*n^2+ 7030417680*n-1551950400)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/( 2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 297/64*(72*n^11+166*n^10-132880*n^9+4052865*n^ 8-58839564*n^7+498478008*n^6-2629430400*n^5+8763233285*n^4-18089234108*n^3+ 21773243676*n^2-13365271920*n+2962814400)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n -9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), -33/128*(965*n^11-\ 136387*n^10+5647680*n^9-115616640*n^8+1385449065*n^7-10452444471*n^6+ 51122099890*n^5-162194584910*n^4+325039643520*n^3-385939361592*n^2+237019517280 *n-53330659200)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n -15)/(2*n-17)/(2*n-19)/(2*n-21), -11/512*(75163*n^12-6971826*n^11+269548691*n^ 10-5839045890*n^9+79535795469*n^8-719819179758*n^7+4437283475873*n^6-\ 18715435263510*n^5+53271103870468*n^4-98744750886216*n^3+111163335579936*n^2-\ 66124906848000*n+14719261939200)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/( 2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -11/512*(119737*n^12 -9735414*n^11+342516329*n^10-6914616510*n^9+89273565231*n^8-775744662042*n^7+ 4638827090147*n^6-19139962942650*n^5+53670176497132*n^4-98590314100344*n^3+ 110552908557024*n^2-65797812138240*n+14719261939200)/(2*n-23)/(2*n-21)/(2*n-19) /(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -13/512*(259778*n^13-23065591*n^12+911020276*n^11-21168067415*n^10+322243825464 *n^9-3385336141473*n^8+25163989278868*n^7-133389510614525*n^6+501120311883158*n ^5-1306582491883636*n^4+2270089312634856*n^3-2442506622176160*n^2+ 1413012145708800*n-311369002560000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2 *n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), -13/512* (296923*n^13-25591451*n^12+986090321*n^11-22451055715*n^10+336182226699*n^9-\ 3485590645053*n^8+25644453279203*n^7-134882770073425*n^6+503877958763278*n^5-\ 1308762610978196*n^4+2268724273109976*n^3-2438731703604960*n^2+1411125060844800 *n-311369002560000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/ (2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 1/1024*(-16442911*n^14+ 1618425683*n^13-72018966929*n^12+1916626789987*n^11-33999501258723*n^10+ 424097213448909*n^9-3823144318918067*n^8+25199326059308401*n^7-\ 121407312972291346*n^6+422655867161636108*n^5-1037335654412324904*n^4+ 1718951148770349312*n^3-1784196056987637120*n^2+1006240516641561600*n-\ 218581039797120000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n)] The limits, as n goes to infinity are 1023 16263 15899 15015 3333 42669 14067 2673 -31845 -826793 [----, -----, -----, -----, ----, -----, -----, -----, ------, -------, 1024 16384 16384 16384 4096 65536 32768 16384 262144 2097152 -1317107 -1688557 -3859999 -16442911 --------, --------, --------, ---------] 2097152 2097152 4194304 16777216 and in Maple notation [1023/1024, 16263/16384, 15899/16384, 15015/16384, 3333/4096, 42669/65536, 14067/32768, 2673/16384, -31845/262144, -826793/2097152, -1317107/2097152, -\ 1688557/2097152, -3859999/4194304, -16442911/16777216] and in floating point [.9990234375, .9926147461, .9703979492, .9164428711, .8137207031, .6510772705, .4292907715, .1631469727, -.1214790344, -.3942456245, -.6280455589, -.805166721\ 3, -.9202954769, -.9800738692] The cut off is at j=, 9 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 16], vs. those in the, 2, -th row from j=1 to j=, 15, are as follws 8 7 6 5 4 [3 (10917 n - 349004 n + 4643114 n - 33167400 n + 135329173 n 3 2 - 294069916 n + 162719196 n + 716775120 n - 1383782400)/(128 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 8 7 6 5 4 3 (10877 n - 345684 n + 4526074 n - 30902200 n + 109488813 n 3 2 - 122883236 n - 416728164 n + 1235211120 n - 86486400)/(128 (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 9 8 7 6 5 39 (3301 n - 131013 n + 2162814 n - 18753882 n + 84942669 n 4 3 2 - 128288517 n - 469858624 n + 2063179812 n - 1772757360 n + 226195200)/ (256 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 9 8 7 (2 n - 5) (2 n - 3) (-1 + 2 n)), 39 (1591 n - 61044 n + 942534 n 6 5 4 3 2 - 7083216 n + 21307839 n + 38462004 n - 428041084 n + 926685456 n - 671964480 n + 113097600)/(128 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 429 ( 10 9 8 7 6 5 1069 n - 48605 n + 888420 n - 7856370 n + 26310417 n + 95331075 n 4 3 2 - 1166766970 n + 4047826220 n - 6263184936 n + 4090027680 n - 781401600 )/(512 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 10 9 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 33 (11959 n - 496205 n 8 7 6 5 4 + 7644390 n - 42985050 n - 156501513 n + 3490852995 n - 19788521140 n 3 2 + 54501174500 n - 76298409696 n + 48980093760 n - 10158220800)/(512 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 11 10 9 (2 n - 5) (2 n - 3) (-1 + 2 n)), 297 (4059 n - 183392 n + 2830355 n 8 7 6 5 - 6317520 n - 345431163 n + 5033671104 n - 33917696175 n 4 3 2 + 129766532720 n - 289801364996 n + 361830425088 n - 222224882880 n + 47405030400)/(1024 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), 297 ( 11 10 9 8 7 6 611 n - 18393 n - 66400 n + 10116450 n - 188738697 n + 1782506271 n 5 4 3 2 - 9975956930 n + 34416306600 n - 72437875064 n + 87947121072 n - 53968998720 n + 11851257600)/(256 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 12 11 10 9 (2 n - 21)), 297 (2347 n + 143034 n - 14950513 n + 474732430 n 8 7 6 5 - 8056274259 n + 84194100942 n - 572992039699 n + 2587788202290 n 4 3 2 - 7709380983188 n + 14688153754424 n - 16739411805888 n + 9944686577280 n - 2180631398400)/(2048 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) 12 11 10 (2 n - 3) (-1 + 2 n)), - 11 (144643 n - 18214746 n + 822808031 n 9 8 7 6 - 19591686390 n + 284060987709 n - 2683537531038 n + 17043731253533 n 5 4 3 - 73375894177050 n + 211687661288548 n - 395519532295416 n 2 + 446789836899936 n - 265644458876160 n + 58877047756800)/(2048 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), - 143 ( 13 12 11 10 104509 n - 10781123 n + 477257303 n - 12114125995 n 9 8 7 + 197582809917 n - 2190158462469 n + 16966096455029 n 6 5 4 - 92771120962825 n + 356459049288274 n - 943731931873508 n 3 2 + 1654762602552168 n - 1787392956120480 n + 1033220674886400 n - 226450183680000)/(4096 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21) 13 12 11 (2 n - 23) (2 n - 25)), - 13 (853387 n - 79633064 n + 3268730879 n 10 9 8 - 78257328160 n + 1219283295681 n - 13040072047992 n 7 6 5 + 98253637113797 n - 526091745163600 n + 1990693013132032 n 4 3 2 - 5215429372061744 n + 9087182448163824 n - 9788901081560640 n + 5661484007155200 n - 1245476010240000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) 14 (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), - 13 (8498621 n 13 12 11 10 - 867966883 n + 39833325469 n - 1087488314087 n + 19699302964803 n 9 8 7 - 249923045812209 n + 2283649548025087 n - 15211636645475501 n 6 5 4 + 73875496827087956 n - 258677797398089608 n + 637373985378187344 n 3 2 - 1058635640039180112 n + 1099860200011350720 n - 620130298571481600 n + 134511409105920000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 14 13 (2 n - 5) (2 n - 3) (-1 + 2 n)), (-124522883 n + 12406834279 n 12 11 10 - 557624886637 n + 14960560773431 n - 267126004264119 n 9 8 7 + 3349411465417017 n - 30317932684647751 n + 200463762205720013 n 6 5 4 - 968091341532252938 n + 3375933838260787804 n - 8295051862151730312 n 3 2 + 13754734089472486656 n - 14280641101700415360 n + 8053286918360140800 n - 1748648318376960000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), ( (38779380 n + 38779380) (2 n - 31)! - (n - 16)! (n - 14)! binomial(2 n, n)) /((n - 16)! (n - 14)! binomial(2 n, n))] and in Maple notation [3/128*(10917*n^8-349004*n^7+4643114*n^6-33167400*n^5+135329173*n^4-294069916*n ^3+162719196*n^2+716775120*n-1383782400)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2* n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 3/128*(10877*n^8-345684*n^7+4526074*n^6-30902200 *n^5+109488813*n^4-122883236*n^3-416728164*n^2+1235211120*n-86486400)/(2*n-15)/ (2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 39/256*(3301*n^9-\ 131013*n^8+2162814*n^7-18753882*n^6+84942669*n^5-128288517*n^4-469858624*n^3+ 2063179812*n^2-1772757360*n+226195200)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n -9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 39/128*(1591*n^9-61044*n^8+942534*n^7-\ 7083216*n^6+21307839*n^5+38462004*n^4-428041084*n^3+926685456*n^2-671964480*n+ 113097600)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/ (-1+2*n), 429/512*(1069*n^10-48605*n^9+888420*n^8-7856370*n^7+26310417*n^6+ 95331075*n^5-1166766970*n^4+4047826220*n^3-6263184936*n^2+4090027680*n-\ 781401600)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5) /(2*n-3)/(-1+2*n), 33/512*(11959*n^10-496205*n^9+7644390*n^8-42985050*n^7-\ 156501513*n^6+3490852995*n^5-19788521140*n^4+54501174500*n^3-76298409696*n^2+ 48980093760*n-10158220800)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9) /(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 297/1024*(4059*n^11-183392*n^10+2830355*n^9-\ 6317520*n^8-345431163*n^7+5033671104*n^6-33917696175*n^5+129766532720*n^4-\ 289801364996*n^3+361830425088*n^2-222224882880*n+47405030400)/(-1+2*n)/(2*n-3)/ (2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), 297/256*(611*n^11-18393*n^10-66400*n^9+10116450*n^8-188738697*n^7+1782506271*n^ 6-9975956930*n^5+34416306600*n^4-72437875064*n^3+87947121072*n^2-53968998720*n+ 11851257600)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15 )/(2*n-17)/(2*n-19)/(2*n-21), 297/2048*(2347*n^12+143034*n^11-14950513*n^10+ 474732430*n^9-8056274259*n^8+84194100942*n^7-572992039699*n^6+2587788202290*n^5 -7709380983188*n^4+14688153754424*n^3-16739411805888*n^2+9944686577280*n-\ 2180631398400)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/( 2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -11/2048*(144643*n^12-18214746*n^11+ 822808031*n^10-19591686390*n^9+284060987709*n^8-2683537531038*n^7+ 17043731253533*n^6-73375894177050*n^5+211687661288548*n^4-395519532295416*n^3+ 446789836899936*n^2-265644458876160*n+58877047756800)/(2*n-23)/(2*n-21)/(2*n-19 )/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), -143/4096*(104509*n^13-10781123*n^12+477257303*n^11-12114125995*n^10+ 197582809917*n^9-2190158462469*n^8+16966096455029*n^7-92771120962825*n^6+ 356459049288274*n^5-943731931873508*n^4+1654762602552168*n^3-1787392956120480*n ^2+1033220674886400*n-226450183680000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9) /(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), -13/ 2048*(853387*n^13-79633064*n^12+3268730879*n^11-78257328160*n^10+1219283295681* n^9-13040072047992*n^8+98253637113797*n^7-526091745163600*n^6+1990693013132032* n^5-5215429372061744*n^4+9087182448163824*n^3-9788901081560640*n^2+ 5661484007155200*n-1245476010240000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/( 2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), -13/ 8192*(8498621*n^14-867966883*n^13+39833325469*n^12-1087488314087*n^11+ 19699302964803*n^10-249923045812209*n^9+2283649548025087*n^8-15211636645475501* n^7+73875496827087956*n^6-258677797398089608*n^5+637373985378187344*n^4-\ 1058635640039180112*n^3+1099860200011350720*n^2-620130298571481600*n+ 134511409105920000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 1/8192*(-\ 124522883*n^14+12406834279*n^13-557624886637*n^12+14960560773431*n^11-\ 267126004264119*n^10+3349411465417017*n^9-30317932684647751*n^8+ 200463762205720013*n^7-968091341532252938*n^6+3375933838260787804*n^5-\ 8295051862151730312*n^4+13754734089472486656*n^3-14280641101700415360*n^2+ 8053286918360140800*n-1748648318376960000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/ (2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-\ 1+2*n), ((38779380*n+38779380)*(2*n-31)!-(n-16)!*(n-14)!*binomial(2*n,n))/(n-16 )!/(n-14)!/binomial(2*n,n)] The limits, as n goes to infinity are 32751 32631 128739 62049 458601 394647 1205523 181467 697059 [-----, -----, ------, -----, ------, ------, -------, ------, -------, 32768 32768 131072 65536 524288 524288 2097152 524288 8388608 -1591073 -14944787 -11094031 -110482073 -124522883 -527176067 --------, ---------, ---------, ----------, ----------, ----------] 8388608 33554432 16777216 134217728 134217728 536870912 and in Maple notation [32751/32768, 32631/32768, 128739/131072, 62049/65536, 458601/524288, 394647/ 524288, 1205523/2097152, 181467/524288, 697059/8388608, -1591073/8388608, -\ 14944787/33554432, -11094031/16777216, -110482073/134217728, -124522883/ 134217728, -527176067/536870912] and in floating point [.9994812012, .9958190918, .9822006226, .9467926025, .8747119904, .7527294159, .5748381615, .3461208344, .8309590816e-1, -.1896706820, -.4453893602, -.6612557\ 769, -.8231555894, -.9277677760, -.9819419440] The cut off is at j=, 10 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 17], vs. those in the, 2, -th row from j=1 to j=, 16, are as follws 8 7 6 5 4 3 [17 (1927 n - 61628 n + 820750 n - 5879720 n + 24185623 n - 53908652 n 2 + 35532180 n + 120390480 n - 259459200)/(128 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15)), 51 ( 9 8 7 6 5 4 1282 n - 51741 n + 885048 n - 8294874 n + 45303258 n - 134015469 n 3 2 + 107941532 n + 531550884 n - 1321697520 n + 86486400)/(128 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) 9 8 7 6 (2 n - 17)), 3 (21614 n - 863397 n + 14425896 n - 127971858 n 5 4 3 2 + 608148366 n - 1105480173 n - 2542902836 n + 13688357028 n - 12090489840 n + 1470268800)/(128 (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 39 ( 10 9 8 7 6 6497 n - 314875 n + 6420180 n - 69897150 n + 411954501 n 5 4 3 2 - 979818315 n - 2283348530 n + 19343472100 n - 38850168648 n + 26945598240 n - 4297708800)/(256 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 10 9 8 7 6 39 (12329 n - 574555 n + 10921170 n - 103816470 n + 436043097 n 5 4 3 2 + 386421525 n - 11475451220 n + 43973313820 n - 70401797376 n + 46222735680 n - 8595417600)/(512 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 11 10 9 8 7 429 (2024 n - 109285 n + 2400255 n - 26066370 n + 115214022 n 6 5 4 3 + 386427195 n - 7388608865 n + 38416052620 n - 99551021196 n 2 + 133132047840 n - 82277173440 n + 16409433600)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) 11 10 9 (2 n - 19) (2 n - 21)), 33 (21923 n - 1073974 n + 19748025 n 8 7 6 5 - 134379300 n - 626989671 n + 17903894778 n - 139072805465 n 4 3 2 + 566061179200 n - 1304219071452 n + 1650978955296 n - 1014390336960 n + 213322636800)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), 297 ( 12 11 10 9 8 7073 n - 365046 n + 6241081 n - 1386550 n - 1506791121 n 7 6 5 4 + 25463398422 n - 218681886797 n + 1133838719310 n - 3681751864252 n 3 2 + 7381852414024 n - 8612993731584 n + 5111495975040 n - 1090315699200)/( 1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 297 ( 12 11 10 9 8 7591 n - 218802 n - 4383853 n + 303961570 n - 6378755367 n 7 6 5 4 + 73850483994 n - 533337934279 n + 2499411411150 n - 7620279098924 n 3 2 + 14714892050408 n - 16869855886368 n + 10016839822080 n - 2180631398400 )/(2048 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 1287 ( 13 12 11 10 9 142 n + 140951 n - 11245636 n + 383425915 n - 7524494004 n 8 7 6 5 + 94721196153 n - 803724131548 n + 4693884681025 n - 18900278329538 n 4 3 2 + 51651362832596 n - 92321838381816 n + 100565108408160 n - 58055872492800 n + 12580565760000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 13 12 (2 n - 21) (2 n - 23) (2 n - 25)), - 143 (29312 n - 3715759 n 11 10 9 8 + 185608534 n - 5099328335 n + 87873229806 n - 1013203895577 n 7 6 5 + 8076215918362 n - 45080218609925 n + 175745081181482 n 4 3 2 - 469815069144964 n + 828535219642104 n - 897135551468640 n + 518358666067200 n - 113225091840000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) 14 (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), - 13 (2528623 n 13 12 11 10 - 291166379 n + 14684052047 n - 432148969231 n + 8312487349089 n 9 8 7 - 110632241367417 n + 1049905849082981 n - 7202793450511813 n 6 5 4 + 35772408750229228 n - 127321748842669304 n + 317235363972702672 n 3 2 - 530441189299735056 n + 552640529708775360 n - 311376395753740800 n + 67255704552960000)/(4096 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 14 13 (2 n - 5) (2 n - 3) (-1 + 2 n)), - 13 (7122071 n - 757842883 n 12 11 10 + 35932182769 n - 1006781187587 n + 18617433776403 n 9 8 7 - 240078744845709 n + 2221836944071987 n - 14945706663658001 n 6 5 4 + 73119625966292906 n - 257391280526950108 n + 636472999864923144 n 3 2 - 1059373799944754112 n + 1101579954761070720 n - 620954510637081600 n + 134511409105920000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 15 14 (2 n - 5) (2 n - 3) (-1 + 2 n)), (-225093796 n + 26221250925 n 13 12 11 - 1384174193185 n + 43873380662475 n - 932432099066377 n 10 9 8 + 14048291931138465 n - 154637768515799855 n + 1262613918877193025 n 7 6 - 7680953453849058383 n + 34646305351665077070 n 5 4 - 114207883163241582260 n + 267890690388532575000 n 3 2 - 427632622319031922944 n + 430618591867715303040 n - 237214712748495283200 n + 50710801232931840000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) 15 14 13 (2 n - 29)), (-250756621 n + 28607893650 n - 1483514988760 n 12 11 10 + 46318457640000 n - 972018058804702 n + 14491649542510140 n 9 8 7 - 158150966478370580 n + 1282376795508408600 n - 7758671148029935133 n 6 5 + 34850966253444511170 n - 114531914142936568460 n 4 3 + 268091773967545109400 n - 427424001309635612544 n 2 + 430192638705217127040 n - 237018032200726963200 n + 50710801232931840000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), ( (141430680 n + 141430680) (2 n - 33)! - (n - 17)! (n - 15)! binomial(2 n, n))/((n - 17)! (n - 15)! binomial(2 n, n))] and in Maple notation [17/128*(1927*n^8-61628*n^7+820750*n^6-5879720*n^5+24185623*n^4-53908652*n^3+ 35532180*n^2+120390480*n-259459200)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-7)/(2*n-9)/(2 *n-11)/(2*n-13)/(2*n-15), 51/128*(1282*n^9-51741*n^8+885048*n^7-8294874*n^6+ 45303258*n^5-134015469*n^4+107941532*n^3+531550884*n^2-1321697520*n+86486400)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17), 3/ 128*(21614*n^9-863397*n^8+14425896*n^7-127971858*n^6+608148366*n^5-1105480173*n ^4-2542902836*n^3+13688357028*n^2-12090489840*n+1470268800)/(2*n-17)/(2*n-15)/( 2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 39/256*(6497*n^10-\ 314875*n^9+6420180*n^8-69897150*n^7+411954501*n^6-979818315*n^5-2283348530*n^4+ 19343472100*n^3-38850168648*n^2+26945598240*n-4297708800)/(2*n-19)/(2*n-17)/(2* n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 39/512*(12329 *n^10-574555*n^9+10921170*n^8-103816470*n^7+436043097*n^6+386421525*n^5-\ 11475451220*n^4+43973313820*n^3-70401797376*n^2+46222735680*n-8595417600)/(2*n-\ 19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n ), 429/512*(2024*n^11-109285*n^10+2400255*n^9-26066370*n^8+115214022*n^7+ 386427195*n^6-7388608865*n^5+38416052620*n^4-99551021196*n^3+133132047840*n^2-\ 82277173440*n+16409433600)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2 *n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), 33/512*(21923*n^11-1073974*n^10+ 19748025*n^9-134379300*n^8-626989671*n^7+17903894778*n^6-139072805465*n^5+ 566061179200*n^4-1304219071452*n^3+1650978955296*n^2-1014390336960*n+ 213322636800)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-\ 15)/(2*n-17)/(2*n-19)/(2*n-21), 297/1024*(7073*n^12-365046*n^11+6241081*n^10-\ 1386550*n^9-1506791121*n^8+25463398422*n^7-218681886797*n^6+1133838719310*n^5-\ 3681751864252*n^4+7381852414024*n^3-8612993731584*n^2+5111495975040*n-\ 1090315699200)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/( 2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 297/2048*(7591*n^12-218802*n^11-\ 4383853*n^10+303961570*n^9-6378755367*n^8+73850483994*n^7-533337934279*n^6+ 2499411411150*n^5-7620279098924*n^4+14714892050408*n^3-16869855886368*n^2+ 10016839822080*n-2180631398400)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2 *n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 1287/2048*(142*n^13+ 140951*n^12-11245636*n^11+383425915*n^10-7524494004*n^9+94721196153*n^8-\ 803724131548*n^7+4693884681025*n^6-18900278329538*n^5+51651362832596*n^4-\ 92321838381816*n^3+100565108408160*n^2-58055872492800*n+12580565760000)/(-1+2*n )/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/ (2*n-21)/(2*n-23)/(2*n-25), -143/2048*(29312*n^13-3715759*n^12+185608534*n^11-\ 5099328335*n^10+87873229806*n^9-1013203895577*n^8+8076215918362*n^7-\ 45080218609925*n^6+175745081181482*n^5-469815069144964*n^4+828535219642104*n^3-\ 897135551468640*n^2+518358666067200*n-113225091840000)/(-1+2*n)/(2*n-3)/(2*n-5) /(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23) /(2*n-25), -13/4096*(2528623*n^14-291166379*n^13+14684052047*n^12-432148969231* n^11+8312487349089*n^10-110632241367417*n^9+1049905849082981*n^8-\ 7202793450511813*n^7+35772408750229228*n^6-127321748842669304*n^5+ 317235363972702672*n^4-530441189299735056*n^3+552640529708775360*n^2-\ 311376395753740800*n+67255704552960000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2* n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2 *n), -13/8192*(7122071*n^14-757842883*n^13+35932182769*n^12-1006781187587*n^11+ 18617433776403*n^10-240078744845709*n^9+2221836944071987*n^8-14945706663658001* n^7+73119625966292906*n^6-257391280526950108*n^5+636472999864923144*n^4-\ 1059373799944754112*n^3+1101579954761070720*n^2-620954510637081600*n+ 134511409105920000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 1/8192*(-\ 225093796*n^15+26221250925*n^14-1384174193185*n^13+43873380662475*n^12-\ 932432099066377*n^11+14048291931138465*n^10-154637768515799855*n^9+ 1262613918877193025*n^8-7680953453849058383*n^7+34646305351665077070*n^6-\ 114207883163241582260*n^5+267890690388532575000*n^4-427632622319031922944*n^3+ 430618591867715303040*n^2-237214712748495283200*n+50710801232931840000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/ (2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 1/8192*(-250756621*n^15+ 28607893650*n^14-1483514988760*n^13+46318457640000*n^12-972018058804702*n^11+ 14491649542510140*n^10-158150966478370580*n^9+1282376795508408600*n^8-\ 7758671148029935133*n^7+34850966253444511170*n^6-114531914142936568460*n^5+ 268091773967545109400*n^4-427424001309635612544*n^3+430192638705217127040*n^2-\ 237018032200726963200*n+50710801232931840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17) /(2*n-23)/(2*n-29), ((141430680*n+141430680)*(2*n-33)!-(n-17)!*(n-15)!*binomial (2*n,n))/(n-17)!/(n-15)!/binomial(2*n,n)] The limits, as n goes to infinity are 32759 32691 32421 253383 480831 108537 723459 2100681 2254527 [-----, -----, -----, ------, ------, ------, -------, -------, -------, 32768 32768 32768 262144 524288 131072 1048576 4194304 8388608 91377 -32747 -32872099 -92586923 -56273449 -250756621 -1056062989 -------, ------, ---------, ---------, ---------, ----------, -----------] 8388608 131072 67108864 134217728 67108864 268435456 1073741824 and in Maple notation [32759/32768, 32691/32768, 32421/32768, 253383/262144, 480831/524288, 108537/ 131072, 723459/1048576, 2100681/4194304, 2254527/8388608, 91377/8388608, -32747 /131072, -32872099/67108864, -92586923/134217728, -56273449/67108864, -\ 250756621/268435456, -1056062989/1073741824] and in floating point [.9997253418, .9976501465, .9894104004, .9665794373, .9171123505, .8280715942, .6899442673, .5008413792, .2687605619, .1089298725e-1, -.2498397827, -.48983244\ 60, -.6898263320, -.8385397345, -.9341412075, -.9835353019] The cut off is at j=, 11 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 18], vs. those in the, 2, -th row from j=1 to j=, 17, are as follws 9 8 7 6 5 [17 (7709 n - 312129 n + 5382606 n - 51527826 n + 298205061 n 4 3 2 - 1052561601 n + 2057983744 n - 971272044 n - 4956171120 n + 9340531200 )/(256 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 7) (2 n - 9) (2 n - 11) 9 8 7 (2 n - 13) (2 n - 15) (2 n - 17)), 17 (3850 n - 155583 n + 2668920 n 6 5 4 3 2 - 25174422 n + 139509930 n - 428108247 n + 421111460 n + 1491275052 n - 4195722960 n + 259459200)/(128 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 7) 10 (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17)), 51 (2554 n 9 8 7 6 5 - 126750 n + 2701635 n - 32026800 n + 225443232 n - 892630830 n 4 3 2 + 1281022415 n + 3891782700 n - 17076721836 n + 14330281680 n - 1643241600)/(128 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) 10 (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), (128354 n 9 8 7 6 - 6278050 n + 129991485 n - 1452939000 n + 9002777532 n 5 4 3 2 - 24971602530 n - 27253410935 n + 363981781900 n - 775824638436 n + 544610317680 n - 83805321600)/(128 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 11 10 9 8 7 143 (13873 n - 804404 n + 19821945 n - 264910260 n + 1980362559 n 6 5 4 3 - 6806542092 n - 8069638765 n + 162863472260 n - 575048647932 n 2 + 875658896496 n - 553477034880 n + 98456601600)/(1024 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) 11 10 9 (2 n - 17) (2 n - 19) (2 n - 21)), 429 (2157 n - 119792 n + 2747385 n 8 7 6 5 4 - 32262840 n + 179601051 n - 3886176 n - 6084945525 n + 36396280040 n 3 2 - 99246300108 n + 135548340768 n - 83950292160 n + 16409433600)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) 12 11 (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), 143 (22825 n - 1441506 n 10 9 8 7 + 37407689 n - 488046570 n + 2703763095 n + 9320956722 n 6 5 4 - 255079838893 n + 1832664978210 n - 7010381910620 n 3 2 + 15386140712184 n - 18729822266496 n + 11127864101760 n - 2264501836800 )/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 11 ( 12 11 10 9 8 239041 n - 13605702 n + 292193297 n - 2286824430 n - 16005424017 n 7 6 5 + 519404718294 n - 5200600431829 n + 28914262956150 n 4 3 2 - 97524280951724 n + 199667975548008 n - 235162462315968 n + 139522173598080 n - 29438523878400)/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 13 12 (2 n - 5) (2 n - 3) (-1 + 2 n)), 1287 (11209 n - 646723 n 11 10 9 8 + 11583803 n + 58857205 n - 5861794983 n + 111975893931 n 7 6 5 - 1180048311871 n + 7881847533175 n - 34669034695526 n 4 3 2 + 100388367639692 n - 185806711797432 n + 205628672092320 n - 118516853145600 n + 25161131520000)/(4096 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 13 12 (2 n - 21) (2 n - 23) (2 n - 25)), 143 (23128 n - 479441 n 11 10 9 8 - 40926574 n + 2273336735 n - 53325600486 n + 737695672377 n 7 6 5 - 6631589516482 n + 40219778928725 n - 166072093383242 n 4 3 2 + 461324548599364 n - 832583206717944 n + 910764850105440 n - 525462985555200 n + 113225091840000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) 14 (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), - 143 (12739 n 13 12 11 10 - 4139177 n + 311908646 n - 11597974003 n + 260944465152 n 9 8 7 - 3885852719721 n + 40075261657958 n - 292588949379769 n 6 5 4 + 1521916238018029 n - 5601143345801402 n + 14278756445592996 n 3 2 - 24208525567803528 n + 25372924228104480 n - 14278080683462400 n + 3057077479680000)/(2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 14 13 (2 n - 5) (2 n - 3) (-1 + 2 n)), - 13 (782519 n - 104630827 n 12 11 10 + 5810407666 n - 182828393153 n + 3691336556292 n 9 8 7 - 50926126191171 n + 496480367163418 n - 3475368780641219 n 6 5 4 + 17518749066419909 n - 63019725091625302 n + 158150485205130516 n 3 2 - 265572596335967928 n + 277183855940899680 n - 156107796746630400 n + 33627852276480000)/(2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 15 14 (2 n - 5) (2 n - 3) (-1 + 2 n)), - 13 (21834959 n - 2805732450 n 13 12 11 + 160311839240 n - 5419418537400 n + 121392658459058 n 10 9 8 - 1908880799839860 n + 21751861450156420 n - 182583866459649600 n 7 6 + 1135110862602939607 n - 5205831250040797530 n 5 4 + 17370740868673900540 n - 41086820751795735000 n 3 2 + 65915396729828057376 n - 66512433477652508160 n + 36617146319780812800 n - 7801661728143360000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) 15 14 13 (2 n - 29)), (-191827171 n + 23127454800 n - 1255399087810 n 12 11 10 + 40703836432350 n - 881116966072252 n + 13473569101582590 n 9 8 7 - 150083623008763730 n + 1236995375095987650 n - 7580208294725699633 n 6 5 + 34381004182691736570 n - 113787843004377711260 n 4 3 + 267630026489812623000 n - 427903056960841954944 n 2 + 431170753374657383040 n - 237469669014120883200 n + 50710801232931840000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 16 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), (-914632309 n 15 14 13 + 120608702552 n - 7258706952340 n + 264381567983120 n 12 11 10 - 6513979856250958 n + 114932025846257984 n - 1499132553746613500 n 9 8 + 14709481775416645360 n - 109389413935642511957 n 7 6 + 616256507446072268776 n - 2608607197197788575880 n 5 4 + 8158143764313751237120 n - 18323100498887365261776 n 3 2 + 28231867393220960546688 n - 27639766419800545601280 n + 14906768119237980518400 n - 3144069676441774080000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 16 15 (2 n - 23) (2 n - 29)), (-1008919429 n + 130697424392 n 14 13 12 - 7746171362740 n + 278452977771920 n - 6784442648545198 n 11 10 + 118582132385216864 n - 1534649086229359100 n 9 8 + 14961053258805699760 n - 110680731597353606117 n 7 6 + 620960451388334553496 n - 2620197252267797045480 n 5 4 + 8175335682026993945920 n - 18332530759959123026256 n 3 2 + 28219697469470034661248 n - 27617418998514234535680 n + 14896786945661974886400 n - 3144069676441774080000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), ((518579160 n + 518579160) (2 n - 35)! - (n - 18)! (n - 16)! binomial(2 n, n))/((n - 18)! (n - 16)! binomial(2 n, n))] and in Maple notation [17/256*(7709*n^9-312129*n^8+5382606*n^7-51527826*n^6+298205061*n^5-1052561601* n^4+2057983744*n^3-971272044*n^2-4956171120*n+9340531200)/(2*n-5)/(-1+2*n)/(2*n -3)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17), 17/128*(3850*n^9-\ 155583*n^8+2668920*n^7-25174422*n^6+139509930*n^5-428108247*n^4+421111460*n^3+ 1491275052*n^2-4195722960*n+259459200)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-7)/(2*n-9) /(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17), 51/128*(2554*n^10-126750*n^9+2701635*n^8-\ 32026800*n^7+225443232*n^6-892630830*n^5+1281022415*n^4+3891782700*n^3-\ 17076721836*n^2+14330281680*n-1643241600)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/( 2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 1/128*(128354*n^10-6278050*n^ 9+129991485*n^8-1452939000*n^7+9002777532*n^6-24971602530*n^5-27253410935*n^4+ 363981781900*n^3-775824638436*n^2+544610317680*n-83805321600)/(2*n-19)/(2*n-17) /(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 143/1024* (13873*n^11-804404*n^10+19821945*n^9-264910260*n^8+1980362559*n^7-6806542092*n^ 6-8069638765*n^5+162863472260*n^4-575048647932*n^3+875658896496*n^2-\ 553477034880*n+98456601600)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/( 2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), 429/512*(2157*n^11-119792*n^10+ 2747385*n^9-32262840*n^8+179601051*n^7-3886176*n^6-6084945525*n^5+36396280040*n ^4-99246300108*n^3+135548340768*n^2-83950292160*n+16409433600)/(-1+2*n)/(2*n-3) /(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), 143/1024*(22825*n^12-1441506*n^11+37407689*n^10-488046570*n^9+2703763095*n^8+ 9320956722*n^7-255079838893*n^6+1832664978210*n^5-7010381910620*n^4+ 15386140712184*n^3-18729822266496*n^2+11127864101760*n-2264501836800)/(2*n-23)/ (2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/( 2*n-3)/(-1+2*n), 11/1024*(239041*n^12-13605702*n^11+292193297*n^10-2286824430*n ^9-16005424017*n^8+519404718294*n^7-5200600431829*n^6+28914262956150*n^5-\ 97524280951724*n^4+199667975548008*n^3-235162462315968*n^2+139522173598080*n-\ 29438523878400)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/ (2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 1287/4096*(11209*n^13-646723*n^12+ 11583803*n^11+58857205*n^10-5861794983*n^9+111975893931*n^8-1180048311871*n^7+ 7881847533175*n^6-34669034695526*n^5+100388367639692*n^4-185806711797432*n^3+ 205628672092320*n^2-118516853145600*n+25161131520000)/(-1+2*n)/(2*n-3)/(2*n-5)/ (2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/ (2*n-25), 143/2048*(23128*n^13-479441*n^12-40926574*n^11+2273336735*n^10-\ 53325600486*n^9+737695672377*n^8-6631589516482*n^7+40219778928725*n^6-\ 166072093383242*n^5+461324548599364*n^4-832583206717944*n^3+910764850105440*n^2 -525462985555200*n+113225091840000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2 *n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), -143/ 2048*(12739*n^14-4139177*n^13+311908646*n^12-11597974003*n^11+260944465152*n^10 -3885852719721*n^9+40075261657958*n^8-292588949379769*n^7+1521916238018029*n^6-\ 5601143345801402*n^5+14278756445592996*n^4-24208525567803528*n^3+ 25372924228104480*n^2-14278080683462400*n+3057077479680000)/(2*n-27)/(2*n-25)/( 2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/( 2*n-5)/(2*n-3)/(-1+2*n), -13/2048*(782519*n^14-104630827*n^13+5810407666*n^12-\ 182828393153*n^11+3691336556292*n^10-50926126191171*n^9+496480367163418*n^8-\ 3475368780641219*n^7+17518749066419909*n^6-63019725091625302*n^5+ 158150485205130516*n^4-265572596335967928*n^3+277183855940899680*n^2-\ 156107796746630400*n+33627852276480000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2* n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2 *n), -13/16384*(21834959*n^15-2805732450*n^14+160311839240*n^13-5419418537400*n ^12+121392658459058*n^11-1908880799839860*n^10+21751861450156420*n^9-\ 182583866459649600*n^8+1135110862602939607*n^7-5205831250040797530*n^6+ 17370740868673900540*n^5-41086820751795735000*n^4+65915396729828057376*n^3-\ 66512433477652508160*n^2+36617146319780812800*n-7801661728143360000)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2* n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 1/8192*(-191827171*n^15+23127454800* n^14-1255399087810*n^13+40703836432350*n^12-881116966072252*n^11+ 13473569101582590*n^10-150083623008763730*n^9+1236995375095987650*n^8-\ 7580208294725699633*n^7+34381004182691736570*n^6-113787843004377711260*n^5+ 267630026489812623000*n^4-427903056960841954944*n^3+431170753374657383040*n^2-\ 237469669014120883200*n+50710801232931840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17) /(2*n-23)/(2*n-29), 1/16384*(-914632309*n^16+120608702552*n^15-7258706952340*n^ 14+264381567983120*n^13-6513979856250958*n^12+114932025846257984*n^11-\ 1499132553746613500*n^10+14709481775416645360*n^9-109389413935642511957*n^8+ 616256507446072268776*n^7-2608607197197788575880*n^6+8158143764313751237120*n^5 -18323100498887365261776*n^4+28231867393220960546688*n^3-\ 27639766419800545601280*n^2+14906768119237980518400*n-3144069676441774080000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2* n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 1/16384*(-\ 1008919429*n^16+130697424392*n^15-7746171362740*n^14+278452977771920*n^13-\ 6784442648545198*n^12+118582132385216864*n^11-1534649086229359100*n^10+ 14961053258805699760*n^9-110680731597353606117*n^8+620960451388334553496*n^7-\ 2620197252267797045480*n^6+8175335682026993945920*n^5-18332530759959123026256*n ^4+28219697469470034661248*n^3-27617418998514234535680*n^2+ 14896786945661974886400*n-3144069676441774080000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-17)/(2*n-23)/(2*n-29), ((518579160*n+518579160)*(2*n-35)!-(n-18)!*(n-\ 16)!*binomial(2*n,n))/(n-18)!/(n-16)!/binomial(2*n,n)] The limits, as n goes to infinity are 131053 32725 65127 64177 1983839 925353 3263975 2629451 14425983 [------, -----, -----, -----, -------, -------, -------, -------, --------, 131072 32768 65536 65536 2097152 1048576 4194304 4194304 33554432 413413 -1821677 -10172747 -283854467 -191827171 -914632309 -------, --------, ---------, ----------, ----------, ----------, 2097152 33554432 33554432 536870912 268435456 1073741824 -1008919429 -4230144901 -----------, -----------] 1073741824 4294967296 and in Maple notation [131053/131072, 32725/32768, 65127/65536, 64177/65536, 1983839/2097152, 925353/ 1048576, 3263975/4194304, 2629451/4194304, 14425983/33554432, 413413/2097152, -\ 1821677/33554432, -10172747/33554432, -283854467/536870912, -191827171/ 268435456, -914632309/1073741824, -1008919429/1073741824, -4230144901/ 4294967296] and in floating point [.9998550415, .9986877441, .9937591553, .9792633057, .9459681511, .8824853897, .7781922817, .6269099712, .4299277961, .1971306801, -.5429020524e-1, -.30317148\ 57, -.5287201460, -.7146118991, -.8518177168, -.9396294402, -.9849073600] The cut off is at j=, 11 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 19], vs. those in the, 2, -th row from j=1 to j=, 18, are as follws 9 8 7 6 5 [19 (3449 n - 139662 n + 2409186 n - 23082948 n + 133912401 n 4 3 2 - 476049798 n + 952670644 n - 538564392 n - 2126832480 n + 4410806400)/ (128 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 7) (2 n - 9) (2 n - 11) 10 9 8 (2 n - 13) (2 n - 15) (2 n - 17)), 323 (811 n - 40505 n + 875400 n 7 6 5 4 3 - 10699770 n + 80729103 n - 379481865 n + 1008858950 n - 736366180 n 2 - 3685056264 n + 8910364320 n - 518918400)/(256 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) 10 9 8 7 (2 n - 17)), 17 (7682 n - 382210 n + 8186925 n - 97979640 n 6 5 4 3 + 702591036 n - 2894235330 n + 4817652025 n + 9933233740 n 2 - 52231093668 n + 44951203440 n - 4929724800)/(128 (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 11 10 9 8 (-1 + 2 n)), 187 (1384 n - 82682 n + 2143560 n - 31295955 n 7 6 5 4 + 276329172 n - 1429050336 n + 3302995880 n + 4880750705 n 3 2 - 47778182556 n + 95174174268 n - 64257233040 n + 9411292800)/(128 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) 11 10 (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), 11 (23003 n - 1349419 n 9 8 7 6 + 33857520 n - 465676485 n + 3663528099 n - 14407616937 n 5 4 3 2 + 919748710 n + 237261263485 n - 924373878852 n + 1453719811356 n - 923960353680 n + 159991977600)/(128 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 12 11 10 9 (2 n - 21)), 143 (27011 n - 1843362 n + 53954947 n - 868805130 n 8 7 6 5 + 8045697693 n - 37414092366 n - 5210554319 n + 1071566793090 n 4 3 2 - 5920106581204 n + 15299148532728 n - 20075325366528 n + 12023070675840 n - 2264501836800)/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) 12 11 10 (2 n - 3) (-1 + 2 n)), 143 (24757 n - 1613454 n + 43908869 n 9 8 7 6 - 623953110 n + 4417066491 n - 4016966562 n - 191960519833 n 5 4 3 + 1662199295550 n - 6796385677148 n + 15397925286216 n 2 - 19008657625536 n + 11300019212160 n - 2264501836800)/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) 13 12 (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 143 (42614 n - 3107533 n 11 10 9 8 + 93881788 n - 1442314445 n + 9577171032 n + 39315777501 n 7 6 5 - 1366360940516 n + 12789888107425 n - 66870534176446 n 4 3 2 + 214420305459932 n - 421212027079272 n + 479276509158720 n - 275962369737600 n + 56612545920000)/(1024 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 13 12 (2 n - 21) (2 n - 23) (2 n - 25)), 143 (66253 n - 4317566 n 11 10 9 8 + 106000301 n - 897946390 n - 10776879861 n + 368531225502 n 7 6 5 - 4545212495857 n + 32717465724350 n - 150185613509492 n 4 3 2 + 446203136671864 n - 838183727302944 n + 933760914805440 n - 537878017555200 n + 113225091840000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) 14 (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), 143 (170137 n 13 12 11 10 - 10696301 n + 181920893 n + 3479325311 n - 213808411509 n 9 8 7 + 4553167534377 n - 57290477702161 n + 475436956740053 n 6 5 4 - 2699527461768368 n + 10557207385441024 n - 28034191056534432 n 3 2 + 48724592976074736 n - 51636644237194560 n + 29006678048140800 n - 6114154959360000)/(4096 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 14 13 (2 n - 5) (2 n - 3) (-1 + 2 n)), 143 (30731 n + 139937 n 12 11 10 9 - 150895766 n + 7859206243 n - 205391630892 n + 3332200716201 n 8 7 6 - 36309303760718 n + 275205392709289 n - 1469291921972959 n 5 4 3 + 5505922784972162 n - 14205391943845116 n + 24257327554185768 n 2 - 25502734717685280 n + 14342440209350400 n - 3057077479680000)/(2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 15 14 13 (-1 + 2 n)), - 13 (584548 n - 131944725 n + 10121323555 n 12 11 10 - 413511275100 n + 10588371635401 n - 183882833596770 n 9 8 7 + 2259995836061315 n - 20107139705076600 n + 130715264304965879 n 6 5 4 - 620065713618335985 n + 2120740694474264030 n - 5102062248585987300 n 3 2 + 8269908512484963672 n - 8381278557735383520 n + 4608898458822201600 n - 975207716017920000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 15 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), - 13 (3620321 n 14 13 12 11 - 521268075 n + 32216881235 n - 1153047731325 n + 26956307758877 n 10 9 8 - 437952266705415 n + 5117567125280005 n - 43796483974081575 n 7 6 5 + 276336063994702558 n - 1281448826183957970 n + 4310336840821204060 n 4 3 - 10250863918359552600 n + 16499002616569029744 n 2 - 16670847577889903040 n + 9174246971793523200 n - 1950415432035840000)/( 4096 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 16 15 (2 n - 17) (2 n - 23) (2 n - 29)), (-604477309 n + 86491652552 n 14 13 12 - 5567741892340 n + 214420559963120 n - 5533192207500958 n 11 10 + 101440452070577984 n - 1365581443402533500 n 9 8 + 13748817252281185360 n - 104389692881350976957 n 7 6 + 597815293883310278776 n - 2562641795401177915880 n 5 4 + 8089138728349366717120 n - 18284307596412386701776 n 3 2 + 28279878239220687266688 n - 27729658978715694401280 n + 14947177728857436518400 n - 3144069676441774080000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 16 15 (2 n - 23) (2 n - 29)), (-790570309 n + 107334068552 n 14 13 12 - 6617306412340 n + 245866555103120 n - 6158107761126958 n 11 10 + 110129254084469984 n - 1452400274164053500 n 9 8 + 14378466665694205360 n - 107690311749180545957 n 7 6 + 610067107522042946776 n - 2593357124737251115880 n 5 4 + 8135522819954221357120 n - 18310692260635052413776 n 3 2 + 28247880450787968290688 n - 27669170921493060161280 n + 14919901242364303718400 n - 3144069676441774080000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 (2 n - 23) (2 n - 29)), - 17 (109063354 n - 16169059253 n 15 14 13 + 1100787869488 n - 45660983031380 n + 1290936762351628 n 12 11 10 - 26360864536998446 n + 401886277056709016 n - 4662645523038877660 n 9 8 + 41571543918911459882 n - 285577389951988368949 n 7 6 + 1505739952594509590504 n - 6029735276662704628840 n 5 4 + 18000983829444228301536 n - 38894396647686872822352 n 3 2 + 58048211160918355012992 n - 55395831279770476773120 n + 29301459333829598361600 n - 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 17 16 (2 n - 17) (2 n - 23) (2 n - 29)), - 17 (119298469 n - 17417743283 n 15 14 13 + 1170100068268 n - 47979441281180 n + 1343096771182558 n 12 11 10 - 27194802506122706 n + 411642642955906076 n - 4747318892330159860 n 9 8 + 42117716553531438677 n - 288170514560570266339 n 7 6 + 1514579315482593358544 n - 6050270151191330286040 n 5 4 + 18029682941677140615696 n - 38908237844698817466672 n 3 2 + 58026177741614347274112 n - 55358776573994357233920 n + 29285398296782759577600 n - 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), ((1910554800 n + 1910554800) (2 n - 37)! - (n - 19)! (n - 17)! binomial(2 n, n))/((n - 19)! (n - 17)! binomial(2 n, n))] and in Maple notation [19/128*(3449*n^9-139662*n^8+2409186*n^7-23082948*n^6+133912401*n^5-476049798*n ^4+952670644*n^3-538564392*n^2-2126832480*n+4410806400)/(2*n-5)/(-1+2*n)/(2*n-3 )/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17), 323/256*(811*n^10-40505* n^9+875400*n^8-10699770*n^7+80729103*n^6-379481865*n^5+1008858950*n^4-736366180 *n^3-3685056264*n^2+8910364320*n-518918400)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17), 17/128*(7682*n^10-382210*n^ 9+8186925*n^8-97979640*n^7+702591036*n^6-2894235330*n^5+4817652025*n^4+ 9933233740*n^3-52231093668*n^2+44951203440*n-4929724800)/(2*n-19)/(2*n-17)/(2*n -15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 187/128*(1384* n^11-82682*n^10+2143560*n^9-31295955*n^8+276329172*n^7-1429050336*n^6+ 3302995880*n^5+4880750705*n^4-47778182556*n^3+95174174268*n^2-64257233040*n+ 9411292800)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15) /(2*n-17)/(2*n-19)/(2*n-21), 11/128*(23003*n^11-1349419*n^10+33857520*n^9-\ 465676485*n^8+3663528099*n^7-14407616937*n^6+919748710*n^5+237261263485*n^4-\ 924373878852*n^3+1453719811356*n^2-923960353680*n+159991977600)/(-1+2*n)/(2*n-3 )/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21) , 143/1024*(27011*n^12-1843362*n^11+53954947*n^10-868805130*n^9+8045697693*n^8-\ 37414092366*n^7-5210554319*n^6+1071566793090*n^5-5920106581204*n^4+ 15299148532728*n^3-20075325366528*n^2+12023070675840*n-2264501836800)/(2*n-23)/ (2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/( 2*n-3)/(-1+2*n), 143/1024*(24757*n^12-1613454*n^11+43908869*n^10-623953110*n^9+ 4417066491*n^8-4016966562*n^7-191960519833*n^6+1662199295550*n^5-6796385677148* n^4+15397925286216*n^3-19008657625536*n^2+11300019212160*n-2264501836800)/(2*n-\ 23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-\ 5)/(2*n-3)/(-1+2*n), 143/1024*(42614*n^13-3107533*n^12+93881788*n^11-1442314445 *n^10+9577171032*n^9+39315777501*n^8-1366360940516*n^7+12789888107425*n^6-\ 66870534176446*n^5+214420305459932*n^4-421212027079272*n^3+479276509158720*n^2-\ 275962369737600*n+56612545920000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n -11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 143/2048*( 66253*n^13-4317566*n^12+106000301*n^11-897946390*n^10-10776879861*n^9+ 368531225502*n^8-4545212495857*n^7+32717465724350*n^6-150185613509492*n^5+ 446203136671864*n^4-838183727302944*n^3+933760914805440*n^2-537878017555200*n+ 113225091840000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2* n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 143/4096*(170137*n^14-\ 10696301*n^13+181920893*n^12+3479325311*n^11-213808411509*n^10+4553167534377*n^ 9-57290477702161*n^8+475436956740053*n^7-2699527461768368*n^6+10557207385441024 *n^5-28034191056534432*n^4+48724592976074736*n^3-51636644237194560*n^2+ 29006678048140800*n-6114154959360000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-\ 19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n ), 143/2048*(30731*n^14+139937*n^13-150895766*n^12+7859206243*n^11-205391630892 *n^10+3332200716201*n^9-36309303760718*n^8+275205392709289*n^7-1469291921972959 *n^6+5505922784972162*n^5-14205391943845116*n^4+24257327554185768*n^3-\ 25502734717685280*n^2+14342440209350400*n-3057077479680000)/(2*n-27)/(2*n-25)/( 2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/( 2*n-5)/(2*n-3)/(-1+2*n), -13/2048*(584548*n^15-131944725*n^14+10121323555*n^13-\ 413511275100*n^12+10588371635401*n^11-183882833596770*n^10+2259995836061315*n^9 -20107139705076600*n^8+130715264304965879*n^7-620065713618335985*n^6+ 2120740694474264030*n^5-5102062248585987300*n^4+8269908512484963672*n^3-\ 8381278557735383520*n^2+4608898458822201600*n-975207716017920000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-\ 25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), -13/4096*(3620321*n^15-521268075*n^14+ 32216881235*n^13-1153047731325*n^12+26956307758877*n^11-437952266705415*n^10+ 5117567125280005*n^9-43796483974081575*n^8+276336063994702558*n^7-\ 1281448826183957970*n^6+4310336840821204060*n^5-10250863918359552600*n^4+ 16499002616569029744*n^3-16670847577889903040*n^2+9174246971793523200*n-\ 1950415432035840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11) /(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 1/ 16384*(-604477309*n^16+86491652552*n^15-5567741892340*n^14+214420559963120*n^13 -5533192207500958*n^12+101440452070577984*n^11-1365581443402533500*n^10+ 13748817252281185360*n^9-104389692881350976957*n^8+597815293883310278776*n^7-\ 2562641795401177915880*n^6+8089138728349366717120*n^5-18284307596412386701776*n ^4+28279878239220687266688*n^3-27729658978715694401280*n^2+ 14947177728857436518400*n-3144069676441774080000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-17)/(2*n-23)/(2*n-29), 1/16384*(-790570309*n^16+107334068552*n^15-\ 6617306412340*n^14+245866555103120*n^13-6158107761126958*n^12+ 110129254084469984*n^11-1452400274164053500*n^10+14378466665694205360*n^9-\ 107690311749180545957*n^8+610067107522042946776*n^7-2593357124737251115880*n^6+ 8135522819954221357120*n^5-18310692260635052413776*n^4+28247880450787968290688* n^3-27669170921493060161280*n^2+14919901242364303718400*n-\ 3144069676441774080000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), -17/16384*(109063354*n^17-16169059253*n^16+1100787869488*n^15-\ 45660983031380*n^14+1290936762351628*n^13-26360864536998446*n^12+ 401886277056709016*n^11-4662645523038877660*n^10+41571543918911459882*n^9-\ 285577389951988368949*n^8+1505739952594509590504*n^7-6029735276662704628840*n^6 +18000983829444228301536*n^5-38894396647686872822352*n^4+ 58048211160918355012992*n^3-55395831279770476773120*n^2+29301459333829598361600 *n-6103194077798737920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2 *n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/ (2*n-23)/(2*n-29), -17/16384*(119298469*n^17-17417743283*n^16+1170100068268*n^ 15-47979441281180*n^14+1343096771182558*n^13-27194802506122706*n^12+ 411642642955906076*n^11-4747318892330159860*n^10+42117716553531438677*n^9-\ 288170514560570266339*n^8+1514579315482593358544*n^7-6050270151191330286040*n^6 +18029682941677140615696*n^5-38908237844698817466672*n^4+ 58026177741614347274112*n^3-55358776573994357233920*n^2+29285398296782759577600 *n-6103194077798737920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2 *n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/ (2*n-23)/(2*n-29), ((1910554800*n+1910554800)*(2*n-37)!-(n-19)!*(n-17)!* binomial(2*n,n))/(n-19)!/(n-17)!/binomial(2*n,n)] The limits, as n goes to infinity are 65531 261953 65297 32351 253033 3862573 3540251 3046901 9474179 [-----, ------, -----, -----, ------, -------, -------, -------, --------, 65536 262144 65536 32768 262144 4194304 4194304 4194304 16777216 24329591 4394533 -1899781 -47064173 -604477309 -790570309 -927038509 --------, --------, --------, ---------, ----------, ----------, ----------, 67108864 33554432 16777216 134217728 1073741824 1073741824 1073741824 -2028073973 -8470524917 -----------, -----------] 2147483648 8589934592 and in Maple notation [65531/65536, 261953/262144, 65297/65536, 32351/32768, 253033/262144, 3862573/ 4194304, 3540251/4194304, 3046901/4194304, 9474179/16777216, 24329591/67108864, 4394533/33554432, -1899781/16777216, -47064173/134217728, -604477309/1073741824 , -790570309/1073741824, -927038509/1073741824, -2028073973/2147483648, -\ 8470524917/8589934592] and in floating point [.9999237061, .9992713928, .9963531494, .9872741699, .9652442932, .9209091663, .8440616131, .7264378071, .5647050738, .3625391573, .1309672892, -.1132357717, -.3506554142, -.5629633637, -.7362759756, -.8633718910, -.9443955370, -.9860988\ 843] The cut off is at j=, 12 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 20], vs. those in the, 2, -th row from j=1 to j=, 19, are as follws 10 9 8 7 6 [19 (27593 n - 1379545 n + 29894700 n - 367982250 n + 2830092909 n 5 4 3 2 - 14045927385 n + 44328924550 n - 79582319300 n + 32205575448 n + 191035349280 n - 352864512000)/(512 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17)) 10 9 8 7 6 , 19 (27583 n - 1378205 n + 29815350 n - 365259450 n + 2770523679 n 5 4 3 2 - 13186720365 n + 36207807500 n - 31807935100 n - 118402689312 n + 318831289920 n - 17643225600)/(512 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17)) 11 10 9 8 7 , 3553 (589 n - 35540 n + 939285 n - 14234340 n + 135278787 n 6 5 4 3 - 817485900 n + 2915424455 n - 3984084460 n - 10610554476 n 2 + 46329312240 n - 38105907840 n + 3962649600)/(1024 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) 11 10 9 (2 n - 19) (2 n - 21)), 187 (1391 n - 83431 n + 2178000 n 8 7 6 5 - 32180685 n + 290116743 n - 1560866853 n + 4039411630 n 4 3 2 + 2823570685 n - 46282918884 n + 97364056284 n - 66502734480 n + 9411292800)/(128 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), 187 ( 12 11 10 9 8 10967 n - 775134 n + 23980219 n - 422422410 n + 4577815821 n 7 6 5 4 - 30120726762 n + 101373663337 n + 20091518730 n - 1563410587288 n 3 2 + 5640003925896 n - 8480890079856 n + 5212257785280 n - 865838937600)/( 512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 11 ( 12 11 10 9 8 180643 n - 12499146 n + 373310831 n - 6198150390 n + 60461455209 n 7 6 5 4 - 318335133438 n + 386318819333 n + 5636261024550 n - 36509064811952 n 3 2 + 99222217767384 n - 132889617299664 n + 79776825186240 n - 14719261939200)/(512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 13 12 11 10 (-1 + 2 n)), 143 (104663 n - 8294611 n + 284297221 n - 5428961915 n 9 8 7 6 + 60998149719 n - 365771895333 n + 317345235103 n + 12435867348775 n 5 4 3 - 100721903465482 n + 390210554419244 n - 847362854746824 n 2 + 1010773933547040 n - 583097111251200 n + 113225091840000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), 143 ( 13 12 11 10 9 47099 n - 3556033 n + 113207653 n - 1911893945 n + 16651823487 n 8 7 6 5 - 29205962799 n - 937954067921 n + 11102797859725 n - 62983641684886 n 4 3 2 + 210316435736732 n - 422214286154232 n + 485136460897920 n - 279282504009600 n + 56612545920000)/(1024 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 14 13 (2 n - 21) (2 n - 23) (2 n - 25)), 143 (632489 n - 52659607 n 12 11 10 9 + 1827546721 n - 32450340923 n + 247488368127 n + 1415294419539 n 8 7 6 - 55417415456117 n + 645567006755671 n - 4376365999000996 n 5 4 3 + 19074309906419768 n - 54262002870579504 n + 98288091581267952 n 2 - 106162196750550720 n + 59527592719257600 n - 12228309918720000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 14 13 12 (-1 + 2 n)), 143 (473099 n - 34807927 n + 955045861 n 11 10 9 8 - 8000233703 n - 186869783793 n + 6525938082279 n - 95846574514097 n 7 6 5 + 859239233984131 n - 5106874900875286 n + 20558745878711348 n 4 3 2 - 55608101986100664 n + 97705776679342272 n - 104044724343285120 n + 58406664310041600 n - 12228309918720000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 143 ( 15 14 13 12 1122011 n - 74225250 n + 931690760 n + 64518623400 n 11 10 9 - 3328267460518 n + 78438826033260 n - 1154909863901420 n 8 7 6 + 11592068172933600 n - 82056160986473597 n + 413854845837493830 n 5 4 3 - 1478744005563287540 n + 3665327665685385000 n - 6050363318598748896 n 2 + 6180927248758556160 n - 3393157204500940800 n + 709241975285760000)/( 16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 15 14 (2 n - 17) (2 n - 23) (2 n - 29)), 13 (721849 n + 65340225 n 13 12 11 - 11005380785 n + 575517410475 n - 16718222094287 n 10 9 8 + 313621718998665 n - 4059004893972655 n + 37450864460837025 n 7 6 5 - 249931972741679998 n + 1208258005191622770 n - 4188364326075320860 n 4 3 + 10168068372255843000 n - 16571302425736334064 n 2 + 16833304473033147840 n - 9251392590157939200 n + 1950415432035840000)/( 4096 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 16 15 (2 n - 17) (2 n - 23) (2 n - 29)), - 13 (13764973 n - 2856143384 n 14 13 12 + 230578817380 n - 10381276233440 n + 300560163576526 n 11 10 9 - 6016877080683728 n + 86754924177425900 n - 921985881125684320 n 8 7 + 7304731854098241629 n - 43244537158100239192 n 6 5 + 190137598763564924360 n - 611502953379626237440 n 4 3 + 1400164567616229505872 n - 2182595246863621485696 n 2 + 2147103675559986543360 n - 1156180054265713612800 n + 241851513572444160000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 16 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), (-422106169 n 15 14 13 + 66066084872 n - 4539168662740 n + 183603484725920 n 12 11 10 - 4920774964947478 n + 92925426096963824 n - 1280498989256243900 n 9 8 + 13131760827136425760 n - 101155086390877999337 n 7 6 + 585808516517352264136 n - 2532540772651826179880 n 5 4 + 8043682318576609169920 n - 18258450625474174304016 n 3 2 + 28311236071884751863168 n - 27788937274793875956480 n + 14973908685620706662400 n - 3144069676441774080000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 (2 n - 23) (2 n - 29)), - 17 (74946304 n - 11904428003 n 15 14 13 + 858693282688 n - 37394421816380 n + 1101422622436528 n 12 11 10 - 23278535431642946 n + 365261179950654416 n - 4340291583931534660 n 9 8 + 39465766024508341232 n - 275464749149893573699 n 7 6 + 1470908653043916645104 n - 5948026658284173484840 n 5 4 + 17885597889345169362336 n - 38837379459168851354352 n 3 2 + 58135562033840484849792 n - 55544995172008162725120 n + 29366468293304898201600 n - 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 17 16 (2 n - 17) (2 n - 23) (2 n - 29)), - 17 (95416534 n - 14504147213 n 15 14 13 + 1008371604448 n - 42569705364980 n + 1221390083910388 n 12 11 10 - 25248947244832766 n + 388877789191112936 n - 4549747697317168060 n 9 8 + 40843313739418154822 n - 282119890473879172429 n 7 6 + 1493954135410397899784 n - 6002355443957870419240 n 5 4 + 17962718346467011882656 n - 38875941718337613296592 n 3 2 + 58077589053323698664832 n - 55445237554138636158720 n + 29322874049892050073600 n - 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), - 17 (882741947 n - 146228256639 n 16 15 14 + 11182987391154 n - 524142551186460 n + 16853337770780154 n 13 12 - 394277601303422298 n + 6944555446452000728 n 11 10 - 93991389148013462820 n + 988852465136549407251 n 9 8 - 8126344435083792490287 n + 52126352938222898553582 n 7 6 - 259371388152144537239880 n + 988918229089067284766048 n 5 4 - 2832311219428850820433776 n + 5910063628439623090045536 n 3 2 - 8569107988385317125603840 n + 7988515078099292512473600 n - 4150467723061894316544000 n + 854447170891823308800000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 18 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), (-16296237589 n 17 16 15 + 2663848542483 n - 201357600826908 n + 9341880139544220 n 14 13 - 297733957567553598 n + 6912414841492620306 n 12 11 - 120957879721809925756 n + 1628057575652076365940 n 10 9 - 17048730190322221656837 n + 139566724261051853161539 n 8 7 - 892444125242763262251264 n + 4429557672810259000316760 n 6 5 - 16856225570418777608069776 n + 48208306895959304869626672 n 4 3 - 100495895298885203012883072 n + 145626168048614174419534080 n 2 - 135728789761120157899315200 n + 70525909523143760007168000 n - 14525601905160996249600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), ((7069052760 n + 7069052760) (2 n - 39)! - (n - 20)! (n - 18)! binomial(2 n, n))/((n - 20)! (n - 18)! binomial(2 n, n))] and in Maple notation [19/512*(27593*n^10-1379545*n^9+29894700*n^8-367982250*n^7+2830092909*n^6-\ 14045927385*n^5+44328924550*n^4-79582319300*n^3+32205575448*n^2+191035349280*n-\ 352864512000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-\ 13)/(2*n-15)/(2*n-17), 19/512*(27583*n^10-1378205*n^9+29815350*n^8-365259450*n^ 7+2770523679*n^6-13186720365*n^5+36207807500*n^4-31807935100*n^3-118402689312*n ^2+318831289920*n-17643225600)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17), 3553/1024*(589*n^11-35540*n^10+939285*n^ 9-14234340*n^8+135278787*n^7-817485900*n^6+2915424455*n^5-3984084460*n^4-\ 10610554476*n^3+46329312240*n^2-38105907840*n+3962649600)/(-1+2*n)/(2*n-3)/(2*n -5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), 187/ 128*(1391*n^11-83431*n^10+2178000*n^9-32180685*n^8+290116743*n^7-1560866853*n^6 +4039411630*n^5+2823570685*n^4-46282918884*n^3+97364056284*n^2-66502734480*n+ 9411292800)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15) /(2*n-17)/(2*n-19)/(2*n-21), 187/512*(10967*n^12-775134*n^11+23980219*n^10-\ 422422410*n^9+4577815821*n^8-30120726762*n^7+101373663337*n^6+20091518730*n^5-\ 1563410587288*n^4+5640003925896*n^3-8480890079856*n^2+5212257785280*n-\ 865838937600)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2 *n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 11/512*(180643*n^12-12499146*n^11+ 373310831*n^10-6198150390*n^9+60461455209*n^8-318335133438*n^7+386318819333*n^6 +5636261024550*n^5-36509064811952*n^4+99222217767384*n^3-132889617299664*n^2+ 79776825186240*n-14719261939200)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/( 2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 143/2048*(104663*n^ 13-8294611*n^12+284297221*n^11-5428961915*n^10+60998149719*n^9-365771895333*n^8 +317345235103*n^7+12435867348775*n^6-100721903465482*n^5+390210554419244*n^4-\ 847362854746824*n^3+1010773933547040*n^2-583097111251200*n+113225091840000)/(-1 +2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-\ 19)/(2*n-21)/(2*n-23)/(2*n-25), 143/1024*(47099*n^13-3556033*n^12+113207653*n^ 11-1911893945*n^10+16651823487*n^9-29205962799*n^8-937954067921*n^7+ 11102797859725*n^6-62983641684886*n^5+210316435736732*n^4-422214286154232*n^3+ 485136460897920*n^2-279282504009600*n+56612545920000)/(-1+2*n)/(2*n-3)/(2*n-5)/ (2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/ (2*n-25), 143/8192*(632489*n^14-52659607*n^13+1827546721*n^12-32450340923*n^11+ 247488368127*n^10+1415294419539*n^9-55417415456117*n^8+645567006755671*n^7-\ 4376365999000996*n^6+19074309906419768*n^5-54262002870579504*n^4+ 98288091581267952*n^3-106162196750550720*n^2+59527592719257600*n-\ 12228309918720000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 143/8192*( 473099*n^14-34807927*n^13+955045861*n^12-8000233703*n^11-186869783793*n^10+ 6525938082279*n^9-95846574514097*n^8+859239233984131*n^7-5106874900875286*n^6+ 20558745878711348*n^5-55608101986100664*n^4+97705776679342272*n^3-\ 104044724343285120*n^2+58406664310041600*n-12228309918720000)/(2*n-27)/(2*n-25) /(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7) /(2*n-5)/(2*n-3)/(-1+2*n), 143/16384*(1122011*n^15-74225250*n^14+931690760*n^13 +64518623400*n^12-3328267460518*n^11+78438826033260*n^10-1154909863901420*n^9+ 11592068172933600*n^8-82056160986473597*n^7+413854845837493830*n^6-\ 1478744005563287540*n^5+3665327665685385000*n^4-6050363318598748896*n^3+ 6180927248758556160*n^2-3393157204500940800*n+709241975285760000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-\ 25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 13/4096*(721849*n^15+65340225*n^14-\ 11005380785*n^13+575517410475*n^12-16718222094287*n^11+313621718998665*n^10-\ 4059004893972655*n^9+37450864460837025*n^8-249931972741679998*n^7+ 1208258005191622770*n^6-4188364326075320860*n^5+10168068372255843000*n^4-\ 16571302425736334064*n^3+16833304473033147840*n^2-9251392590157939200*n+ 1950415432035840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11) /(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), -13/ 16384*(13764973*n^16-2856143384*n^15+230578817380*n^14-10381276233440*n^13+ 300560163576526*n^12-6016877080683728*n^11+86754924177425900*n^10-\ 921985881125684320*n^9+7304731854098241629*n^8-43244537158100239192*n^7+ 190137598763564924360*n^6-611502953379626237440*n^5+1400164567616229505872*n^4-\ 2182595246863621485696*n^3+2147103675559986543360*n^2-1156180054265713612800*n+ 241851513572444160000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 1/16384*(-422106169*n^16+66066084872*n^15-4539168662740*n^14+ 183603484725920*n^13-4920774964947478*n^12+92925426096963824*n^11-\ 1280498989256243900*n^10+13131760827136425760*n^9-101155086390877999337*n^8+ 585808516517352264136*n^7-2532540772651826179880*n^6+8043682318576609169920*n^5 -18258450625474174304016*n^4+28311236071884751863168*n^3-\ 27788937274793875956480*n^2+14973908685620706662400*n-3144069676441774080000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2* n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), -17/16384*( 74946304*n^17-11904428003*n^16+858693282688*n^15-37394421816380*n^14+ 1101422622436528*n^13-23278535431642946*n^12+365261179950654416*n^11-\ 4340291583931534660*n^10+39465766024508341232*n^9-275464749149893573699*n^8+ 1470908653043916645104*n^7-5948026658284173484840*n^6+17885597889345169362336*n ^5-38837379459168851354352*n^4+58135562033840484849792*n^3-\ 55544995172008162725120*n^2+29366468293304898201600*n-6103194077798737920000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2* n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), -17/16384 *(95416534*n^17-14504147213*n^16+1008371604448*n^15-42569705364980*n^14+ 1221390083910388*n^13-25248947244832766*n^12+388877789191112936*n^11-\ 4549747697317168060*n^10+40843313739418154822*n^9-282119890473879172429*n^8+ 1493954135410397899784*n^7-6002355443957870419240*n^6+17962718346467011882656*n ^5-38875941718337613296592*n^4+58077589053323698664832*n^3-\ 55445237554138636158720*n^2+29322874049892050073600*n-6103194077798737920000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2* n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), -17/65536 *(882741947*n^18-146228256639*n^17+11182987391154*n^16-524142551186460*n^15+ 16853337770780154*n^14-394277601303422298*n^13+6944555446452000728*n^12-\ 93991389148013462820*n^11+988852465136549407251*n^10-8126344435083792490287*n^9 +52126352938222898553582*n^8-259371388152144537239880*n^7+ 988918229089067284766048*n^6-2832311219428850820433776*n^5+ 5910063628439623090045536*n^4-8569107988385317125603840*n^3+ 7988515078099292512473600*n^2-4150467723061894316544000*n+ 854447170891823308800000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25) /(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19) /(2*n-3)/(-1+2*n)/(2*n-5), 1/65536*(-16296237589*n^18+2663848542483*n^17-\ 201357600826908*n^16+9341880139544220*n^15-297733957567553598*n^14+ 6912414841492620306*n^13-120957879721809925756*n^12+1628057575652076365940*n^11 -17048730190322221656837*n^10+139566724261051853161539*n^9-\ 892444125242763262251264*n^8+4429557672810259000316760*n^7-\ 16856225570418777608069776*n^6+48208306895959304869626672*n^5-\ 100495895298885203012883072*n^4+145626168048614174419534080*n^3-\ 135728789761120157899315200*n^2+70525909523143760007168000*n-\ 14525601905160996249600000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-\ 25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-\ 19)/(2*n-3)/(-1+2*n)/(2*n-5), ((7069052760*n+7069052760)*(2*n-39)!-(n-20)!*(n-\ 18)!*binomial(2*n,n))/(n-20)!/(n-18)!/binomial(2*n,n)] The limits, as n goes to infinity are 524267 524077 2092717 260117 2050829 1987073 14966809 6735157 [------, ------, -------, ------, -------, -------, --------, -------, 524288 524288 2097152 262144 2097152 2097152 16777216 8388608 90445927 67653157 160447573 9384037 -178944649 -422106169 ---------, ---------, ---------, ---------, ----------, ----------, 134217728 134217728 536870912 134217728 1073741824 1073741824 -4976903 -811040539 -15006613099 -16296237589 -67835845141 --------, ----------, ------------, ------------, ------------] 8388608 1073741824 17179869184 17179869184 68719476736 and in Maple notation [524267/524288, 524077/524288, 2092717/2097152, 260117/262144, 2050829/2097152, 1987073/2097152, 14966809/16777216, 6735157/8388608, 90445927/134217728, 67653157/134217728, 160447573/536870912, 9384037/134217728, -178944649/ 1073741824, -422106169/1073741824, -4976903/8388608, -811040539/1073741824, -\ 15006613099/17179869184, -16296237589/17179869184, -67835845141/68719476736] and in floating point [.9999599457, .9995975494, .9978852272, .9922676086, .9779114723, .9475102425, .8920913339, .8028932810, .6738746688, .5040552989, .2988568936, .6991652399e-1 , -.1666551912, -.3931170041, -.5932930708, -.7553403629, -.8734998467, -.94856\ 58717, -.9871414679] The cut off is at j=, 13 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 21], vs. those in the, 2, -th row from j=1 to j=, 20, are as follws 10 9 8 7 6 [3 (174759 n - 8737565 n + 189359550 n - 2331461850 n + 17943778167 n 5 4 3 2 - 89243942445 n + 283456894500 n - 519946150300 n + 254171399424 n + 1167291898560 n - 2346549004800)/(512 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17)) 11 10 9 8 7 , 627 (1672 n - 101121 n + 2686435 n - 41192850 n + 402153486 n 6 5 4 3 - 2588292273 n + 10832056355 n - 26416794300 n + 17629379492 n 2 + 90797670144 n - 214120146240 n + 11227507200)/(512 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) 11 10 9 (2 n - 15) (2 n - 17)), 209 (5011 n - 302648 n + 8014095 n 8 7 6 5 - 121920960 n + 1167681333 n - 7166151384 n + 26429520125 n 4 3 2 - 40885158640 n - 78142369044 n + 401038045632 n - 337613814720 n + 33682521600)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21)), 3553 ( 12 11 10 9 8 1175 n - 84186 n + 2667319 n - 48990210 n + 571827225 n 7 6 5 4 - 4328257878 n + 20198757517 n - 45066982950 n - 45265486900 n 3 2 + 516601391064 n - 1021647260736 n + 674086008960 n - 91140940800)/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 187 ( 12 11 10 9 8 11059 n - 785898 n + 24525503 n - 438026070 n + 4853393817 n 7 6 5 4 - 33195603294 n + 122596416629 n - 63951421050 n - 1405774320176 n 3 2 + 5586953909592 n - 8660025223632 n + 5352441232320 n - 865838937600)/( 512 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 2431 ( 13 12 11 10 9 1666 n - 137327 n + 4991972 n - 104357155 n + 1363014108 n 8 7 6 5 - 11137989981 n + 50883191996 n - 50024863825 n - 800402955974 n 4 3 2 + 4795934047408 n - 12415632622968 n + 16045467803280 n - 9348027998400 n + 1665074880000)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 13 12 (2 n - 21) (2 n - 23) (2 n - 25)), 143 (27172 n - 2187359 n 11 10 9 8 + 76639874 n - 1511584135 n + 17912470386 n - 120996817077 n 7 6 5 + 290226055982 n + 2169569878475 n - 22753035490958 n 4 3 2 + 94627415811136 n - 212122856228856 n + 256587811439760 n - 148133133892800 n + 28306272960000)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 14 13 (2 n - 21) (2 n - 23) (2 n - 25)), 143 (201851 n - 18368623 n 12 11 10 9 + 728398489 n - 16271101847 n + 217839555843 n - 1624842730329 n 8 7 6 + 3016265534347 n + 65580248625619 n - 743660829571114 n 5 4 3 + 4008517912742252 n - 12815030877402936 n + 24814825918410528 n 2 - 27658476556476480 n + 15496953053702400 n - 3057077479680000)/(2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 14 13 12 (-1 + 2 n)), 143 (713219 n - 61701367 n + 2269462741 n 11 10 9 - 44834161463 n + 467487951567 n - 1173213409641 n 8 7 6 - 34940308920257 n + 537343411146451 n - 4006367983765966 n 5 4 3 + 18322452725648708 n - 53580212409471384 n + 98583030297827712 n 2 - 107234682775009920 n + 60095335679769600 n - 12228309918720000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 15 14 13 (-1 + 2 n)), 143 (1166308 n - 109748925 n + 4321622305 n 12 11 10 - 87100288275 n + 728210604121 n + 6614920967655 n 9 8 7 - 268654165679185 n + 3756312010254375 n - 31762320212463841 n 6 5 4 + 179096544433696890 n - 689161452553794220 n + 1793979469877267400 n 3 2 - 3050914294152137088 n + 3159051311830250880 n - 1730633474873318400 n + 354620987642880000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 15 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 143 (836143 n 14 13 12 11 - 68478300 n + 2045134630 n - 14079346050 n - 783495080984 n 10 9 8 + 27821905911030 n - 474671540938210 n + 5147362691356050 n 7 6 5 - 38202124616859511 n + 198760978786221390 n - 725192093590754020 n 4 3 2 + 1822371872971461000 n - 3033003497442363648 n + 3109582143655274880 n - 1705866294783974400 n + 354620987642880000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 143 ( 16 15 14 13 1793797 n - 117950216 n - 11022380 n + 256094764240 n 12 11 10 - 12448291038386 n + 323376806451328 n - 5487778789665700 n 9 8 7 + 65261725517444720 n - 560996057538124219 n + 3528166988928758792 n 6 5 - 16224453158578452760 n + 53907869013487106240 n 4 3 - 126235167957223830192 n + 199495052007645972096 n 2 - 197410030736734460160 n + 106135217977946572800 n - 21986501233858560000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 16 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 13 (1122467 n 15 14 13 + 1084538024 n - 136192447780 n + 7402894987040 n 12 11 10 - 238509581630446 n + 5116508261413808 n - 77406864486627500 n 9 8 7 + 851826021584713120 n - 6925533550775740109 n + 41797831027770699112 n 6 5 - 186418089249759893960 n + 605739039626066967040 n 4 3 - 1396716940281748440912 n + 2186421908333559901056 n 2 - 2154659420916269141760 n + 1159635075888177100800 n - 241851513572444160000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 17 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), - 17 (27182434 n 16 15 14 - 5647361033 n + 487663540528 n - 24204906754580 n 13 12 11 + 787675994039788 n - 18000904731561206 n + 300605288900784296 n 10 9 - 3755295069808513660 n + 35548016875687924322 n 8 7 - 256224098970503324089 n + 1403277040669438212824 n 6 5 - 5786332985376677155240 n + 17652626072135010701856 n 4 3 - 38716923012340716102672 n + 58307129850117853844352 n 2 - 55846976229263779787520 n + 29499486625769742489600 n - 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), - 17 ( 17 16 15 14 54476074 n - 9304708793 n + 709014960928 n - 32219138267780 n 13 12 11 + 981455160962668 n - 21308123618453126 n + 341644570710195896 n 10 9 - 4130835470545901260 n + 38088218309598527642 n 8 7 - 268809607825907974969 n + 1447863170677435390424 n 6 5 - 5893697872610476550440 n + 17808477432223326842016 n 4 3 - 38798817200000089412112 n + 58193535014357271034752 n 2 - 55644752789877689291520 n + 29410062536717746329600 n - 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), - 17 ( 18 17 16 15 156716018 n - 27537369066 n + 2214264376851 n - 108314450130240 n 14 13 12 + 3612017470337976 n - 87163286970156012 n + 1576131440909512682 n 11 10 - 21810070278403201080 n + 233744574714422723094 n 9 8 - 1950521079606188781978 n + 12668408669236100773083 n 7 6 - 63664829729272276497720 n + 244611671470536822435512 n 5 4 - 704583277798229713774944 n + 1476009973835387394366384 n 3 2 - 2145125057549780947512960 n + 2001629397317280065798400 n - 1039524178914785684736000 n + 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 18 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 17 (780390797 n 17 16 15 - 132103797939 n + 10290383012004 n - 489899950442460 n 14 13 12 + 15962288924407854 n - 377635091832434898 n + 6714362023267404428 n 11 10 - 91594249454692232820 n + 969944664898384190301 n 9 8 - 8013735795034000361187 n + 51626660454653534518632 n 7 6 - 257764715594699944601880 n + 985377302429969366746448 n 5 4 - 2827627396756720588508976 n + 5908094293883146067655936 n 3 2 - 8572970508538985118915840 n + 7994544170684039719833600 n - 4153010720594310457344000 n + 854447170891823308800000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 19 (1595804192 n 18 17 16 - 293723145481 n + 25076040672543 n - 1318776457773324 n 15 14 + 47851222322342604 n - 1271316939605124342 n 13 12 + 25613336590529160026 n - 399819762942327372268 n 11 10 + 4898124920811384217956 n - 47406342268964433542073 n 9 8 + 363040278213869423455119 n - 2192967846245455296955392 n 7 6 + 10366146651083202375937208 n - 37828170625514763663607504 n 5 4 + 104367985931998433575011312 n - 210999300129943771127078016 n 3 2 + 297954295280785738344695040 n - 271853422549423958351001600 n + 138916480467902911893504000 n - 28286698446892466380800000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 19 18 17 - 19 (1722037277 n - 313289273656 n + 26472683524983 n 16 15 14 - 1379678871962424 n + 49665689364613674 n - 1310444809355420592 n 13 12 + 26244406985747975006 n - 407574729448878507568 n 11 10 + 4971382850880966366561 n - 47938903049909097417048 n 9 8 + 366000012884529668295339 n - 2205343673200556699347992 n 7 6 + 10403921310215078748289448 n - 37907546062973486662823104 n 5 4 + 104467728914335578860058672 n - 211035406826256798179846016 n 3 2 + 297866566980117108719735040 n - 271725024233760863951001600 n + 138863657501615531003904000 n - 28286698446892466380800000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), ((26256481680 n + 26256481680) (2 n - 41)! - (n - 21)! (n - 19)! binomial(2 n, n))/((n - 21)! (n - 19)! binomial(2 n, n))] and in Maple notation [3/512*(174759*n^10-8737565*n^9+189359550*n^8-2331461850*n^7+17943778167*n^6-\ 89243942445*n^5+283456894500*n^4-519946150300*n^3+254171399424*n^2+ 1167291898560*n-2346549004800)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17), 627/512*(1672*n^11-101121*n^10+2686435*n ^9-41192850*n^8+402153486*n^7-2588292273*n^6+10832056355*n^5-26416794300*n^4+ 17629379492*n^3+90797670144*n^2-214120146240*n+11227507200)/(2*n-5)/(-1+2*n)/(2 *n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-15)/(2*n-17), 209/512*(5011*n^11-302648*n^10+8014095*n^9-121920960*n^8+1167681333*n^7-\ 7166151384*n^6+26429520125*n^5-40885158640*n^4-78142369044*n^3+401038045632*n^2 -337613814720*n+33682521600)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/ (2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21), 3553/1024*(1175*n^12-84186*n^11+ 2667319*n^10-48990210*n^9+571827225*n^8-4328257878*n^7+20198757517*n^6-\ 45066982950*n^5-45265486900*n^4+516601391064*n^3-1021647260736*n^2+674086008960 *n-91140940800)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/ (2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 187/512*(11059*n^12-785898*n^11+ 24525503*n^10-438026070*n^9+4853393817*n^8-33195603294*n^7+122596416629*n^6-\ 63951421050*n^5-1405774320176*n^4+5586953909592*n^3-8660025223632*n^2+ 5352441232320*n-865838937600)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n -13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 2431/512*(1666*n^13-\ 137327*n^12+4991972*n^11-104357155*n^10+1363014108*n^9-11137989981*n^8+ 50883191996*n^7-50024863825*n^6-800402955974*n^5+4795934047408*n^4-\ 12415632622968*n^3+16045467803280*n^2-9348027998400*n+1665074880000)/(-1+2*n)/( 2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2* n-21)/(2*n-23)/(2*n-25), 143/512*(27172*n^13-2187359*n^12+76639874*n^11-\ 1511584135*n^10+17912470386*n^9-120996817077*n^8+290226055982*n^7+2169569878475 *n^6-22753035490958*n^5+94627415811136*n^4-212122856228856*n^3+256587811439760* n^2-148133133892800*n+28306272960000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/ (2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 143/ 2048*(201851*n^14-18368623*n^13+728398489*n^12-16271101847*n^11+217839555843*n^ 10-1624842730329*n^9+3016265534347*n^8+65580248625619*n^7-743660829571114*n^6+ 4008517912742252*n^5-12815030877402936*n^4+24814825918410528*n^3-\ 27658476556476480*n^2+15496953053702400*n-3057077479680000)/(2*n-27)/(2*n-25)/( 2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/( 2*n-5)/(2*n-3)/(-1+2*n), 143/8192*(713219*n^14-61701367*n^13+2269462741*n^12-\ 44834161463*n^11+467487951567*n^10-1173213409641*n^9-34940308920257*n^8+ 537343411146451*n^7-4006367983765966*n^6+18322452725648708*n^5-\ 53580212409471384*n^4+98583030297827712*n^3-107234682775009920*n^2+ 60095335679769600*n-12228309918720000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n -19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2* n), 143/8192*(1166308*n^15-109748925*n^14+4321622305*n^13-87100288275*n^12+ 728210604121*n^11+6614920967655*n^10-268654165679185*n^9+3756312010254375*n^8-\ 31762320212463841*n^7+179096544433696890*n^6-689161452553794220*n^5+ 1793979469877267400*n^4-3050914294152137088*n^3+3159051311830250880*n^2-\ 1730633474873318400*n+354620987642880000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 143/8192*(836143*n^15-68478300*n^14+2045134630*n^13-14079346050 *n^12-783495080984*n^11+27821905911030*n^10-474671540938210*n^9+ 5147362691356050*n^8-38202124616859511*n^7+198760978786221390*n^6-\ 725192093590754020*n^5+1822371872971461000*n^4-3033003497442363648*n^3+ 3109582143655274880*n^2-1705866294783974400*n+354620987642880000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-\ 25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 143/16384*(1793797*n^16-117950216*n^15 -11022380*n^14+256094764240*n^13-12448291038386*n^12+323376806451328*n^11-\ 5487778789665700*n^10+65261725517444720*n^9-560996057538124219*n^8+ 3528166988928758792*n^7-16224453158578452760*n^6+53907869013487106240*n^5-\ 126235167957223830192*n^4+199495052007645972096*n^3-197410030736734460160*n^2+ 106135217977946572800*n-21986501233858560000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-17)/(2*n-23)/(2*n-29), 13/16384*(1122467*n^16+1084538024*n^15-\ 136192447780*n^14+7402894987040*n^13-238509581630446*n^12+5116508261413808*n^11 -77406864486627500*n^10+851826021584713120*n^9-6925533550775740109*n^8+ 41797831027770699112*n^7-186418089249759893960*n^6+605739039626066967040*n^5-\ 1396716940281748440912*n^4+2186421908333559901056*n^3-2154659420916269141760*n^ 2+1159635075888177100800*n-241851513572444160000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-17)/(2*n-23)/(2*n-29), -17/16384*(27182434*n^17-5647361033*n^16+ 487663540528*n^15-24204906754580*n^14+787675994039788*n^13-18000904731561206*n^ 12+300605288900784296*n^11-3755295069808513660*n^10+35548016875687924322*n^9-\ 256224098970503324089*n^8+1403277040669438212824*n^7-5786332985376677155240*n^6 +17652626072135010701856*n^5-38716923012340716102672*n^4+ 58307129850117853844352*n^3-55846976229263779787520*n^2+29499486625769742489600 *n-6103194077798737920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2 *n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/ (2*n-23)/(2*n-29), -17/16384*(54476074*n^17-9304708793*n^16+709014960928*n^15-\ 32219138267780*n^14+981455160962668*n^13-21308123618453126*n^12+ 341644570710195896*n^11-4130835470545901260*n^10+38088218309598527642*n^9-\ 268809607825907974969*n^8+1447863170677435390424*n^7-5893697872610476550440*n^6 +17808477432223326842016*n^5-38798817200000089412112*n^4+ 58193535014357271034752*n^3-55644752789877689291520*n^2+29410062536717746329600 *n-6103194077798737920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2 *n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/ (2*n-23)/(2*n-29), -17/16384*(156716018*n^18-27537369066*n^17+2214264376851*n^ 16-108314450130240*n^15+3612017470337976*n^14-87163286970156012*n^13+ 1576131440909512682*n^12-21810070278403201080*n^11+233744574714422723094*n^10-\ 1950521079606188781978*n^9+12668408669236100773083*n^8-63664829729272276497720* n^7+244611671470536822435512*n^6-704583277798229713774944*n^5+ 1476009973835387394366384*n^4-2145125057549780947512960*n^3+ 2001629397317280065798400*n^2-1039524178914785684736000*n+ 213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25) /(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19) /(2*n-3)/(-1+2*n)/(2*n-5), -17/65536*(780390797*n^18-132103797939*n^17+ 10290383012004*n^16-489899950442460*n^15+15962288924407854*n^14-\ 377635091832434898*n^13+6714362023267404428*n^12-91594249454692232820*n^11+ 969944664898384190301*n^10-8013735795034000361187*n^9+51626660454653534518632*n ^8-257764715594699944601880*n^7+985377302429969366746448*n^6-\ 2827627396756720588508976*n^5+5908094293883146067655936*n^4-\ 8572970508538985118915840*n^3+7994544170684039719833600*n^2-\ 4153010720594310457344000*n+854447170891823308800000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-\ 33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/65536*(1595804192*n^ 19-293723145481*n^18+25076040672543*n^17-1318776457773324*n^16+ 47851222322342604*n^15-1271316939605124342*n^14+25613336590529160026*n^13-\ 399819762942327372268*n^12+4898124920811384217956*n^11-47406342268964433542073* n^10+363040278213869423455119*n^9-2192967846245455296955392*n^8+ 10366146651083202375937208*n^7-37828170625514763663607504*n^6+ 104367985931998433575011312*n^5-210999300129943771127078016*n^4+ 297954295280785738344695040*n^3-271853422549423958351001600*n^2+ 138916480467902911893504000*n-28286698446892466380800000)/(2*n-29)/(2*n-35)/(2* n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/( 2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/65536*( 1722037277*n^19-313289273656*n^18+26472683524983*n^17-1379678871962424*n^16+ 49665689364613674*n^15-1310444809355420592*n^14+26244406985747975006*n^13-\ 407574729448878507568*n^12+4971382850880966366561*n^11-47938903049909097417048* n^10+366000012884529668295339*n^9-2205343673200556699347992*n^8+ 10403921310215078748289448*n^7-37907546062973486662823104*n^6+ 104467728914335578860058672*n^5-211035406826256798179846016*n^4+ 297866566980117108719735040*n^3-271725024233760863951001600*n^2+ 138863657501615531003904000*n-28286698446892466380800000)/(2*n-29)/(2*n-35)/(2* n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/( 2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), (( 26256481680*n+26256481680)*(2*n-41)!-(n-21)!*(n-19)!*binomial(2*n,n))/(n-21)!/( n-19)!/binomial(2*n,n)] The limits, as n goes to infinity are 524277 131043 1047299 4174775 2068033 2025023 971399 28864693 [------, ------, -------, -------, -------, -------, -------, --------, 524288 131072 1048576 4194304 2097152 2097152 1048576 33554432 101990317 41695511 119568449 256512971 14592071 -231050689 ---------, --------, ---------, ----------, ----------, ----------, 134217728 67108864 268435456 1073741824 1073741824 1073741824 -463046629 -1332086153 -13266643549 -947508739 -32718708263 ----------, -----------, ------------, ----------, ------------, 1073741824 2147483648 17179869184 1073741824 34359738368 -135797923367 -------------] 137438953472 and in Maple notation [524277/524288, 131043/131072, 1047299/1048576, 4174775/4194304, 2068033/ 2097152, 2025023/2097152, 971399/1048576, 28864693/33554432, 101990317/ 134217728, 41695511/67108864, 119568449/268435456, 256512971/1073741824, 14592071/1073741824, -231050689/1073741824, -463046629/1073741824, -1332086153/ 2147483648, -13266643549/17179869184, -947508739/1073741824, -32718708263/ 34359738368, -135797923367/137438953472] and in floating point [.9999790192, .9997787476, .9987821579, .9953439236, .9861149788, .9656062126, .9263982773, .8602348864, .7598870769, .6213115305, .4454271831, .2388963206, .\ 1358992513e-1, -.2151827225, -.4312457787, -.6203009528, -.7722202892, -.882436\ 2783, -.9522397380, -.9880599345] The cut off is at j=, 14 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 22], vs. those in the, 2, -th row from j=1 to j=, 21, are as follws 11 10 9 8 [3 (699043 n - 42291216 n + 1124490235 n - 17281965360 n 7 6 5 + 169776328029 n - 1112049866928 n + 4899786840905 n 4 3 2 - 14132111877840 n + 23628691074428 n - 8204004046656 n - 56857967490240 n + 103248156211200)/(1024 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) (2 n - 15) 11 10 9 8 (2 n - 17)), 3 (349483 n - 21139294 n + 561781385 n - 8620908780 n 7 6 5 4 + 84321532449 n - 545144163102 n + 2306366320775 n - 5789666578520 n 3 2 + 4601753866468 n + 18076197813696 n - 46884336932160 n + 2346549004800) /(512 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) 12 (2 n - 11) (2 n - 21) (2 n - 13) (2 n - 15) (2 n - 17)), 57 (73533 n 11 10 9 8 - 5289546 n + 168938781 n - 3152939570 n + 38000009619 n 7 6 5 4 - 306940006758 n + 1647425032143 n - 5398072415670 n + 7048575954948 n 3 2 + 17071058420504 n - 73943531826624 n + 59774597850240 n - 5681118643200 )/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 19 ( 12 11 10 9 8 220139 n - 15798258 n + 502044763 n - 9270692970 n + 109247550357 n 7 6 5 4 - 841075609374 n + 4054334774809 n - 9884265479550 n - 4561637271796 n 3 2 + 93887497351032 n - 195230532771072 n + 130259587086720 n - 17043355929600)/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 13 12 11 10 (-1 + 2 n)), 4199 (3961 n - 332117 n + 12410987 n - 271317805 n 9 8 7 6 + 3813695193 n - 35396491251 n + 210350867441 n - 684561927775 n 5 4 3 + 202646843146 n + 7520966932468 n - 27765694751928 n 2 + 41321679878880 n - 24783747206400 n + 3855962880000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) 13 (2 n - 17) (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), 221 (18556 n 12 11 10 9 - 1540037 n + 56556422 n - 1200426205 n + 16044909078 n 8 7 6 5 - 136187692311 n + 674672753546 n - 1154272014775 n - 6975195467834 n 4 3 2 + 50135967120448 n - 136360538524968 n + 179737174079280 n - 104984976398400 n + 18315823680000)/(512 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 14 13 (2 n - 21) (2 n - 23) (2 n - 25)), 221 (36077 n - 3426871 n 12 11 10 9 + 144510478 n - 3536210069 n + 54807419436 n - 544683075783 n 8 7 6 + 3217307474194 n - 6943217937887 n - 48094472702353 n 5 4 3 + 473251996201754 n - 1874868130174872 n + 4046621482204056 n 2 - 4747269476082960 n + 2667420844804800 n - 494527239360000)/(512 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 14 13 12 (-1 + 2 n)), 13 (582259 n - 53909807 n + 2188902926 n 11 10 9 - 50586947173 n + 715559818512 n - 6004709098311 n 8 7 6 + 21853971870698 n + 101012294303921 n - 1747454397047051 n 5 4 3 + 10363314334652818 n - 34572078499437024 n + 68435850134036952 n 2 - 77041355756530320 n + 43162527590481600 n - 8406963069120000)/(512 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 15 14 13 (-1 + 2 n)), 13 (34113001 n - 3530434050 n + 160291582360 n 12 11 10 - 4138636101600 n + 64996702687462 n - 586322936138940 n 9 8 7 + 1655502298401380 n + 28417286774190600 n - 433355721719447527 n 6 5 + 3086447535312771630 n - 13515430248724707940 n 4 3 + 38070462731574273000 n - 67850160347480823936 n 2 + 71804846560414239360 n - 39263042536729804800 n + 7801661728143360000)/ (16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 15 14 (2 n - 17) (2 n - 23) (2 n - 29)), 13 (14780363 n - 1451110050 n 13 12 11 + 60989817980 n - 1389590556900 n + 16943122966406 n 10 9 8 - 52549871293920 n - 1737820423213160 n + 33099587179015500 n 7 6 5 - 311332132674764201 n + 1853863058505748290 n - 7367867644691519420 n 4 3 + 19566000877638797400 n - 33665893761685805568 n 2 + 35041882242075799680 n - 19183319742316262400 n + 3900830864071680000)/ (8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 16 15 (2 n - 17) (2 n - 23) (2 n - 29)), 143 (4275037 n - 450436376 n 14 13 12 + 19888522420 n - 446642028560 n + 3851277232894 n 11 10 9 + 61756825993408 n - 2505163654579300 n + 40873502775282320 n 8 7 6 - 418514342007546499 n + 2944820832867025112 n - 14623646599426915960 n 5 4 + 51257514263127885440 n - 124449411408065760432 n 3 2 + 201078241461088649856 n - 200942074495180028160 n + 107810379976716748800 n - 21986501233858560000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 16 15 14 (2 n - 23) (2 n - 29)), 13 (32137967 n - 2885445976 n + 90592888220 n 13 12 11 - 240192646960 n - 69212281283446 n + 2517569277331808 n 10 9 8 - 49004701486119500 n + 628580953430011120 n - 5667905354348450609 n 7 6 + 36817840778513445112 n - 173169581740765037960 n 5 4 + 584484905496552303040 n - 1383166785694604592912 n 3 2 + 2199794681251136221056 n - 2182744282895952341760 n + 1172722279003569100800 n - 241851513572444160000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 15 (2 n - 23) (2 n - 29)), 221 (1774052 n - 102823789 n - 3809706256 n 14 13 12 + 598599238460 n - 29016503526136 n + 828749668089602 n 11 10 9 - 16021591703355392 n + 222116270363757220 n - 2271725544842994284 n 8 7 + 17365044638951713363 n - 99467002481777098448 n 6 5 + 424293298790505774280 n - 1327071242424298950432 n 4 3 + 2961317165363406643824 n - 4506994626147396203904 n 2 + 4336049979906820727040 n - 2287130814252771379200 n + 469476467522979840000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), - 17 ( 17 16 15 14 4851274 n - 2654985593 n + 306557832928 n - 17647808243780 n 13 12 11 + 629129402921068 n - 15294998369558726 n + 267027694693083896 n 10 9 - 3448034741932469260 n + 33469670247942885242 n 8 7 - 245926864452444973369 n + 1366797479753804158424 n 6 5 - 5698488986730841286440 n + 17525111322971842950816 n 4 3 - 38649918676983047031312 n + 58400071079376512506752 n 2 - 56012431770579672011520 n + 29572651789539557529600 n - 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), - 17 ( 18 17 16 15 65530448 n - 14133090276 n + 1315904141211 n - 71925936566040 n 14 13 12 + 2616546243988236 n - 67697987780093832 n + 1295384401845104402 n 11 10 - 18773013313717256880 n + 208947590246377123284 n 9 8 - 1798148471834651358708 n + 11972838929830330933563 n 7 6 - 61369676709530205645720 n + 239427933983337433416632 n 5 4 - 697540768827441076949184 n + 1472832670386290650699824 n 3 2 - 2150734297929967971423360 n + 2010711086720858756102400 n - 1043408029691566699776000 n + 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 18 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 17 (117636488 n 17 16 15 - 21948996276 n + 1849417884771 n - 93896448332040 n 14 13 12 + 3226579895741916 n - 79786064873331432 n + 1471809483912949922 n 11 10 - 20701787427946619280 n + 224844045176791701804 n 9 8 - 1896646158287471813508 n + 12425820289455594591843 n 7 6 - 62874365120425881214920 n + 242847514531091202834392 n 5 4 - 702217459654716545270784 n + 1474977945115708287772464 n 3 2 - 2147041713972361700165760 n + 2004677176593482385926400 n - 1040818795840379356416000 n + 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 323 (137113427 n 18 17 16 - 26556401836 n + 2368859578233 n - 129381408560244 n 15 14 13 + 4850032414041174 n - 132519252296564952 n + 2734822838640319106 n 12 11 - 43574553637473183208 n + 543182183439443287311 n 10 9 - 5334458824983989917788 n + 41349404181191688256389 n 8 7 - 252257990969160792777252 n + 1201900248003162558681848 n 6 5 - 4413025592753056715561824 n + 12231275161037703128440272 n 4 3 - 24805874957451843651103296 n + 35094640906554337689258240 n 2 - 32043659597009935349529600 n + 16368273346143496909824000 n - 3327846876104996044800000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 19 18 (2 n - 3) (-1 + 2 n) (2 n - 5)), - 19 (1423668167 n - 267042061606 n 17 16 15 + 23171527691943 n - 1235727711151824 n + 45376949082882054 n 14 13 - 1217960753581993092 n + 24752786051594412326 n 12 11 - 389244808615212187768 n + 4798227743443772197131 n 10 9 - 46680123022221710076198 n + 359004276390241816854819 n 8 7 - 2176091718579407930056392 n + 10314635752267007322729608 n 6 5 - 37719931392616505028313504 n + 104231972774265962731764912 n 4 3 - 210950063725880552418758016 n + 298073924781697506015095040 n 2 - 272028511161691814351001600 n + 138988511785567522197504000 n - 28286698446892466380800000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), - 19 (6440595443 n - 1310496771850 n 18 17 16 + 124196259788265 n - 7283478587839950 n + 296173844448256038 n 15 14 - 8867614047017251380 n + 202596548249261960930 n 13 12 - 3611831447612440752700 n + 50949214658122106647503 n 11 10 - 573219849287172233928210 n + 5160641993983568117358645 n 9 8 - 37147106485259477558757750 n + 212752638570742336787025608 n 7 6 - 960577842715458795214329040 n + 3369521700265066819268001360 n 5 4 - 8986484686400862763890837600 n + 17650362788011275021942225408 n 3 2 - 24326235175716374492575019520 n + 21757932718327976734184140800 n - 10947695863154357059264512000 n + 2206362478857612377702400000)/(131072 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), (-131322386717 n + 26447974346050 n - 2483710252255335 n 17 16 + 144483626958432150 n - 5833474279475293722 n 15 14 + 173566644720817594020 n - 3943807720209840782270 n 13 12 + 69976181162437563383500 n - 983073428526576330616257 n 11 10 + 11021883811448735616795690 n - 98937436912883137453000155 n 9 8 + 710424475014677741294123550 n - 4060661229809525674086148952 n 7 6 + 18304459482449305799705545360 n - 64128504621957388268361797040 n 5 4 + 170872038115407941533215832800 n - 335396330156910342013909604352 n 3 2 + 462078256748078794091321018880 n - 413234430677592085076472115200 n + 207939361930876492841852928000 n - 41920887098294635176345600000)/( 131072 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), ((97865068080 n + 97865068080) (2 n - 43)! - (n - 22)! (n - 20)! binomial(2 n, n))/((n - 22)! (n - 20)! binomial(2 n, n))] and in Maple notation [3/1024*(699043*n^11-42291216*n^10+1124490235*n^9-17281965360*n^8+169776328029* n^7-1112049866928*n^6+4899786840905*n^5-14132111877840*n^4+23628691074428*n^3-\ 8204004046656*n^2-56857967490240*n+103248156211200)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-15)/(2*n-17), 3/512*( 349483*n^11-21139294*n^10+561781385*n^9-8620908780*n^8+84321532449*n^7-\ 545144163102*n^6+2306366320775*n^5-5789666578520*n^4+4601753866468*n^3+ 18076197813696*n^2-46884336932160*n+2346549004800)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-15)/(2*n-17), 57/1024*( 73533*n^12-5289546*n^11+168938781*n^10-3152939570*n^9+38000009619*n^8-\ 306940006758*n^7+1647425032143*n^6-5398072415670*n^5+7048575954948*n^4+ 17071058420504*n^3-73943531826624*n^2+59774597850240*n-5681118643200)/(2*n-23)/ (2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/( 2*n-3)/(-1+2*n), 19/1024*(220139*n^12-15798258*n^11+502044763*n^10-9270692970*n ^9+109247550357*n^8-841075609374*n^7+4054334774809*n^6-9884265479550*n^5-\ 4561637271796*n^4+93887497351032*n^3-195230532771072*n^2+130259587086720*n-\ 17043355929600)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/ (2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 4199/2048*(3961*n^13-332117*n^12+ 12410987*n^11-271317805*n^10+3813695193*n^9-35396491251*n^8+210350867441*n^7-\ 684561927775*n^6+202646843146*n^5+7520966932468*n^4-27765694751928*n^3+ 41321679878880*n^2-24783747206400*n+3855962880000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2* n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2* n-25), 221/512*(18556*n^13-1540037*n^12+56556422*n^11-1200426205*n^10+ 16044909078*n^9-136187692311*n^8+674672753546*n^7-1154272014775*n^6-\ 6975195467834*n^5+50135967120448*n^4-136360538524968*n^3+179737174079280*n^2-\ 104984976398400*n+18315823680000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n -11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 221/512*( 36077*n^14-3426871*n^13+144510478*n^12-3536210069*n^11+54807419436*n^10-\ 544683075783*n^9+3217307474194*n^8-6943217937887*n^7-48094472702353*n^6+ 473251996201754*n^5-1874868130174872*n^4+4046621482204056*n^3-4747269476082960* n^2+2667420844804800*n-494527239360000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2* n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2 *n), 13/512*(582259*n^14-53909807*n^13+2188902926*n^12-50586947173*n^11+ 715559818512*n^10-6004709098311*n^9+21853971870698*n^8+101012294303921*n^7-\ 1747454397047051*n^6+10363314334652818*n^5-34572078499437024*n^4+ 68435850134036952*n^3-77041355756530320*n^2+43162527590481600*n-\ 8406963069120000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15 )/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 13/16384*( 34113001*n^15-3530434050*n^14+160291582360*n^13-4138636101600*n^12+ 64996702687462*n^11-586322936138940*n^10+1655502298401380*n^9+28417286774190600 *n^8-433355721719447527*n^7+3086447535312771630*n^6-13515430248724707940*n^5+ 38070462731574273000*n^4-67850160347480823936*n^3+71804846560414239360*n^2-\ 39263042536729804800*n+7801661728143360000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/( 2*n-23)/(2*n-29), 13/8192*(14780363*n^15-1451110050*n^14+60989817980*n^13-\ 1389590556900*n^12+16943122966406*n^11-52549871293920*n^10-1737820423213160*n^9 +33099587179015500*n^8-311332132674764201*n^7+1853863058505748290*n^6-\ 7367867644691519420*n^5+19566000877638797400*n^4-33665893761685805568*n^3+ 35041882242075799680*n^2-19183319742316262400*n+3900830864071680000)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2* n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 143/16384*(4275037*n^16-450436376*n^ 15+19888522420*n^14-446642028560*n^13+3851277232894*n^12+61756825993408*n^11-\ 2505163654579300*n^10+40873502775282320*n^9-418514342007546499*n^8+ 2944820832867025112*n^7-14623646599426915960*n^6+51257514263127885440*n^5-\ 124449411408065760432*n^4+201078241461088649856*n^3-200942074495180028160*n^2+ 107810379976716748800*n-21986501233858560000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-17)/(2*n-23)/(2*n-29), 13/16384*(32137967*n^16-2885445976*n^15+ 90592888220*n^14-240192646960*n^13-69212281283446*n^12+2517569277331808*n^11-\ 49004701486119500*n^10+628580953430011120*n^9-5667905354348450609*n^8+ 36817840778513445112*n^7-173169581740765037960*n^6+584484905496552303040*n^5-\ 1383166785694604592912*n^4+2199794681251136221056*n^3-2182744282895952341760*n^ 2+1172722279003569100800*n-241851513572444160000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-17)/(2*n-23)/(2*n-29), 221/16384*(1774052*n^17-102823789*n^16-\ 3809706256*n^15+598599238460*n^14-29016503526136*n^13+828749668089602*n^12-\ 16021591703355392*n^11+222116270363757220*n^10-2271725544842994284*n^9+ 17365044638951713363*n^8-99467002481777098448*n^7+424293298790505774280*n^6-\ 1327071242424298950432*n^5+2961317165363406643824*n^4-4506994626147396203904*n^ 3+4336049979906820727040*n^2-2287130814252771379200*n+469476467522979840000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n -27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), -17/16384* (4851274*n^17-2654985593*n^16+306557832928*n^15-17647808243780*n^14+ 629129402921068*n^13-15294998369558726*n^12+267027694693083896*n^11-\ 3448034741932469260*n^10+33469670247942885242*n^9-245926864452444973369*n^8+ 1366797479753804158424*n^7-5698488986730841286440*n^6+17525111322971842950816*n ^5-38649918676983047031312*n^4+58400071079376512506752*n^3-\ 56012431770579672011520*n^2+29572651789539557529600*n-6103194077798737920000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2* n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), -17/16384 *(65530448*n^18-14133090276*n^17+1315904141211*n^16-71925936566040*n^15+ 2616546243988236*n^14-67697987780093832*n^13+1295384401845104402*n^12-\ 18773013313717256880*n^11+208947590246377123284*n^10-1798148471834651358708*n^9 +11972838929830330933563*n^8-61369676709530205645720*n^7+ 239427933983337433416632*n^6-697540768827441076949184*n^5+ 1472832670386290650699824*n^4-2150734297929967971423360*n^3+ 2010711086720858756102400*n^2-1043408029691566699776000*n+ 213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25) /(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19) /(2*n-3)/(-1+2*n)/(2*n-5), -17/16384*(117636488*n^18-21948996276*n^17+ 1849417884771*n^16-93896448332040*n^15+3226579895741916*n^14-79786064873331432* n^13+1471809483912949922*n^12-20701787427946619280*n^11+224844045176791701804*n ^10-1896646158287471813508*n^9+12425820289455594591843*n^8-\ 62874365120425881214920*n^7+242847514531091202834392*n^6-\ 702217459654716545270784*n^5+1474977945115708287772464*n^4-\ 2147041713972361700165760*n^3+2004677176593482385926400*n^2-\ 1040818795840379356416000*n+213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-\ 33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -323/131072*(137113427*n ^19-26556401836*n^18+2368859578233*n^17-129381408560244*n^16+4850032414041174*n ^15-132519252296564952*n^14+2734822838640319106*n^13-43574553637473183208*n^12+ 543182183439443287311*n^11-5334458824983989917788*n^10+41349404181191688256389* n^9-252257990969160792777252*n^8+1201900248003162558681848*n^7-\ 4413025592753056715561824*n^6+12231275161037703128440272*n^5-\ 24805874957451843651103296*n^4+35094640906554337689258240*n^3-\ 32043659597009935349529600*n^2+16368273346143496909824000*n-\ 3327846876104996044800000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15 )/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7 )/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/65536*(1423668167*n^19-267042061606*n^ 18+23171527691943*n^17-1235727711151824*n^16+45376949082882054*n^15-\ 1217960753581993092*n^14+24752786051594412326*n^13-389244808615212187768*n^12+ 4798227743443772197131*n^11-46680123022221710076198*n^10+ 359004276390241816854819*n^9-2176091718579407930056392*n^8+ 10314635752267007322729608*n^7-37719931392616505028313504*n^6+ 104231972774265962731764912*n^5-210950063725880552418758016*n^4+ 298073924781697506015095040*n^3-272028511161691814351001600*n^2+ 138988511785567522197504000*n-28286698446892466380800000)/(2*n-29)/(2*n-35)/(2* n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/( 2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/131072* (6440595443*n^20-1310496771850*n^19+124196259788265*n^18-7283478587839950*n^17+ 296173844448256038*n^16-8867614047017251380*n^15+202596548249261960930*n^14-\ 3611831447612440752700*n^13+50949214658122106647503*n^12-\ 573219849287172233928210*n^11+5160641993983568117358645*n^10-\ 37147106485259477558757750*n^9+212752638570742336787025608*n^8-\ 960577842715458795214329040*n^7+3369521700265066819268001360*n^6-\ 8986484686400862763890837600*n^5+17650362788011275021942225408*n^4-\ 24326235175716374492575019520*n^3+21757932718327976734184140800*n^2-\ 10947695863154357059264512000*n+2206362478857612377702400000)/(2*n-29)/(2*n-35) /(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 1/131072*(-131322386717*n^20+26447974346050*n^19-2483710252255335*n^18+ 144483626958432150*n^17-5833474279475293722*n^16+173566644720817594020*n^15-\ 3943807720209840782270*n^14+69976181162437563383500*n^13-\ 983073428526576330616257*n^12+11021883811448735616795690*n^11-\ 98937436912883137453000155*n^10+710424475014677741294123550*n^9-\ 4060661229809525674086148952*n^8+18304459482449305799705545360*n^7-\ 64128504621957388268361797040*n^6+170872038115407941533215832800*n^5-\ 335396330156910342013909604352*n^4+462078256748078794091321018880*n^3-\ 413234430677592085076472115200*n^2+207939361930876492841852928000*n-\ 41920887098294635176345600000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2* n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/( 2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), ((97865068080*n+97865068080)* (2*n-43)!-(n-22)!*(n-20)!*binomial(2*n,n))/(n-22)!/(n-20)!/binomial(2*n,n)] The limits, as n goes to infinity are 2097129 1048449 4191381 4182641 16632239 1025219 7973017 7569367 [-------, -------, -------, -------, --------, -------, -------, -------, 2097152 1048576 4194304 4194304 16777216 1048576 8388608 8388608 443469013 192144719 611330291 417793571 98016373 -41235829 ---------, ---------, ----------, ----------, ---------, ----------, 536870912 268435456 1073741824 1073741824 536870912 1073741824 -69626101 -249977537 -44287636921 -27049695173 -122371313417 ---------, ----------, ------------, ------------, -------------, 268435456 536870912 68719476736 34359738368 137438953472 -131322386717 -543639247133 -------------, -------------] 137438953472 549755813888 and in Maple notation [2097129/2097152, 1048449/1048576, 4191381/4194304, 4182641/4194304, 16632239/ 16777216, 1025219/1048576, 7973017/8388608, 7569367/8388608, 443469013/ 536870912, 192144719/268435456, 611330291/1073741824, 417793571/1073741824, 98016373/536870912, -41235829/1073741824, -69626101/268435456, -249977537/ 536870912, -44287636921/68719476736, -27049695173/34359738368, -122371313417/ 137438953472, -131322386717/137438953472, -543639247133/549755813888] and in floating point [.9999890327, .9998788834, .9993031025, .9972193241, .9913586974, .9777250290, .9504576921, .9023388624, .8260254059, .7157948576, .5693457006, .3891005842, .\ 1825697217, -.3840385843e-1, -.2593774386, -.4656194467, -.6444699381, -.787249\ 7422, -.8903684896, -.9554961195, -.9888740299] The cut off is at j=, 14 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 23], vs. those in the, 2, -th row from j=1 to j=, 22, are as follws 11 10 9 8 7 [23 (22795 n - 1379081 n + 36669600 n - 563604690 n + 5537944335 n 6 5 4 3 - 36296286873 n + 160221394190 n - 464793677260 n + 792900148920 n 2 - 336398268096 n - 1796749395840 n + 3519823507200)/(256 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) 12 11 (2 n - 13) (2 n - 15) (2 n - 17)), 69 (60783 n - 4375914 n 10 9 8 7 + 140020023 n - 2624596810 n + 31955704209 n - 264396899382 n 6 5 4 + 1503876129789 n - 5730621556110 n + 13006820061708 n 3 2 - 7988360272904 n - 42701380034112 n + 98461771873920 n - 4693098009600) /(1024 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 23)), 3 ( 12 11 10 9 1397547 n - 100565634 n + 3214193499 n - 60077618810 n 8 7 6 + 726339402261 n - 5904670242702 n + 32111803730457 n 5 4 3 - 108324636207150 n + 157394885225292 n + 285386598516536 n 2 - 1429214118483456 n + 1180515112882560 n - 107941254220800)/(1024 (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) (-1 + 2 n)), 741 ( 13 12 11 10 9 11302 n - 953269 n + 36009884 n - 802752885 n + 11688131576 n 8 7 6 5 - 115650004107 n + 776258198012 n - 3331175605175 n + 7170344540722 n 4 3 2 + 5116544036076 n - 68558752449896 n + 134677932716160 n - 87108659164800 n + 10925228160000)/(1024 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 13 12 (2 n - 21) (2 n - 23) (2 n - 25)), 247 (33781 n - 2839307 n 11 10 9 8 + 106535027 n - 2344607155 n + 33316471353 n - 314759584821 n 7 6 5 + 1929066663161 n - 6730688187025 n + 5173701633166 n 4 3 2 + 57662404060228 n - 234063410411688 n + 358425933468480 n - 216200035094400 n + 32775684480000)/(1024 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 14 13 (2 n - 21) (2 n - 23) (2 n - 25)), 4199 (7877 n - 763441 n 12 11 10 9 + 33196303 n - 851120249 n + 14178163461 n - 158486317143 n 8 7 6 + 1169967663469 n - 5160122225027 n + 7659336899522 n 5 4 3 + 49349792483684 n - 327046295222472 n + 850446402041376 n 2 - 1085367709028160 n + 617060714860800 n - 104110997760000)/(2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 14 13 12 11 (-1 + 2 n)), 221 (36707 n - 3515071 n + 149943598 n - 3729544469 n 10 9 8 7 + 59193402576 n - 610724589783 n + 3883631492554 n - 11387649806687 n 6 5 4 - 29227984216123 n + 426219495347954 n - 1820099312768352 n 3 2 + 4053859179143256 n - 4821575497410960 n + 2711726315524800 n - 494527239360000)/(512 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 15 14 (2 n - 5) (2 n - 3) (-1 + 2 n)), 221 (70804 n - 7671825 n 13 12 11 10 + 371276815 n - 10511306400 n + 190752043573 n - 2264706261510 n 9 8 7 + 16698578593895 n - 56257291767600 n - 211453509523333 n 6 5 4 + 3614445129959895 n - 20514139883415010 n + 65852400184872000 n 3 2 - 126237563580587544 n + 138293127733375440 n - 75590990440459200 n + 14341289941440000)/(512 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 15 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 13 (4521047 n 14 13 12 11 - 476759475 n + 22198702295 n - 594054606075 n + 9880403748089 n 10 9 8 - 100449702773805 n + 496529896226485 n + 1419821873605575 n 7 6 5 - 43485245570256644 n + 350742142361701860 n - 1620557050116978680 n 4 3 2 + 4698896567115702000 n - 8510526034762336992 n + 9072740469004729920 n - 4958471626469625600 n + 975207716017920000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 13 ( 16 15 14 13 65257127 n - 7616631256 n + 392339721020 n - 11585142565360 n 12 11 10 + 210517062757274 n - 2244730164537952 n + 8411540305210900 n 9 8 7 + 133892692563263920 n - 2634960632609977529 n + 23839429031470524472 n 6 5 - 136067217400831513160 n + 520482955750209000640 n 4 3 - 1337008145967694825872 n + 2235362072833832996736 n 2 - 2269095927957449429760 n + 1214634710824248268800 n - 241851513572444160000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 16 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 13 (55440047 n 15 14 13 - 6124435096 n + 291223797020 n - 7554249517360 n 12 11 10 + 105428932213514 n - 359682500518432 n - 15434073386689100 n 9 8 7 + 348450204540071920 n - 4002126514280806769 n + 29892048682376832952 n 6 5 - 153902586554048401160 n + 552159646814933160640 n 4 3 - 1360904291459312235792 n + 2218680554883753672576 n 2 - 2225973296627996117760 n + 1193370471466671052800 n - 241851513572444160000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 17 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 (5037422 n 16 15 14 13 - 589065919 n + 28823993744 n - 706683494140 n + 5684417360804 n 12 11 10 + 181222661780342 n - 7290665023691192 n + 135870575206357420 n 9 8 - 1646126649235342274 n + 14062251536967478273 n 7 6 - 87069415845297915848 n + 392783552868742988680 n 5 4 - 1278683135008524022752 n + 2932784263378271180304 n 3 2 - 4539497655958790200704 n + 4399482315443635895040 n - 2316047843049476659200 n + 469476467522979840000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 17 16 (2 n - 17) (2 n - 23) (2 n - 29)), 221 (3257402 n - 314942839 n 15 14 13 + 9848980544 n + 74318081060 n - 15629723681236 n 12 11 10 + 588489863604902 n - 12899770311837992 n + 192330011913323020 n 9 8 - 2062500328346262134 n + 16292609249546883913 n 7 6 - 95549171938792940648 n + 414585873056221095880 n 5 4 - 1312554156018426860832 n + 2953197596282023227024 n 3 2 - 4517136895064961515904 n + 4354979437154641847040 n - 2295635822722390579200 n + 469476467522979840000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 17 (32775202 n - 1104285474 n 16 15 14 - 242013220611 n + 26318013305040 n - 1312421328363936 n 13 12 + 41129174366227332 n - 897420714204346202 n 11 10 + 14316297318866279880 n - 171393465541784748834 n 9 8 + 1560684145217421038958 n - 10860314997798612836763 n 7 6 + 57610595350410301737720 n - 230745796136664422511032 n 5 4 + 685457619942086754161184 n - 1467047554360560012725424 n 3 2 + 2160054674550379148319360 n - 2026328817918885261062400 n + 1050171830972678405376000 n - 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 18 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 17 (21839048 n 17 16 15 - 7579380276 n + 868547896611 n - 53503457756040 n 14 13 12 + 2105027343449436 n - 57562022515577832 n + 1147450326819271202 n 11 10 - 17155718248990772880 n + 195618264683607135084 n 9 8 - 1715557244063652840708 n + 11593009839461321033763 n 7 6 - 60107981458158055323720 n + 236560583834599800675032 n 5 4 - 693619319997116926493184 n + 1471033837538642321477424 n 3 2 - 2153830563981376956639360 n + 2015770576579217824262400 n - 1045579126399084037376000 n + 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 323 (7971064 n 18 17 16 - 1815979472 n + 182982084006 n - 10992728120613 n 15 14 13 + 444797671681968 n - 12935291055815604 n + 281033925621037492 n 12 11 - 4672858713404655566 n + 60349595779676984352 n 10 9 - 610318598796257055276 n + 4846342110134782868598 n 8 7 - 30151661433357608609829 n + 145929330226091008766536 n 6 5 - 542381825459336223817448 n + 1517011768027389510527904 n 4 3 - 3096099595956040016326992 n + 4396987784619068583070080 n 2 - 4020776450773880668819200 n + 2052412267414951588608000 n - 415980859513124505600000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 19 18 (2 n - 3) (-1 + 2 n) (2 n - 5)), - 323 (26419973 n - 5381759059 n 17 16 15 + 499808160042 n - 28206655754286 n + 1086152821141926 n 14 13 - 30343596711409338 n + 637846825042419344 n 12 11 - 10319604849034248352 n + 130281396775254170589 n 10 9 - 1292916614641629781947 n + 10108045019424933947886 n 8 7 - 62093664840591118454538 n + 297477733697830296115952 n 6 5 - 1096889020016487393576256 n + 3049719062702389247413728 n 4 3 - 6198421408437834088033824 n + 8780676141717282467554560 n 2 - 8021341716166607545382400 n + 4096384343477462969856000 n - 831961719026249011200000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), - 19 (4819024193 n - 1025100231850 n 18 17 16 + 100976981059515 n - 6124372972054950 n + 256436264781113538 n 15 14 - 7875507516094471380 n + 183934667241677503430 n 13 12 - 3341972966901075882700 n + 47916152457338482271253 n 11 10 - 546616385787887098608210 n + 4978956575093769282584895 n 9 8 - 36189702023296468593852750 n + 208928852674045621674615608 n 7 6 - 949367998219445308413969040 n + 3346827872995685132023461360 n 5 4 - 8959113914434350891851397600 n + 17641749487433444784510225408 n 3 2 - 24351550397244196740415019520 n + 21793248369254617764584140800 n - 10961945536866269097664512000 n + 2206362478857612377702400000)/(131072 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), - 19 (5791966943 n - 1198284041350 n + 115212106434765 n 17 16 15 - 6841628313240450 n + 281233899886051038 n - 8499354784155670380 n 14 13 + 195750656693184893930 n - 3513905095597861833700 n 12 11 + 49859479149262954161003 n - 563748351099312583771710 n 10 9 + 5096494238389702666403145 n - 36811638963900950915444250 n 8 7 + 211422124297885722646977608 n - 956702446276648020530757040 n 6 5 + 3361725155560637927799177360 n - 8977149246271040371188669600 n 4 3 + 17647505021004310051144593408 n - 24334946160537557193126059520 n 2 + 21769982788664170420635340800 n - 10952540752216407152320512000 n + 2206362478857612377702400000)/(131072 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), (-246688512334 n 20 19 18 + 55232520677397 n - 5781089483137220 n + 375951674399255535 n 17 16 - 17027603326423403994 n + 570631149581582964042 n 15 14 - 14672373740098599004360 n + 296208129812889032165070 n 13 12 - 4765001002523102063454914 n + 61634244453742920673985817 n 11 10 - 644000095909971092011820100 n + 5440626736255180741052645955 n 9 8 - 37067450265651010435726964854 n + 202411542914673510438514072152 n 7 6 - 876832027481501952564632201840 n + 2967330803908754058066396744240 n 5 4 - 7672342189087689593119526615904 n + 14674354920925398616117299950592 n 3 - 19775831043835504792387806036480 n 2 + 17363945337058821470468831539200 n - 8611500930239201258073858048000 n + 1718756371030080042230169600000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) 21 (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), (-263442586489 n 20 19 18 + 58449302915157 n - 6068170543783145 n + 391766180075643135 n 17 16 - 17629813208701101324 n + 587452374601926577002 n 15 14 - 15029258776097974301410 n + 302082963042024546372270 n 13 12 - 4840979313848365486622369 n + 62410920760182562997260377 n 11 10 - 650279337734182174218543525 n + 5480586169187216788198252755 n 9 8 - 37265421265501629177657673834 n + 203160330680909987203471373112 n 7 6 - 878922333473313623092261372640 n + 2971371092533613214845629358640 n 5 4 - 7676963681716867415680555197984 n + 14675520445593441112793391524352 n 3 - 19771274767201384970557836449280 n 2 + 17357949993998774592680764723200 n - 8609141081674857898748141568000 n + 1718756371030080042230169600000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), ( (365930254560 n + 365930254560) (2 n - 45)! - (n - 23)! (n - 21)! binomial(2 n, n))/((n - 23)! (n - 21)! binomial(2 n, n))] and in Maple notation [23/256*(22795*n^11-1379081*n^10+36669600*n^9-563604690*n^8+5537944335*n^7-\ 36296286873*n^6+160221394190*n^5-464793677260*n^4+792900148920*n^3-336398268096 *n^2-1796749395840*n+3519823507200)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/( 2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-15)/(2*n-17), 69/1024*(60783*n^12-\ 4375914*n^11+140020023*n^10-2624596810*n^9+31955704209*n^8-264396899382*n^7+ 1503876129789*n^6-5730621556110*n^5+13006820061708*n^4-7988360272904*n^3-\ 42701380034112*n^2+98461771873920*n-4693098009600)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-23), 3/ 1024*(1397547*n^12-100565634*n^11+3214193499*n^10-60077618810*n^9+726339402261* n^8-5904670242702*n^7+32111803730457*n^6-108324636207150*n^5+157394885225292*n^ 4+285386598516536*n^3-1429214118483456*n^2+1180515112882560*n-107941254220800)/ (2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/ (2*n-5)/(2*n-3)/(-1+2*n), 741/1024*(11302*n^13-953269*n^12+36009884*n^11-\ 802752885*n^10+11688131576*n^9-115650004107*n^8+776258198012*n^7-3331175605175* n^6+7170344540722*n^5+5116544036076*n^4-68558752449896*n^3+134677932716160*n^2-\ 87108659164800*n+10925228160000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 247/1024*( 33781*n^13-2839307*n^12+106535027*n^11-2344607155*n^10+33316471353*n^9-\ 314759584821*n^8+1929066663161*n^7-6730688187025*n^6+5173701633166*n^5+ 57662404060228*n^4-234063410411688*n^3+358425933468480*n^2-216200035094400*n+ 32775684480000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n -15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 4199/2048*(7877*n^14-763441* n^13+33196303*n^12-851120249*n^11+14178163461*n^10-158486317143*n^9+ 1169967663469*n^8-5160122225027*n^7+7659336899522*n^6+49349792483684*n^5-\ 327046295222472*n^4+850446402041376*n^3-1085367709028160*n^2+617060714860800*n-\ 104110997760000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15) /(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 221/512*(36707*n^ 14-3515071*n^13+149943598*n^12-3729544469*n^11+59193402576*n^10-610724589783*n^ 9+3883631492554*n^8-11387649806687*n^7-29227984216123*n^6+426219495347954*n^5-\ 1820099312768352*n^4+4053859179143256*n^3-4821575497410960*n^2+2711726315524800 *n-494527239360000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 221/512*(70804* n^15-7671825*n^14+371276815*n^13-10511306400*n^12+190752043573*n^11-\ 2264706261510*n^10+16698578593895*n^9-56257291767600*n^8-211453509523333*n^7+ 3614445129959895*n^6-20514139883415010*n^5+65852400184872000*n^4-\ 126237563580587544*n^3+138293127733375440*n^2-75590990440459200*n+ 14341289941440000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/( 2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 13/2048 *(4521047*n^15-476759475*n^14+22198702295*n^13-594054606075*n^12+9880403748089* n^11-100449702773805*n^10+496529896226485*n^9+1419821873605575*n^8-\ 43485245570256644*n^7+350742142361701860*n^6-1620557050116978680*n^5+ 4698896567115702000*n^4-8510526034762336992*n^3+9072740469004729920*n^2-\ 4958471626469625600*n+975207716017920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 13/16384*(65257127*n^16-7616631256*n^15+392339721020*n^14-\ 11585142565360*n^13+210517062757274*n^12-2244730164537952*n^11+8411540305210900 *n^10+133892692563263920*n^9-2634960632609977529*n^8+23839429031470524472*n^7-\ 136067217400831513160*n^6+520482955750209000640*n^5-1337008145967694825872*n^4+ 2235362072833832996736*n^3-2269095927957449429760*n^2+1214634710824248268800*n-\ 241851513572444160000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 13/16384*(55440047*n^16-6124435096*n^15+291223797020*n^14-7554249517360* n^13+105428932213514*n^12-359682500518432*n^11-15434073386689100*n^10+ 348450204540071920*n^9-4002126514280806769*n^8+29892048682376832952*n^7-\ 153902586554048401160*n^6+552159646814933160640*n^5-1360904291459312235792*n^4+ 2218680554883753672576*n^3-2225973296627996117760*n^2+1193370471466671052800*n-\ 241851513572444160000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 221/16384*(5037422*n^17-589065919*n^16+28823993744*n^15-706683494140*n^ 14+5684417360804*n^13+181222661780342*n^12-7290665023691192*n^11+ 135870575206357420*n^10-1646126649235342274*n^9+14062251536967478273*n^8-\ 87069415845297915848*n^7+392783552868742988680*n^6-1278683135008524022752*n^5+ 2932784263378271180304*n^4-4539497655958790200704*n^3+4399482315443635895040*n^ 2-2316047843049476659200*n+469476467522979840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n -25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 221/16384*(3257402*n^17-314942839*n^ 16+9848980544*n^15+74318081060*n^14-15629723681236*n^13+588489863604902*n^12-\ 12899770311837992*n^11+192330011913323020*n^10-2062500328346262134*n^9+ 16292609249546883913*n^8-95549171938792940648*n^7+414585873056221095880*n^6-\ 1312554156018426860832*n^5+2953197596282023227024*n^4-4517136895064961515904*n^ 3+4354979437154641847040*n^2-2295635822722390579200*n+469476467522979840000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n -27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 17/16384*( 32775202*n^18-1104285474*n^17-242013220611*n^16+26318013305040*n^15-\ 1312421328363936*n^14+41129174366227332*n^13-897420714204346202*n^12+ 14316297318866279880*n^11-171393465541784748834*n^10+1560684145217421038958*n^9 -10860314997798612836763*n^8+57610595350410301737720*n^7-\ 230745796136664422511032*n^6+685457619942086754161184*n^5-\ 1467047554360560012725424*n^4+2160054674550379148319360*n^3-\ 2026328817918885261062400*n^2+1050171830972678405376000*n-\ 213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25) /(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19) /(2*n-3)/(-1+2*n)/(2*n-5), -17/16384*(21839048*n^18-7579380276*n^17+ 868547896611*n^16-53503457756040*n^15+2105027343449436*n^14-57562022515577832*n ^13+1147450326819271202*n^12-17155718248990772880*n^11+195618264683607135084*n^ 10-1715557244063652840708*n^9+11593009839461321033763*n^8-\ 60107981458158055323720*n^7+236560583834599800675032*n^6-\ 693619319997116926493184*n^5+1471033837538642321477424*n^4-\ 2153830563981376956639360*n^3+2015770576579217824262400*n^2-\ 1045579126399084037376000*n+213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-\ 33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -323/16384*(7971064*n^19 -1815979472*n^18+182982084006*n^17-10992728120613*n^16+444797671681968*n^15-\ 12935291055815604*n^14+281033925621037492*n^13-4672858713404655566*n^12+ 60349595779676984352*n^11-610318598796257055276*n^10+4846342110134782868598*n^9 -30151661433357608609829*n^8+145929330226091008766536*n^7-\ 542381825459336223817448*n^6+1517011768027389510527904*n^5-\ 3096099595956040016326992*n^4+4396987784619068583070080*n^3-\ 4020776450773880668819200*n^2+2052412267414951588608000*n-\ 415980859513124505600000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15) /(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7) /(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -323/32768*(26419973*n^19-5381759059*n^18+ 499808160042*n^17-28206655754286*n^16+1086152821141926*n^15-30343596711409338*n ^14+637846825042419344*n^13-10319604849034248352*n^12+130281396775254170589*n^ 11-1292916614641629781947*n^10+10108045019424933947886*n^9-\ 62093664840591118454538*n^8+297477733697830296115952*n^7-\ 1096889020016487393576256*n^6+3049719062702389247413728*n^5-\ 6198421408437834088033824*n^4+8780676141717282467554560*n^3-\ 8021341716166607545382400*n^2+4096384343477462969856000*n-\ 831961719026249011200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15) /(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7) /(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/131072*(4819024193*n^20-1025100231850*n ^19+100976981059515*n^18-6124372972054950*n^17+256436264781113538*n^16-\ 7875507516094471380*n^15+183934667241677503430*n^14-3341972966901075882700*n^13 +47916152457338482271253*n^12-546616385787887098608210*n^11+ 4978956575093769282584895*n^10-36189702023296468593852750*n^9+ 208928852674045621674615608*n^8-949367998219445308413969040*n^7+ 3346827872995685132023461360*n^6-8959113914434350891851397600*n^5+ 17641749487433444784510225408*n^4-24351550397244196740415019520*n^3+ 21793248369254617764584140800*n^2-10961945536866269097664512000*n+ 2206362478857612377702400000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n -37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2 *n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/131072*(5791966943*n^20-\ 1198284041350*n^19+115212106434765*n^18-6841628313240450*n^17+ 281233899886051038*n^16-8499354784155670380*n^15+195750656693184893930*n^14-\ 3513905095597861833700*n^13+49859479149262954161003*n^12-\ 563748351099312583771710*n^11+5096494238389702666403145*n^10-\ 36811638963900950915444250*n^9+211422124297885722646977608*n^8-\ 956702446276648020530757040*n^7+3361725155560637927799177360*n^6-\ 8977149246271040371188669600*n^5+17647505021004310051144593408*n^4-\ 24334946160537557193126059520*n^3+21769982788664170420635340800*n^2-\ 10952540752216407152320512000*n+2206362478857612377702400000)/(2*n-29)/(2*n-35) /(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 1/131072*(-246688512334*n^21+55232520677397*n^20-5781089483137220*n^19+ 375951674399255535*n^18-17027603326423403994*n^17+570631149581582964042*n^16-\ 14672373740098599004360*n^15+296208129812889032165070*n^14-\ 4765001002523102063454914*n^13+61634244453742920673985817*n^12-\ 644000095909971092011820100*n^11+5440626736255180741052645955*n^10-\ 37067450265651010435726964854*n^9+202411542914673510438514072152*n^8-\ 876832027481501952564632201840*n^7+2967330803908754058066396744240*n^6-\ 7672342189087689593119526615904*n^5+14674354920925398616117299950592*n^4-\ 19775831043835504792387806036480*n^3+17363945337058821470468831539200*n^2-\ 8611500930239201258073858048000*n+1718756371030080042230169600000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29), 1/131072*(-263442586489*n^21+58449302915157*n^20-\ 6068170543783145*n^19+391766180075643135*n^18-17629813208701101324*n^17+ 587452374601926577002*n^16-15029258776097974301410*n^15+ 302082963042024546372270*n^14-4840979313848365486622369*n^13+ 62410920760182562997260377*n^12-650279337734182174218543525*n^11+ 5480586169187216788198252755*n^10-37265421265501629177657673834*n^9+ 203160330680909987203471373112*n^8-878922333473313623092261372640*n^7+ 2971371092533613214845629358640*n^6-7676963681716867415680555197984*n^5+ 14675520445593441112793391524352*n^4-19771274767201384970557836449280*n^3+ 17357949993998774592680764723200*n^2-8609141081674857898748141568000*n+ 1718756371030080042230169600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2* n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), ((365930254560*n+ 365930254560)*(2*n-45)!-(n-23)!*(n-21)!*binomial(2*n,n))/(n-23)!/(n-21)!/ binomial(2*n,n)] The limits, as n goes to infinity are 524285 4194027 4192641 4187391 8343907 33075523 8112247 3911921 [------, -------, -------, -------, -------, --------, -------, -------, 524288 4194304 4194304 4194304 8388608 33554432 8388608 4194304 58773611 848342651 720720611 556635131 359942921 278589217 --------, ----------, ----------, ----------, ----------, ----------, 67108864 1073741824 1073741824 1073741824 1073741824 2147483648 -46407977 -321831709 -8533651279 -91561459667 -110047371917 ---------, ----------, -----------, ------------, -------------, 536870912 1073741824 17179869184 137438953472 137438953472 -123344256167 -263442586489 -1088076307321 -------------, -------------, --------------] 137438953472 274877906944 1099511627776 and in Maple notation [524285/524288, 4194027/4194304, 4192641/4194304, 4187391/4194304, 8343907/ 8388608, 33075523/33554432, 8112247/8388608, 3911921/4194304, 58773611/67108864 , 848342651/1073741824, 720720611/1073741824, 556635131/1073741824, 359942921/ 1073741824, 278589217/2147483648, -46407977/536870912, -321831709/1073741824, -\ 8533651279/17179869184, -91561459667/137438953472, -110047371917/137438953472, -123344256167/137438953472, -263442586489/274877906944, -1088076307321/ 1099511627776] and in floating point [.9999942780, .9999339581, .9996035099, .9983518124, .9946712255, .9857273996, .9670552015, .9326746464, .8757950515, .7900806619, .6712233750, .5184068633, .\ 3352229679, .1297282134, -.8644159324e-1, -.2997291358, -.4967238800, -.6661972\ 996, -.8007000136, -.8974475798, -.9583985465, -.9895996366] The cut off is at j=, 15 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 24], vs. those in the, 2, -th row from j=1 to j=, 23, are as follws 12 11 10 9 [23 (364721 n - 26259762 n + 840480157 n - 15765667830 n 8 7 6 + 192336560823 n - 1600328754486 n + 9252004651351 n 5 4 3 - 37032603823050 n + 99125144390756 n - 155973537784152 n 2 + 46301907868992 n + 377653084663680 n - 675806113382400)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) 12 11 (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 23)), 23 (364709 n - 26257398 n 10 9 8 7 + 840269353 n - 15754440570 n + 191939982267 n - 1590582787794 n 6 5 4 + 9082241237179 n - 34942768850550 n + 81416205517724 n 3 2 - 59099658894408 n - 245563461688032 n + 616582670296320 n - 28158588057600)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) (2 n - 15) (2 n - 17) 13 12 11 10 (2 n - 23)), 897 (37399 n - 3159253 n + 119720933 n - 2686511245 n 9 8 7 6 + 39651120487 n - 403486799859 n + 2870966481119 n - 13999771430975 n 5 4 3 + 42742908087814 n - 53517500518988 n - 120919289428552 n 2 + 518460857341920 n - 412816549017600 n + 36100753920000)/(4096 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21) (2 n - 23) (2 n - 25)), 13 ( 13 12 11 10 1289303 n - 108816166 n + 4115589151 n - 91959389390 n 9 8 7 + 1344682463289 n - 13409433460698 n + 91291390089493 n 6 5 4 - 402593889389450 n + 931780601565308 n + 287096616448664 n 3 2 - 7616987707854144 n + 15672539382402240 n - 10246268741587200 n + 1245476010240000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) (2 n - 21) 14 13 12 (2 n - 23) (2 n - 25)), 247 (541621 n - 52913483 n + 2331855869 n 11 10 9 - 61138498687 n + 1056948588003 n - 12569418948009 n 8 7 6 + 103319649852287 n - 564527072025301 n + 1771740505398556 n 5 4 3 - 1016843999071208 n - 14802164623334256 n + 55936529233866288 n 2 - 82516748018662080 n + 48442018976870400 n - 7079547847680000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 14 13 12 (-1 + 2 n)), 247 (538471 n - 52359083 n + 2288303969 n 11 10 9 - 59129283787 n + 996726610803 n - 11345411232309 n 8 7 6 + 86252660751587 n - 403380034350601 n + 783274247122906 n 5 4 3 + 2588553964688692 n - 21172630433281656 n + 57773400261026688 n 2 - 75140688695806080 n + 42837376930790400 n - 7079547847680000)/(8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 15 14 13 (-1 + 2 n)), 4199 (125087 n - 13866750 n + 693638120 n 12 11 10 - 20602528800 n + 401326648394 n - 5320423318980 n 9 8 7 + 47762718870460 n - 271780236030600 n + 724797977808751 n 6 5 4 + 1805288094019410 n - 24453973727673980 n + 99810538488053400 n 3 2 - 214774786209359232 n + 248474247167272320 n - 136343794716057600 n + 24153751480320000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 15 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 (289741 n 14 13 12 11 - 31685625 n + 1552843285 n - 44727764025 n + 831691179067 n 10 9 8 - 10248832224015 n + 80993724334655 n - 341591782932675 n 7 6 5 - 199004995367332 n + 12125338885963380 n - 77043325354887640 n 4 3 2 + 258499970638501200 n - 506586438516074976 n + 560532337974941760 n - 306402088950316800 n + 57365159765760000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 ( 16 15 14 13 4433581 n - 543225368 n + 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(2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 17 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 (7316387 n 16 15 14 13 - 955979284 n + 55414957364 n - 1853686578640 n + 38501563498034 n 12 11 10 - 475936818681688 n + 2183327024254948 n + 36242281319735920 n 9 8 7 - 880476910610211629 n + 9800778972373388428 n - 70283410521229029488 n 6 5 + 348166853319318167680 n - 1206954157402028432592 n 4 3 + 2886730883214553220544 n - 4584360723948829021824 n 2 + 4494514154322109847040 n - 2360474914564414771200 n + 469476467522979840000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 ( 17 16 15 14 6089252 n - 751047739 n + 40036501544 n - 1168184424940 n 13 12 11 + 18279137067464 n - 59435230206898 n - 3976193717058992 n 10 9 + 102508180788605020 n - 1400087657033435084 n 8 7 + 12744312888625102213 n - 82058650880959946648 n 6 5 + 379900363667051379880 n - 1258668440775399618432 n 4 3 + 2920721839389690425424 n - 4552710832850598059904 n 2 + 4425779470796223287040 n - 2328109491424572979200 n + 469476467522979840000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 ( 18 17 16 15 18221333 n - 2344424121 n + 125580491556 n - 3278859248340 n 14 13 12 + 19502218177806 n + 1576348748352378 n - 63366991884062308 n 11 10 + 1327671817758592020 n - 18687271988895664011 n 9 8 + 189681594449260606407 n - 1426389403338837216552 n 7 6 + 8011136586060613703880 n - 33472946154048711604528 n 5 4 + 102545347161380980308336 n - 224202670441946427453696 n 3 2 + 334466708210185776445440 n - 315518439629440385049600 n + 163227299112819938304000 n - 32863352726608588800000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 18 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 17 (143426279 n 17 16 15 - 14590835073 n + 409365596778 n + 13871596559580 n 14 13 12 - 1494645190772622 n + 58826324519491914 n - 1438317721531854904 n 11 10 + 24583790718851338260 n - 308244161936896835793 n 9 8 + 2900530723561511545791 n - 20675861425078178800026 n 7 6 + 111660103102735843732440 n - 453200135741087657640064 n 5 4 + 1359277260176074929256368 n - 2928410964189410066674848 n 3 2 + 4328984469726928088478720 n - 4067712905896185200524800 n + 2106892518466963594752000 n - 427223585445911654400000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (8530289 n 18 17 16 + 312616148 n - 184755811269 n + 17826420005592 n 15 14 13 - 931138259945982 n + 31955986727620536 n - 781029789952353658 n 12 11 + 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3093432931730716849735632 n 3 2 + 4402687539270863453892480 n - 4029438387365803722643200 n + 2056029492280283106048000 n - 415980859513124505600000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 20 19 18 - 323 (286507583 n - 69560595250 n + 7594856468265 n 17 16 15 - 500271465886050 n + 22410540617447478 n - 727899329615006580 n 14 13 + 17815386868277210930 n - 336684450617791474900 n 12 11 + 4989742346162186832843 n - 58526299189406334639210 n 10 9 + 545624527365390092472645 n - 4042938882014353769875050 n 8 7 + 23710702532330485333292048 n - 109109928152813842064497840 n 6 5 + 388456390427889447353193360 n - 1047545573587031993741892000 n 4 3 + 2073357858035329825229970048 n - 2870768557450979516126661120 n 2 + 2572251668576527673862604800 n - 1293029139983114271209472000 n + 259572056336189691494400000)/(262144 (2 n 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(2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 19 (39687293419 n 20 19 18 - 9239708695077 n + 1000823891658395 n - 67072920065365935 n 17 16 + 3119126573775053604 n - 106973582911449300522 n 15 14 + 2806686585921932641510 n - 57666854808420404745870 n 13 12 + 941904087527059114793699 n - 12344123175600324852768297 n 11 10 + 130433346151205413087200975 n - 1112414305826777287329241155 n 9 8 + 7639200087099713678305991614 n - 41987094173039465749262947032 n 7 6 + 182839185953683152960736641440 n - 621286310389933548577157409840 n 5 4 + 1611298082695388760779083949664 n - 3088315253863353827698503939072 n 3 2 + 4167178872935595427298140807680 n - 3660664454332352895951329587200 n + 1814960105677843967986311168000 n - 361843446532648429943193600000)/( 524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 21 20 (2 n - 39) (2 n - 35) (2 n - 29)), (-446837929793 n + 101529535138794 n 19 18 - 10764731575591315 n + 707974166364101070 n 17 16 - 32382401424297648738 n + 1094536674106655892084 n 15 14 - 28353400157976711072470 n + 576097278433734969310140 n 13 12 - 9318951140253805729222453 n + 121111054722931279338875634 n 11 10 - 1270557853419355844560519575 n + 10770255047699150240034161910 n 9 8 - 73584981087272524366090849208 n + 402743119812023474307702308304 n 7 6 - 1747857649430193709219183373680 n + 5923438583859566013968258448480 n 5 - 15331846898649885234680640503808 n 4 + 29345472273328456963689901085184 n 3 - 39564318411654675756525527592960 n 2 + 34744544404839995379237848678400 n - 17229556995379356292052484096000 n + 3437512742060160084460339200000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), ( 22 21 20 -3974939654269 n + 975143191458623 n - 112206423338002763 n 19 18 + 8050853224727562865 n - 403906085260055641494 n 17 16 + 15058787687525452728858 n - 432855424817432374790758 n 15 14 + 9821994793228656992108210 n - 178683272929633610970191129 n 13 12 + 2632047828950210456478055363 n - 31572308831648961687552791823 n 11 10 + 309102672158734801812299639805 n - 2467848512548745289842394648484 n 9 + 16005223479285078894686264042708 n 8 - 83716030168918352450917611236528 n 7 + 349264049393298004927074636938800 n 6 - 1143816528075844632071532957541824 n 5 + 2874408541293842746002327822752448 n 4 - 5364467324973191379847959359728128 n 3 + 7080372479075142793414778649400320 n 2 - 6110547611498484540796772128972800 n + 2989687299458321906034746867712000 n - 591252191634347534527178342400000 )/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 22 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), (-4226516704279 n 21 20 19 + 1028477526060743 n - 117482497230812483 n + 8374408984515923965 n 18 17 - 417688054938395514504 n + 15491737214721539249478 n 16 15 - 443249724525738568047478 n + 10016991256010197978988810 n 14 13 - 181581149859962331407198039 n + 2666424280261682540378061283 n 12 11 - 31898703912822767940485636343 n + 311578615264149506066247358905 n 10 - 2482755540161794040088553993594 n 9 + 16075577546177277132701372390528 n 8 - 83970937082632959854205221728448 n 7 + 349948549844647594805601085464400 n 6 - 1145091444452455788978207562058784 n 5 + 2875804621160341525463402261635968 n 4 - 5364746150200728665495713086325248 n 3 + 7078922905832123320331716440913920 n 2 - 6108723726649642250999526655180800 n + 2988983427801614349446024036352000 n - 591252191634347534527178342400000 )/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), ( (1372238454600 n + 1372238454600) (2 n - 47)! - (n - 24)! (n - 22)! binomial(2 n, n))/((n - 24)! (n - 22)! binomial(2 n, n))] and in Maple notation [23/2048*(364721*n^12-26259762*n^11+840480157*n^10-15765667830*n^9+192336560823 *n^8-1600328754486*n^7+9252004651351*n^6-37032603823050*n^5+99125144390756*n^4-\ 155973537784152*n^3+46301907868992*n^2+377653084663680*n-675806113382400)/(2*n-\ 5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-15 )/(2*n-17)/(2*n-23), 23/2048*(364709*n^12-26257398*n^11+840269353*n^10-\ 15754440570*n^9+191939982267*n^8-1590582787794*n^7+9082241237179*n^6-\ 34942768850550*n^5+81416205517724*n^4-59099658894408*n^3-245563461688032*n^2+ 616582670296320*n-28158588057600)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2* n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-23), 897/4096*(37399*n^ 13-3159253*n^12+119720933*n^11-2686511245*n^10+39651120487*n^9-403486799859*n^8 +2870966481119*n^7-13999771430975*n^6+42742908087814*n^5-53517500518988*n^4-\ 120919289428552*n^3+518460857341920*n^2-412816549017600*n+36100753920000)/(-1+2 *n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19 )/(2*n-21)/(2*n-23)/(2*n-25), 13/2048*(1289303*n^13-108816166*n^12+4115589151*n ^11-91959389390*n^10+1344682463289*n^9-13409433460698*n^8+91291390089493*n^7-\ 402593889389450*n^6+931780601565308*n^5+287096616448664*n^4-7616987707854144*n^ 3+15672539382402240*n^2-10246268741587200*n+1245476010240000)/(-1+2*n)/(2*n-3)/ (2*n-5)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/( 2*n-23)/(2*n-25), 247/8192*(541621*n^14-52913483*n^13+2331855869*n^12-\ 61138498687*n^11+1056948588003*n^10-12569418948009*n^9+103319649852287*n^8-\ 564527072025301*n^7+1771740505398556*n^6-1016843999071208*n^5-14802164623334256 *n^4+55936529233866288*n^3-82516748018662080*n^2+48442018976870400*n-\ 7079547847680000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15 )/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 247/8192*(538471* n^14-52359083*n^13+2288303969*n^12-59129283787*n^11+996726610803*n^10-\ 11345411232309*n^9+86252660751587*n^8-403380034350601*n^7+783274247122906*n^6+ 2588553964688692*n^5-21172630433281656*n^4+57773400261026688*n^3-\ 75140688695806080*n^2+42837376930790400*n-7079547847680000)/(2*n-27)/(2*n-25)/( 2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/( 2*n-5)/(2*n-3)/(-1+2*n), 4199/16384*(125087*n^15-13866750*n^14+693638120*n^13-\ 20602528800*n^12+401326648394*n^11-5320423318980*n^10+47762718870460*n^9-\ 271780236030600*n^8+724797977808751*n^7+1805288094019410*n^6-24453973727673980* n^5+99810538488053400*n^4-214774786209359232*n^3+248474247167272320*n^2-\ 136343794716057600*n+24153751480320000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-\ 23)/(2*n-29), 221/2048*(289741*n^15-31685625*n^14+1552843285*n^13-44727764025*n ^12+831691179067*n^11-10248832224015*n^10+80993724334655*n^9-341591782932675*n^ 8-199004995367332*n^7+12125338885963380*n^6-77043325354887640*n^5+ 258499970638501200*n^4-506586438516074976*n^3+560532337974941760*n^2-\ 306402088950316800*n+57365159765760000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-\ 23)/(2*n-29), 221/16384*(4433581*n^16-543225368*n^15+29891905060*n^14-\ 969294882080*n^13+20357721298222*n^12-284379523071056*n^11+2548953154282700*n^ 10-11814291769366240*n^9-21629543192881987*n^8+776839199140312616*n^7-\ 6058218393892795480*n^6+26969378720823465920*n^5-75663220286603616816*n^4+ 133207200191175239808*n^3-138541714617601025280*n^2+74026375930544486400*n-\ 14226559621908480000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11 )/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-\ 29), 13/16384*(69976877*n^16-8334033256*n^15+440953146020*n^14-13523071915360*n ^13+261040202441774*n^12-3151003079931952*n^11+19875777657085900*n^10+ 30740042574413920*n^9-1977669343345155779*n^8+20929515737765568472*n^7-\ 127492520692554163160*n^6+505253777353707000640*n^5-1325519614481340301872*n^4+ 2243382033386755748736*n^3-2289827962250455829760*n^2+1224857902823083468800*n-\ 241851513572444160000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 221/16384*(7316387*n^17-955979284*n^16+55414957364*n^15-1853686578640*n^ 14+38501563498034*n^13-475936818681688*n^12+2183327024254948*n^11+ 36242281319735920*n^10-880476910610211629*n^9+9800778972373388428*n^8-\ 70283410521229029488*n^7+348166853319318167680*n^6-1206954157402028432592*n^5+ 2886730883214553220544*n^4-4584360723948829021824*n^3+4494514154322109847040*n^ 2-2360474914564414771200*n+469476467522979840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n -25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 221/16384*(6089252*n^17-751047739*n^ 16+40036501544*n^15-1168184424940*n^14+18279137067464*n^13-59435230206898*n^12-\ 3976193717058992*n^11+102508180788605020*n^10-1400087657033435084*n^9+ 12744312888625102213*n^8-82058650880959946648*n^7+379900363667051379880*n^6-\ 1258668440775399618432*n^5+2920721839389690425424*n^4-4552710832850598059904*n^ 3+4425779470796223287040*n^2-2328109491424572979200*n+469476467522979840000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n -27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 221/32768* (18221333*n^18-2344424121*n^17+125580491556*n^16-3278859248340*n^15+ 19502218177806*n^14+1576348748352378*n^13-63366991884062308*n^12+ 1327671817758592020*n^11-18687271988895664011*n^10+189681594449260606407*n^9-\ 1426389403338837216552*n^8+8011136586060613703880*n^7-33472946154048711604528*n ^6+102545347161380980308336*n^5-224202670441946427453696*n^4+ 334466708210185776445440*n^3-315518439629440385049600*n^2+ 163227299112819938304000*n-32863352726608588800000)/(2*n-29)/(2*n-35)/(2*n-23)/ (2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33 )/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 17/32768*(143426279*n^18-\ 14590835073*n^17+409365596778*n^16+13871596559580*n^15-1494645190772622*n^14+ 58826324519491914*n^13-1438317721531854904*n^12+24583790718851338260*n^11-\ 308244161936896835793*n^10+2900530723561511545791*n^9-20675861425078178800026*n ^8+111660103102735843732440*n^7-453200135741087657640064*n^6+ 1359277260176074929256368*n^5-2928410964189410066674848*n^4+ 4328984469726928088478720*n^3-4067712905896185200524800*n^2+ 2106892518466963594752000*n-427223585445911654400000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-\ 33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(8530289*n^19+ 312616148*n^18-184755811269*n^17+17826420005592*n^16-931138259945982*n^15+ 31955986727620536*n^14-781029789952353658*n^13+14184978491970918944*n^12-\ 196156802405961748923*n^11+2093625734294485332084*n^10-17353295046807977475777* n^9+111710055636901843930536*n^8-555389523651358465727464*n^7+ 2107549386142484229806432*n^6-5986581668028386875801296*n^5+ 12350656181472069876956928*n^4-17655640180741429319473920*n^3+ 16188074686167583249612800*n^2-8253830887657784174592000*n+ 1663923438052498022400000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15 )/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7 )/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -323/16384*(3480559*n^19-1066065137*n^18+ 125620373136*n^17-8326499757873*n^16+360558752634258*n^15-11018284798767834*n^ 14+248557222563747052*n^13-4255478389593968186*n^12+56242936401434040687*n^11-\ 579341705775207481821*n^10+4668330339068196492588*n^9-29384372015982222366309*n ^8+143521155977956032231016*n^7-537185238703129818414008*n^6+ 1510285113749842270660224*n^5-3093432931730716849735632*n^4+ 4402687539270863453892480*n^3-4029438387365803722643200*n^2+ 2056029492280283106048000*n-415980859513124505600000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -323/262144*( 286507583*n^20-69560595250*n^19+7594856468265*n^18-500271465886050*n^17+ 22410540617447478*n^16-727899329615006580*n^15+17815386868277210930*n^14-\ 336684450617791474900*n^13+4989742346162186832843*n^12-58526299189406334639210* n^11+545624527365390092472645*n^10-4042938882014353769875050*n^9+ 23710702532330485333292048*n^8-109109928152813842064497840*n^7+ 388456390427889447353193360*n^6-1047545573587031993741892000*n^5+ 2073357858035329825229970048*n^4-2870768557450979516126661120*n^3+ 2572251668576527673862604800*n^2-1293029139983114271209472000*n+ 259572056336189691494400000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/262144*(7594868611*n^20-\ 1686514463750*n^19+172060198831005*n^18-10742509727620350*n^17+ 460797495841858326*n^16-14440935769260424860*n^15+343055756635189486810*n^14-\ 6322888463538901268300*n^13+91751318861635573574031*n^12-\ 1057255644421780678373070*n^11+9711084057266078459151465*n^10-\ 71073336471323524312363350*n^9+412621834938027031307271016*n^8-\ 1883333655518764921382683280*n^7+6662371452604969392917919120*n^6-\ 17880353632011653877094524000*n^5+35271412354368072509088278016*n^4-\ 48737969691572336530136855040*n^3+43635354457749174951460761600*n^2-\ 21943641121497248280551424000*n+4412724957715224755404800000)/(2*n-29)/(2*n-35) /(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), -19/524288*(39687293419*n^21-9239708695077*n^20+1000823891658395*n^19-\ 67072920065365935*n^18+3119126573775053604*n^17-106973582911449300522*n^16+ 2806686585921932641510*n^15-57666854808420404745870*n^14+ 941904087527059114793699*n^13-12344123175600324852768297*n^12+ 130433346151205413087200975*n^11-1112414305826777287329241155*n^10+ 7639200087099713678305991614*n^9-41987094173039465749262947032*n^8+ 182839185953683152960736641440*n^7-621286310389933548577157409840*n^6+ 1611298082695388760779083949664*n^5-3088315253863353827698503939072*n^4+ 4167178872935595427298140807680*n^3-3660664454332352895951329587200*n^2+ 1814960105677843967986311168000*n-361843446532648429943193600000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29), 1/262144*(-446837929793*n^21+101529535138794*n^20-\ 10764731575591315*n^19+707974166364101070*n^18-32382401424297648738*n^17+ 1094536674106655892084*n^16-28353400157976711072470*n^15+ 576097278433734969310140*n^14-9318951140253805729222453*n^13+ 121111054722931279338875634*n^12-1270557853419355844560519575*n^11+ 10770255047699150240034161910*n^10-73584981087272524366090849208*n^9+ 402743119812023474307702308304*n^8-1747857649430193709219183373680*n^7+ 5923438583859566013968258448480*n^6-15331846898649885234680640503808*n^5+ 29345472273328456963689901085184*n^4-39564318411654675756525527592960*n^3+ 34744544404839995379237848678400*n^2-17229556995379356292052484096000*n+ 3437512742060160084460339200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2* n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/1048576*(-\ 3974939654269*n^22+975143191458623*n^21-112206423338002763*n^20+ 8050853224727562865*n^19-403906085260055641494*n^18+15058787687525452728858*n^ 17-432855424817432374790758*n^16+9821994793228656992108210*n^15-\ 178683272929633610970191129*n^14+2632047828950210456478055363*n^13-\ 31572308831648961687552791823*n^12+309102672158734801812299639805*n^11-\ 2467848512548745289842394648484*n^10+16005223479285078894686264042708*n^9-\ 83716030168918352450917611236528*n^8+349264049393298004927074636938800*n^7-\ 1143816528075844632071532957541824*n^6+2874408541293842746002327822752448*n^5-\ 5364467324973191379847959359728128*n^4+7080372479075142793414778649400320*n^3-\ 6110547611498484540796772128972800*n^2+2989687299458321906034746867712000*n-\ 591252191634347534527178342400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/ 1048576*(-4226516704279*n^22+1028477526060743*n^21-117482497230812483*n^20+ 8374408984515923965*n^19-417688054938395514504*n^18+15491737214721539249478*n^ 17-443249724525738568047478*n^16+10016991256010197978988810*n^15-\ 181581149859962331407198039*n^14+2666424280261682540378061283*n^13-\ 31898703912822767940485636343*n^12+311578615264149506066247358905*n^11-\ 2482755540161794040088553993594*n^10+16075577546177277132701372390528*n^9-\ 83970937082632959854205221728448*n^8+349948549844647594805601085464400*n^7-\ 1145091444452455788978207562058784*n^6+2875804621160341525463402261635968*n^5-\ 5364746150200728665495713086325248*n^4+7078922905832123320331716440913920*n^3-\ 6108723726649642250999526655180800*n^2+2988983427801614349446024036352000*n-\ 591252191634347534527178342400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), (( 1372238454600*n+1372238454600)*(2*n-47)!-(n-24)!*(n-22)!*binomial(2*n,n))/(n-24 )!/(n-22)!/binomial(2*n,n)] The limits, as n goes to infinity are 8388583 8388307 33546903 16760939 133780387 133002337 525240313 [-------, -------, --------, --------, ---------, ---------, ---------, 8388608 8388608 33554432 16777216 134217728 134217728 536870912 64032761 979821401 909699401 1616921527 336431173 4026914593 --------, ----------, ----------, ----------, ---------, ----------, 67108864 1073741824 1073741824 2147483648 536870912 8589934592 2438246743 2755283347 -1124220557 -92541949309 -144302503609 ----------, -----------, -----------, ------------, -------------, 8589934592 34359738368 8589934592 274877906944 274877906944 -754058574961 -446837929793 -3974939654269 -4226516704279 -------------, -------------, --------------, --------------, 1099511627776 549755813888 4398046511104 4398046511104 -17420656237591 ---------------] 17592186044416 and in Maple notation [8388583/8388608, 8388307/8388608, 33546903/33554432, 16760939/16777216, 133780387/134217728, 133002337/134217728, 525240313/536870912, 64032761/ 67108864, 979821401/1073741824, 909699401/1073741824, 1616921527/2147483648, 336431173/536870912, 4026914593/8589934592, 2438246743/8589934592, 2755283347/ 34359738368, -1124220557/8589934592, -92541949309/274877906944, -144302503609/ 274877906944, -754058574961/1099511627776, -446837929793/549755813888, -\ 3974939654269/4398046511104, -4226516704279/4398046511104, -17420656237591/ 17592186044416] and in floating point [.9999970198, .9999641180, .9997756183, .9990298152, .9967415556, .9909446314, .9783363212, .9541624933, .9125297898, .8472235883, .7529377597, .6266518924, .\ 4687945583, .2838492793, .8018929939e-1, -.1308764979, -.3366656504, -.52496944\ 99, -.6858122788, -.8127934594, -.9037966389, -.9609986374, -.9902496593] The cut off is at j=, 16 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 25], vs. those in the, 2, -th row from j=1 to j=, 24, are as follws 12 11 10 9 [5 (1677719 n - 120795378 n + 3866250883 n - 72524317470 n 8 7 6 + 884827495497 n - 7363461463974 n + 42593174079049 n 5 4 3 - 170767944580530 n + 459517451972084 n - 736853049585048 n 2 + 271361850108768 n + 1689418272326400 n - 3238237626624000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) 13 (2 n - 21) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 23)), 1495 (11222 n 12 11 10 9 - 948229 n + 35956684 n - 808105265 n + 11971347036 n 8 7 6 5 - 122914016907 n + 893827932052 n - 4600483132355 n + 16217085710342 n 4 3 2 - 34622743609564 n + 19665075669864 n + 109743419956320 n - 247977407059200 n + 10830226176000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) (2 n - 25) 13 12 (2 n - 15) (2 n - 17) (2 n - 23)), 299 (56104 n - 4739897 n 11 10 9 8 + 179665874 n - 4033911145 n + 59609716122 n - 608149406271 n 7 6 5 + 4350775170302 n - 21459075574915 n + 67216360574374 n 4 3 2 - 92467395831932 n - 161691664299576 n + 790344772846560 n - 642058550380800 n + 54151130880000)/(2048 (-1 + 2 n) (2 n - 3) (2 n - 5) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 13) (2 n - 15) (2 n - 17) (2 n - 19) 14 13 (2 n - 21) (2 n - 23) (2 n - 25)), 299 (224317 n - 21968513 n 12 11 10 9 + 972657833 n - 25726511797 n + 452085101391 n - 5541505289859 n 8 7 6 + 48184213344179 n - 293360035978471 n + 1173350790283672 n 5 4 3 - 2440049116155728 n - 1274472802542912 n + 20169866772841968 n 2 - 39369538539708480 n + 25023596242982400 n - 2924161067520000)/(4096 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 14 13 12 (-1 + 2 n)), 13 (10304029 n - 1007684657 n + 44488179491 n 11 10 9 - 1170070177633 n + 20334955476297 n - 243959792418111 n 8 7 6 + 2034754701416393 n - 11402831926714459 n + 37814627887330294 n 5 4 3 - 34462707430144532 n - 254485171441571784 n + 1055079197129385792 n 2 - 1598797661510574720 n + 943937857154073600 n - 134511409105920000)/( 8192 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 15 14 13 (-1 + 2 n)), 247 (1080596 n - 120991545 n + 6148113965 n 12 11 10 - 187152152775 n + 3787382702117 n - 53295173102325 n 9 8 7 + 526999350397795 n - 3574208623627125 n + 15158912604823603 n 6 5 4 - 26306590313547390 n - 93294538951150460 n + 697863351337559400 n 3 2 - 1825743132209874816 n + 2305481026071461760 n - 1282448565782092800 n + 205306887582720000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 15 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 247 (1071461 n 14 13 12 11 - 119265030 n + 6001816940 n - 179829125700 n + 3547311275522 n 10 9 8 - 47897232148920 n + 442318971339520 n - 2650484637216900 n 7 6 5 + 8311631027157073 n + 6566635576977630 n - 186617822403367460 n 4 3 2 + 824365642411554600 n - 1829509802958445056 n + 2145176379854824320 n - 1178426409406848000 n + 205306887582720000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 4199 ( 16 15 14 13 247825 n - 31139048 n + 1774511140 n - 60424893200 n 12 11 10 + 1360768454710 n - 21113197987616 n + 226396128341900 n 9 8 7 - 1605129237012400 n + 6245703336725105 n + 2700298279239176 n 6 5 4 - 188691496704140920 n + 1152692410383886400 n - 3735338510309798640 n 3 2 + 7110740294116495488 n - 7676244165547941120 n + 4102214058035251200 n - 748766295889920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 16 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 (4563037 n 15 14 13 12 - 564844520 n + 31504926820 n - 1040213468000 n + 22398706729294 n 11 10 9 - 324782294565680 n + 3111907377459980 n - 17372283833053600 n 8 7 6 + 17009667424705901 n + 591481214683186520 n - 5470151001979756120 n 5 4 + 25846794582430472000 n - 74721343415985725232 n 3 2 + 133714569067892599680 n - 140110863087571303680 n + 74832547642452633600 n - 14226559621908480000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 15 (2 n - 23) (2 n - 29)), 221 (8651402 n - 1189606909 n + 73832824304 n 14 13 12 - 2718108791140 n + 65400009094364 n - 1060971697783438 n 11 10 9 + 11322021099077128 n - 67518235249165580 n - 23475705862129334 n 8 7 + 4701539881708274803 n - 48926931824412908168 n 6 5 + 288074644559314670680 n - 1104627860756359427232 n 4 3 + 2814258282929724938544 n - 4642400499874579707264 n 2 + 4632282548873710583040 n - 2426984080796836915200 n + 469476467522979840000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 221 ( 17 16 15 14 7939394 n - 1060021453 n + 63222481088 n - 2201710748980 n 13 12 11 + 48768333839708 n - 687391471291966 n + 5310468323691256 n 10 9 8 + 2599593897079300 n - 616674531656882798 n + 8306369445660979891 n 7 6 - 64305211569365640776 n + 332055994219699767640 n 5 4 - 1180699213535547676704 n + 2869473936233329716528 n 3 2 - 4600429130198700125568 n + 4529410224419867639040 n - 2376906590927719065600 n + 469476467522979840000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 221 (13896493 n - 2019303351 n 16 15 14 + 130758553731 n - 4912647800580 n + 115353156162426 n 13 12 11 - 1624856451667722 n + 8582196899610022 n + 165177592039462860 n 10 9 - 4732784004111365631 n + 63096195840742904697 n 8 7 - 552473441532651193737 n + 3422573228644641555120 n 6 5 - 15297180468306612901688 n + 49123251800045978807376 n 4 3 - 110899122505425963488016 n + 168735765123674853321600 n 2 - 160566097622475151353600 n + 82875901058863799040000 n - 16431676363304294400000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 18 17 (-1 + 2 n) (2 n - 5)), 221 (22639019 n - 3095430741 n 16 15 14 + 183403583610 n - 5949615496500 n + 102143172185418 n 13 12 11 - 235794089036142 n - 34315687333125640 n + 981443664508172580 n 10 9 - 15599907846872785293 n + 169133558287898396907 n 8 7 - 1325566169296099635450 n + 7655866559546714603640 n 6 5 - 32619586646412685102144 n + 101307380935869749732976 n 4 3 - 223551429606406846991520 n + 335368801232263343969280 n 2 - 317126979704804084928000 n + 163939338421896457728000 n - 32863352726608588800000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 19 18 17 (-1 + 2 n) (2 n - 5)), 4199 (1721296 n - 241540151 n + 13971850509 n 16 15 14 - 375474654654 n + 356844874452 n + 344829716623518 n 13 12 11 - 14019724784698562 n + 326442838040255872 n - 5260546882543896372 n 10 9 + 62359857837724235217 n - 557874808538658510003 n 8 7 + 3802938788625926085318 n - 19747510742006376188696 n 6 5 + 77439680463084299970016 n - 225362742182546011624944 n 4 3 + 472845145315234080154464 n - 682984292626114323079680 n 2 + 628930246945835125670400 n - 320149708685684494848000 n + 63997055309711462400000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 19 18 (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (12497737 n - 1292628206 n 17 16 15 + 24121510833 n + 3237364359966 n - 278166988089606 n 14 13 12 + 11535037459078308 n - 312341891531877494 n + 6052279721636976412 n 11 10 - 87512922029730256059 n + 964774467918249647802 n 9 8 - 8192637876600225623211 n + 53718302595207671935878 n 7 6 - 270839891730175156358312 n + 1038671117892229265491696 n 5 4 - 2973285945458166628726128 n + 6166867057404075847083744 n 3 2 - 8844279765711972344501760 n + 8119821759875393356454400 n - 4137810180625664395776000 n + 831961719026249011200000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 20 19 18 17 323 (7181467 n + 2416559890 n - 589313169975 n + 55121504112750 n 16 15 14 - 3048019046454558 n + 114328332918955140 n - 3110081002756523150 n 13 12 + 63736866782495019100 n - 1006888146746199085953 n 11 10 + 12430277266804399435290 n - 120778202391182271894075 n 9 8 + 925414339837622823218550 n - 5575695801704277837927788 n 7 6 + 26214222405372354822731440 n - 94898442580258692237522000 n 5 4 + 259124379440471633489205600 n - 517363096188419760095823168 n 3 2 + 720144862315130128233738240 n - 646622790145848227272780800 n + 324717377812669634817024000 n - 64893014084047422873600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), - 323 (146403827 n - 43641400390 n + 5386120754925 n 17 16 15 - 385179732925950 n + 18305599199691702 n - 621614618230733340 n 14 13 + 15748257617589827450 n - 305865735033059956300 n 12 11 + 4633563541637012108607 n - 55321859053310009464590 n 10 9 + 523229076916875681135225 n - 3922422428907407529758550 n 8 7 + 23220079520926605186024872 n - 107646020196880319455492240 n 6 5 + 385442017068017223974778000 n - 1043838659705380623162799200 n 4 3 + 2072106664264431600535060992 n - 2874116340684401442961789440 n 2 + 2577030612419191442650214400 n - 1294975930405635693895680000 n + 259572056336189691494400000)/(262144 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 19 (10721984114 n 20 19 18 - 2786400165843 n + 329608813728460 n - 23742394078955685 n 17 16 + 1172413509999630234 n - 42290835183856936278 n 15 14 + 1158042603185182862120 n - 24673593072234525853770 n 13 12 + 415658524854609179229814 n - 5592398985566259022302783 n 11 10 + 60420506937638724592077180 n - 525031327012637182363840305 n 9 8 + 3662114869984761810879145214 n - 20387153382194467168026202728 n 7 6 + 89698379393931515681397083440 n - 307261727181652920296058957840 n 5 4 + 801706074603792744624981702624 n - 1543072216133572028733188602368 n 3 2 + 2087387974880043924268416268800 n - 1835437219341463522302510182400 n + 909508031632397448232869888000 n - 180921723266324214971596800000)/( 262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 21 (2 n - 39) (2 n - 35) (2 n - 29)), - 19 (15934362740 n 20 19 18 - 3849725405547 n + 430025287958350 n - 29572700191067985 n 17 16 + 1405524144375186990 n - 49102683814642948182 n 15 14 + 1308710147824738665980 n - 27250746570000699416370 n 13 12 + 450183754473711885630640 n - 5956916052735383605079847 n 11 10 + 63455973436560059010602310 n - 544877280942081747081863805 n 9 8 + 3762895548999643151510695790 n - 20777066477103414092008032072 n 7 6 + 90810011536110166139205945040 n - 309454911626275725349403973840 n 5 4 + 804276195017247627981898767840 n - 1543794894680587560218221604352 n 3 2 + 2084926765816872638606251768320 n - 1832111230199948447384452608000 n + 908183637318291022606934016000 n - 180921723266324214971596800000)/( 262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 22 (2 n - 39) (2 n - 35) (2 n - 29)), (-1547209989617 n 21 20 19 + 392915730663049 n - 46615171910661739 n + 3436347943269428495 n 18 17 - 176570193653873846442 n + 6723400469100415354254 n 16 15 - 196883005240714927345574 n + 4540922484168867269575630 n 14 13 - 83794411356115980028521397 n + 1249694139927689028264160469 n 12 11 - 15151735364679552470516167119 n + 149706186991453766815771144515 n 10 9 - 1204570981063430265930626117312 n + 7863283788486720847160410557604 n 8 - 41350527482069534509595523677584 n 7 + 173262497174178027750945386483600 n 6 - 569344915928842100665064360360832 n 5 + 1434380464453543060080651619218624 n 4 - 2681648961152390376278778312297984 n 3 + 3543097546802570936423055699317760 n 2 - 3058959336055974736643799552614400 n + 1496269915980910475578732855296000 n - 295626095817173767263589171200000 )/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 22 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), - 5 (724546356851 n 21 20 19 - 180095024603131 n + 20963983977613831 n - 1519575032204771465 n 18 17 + 76922265542075963856 n - 2890531669890186319998 n 16 15 + 83660681045160740846270 n - 1909799949066899922095074 n 14 13 + 34925249045434680471676291 n - 516784159422829907803609415 n 12 11 + 6223071143601126586689361899 n - 61127270362230843171354566613 n 10 9 + 489395734778095407899554313066 n - 3181345557127200272293022471152 n 8 + 16671832097943580417262991309568 n 7 - 69661149752281715819427521800592 n 6 + 228406329029717802480437702243616 n 5 - 574490805896148890951364721663104 n 4 + 1072815393930927835988220828498432 n 3 - 1416480376323074011146213148256256 n 2 + 1222620210057372749302583158456320 n - 598134543955542497051791766323200 n + 118250438326869506905435668480000) /(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 23 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), - 115 (69566910596 n 22 21 20 - 18626867197651 n + 2346343600709779 n - 184897315797027911 n 19 18 + 10224077925274911621 n - 421773711689545091184 n 17 16 + 13472449576173357736670 n - 341348101679763780752062 n 15 14 + 6971224827005707822213726 n - 115980861933645098324589347 n 13 12 + 1582273365600829599324388671 n - 17759930313854957188623130923 n 11 10 + 164092018832277380431069864601 n - 1245320557262257167235641243850 n 9 8 + 7724793781655982267700092188032 n - 38853743640883680476093352727424 n 7 + 156616789271196243705647804651856 n 6 - 497672298342291253071964565767968 n 5 + 1218173386887072198278955577726848 n 4 - 2222295600820375037880007822471680 n 3 + 2876734311697577835387404995737600 n 2 - 2442934446768500226334185093120000 n + 1180105200121589676794082263040000 n - 231359553248222948293243699200000 )/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 23 ( 23 22 21 368418129799 n - 97930309387082 n + 12254973109864694 n 20 19 - 960012906313776700 n + 52802906345072156889 n 18 17 - 2167915825005617850702 n + 68954477287806484084084 n 16 15 - 1740496075739298774822200 n + 35427044871418697254888889 n 14 13 - 587676151326067118964889702 n + 7996882277045856693108780534 n 12 11 - 89560452693671113350236231100 n + 825913154381475959513836238359 n 10 - 6257840980814588531630000096162 n 9 + 38765060725623154524445381507904 n 8 - 194760271958864056638587225616400 n 7 + 784357612670740643193823146662064 n 6 - 2490653289688050729191634090296352 n 5 + 6093273724620429687574478446502784 n 4 - 11111834863065225332999694388953600 n 3 + 14381046338831615710956119572224000 n 2 - 12211500016346826862978691143680000 n + 5899324132798866706316939919360000 n - 1156797766241114741466218496000000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), ((5159616589296 n + 5159616589296) (2 n - 49)! - (n - 25)! (n - 23)! binomial(2 n, n))/((n - 25)! (n - 23)! binomial(2 n, n))] and in Maple notation [5/2048*(1677719*n^12-120795378*n^11+3866250883*n^10-72524317470*n^9+ 884827495497*n^8-7363461463974*n^7+42593174079049*n^6-170767944580530*n^5+ 459517451972084*n^4-736853049585048*n^3+271361850108768*n^2+1689418272326400*n-\ 3238237626624000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2 *n-21)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-23), 1495/2048*(11222*n^13-948229*n^12+ 35956684*n^11-808105265*n^10+11971347036*n^9-122914016907*n^8+893827932052*n^7-\ 4600483132355*n^6+16217085710342*n^5-34622743609564*n^4+19665075669864*n^3+ 109743419956320*n^2-247977407059200*n+10830226176000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/ (2*n-23), 299/2048*(56104*n^13-4739897*n^12+179665874*n^11-4033911145*n^10+ 59609716122*n^9-608149406271*n^8+4350775170302*n^7-21459075574915*n^6+ 67216360574374*n^5-92467395831932*n^4-161691664299576*n^3+790344772846560*n^2-\ 642058550380800*n+54151130880000)/(-1+2*n)/(2*n-3)/(2*n-5)/(2*n-7)/(2*n-9)/(2*n -11)/(2*n-13)/(2*n-15)/(2*n-17)/(2*n-19)/(2*n-21)/(2*n-23)/(2*n-25), 299/4096*( 224317*n^14-21968513*n^13+972657833*n^12-25726511797*n^11+452085101391*n^10-\ 5541505289859*n^9+48184213344179*n^8-293360035978471*n^7+1173350790283672*n^6-\ 2440049116155728*n^5-1274472802542912*n^4+20169866772841968*n^3-\ 39369538539708480*n^2+25023596242982400*n-2924161067520000)/(2*n-27)/(2*n-25)/( 2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/( 2*n-5)/(2*n-3)/(-1+2*n), 13/8192*(10304029*n^14-1007684657*n^13+44488179491*n^ 12-1170070177633*n^11+20334955476297*n^10-243959792418111*n^9+2034754701416393* n^8-11402831926714459*n^7+37814627887330294*n^6-34462707430144532*n^5-\ 254485171441571784*n^4+1055079197129385792*n^3-1598797661510574720*n^2+ 943937857154073600*n-134511409105920000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2 *n-19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+ 2*n), 247/8192*(1080596*n^15-120991545*n^14+6148113965*n^13-187152152775*n^12+ 3787382702117*n^11-53295173102325*n^10+526999350397795*n^9-3574208623627125*n^8 +15158912604823603*n^7-26306590313547390*n^6-93294538951150460*n^5+ 697863351337559400*n^4-1825743132209874816*n^3+2305481026071461760*n^2-\ 1282448565782092800*n+205306887582720000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 247/8192*(1071461*n^15-119265030*n^14+6001816940*n^13-\ 179829125700*n^12+3547311275522*n^11-47897232148920*n^10+442318971339520*n^9-\ 2650484637216900*n^8+8311631027157073*n^7+6566635576977630*n^6-\ 186617822403367460*n^5+824365642411554600*n^4-1829509802958445056*n^3+ 2145176379854824320*n^2-1178426409406848000*n+205306887582720000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-\ 25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 4199/16384*(247825*n^16-31139048*n^15+ 1774511140*n^14-60424893200*n^13+1360768454710*n^12-21113197987616*n^11+ 226396128341900*n^10-1605129237012400*n^9+6245703336725105*n^8+2700298279239176 *n^7-188691496704140920*n^6+1152692410383886400*n^5-3735338510309798640*n^4+ 7110740294116495488*n^3-7676244165547941120*n^2+4102214058035251200*n-\ 748766295889920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/ (2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29 ), 221/16384*(4563037*n^16-564844520*n^15+31504926820*n^14-1040213468000*n^13+ 22398706729294*n^12-324782294565680*n^11+3111907377459980*n^10-\ 17372283833053600*n^9+17009667424705901*n^8+591481214683186520*n^7-\ 5470151001979756120*n^6+25846794582430472000*n^5-74721343415985725232*n^4+ 133714569067892599680*n^3-140110863087571303680*n^2+74832547642452633600*n-\ 14226559621908480000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11 )/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-\ 29), 221/16384*(8651402*n^17-1189606909*n^16+73832824304*n^15-2718108791140*n^ 14+65400009094364*n^13-1060971697783438*n^12+11322021099077128*n^11-\ 67518235249165580*n^10-23475705862129334*n^9+4701539881708274803*n^8-\ 48926931824412908168*n^7+288074644559314670680*n^6-1104627860756359427232*n^5+ 2814258282929724938544*n^4-4642400499874579707264*n^3+4632282548873710583040*n^ 2-2426984080796836915200*n+469476467522979840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n -25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 221/16384*(7939394*n^17-1060021453*n^ 16+63222481088*n^15-2201710748980*n^14+48768333839708*n^13-687391471291966*n^12 +5310468323691256*n^11+2599593897079300*n^10-616674531656882798*n^9+ 8306369445660979891*n^8-64305211569365640776*n^7+332055994219699767640*n^6-\ 1180699213535547676704*n^5+2869473936233329716528*n^4-4600429130198700125568*n^ 3+4529410224419867639040*n^2-2376906590927719065600*n+469476467522979840000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n -27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 221/16384* (13896493*n^18-2019303351*n^17+130758553731*n^16-4912647800580*n^15+ 115353156162426*n^14-1624856451667722*n^13+8582196899610022*n^12+ 165177592039462860*n^11-4732784004111365631*n^10+63096195840742904697*n^9-\ 552473441532651193737*n^8+3422573228644641555120*n^7-15297180468306612901688*n^ 6+49123251800045978807376*n^5-110899122505425963488016*n^4+ 168735765123674853321600*n^3-160566097622475151353600*n^2+ 82875901058863799040000*n-16431676363304294400000)/(2*n-29)/(2*n-35)/(2*n-23)/( 2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33) /(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 221/32768*(22639019*n^18-\ 3095430741*n^17+183403583610*n^16-5949615496500*n^15+102143172185418*n^14-\ 235794089036142*n^13-34315687333125640*n^12+981443664508172580*n^11-\ 15599907846872785293*n^10+169133558287898396907*n^9-1325566169296099635450*n^8+ 7655866559546714603640*n^7-32619586646412685102144*n^6+101307380935869749732976 *n^5-223551429606406846991520*n^4+335368801232263343969280*n^3-\ 317126979704804084928000*n^2+163939338421896457728000*n-32863352726608588800000 )/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 4199/32768*(1721296*n^19-241540151*n^18+13971850509*n^17-375474654654*n^16+ 356844874452*n^15+344829716623518*n^14-14019724784698562*n^13+ 326442838040255872*n^12-5260546882543896372*n^11+62359857837724235217*n^10-\ 557874808538658510003*n^9+3802938788625926085318*n^8-19747510742006376188696*n^ 7+77439680463084299970016*n^6-225362742182546011624944*n^5+ 472845145315234080154464*n^4-682984292626114323079680*n^3+ 628930246945835125670400*n^2-320149708685684494848000*n+63997055309711462400000 )/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-\ 13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2* n)/(2*n-5), 323/32768*(12497737*n^19-1292628206*n^18+24121510833*n^17+ 3237364359966*n^16-278166988089606*n^15+11535037459078308*n^14-\ 312341891531877494*n^13+6052279721636976412*n^12-87512922029730256059*n^11+ 964774467918249647802*n^10-8192637876600225623211*n^9+53718302595207671935878*n ^8-270839891730175156358312*n^7+1038671117892229265491696*n^6-\ 2973285945458166628726128*n^5+6166867057404075847083744*n^4-\ 8844279765711972344501760*n^3+8119821759875393356454400*n^2-\ 4137810180625664395776000*n+831961719026249011200000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*( 7181467*n^20+2416559890*n^19-589313169975*n^18+55121504112750*n^17-\ 3048019046454558*n^16+114328332918955140*n^15-3110081002756523150*n^14+ 63736866782495019100*n^13-1006888146746199085953*n^12+12430277266804399435290*n ^11-120778202391182271894075*n^10+925414339837622823218550*n^9-\ 5575695801704277837927788*n^8+26214222405372354822731440*n^7-\ 94898442580258692237522000*n^6+259124379440471633489205600*n^5-\ 517363096188419760095823168*n^4+720144862315130128233738240*n^3-\ 646622790145848227272780800*n^2+324717377812669634817024000*n-\ 64893014084047422873600000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -323/262144*(146403827*n^20-\ 43641400390*n^19+5386120754925*n^18-385179732925950*n^17+18305599199691702*n^16 -621614618230733340*n^15+15748257617589827450*n^14-305865735033059956300*n^13+ 4633563541637012108607*n^12-55321859053310009464590*n^11+ 523229076916875681135225*n^10-3922422428907407529758550*n^9+ 23220079520926605186024872*n^8-107646020196880319455492240*n^7+ 385442017068017223974778000*n^6-1043838659705380623162799200*n^5+ 2072106664264431600535060992*n^4-2874116340684401442961789440*n^3+ 2577030612419191442650214400*n^2-1294975930405635693895680000*n+ 259572056336189691494400000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -19/262144*(10721984114*n^21-\ 2786400165843*n^20+329608813728460*n^19-23742394078955685*n^18+ 1172413509999630234*n^17-42290835183856936278*n^16+1158042603185182862120*n^15-\ 24673593072234525853770*n^14+415658524854609179229814*n^13-\ 5592398985566259022302783*n^12+60420506937638724592077180*n^11-\ 525031327012637182363840305*n^10+3662114869984761810879145214*n^9-\ 20387153382194467168026202728*n^8+89698379393931515681397083440*n^7-\ 307261727181652920296058957840*n^6+801706074603792744624981702624*n^5-\ 1543072216133572028733188602368*n^4+2087387974880043924268416268800*n^3-\ 1835437219341463522302510182400*n^2+909508031632397448232869888000*n-\ 180921723266324214971596800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-\ 9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), -19/262144*( 15934362740*n^21-3849725405547*n^20+430025287958350*n^19-29572700191067985*n^18 +1405524144375186990*n^17-49102683814642948182*n^16+1308710147824738665980*n^15 -27250746570000699416370*n^14+450183754473711885630640*n^13-\ 5956916052735383605079847*n^12+63455973436560059010602310*n^11-\ 544877280942081747081863805*n^10+3762895548999643151510695790*n^9-\ 20777066477103414092008032072*n^8+90810011536110166139205945040*n^7-\ 309454911626275725349403973840*n^6+804276195017247627981898767840*n^5-\ 1543794894680587560218221604352*n^4+2084926765816872638606251768320*n^3-\ 1832111230199948447384452608000*n^2+908183637318291022606934016000*n-\ 180921723266324214971596800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-\ 9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/524288*(-\ 1547209989617*n^22+392915730663049*n^21-46615171910661739*n^20+ 3436347943269428495*n^19-176570193653873846442*n^18+6723400469100415354254*n^17 -196883005240714927345574*n^16+4540922484168867269575630*n^15-\ 83794411356115980028521397*n^14+1249694139927689028264160469*n^13-\ 15151735364679552470516167119*n^12+149706186991453766815771144515*n^11-\ 1204570981063430265930626117312*n^10+7863283788486720847160410557604*n^9-\ 41350527482069534509595523677584*n^8+173262497174178027750945386483600*n^7-\ 569344915928842100665064360360832*n^6+1434380464453543060080651619218624*n^5-\ 2681648961152390376278778312297984*n^4+3543097546802570936423055699317760*n^3-\ 3058959336055974736643799552614400*n^2+1496269915980910475578732855296000*n-\ 295626095817173767263589171200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), -5/ 1048576*(724546356851*n^22-180095024603131*n^21+20963983977613831*n^20-\ 1519575032204771465*n^19+76922265542075963856*n^18-2890531669890186319998*n^17+ 83660681045160740846270*n^16-1909799949066899922095074*n^15+ 34925249045434680471676291*n^14-516784159422829907803609415*n^13+ 6223071143601126586689361899*n^12-61127270362230843171354566613*n^11+ 489395734778095407899554313066*n^10-3181345557127200272293022471152*n^9+ 16671832097943580417262991309568*n^8-69661149752281715819427521800592*n^7+ 228406329029717802480437702243616*n^6-574490805896148890951364721663104*n^5+ 1072815393930927835988220828498432*n^4-1416480376323074011146213148256256*n^3+ 1222620210057372749302583158456320*n^2-598134543955542497051791766323200*n+ 118250438326869506905435668480000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), -115/ 1048576*(69566910596*n^23-18626867197651*n^22+2346343600709779*n^21-\ 184897315797027911*n^20+10224077925274911621*n^19-421773711689545091184*n^18+ 13472449576173357736670*n^17-341348101679763780752062*n^16+ 6971224827005707822213726*n^15-115980861933645098324589347*n^14+ 1582273365600829599324388671*n^13-17759930313854957188623130923*n^12+ 164092018832277380431069864601*n^11-1245320557262257167235641243850*n^10+ 7724793781655982267700092188032*n^9-38853743640883680476093352727424*n^8+ 156616789271196243705647804651856*n^7-497672298342291253071964565767968*n^6+ 1218173386887072198278955577726848*n^5-2222295600820375037880007822471680*n^4+ 2876734311697577835387404995737600*n^3-2442934446768500226334185093120000*n^2+ 1180105200121589676794082263040000*n-231359553248222948293243699200000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/ (2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23 )/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -23/1048576*(368418129799*n^23-\ 97930309387082*n^22+12254973109864694*n^21-960012906313776700*n^20+ 52802906345072156889*n^19-2167915825005617850702*n^18+68954477287806484084084*n ^17-1740496075739298774822200*n^16+35427044871418697254888889*n^15-\ 587676151326067118964889702*n^14+7996882277045856693108780534*n^13-\ 89560452693671113350236231100*n^12+825913154381475959513836238359*n^11-\ 6257840980814588531630000096162*n^10+38765060725623154524445381507904*n^9-\ 194760271958864056638587225616400*n^8+784357612670740643193823146662064*n^7-\ 2490653289688050729191634090296352*n^6+6093273724620429687574478446502784*n^5-\ 11111834863065225332999694388953600*n^4+14381046338831615710956119572224000*n^3 -12211500016346826862978691143680000*n^2+5899324132798866706316939919360000*n-\ 1156797766241114741466218496000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/( 2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/ (2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45 ), ((5159616589296*n+5159616589296)*(2*n-49)!-(n-25)!*(n-23)!*binomial(2*n,n))/ (n-25)!/(n-23)!/binomial(2*n,n)] The limits, as n goes to infinity are 8388595 8388445 2096887 67070783 133952377 66726803 264650867 [-------, -------, -------, --------, ---------, --------, ---------, 8388608 8388608 2097152 67108864 134217728 67108864 268435456 1040617175 1008431177 955979921 877303037 3071124953 5003223199 ----------, ----------, ----------, ----------, ----------, ----------, 1073741824 1073741824 1073741824 1073741824 4294967296 8589934592 451732619 4036769051 2319613841 -47288436121 -101858849083 ----------, -----------, -----------, ------------, -------------, 1073741824 17179869184 68719476736 274877906944 274877906944 -75688223015 -1547209989617 -3622731784255 -2000048679635 ------------, --------------, --------------, --------------, 137438953472 2199023255552 4398046511104 2199023255552 -8473616985377 -34861896052001 --------------, ---------------] 8796093022208 35184372088832 and in Maple notation [8388595/8388608, 8388445/8388608, 2096887/2097152, 67070783/67108864, 133952377/134217728, 66726803/67108864, 264650867/268435456, 1040617175/ 1073741824, 1008431177/1073741824, 955979921/1073741824, 877303037/1073741824, 3071124953/4294967296, 5003223199/8589934592, 451732619/1073741824, 4036769051/ 17179869184, 2319613841/68719476736, -47288436121/274877906944, -101858849083/ 274877906944, -75688223015/137438953472, -1547209989617/2199023255552, -\ 3622731784255/4398046511104, -2000048679635/2199023255552, -8473616985377/ 8796093022208, -34861896052001/35184372088832] and in floating point [.9999984503, .9999805689, .9998736382, .9994325489, .9980229810, .9943068475, .9859013073, .9691502666, .9391747201, .8903256813, .8170521231, .7150519995, .\ 5824518389, .4207087858, .2349708841, .3375482398e-1, -.1720343284, -.370560334\ 3, -.5507043026, -.7035896440, -.8237138409, -.9095168387, -.9633387191, -.9908\ 346798] The cut off is at j=, 17 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 26], vs. those in the, 2, -th row from j=1 to j=, 25, are as follws 13 12 11 10 [5 (6710881 n - 567068567 n + 21504883907 n - 483416600095 n 9 8 7 + 7165803369153 n - 73703247235761 n + 538713963937721 n 6 5 4 - 2815298880988165 n + 10404744111170266 n - 26140738301878772 n 3 2 + 39018449239233672 n - 9828627105716640 n - 95304421402041600 n + 168388356584448000)/(4096 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) (2 n - 25) (2 n - 15) 13 12 11 (2 n - 17) (2 n - 23)), 5 (3355408 n - 283526711 n + 10751636246 n 10 9 8 - 241656550135 n + 3580667859054 n - 36783394446513 n 7 6 5 + 267846374133338 n - 1383261157008445 n + 4918253967278938 n 4 3 2 - 10755431950339076 n + 7179942838533816 n + 31605144201614880 n - 77134387135276800 n + 3238237626624000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) 14 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23)), 115 (291757 n 13 12 11 10 - 28589351 n + 1267324058 n - 33607060669 n + 593835898656 n 9 8 7 - 7364981767143 n + 65653039159994 n - 421498212302167 n 6 5 4 + 1897565682119107 n - 5454610524168806 n + 6570205618178748 n 3 2 + 14106106920204936 n - 59782649707804320 n + 46968233026579200 n - 3801409387776000)/(2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 14 13 (2 n - 5) (2 n - 3) (-1 + 2 n)), 23 (1458407 n - 142863595 n 12 11 10 + 6328325458 n - 167540168705 n + 2949528377796 n 9 8 7 - 36280485596355 n + 317524032270874 n - 1956643322628995 n 6 5 4 + 8011530338408597 n - 17712213348179350 n - 3620753632609932 n 3 2 + 128074007223981000 n - 260954407920391200 n + 167585460579936000 n - 19007046938880000)/(2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) 15 14 (2 n - 5) (2 n - 3) (-1 + 2 n)), 115 (4662883 n - 523941282 n 13 12 11 + 26789542024 n - 824321506008 n + 16992867912202 n 10 9 8 - 246748723136916 n + 2573698995187892 n - 19161930695285664 n 7 6 5 + 97519149171751307 n - 295354560481822842 n + 219116493539245484 n 4 3 + 1965351941880453672 n - 7597816886408071392 n 2 + 11122147665792599040 n - 6406483270953830400 n + 881926977964032000)/( 16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 15 14 (2 n - 17) (2 n - 23) (2 n - 29)), (267482717 n - 29993682060 n 13 12 11 + 1527800861930 n - 46687932441150 n + 950608027298384 n 10 9 8 - 13503978036338790 n + 135503703428926690 n - 941024141179744050 n 7 6 + 4175630152784421331 n - 8568741038549281590 n 5 4 - 17164384263444492620 n + 164402603442715474200 n 3 2 - 450721253398679154432 n + 579553006151299213440 n - 323318191599817344000 n + 50710801232931840000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) 16 15 14 (2 n - 29)), 19 (55999157 n - 7120317016 n + 413024912660 n 13 12 11 - 14438268868240 n + 337979745062414 n - 5554336200492352 n 10 9 8 + 65067528220268860 n - 536095194367681520 n + 2923938377810236501 n 7 6 - 8438108205030351848 n - 7008463239968972600 n 5 4 + 174894298304127525760 n - 739848595627877217072 n 3 2 + 1590519683003644953216 n - 1818362242463355889920 n + 976611078374383872000 n - 165477351391672320000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 16 15 (2 n - 23) (2 n - 29)), 19 (55351877 n - 6988919176 n 14 13 12 + 401010101300 n - 13785339408400 n + 314553404131694 n 11 10 9 - 4971490624059232 n + 54792020607704380 n - 407234248060468400 n 8 7 + 1786935221308532581 n - 1563936269798710808 n 6 5 - 34160770513210828760 n + 238962467554292372800 n 4 3 - 810510163898067295152 n + 1577031591081028761216 n 2 - 1719401064817453297920 n + 919027384666659072000 n - 165477351391672320000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 17 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 323 (6370631 n 16 15 14 13 - 900130450 n + 57948391844 n - 2241904801000 n + 57794630253962 n 12 11 10 - 1037506331630860 n + 13088972034274708 n - 112665998878810520 n 9 8 7 + 583594048888633783 n - 661742718950357330 n - 15919809690132905648 n 6 5 + 143994181464543613840 n - 658601841137546207376 n 4 3 + 1848518263691414684640 n - 3224294353866009803904 n 2 + 3303538247844521671680 n - 1728010683496677888000 n + 321220740936775680000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 17 ( 17 16 15 14 116502938 n - 16199207401 n + 1019951994176 n - 38261669857060 n 13 12 11 + 944446278003116 n - 15909586687969702 n + 181251740015189272 n 10 9 - 1275071423401206860 n + 3056275836629254954 n 8 7 + 40692650933142243607 n - 548906528495968988072 n 6 5 + 3495742731195575169400 n - 13929091190750605807008 n 4 3 + 36272468976032663792496 n - 60589044259866187157376 n 2 + 60802616307263347595520 n - 31834565492950547712000 n + 6103194077798737920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 17 ( 18 17 16 15 218884384 n - 33552868788 n + 2332184985003 n - 96718803162840 n 14 13 12 + 2642142079214988 n - 49218912966867336 n + 614876355095476786 n 11 10 - 4522555218138019920 n + 4208184495691719972 n 9 8 + 340039185059221140636 n - 4614453046720028638581 n 7 6 + 34700574434886765929160 n - 173525345676741186733544 n 5 4 + 598876952505205532623488 n - 1417259831945341364432208 n 3 2 + 2219370676477218401385600 n - 2139759970649350297516800 n + 1101586091682095712000000 n - 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 18 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 17 (198495064 n 17 16 15 - 29515783428 n + 1970254165683 n - 77280237154440 n 14 13 12 + 1944617284440348 n - 31555929767331816 n + 289768840537665346 n 11 10 - 106459120357452720 n - 40291938765812950788 n 9 8 + 671608046138229573516 n - 6418472595279398121741 n 7 6 + 41687113156873846585560 n - 191852949400218608610824 n 5 4 + 628006928776751521009728 n - 1435619876662847417598288 n 3 2 + 2200843623967687425513600 n - 2101223476731144644524800 n + 1083660626558491027200000 n - 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 4199 (5538161 n 18 17 16 15 - 889620688 n + 63922525179 n - 2676140167332 n + 70320700603362 n 14 13 12 - 1110782260882896 n + 6371846686829318 n + 157911315676700456 n 11 10 - 5114900517123446907 n + 79824117007670033856 n 9 8 - 833991722578971731073 n + 6289518310444605520284 n 7 6 - 35045610133668196203016 n + 144576950757903124742528 n 5 4 - 436322793093695392121424 n + 938704080115823577590592 n 3 2 - 1377001592770091939193600 n + 1276510866216783029683200 n - 648405711043932441600000 n + 127994110619422924800000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (28663555 n 18 17 16 - 4315636172 n + 281869313025 n - 10047510882618 n 15 14 13 + 184606419749910 n - 5009865785904 n - 99474759899665550 n 12 11 + 3092878731814639924 n - 56216137846105450785 n 10 9 + 712616859872702136924 n - 6654021823804046543475 n 8 7 + 46714505306546636401206 n - 247730751956205993967880 n 6 5 + 986380673010556484958352 n - 2901989095682600029014000 n 4 3 + 6134335872903294893323488 n - 8900760555866085926284800 n 2 + 8211902315108878719244800 n - 4177305506188229184000000 n + 831961719026249011200000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (79628281 n - 12072802910 n 18 17 16 + 741897037275 n - 19430614302090 n - 200141598115074 n 15 14 + 35650988591542980 n - 1483806285367299050 n 13 12 + 38072884640792348620 n - 694219678479194021739 n 11 10 + 9476910152198730268170 n - 99192407582145831032625 n 9 8 + 804388320468127453140030 n - 5064096577714867558472924 n 7 6 + 24633562379626580789965360 n - 91532524483158142521073200 n 5 4 + 254824889884117531387273440 n - 515720971091713067619188544 n 3 2 + 723848048614969846494566400 n - 652164332656046109311462400 n + 327018346224147946368000000 n - 64893014084047422873600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), 323 (40111837 n - 3971931890 n - 19782420825 n 17 16 15 + 24153454861050 n - 1898467710512538 n + 83434073908629660 n 14 13 - 2488076392975308050 n + 54162491621232135700 n 12 11 - 892937368948346548983 n + 11377144790787022871910 n 10 9 - 113234972632658980797525 n + 883902959237177035086450 n 8 7 - 5403212734093945663954268 n + 25689770502307399236231760 n 6 5 - 93798772251766264796982000 n + 257743993775883001330816800 n 4 3 - 516864000895799620580106048 n + 721360147688417474724418560 n 2 - 648401696638262178227481600 n + 325449504125412733946880000 n - 64893014084047422873600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), - 323 (33283193 n 20 19 18 - 71133303339 n + 13963207468105 n - 1319673430406625 n 17 16 + 77734744190932668 n - 3180119960968144614 n 15 14 + 95760000772942175330 n - 2197740976154377060050 n 13 12 + 39295153369940890043713 n - 554906763047405678763399 n 11 10 + 6237829657509320150958405 n - 56000406924859303606661325 n 9 8 + 401182059819037474115782298 n - 2282438940982859394241380504 n 7 6 + 10218349922839971686847885280 n - 35481780148469898897226885200 n 5 4 + 93527533079422805207036068128 n - 181304027644152386044611558144 n 3 2 + 246321254251108875490203962880 n - 216956250396583214805317836800 n + 107410510843199297196503040000 n - 21284908619567554702540800000)/( 524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 21 20 (2 n - 39) (2 n - 35) (2 n - 29)), - 19 (6082394348 n - 1839923853579 n 19 18 17 + 240227116886470 n - 18552780946196385 n + 964919428852156638 n 16 15 - 36227541347662793814 n + 1023931931582940882860 n 14 13 - 22379643255541558177170 n + 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29698188288615900006184231 n + 367790605004039788039383642 n 11 10 - 3700195956976337887665131535 n + 30230861812352498563848002377 n 9 8 - 199880297736539061478480516466 n + 1062232856414893208576430411512 n 7 6 - 4488859164418697283071120756800 n + 14849200782010588735282682237712 n 5 - 37598011052035238985667213542816 n 4 + 70536786078739318511515799795712 n 3 - 93390771904304780279473066851840 n 2 + 80692851981617215131476688998400 n - 39450975706742693060985937920000 n + 7779634100451941243778662400000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), ( 22 21 20 -631367201149 n + 165592264107413 n - 20187315475822418 n 19 18 + 1522933397734906465 n - 79812151333920581499 n 17 16 + 3090892469982872821728 n - 91835280958410658027348 n 15 14 + 2144688802123766115585170 n - 40002515000958370933131059 n 13 12 + 602088315317252191923307813 n - 7357283262319557160377263298 n 11 10 + 73177407157352776654938887325 n - 592098683518433765524711700389 n 9 8 + 3883137222680363590118280714038 n - 20498081484148158352110555517688 n 7 + 86151814706651795984444698924480 n 6 - 283772739001485139659285313508304 n 5 + 716195935918478901495378017719008 n 4 - 1340614737058925611536126362869248 n 3 + 1772570022783147111280963218946560 n 2 - 1530776720057818140915083745177600 n + 748637642040022901685118279680000 n - 147813047908586883631794585600000) /(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 23 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), - 115 (110105700037 n 22 21 20 - 30403097802722 n + 3936214364120258 n - 317850608530694932 n 19 18 + 17961928014404376387 n - 755426254208754397038 n 17 16 + 24546619726076068963540 n - 631412396923271066306984 n 15 14 + 13068089034347680553411147 n - 219970320293517730373620414 n 13 12 + 3031689433588297321741125042 n - 34330311583065369647515186356 n 11 10 + 319607538186516108668958428797 n - 2441242338031587229403251799570 n 9 + 15225348580335730098979006202264 n 8 - 76922610391709649310944886185808 n 7 + 311190933095200028104606066711632 n 6 - 991652972887832520156860099156256 n 5 + 2432448534309893474321733433168896 n 4 - 4443986328692533685475820195345920 n 3 + 5757693274473768321981433661952000 n 2 - 4891000867540337665559441055744000 n + 2362159583292523356449243136000000 n - 462719106496445896586487398400000 )/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 115 ( 23 22 21 63761286365 n - 17274156751828 n + 2198758827133528 n 20 19 - 174877069627412306 n + 9749521927693537605 n 18 17 - 405119354455267749066 n + 13023359230626264162104 n 16 15 - 331827300635013047888452 n + 6810195901357201679084935 n 14 13 - 113788804030151305997374664 n + 1558153623563424582366562800 n 12 11 - 17545345381332916125076301658 n + 162553976206098415535086375655 n 10 9 - 1236509784453633330056930150362 n + 7684998653687375215254088950976 n 8 - 38715100274245162608418350630704 n 7 + 156257550054212816235867667281840 n 6 - 497025893784790250452735748434080 n 5 + 1217494548629745130125706702430592 n 4 - 2222194948292250638403181975226880 n 3 + 2877474758267295992458942418073600 n 2 - 2443829174780100661093533235200000 n + 1180444188477997329465574195200000 n - 231359553248222948293243699200000 )/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 115 ( 24 23 22 279850994447 n - 81490096964268 n + 11193889600653746 n 21 20 - 964770124526021616 n + 58534525086262081241 n 19 18 - 2658783214221458944620 n + 93872484967753795514552 n 17 16 - 2640155770684477331401104 n + 60137577204489296839835585 n 15 14 - 1121880513493746087055793412 n + 17265970792393670346082657658 n 13 12 - 220141112024156013871139052912 n + 2328971579682761377877404073639 n 11 - 20428049847348293844178569086244 n 10 + 148083098813814446627272125118844 n 9 - 882093503052241523067992209558800 n 8 + 4280594189541212876098231439325968 n 7 - 16717448470613172491627541520453056 n 6 + 51662360460576359543132401481275200 n 5 - 123406209693563331317745361865165568 n 4 + 220403811966337963616362885494789120 n 3 - 280175027891158584572537826246758400 n 2 + 234344697559024935959036767457280000 n - 111850981810035641500300310937600000 n + 21747798005332957139564907724800000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), (-33968885488469 n 23 22 + 9826796538292140 n - 1341849533380223606 n 21 20 + 115028890327171629600 n - 6945168761150565413699 n 19 18 + 314088726699868517872860 n - 11045929981668885280249976 n 17 16 + 309577627247579974751521680 n - 7029590230402015296611870939 n 15 14 + 130776623494129819943839748580 n - 2007786048859002182338745050286 n 13 + 25544818552396807825811525235360 n 12 - 269750920946006075325426394285949 n 11 + 2362298381816653351764521683103700 n 10 - 17101146722415447956108663315155076 n 9 + 101751407595947724334560884743473840 n 8 - 493312387999018650688318140261188144 n 7 + 1925123939065549529887808348982966720 n 6 - 5945731376157665808241351379118181056 n 5 + 14196313560302921038989968737302539520 n 4 - 25346886805994764367798997966283852800 n 3 + 32214868676796757892274677774255616000 n 2 - 26943514278167268783963827352023040000 n + 12860581670601091818988427550720000000 n - 2500996770613290071049964388352000000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), ( (19447785605808 n + 19447785605808) (2 n - 51)! - (n - 26)! (n - 24)! binomial(2 n, n))/((n - 26)! (n - 24)! binomial(2 n, n))] and in Maple notation [5/4096*(6710881*n^13-567068567*n^12+21504883907*n^11-483416600095*n^10+ 7165803369153*n^9-73703247235761*n^8+538713963937721*n^7-2815298880988165*n^6+ 10404744111170266*n^5-26140738301878772*n^4+39018449239233672*n^3-\ 9828627105716640*n^2-95304421402041600*n+168388356584448000)/(2*n-5)/(-1+2*n)/( 2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-25)/(2*n-15)/(2 *n-17)/(2*n-23), 5/2048*(3355408*n^13-283526711*n^12+10751636246*n^11-\ 241656550135*n^10+3580667859054*n^9-36783394446513*n^8+267846374133338*n^7-\ 1383261157008445*n^6+4918253967278938*n^5-10755431950339076*n^4+ 7179942838533816*n^3+31605144201614880*n^2-77134387135276800*n+3238237626624000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/ (2*n-25)/(2*n-15)/(2*n-17)/(2*n-23), 115/2048*(291757*n^14-28589351*n^13+ 1267324058*n^12-33607060669*n^11+593835898656*n^10-7364981767143*n^9+ 65653039159994*n^8-421498212302167*n^7+1897565682119107*n^6-5454610524168806*n^ 5+6570205618178748*n^4+14106106920204936*n^3-59782649707804320*n^2+ 46968233026579200*n-3801409387776000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-\ 19)/(2*n-17)/(2*n-15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n ), 23/2048*(1458407*n^14-142863595*n^13+6328325458*n^12-167540168705*n^11+ 2949528377796*n^10-36280485596355*n^9+317524032270874*n^8-1956643322628995*n^7+ 8011530338408597*n^6-17712213348179350*n^5-3620753632609932*n^4+ 128074007223981000*n^3-260954407920391200*n^2+167585460579936000*n-\ 19007046938880000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 115/16384*( 4662883*n^15-523941282*n^14+26789542024*n^13-824321506008*n^12+16992867912202*n ^11-246748723136916*n^10+2573698995187892*n^9-19161930695285664*n^8+ 97519149171751307*n^7-295354560481822842*n^6+219116493539245484*n^5+ 1965351941880453672*n^4-7597816886408071392*n^3+11122147665792599040*n^2-\ 6406483270953830400*n+881926977964032000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 1/8192*(267482717*n^15-29993682060*n^14+1527800861930*n^13-\ 46687932441150*n^12+950608027298384*n^11-13503978036338790*n^10+ 135503703428926690*n^9-941024141179744050*n^8+4175630152784421331*n^7-\ 8568741038549281590*n^6-17164384263444492620*n^5+164402603442715474200*n^4-\ 450721253398679154432*n^3+579553006151299213440*n^2-323318191599817344000*n+ 50710801232931840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11 )/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 19/ 16384*(55999157*n^16-7120317016*n^15+413024912660*n^14-14438268868240*n^13+ 337979745062414*n^12-5554336200492352*n^11+65067528220268860*n^10-\ 536095194367681520*n^9+2923938377810236501*n^8-8438108205030351848*n^7-\ 7008463239968972600*n^6+174894298304127525760*n^5-739848595627877217072*n^4+ 1590519683003644953216*n^3-1818362242463355889920*n^2+976611078374383872000*n-\ 165477351391672320000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 19/16384*(55351877*n^16-6988919176*n^15+401010101300*n^14-13785339408400 *n^13+314553404131694*n^12-4971490624059232*n^11+54792020607704380*n^10-\ 407234248060468400*n^9+1786935221308532581*n^8-1563936269798710808*n^7-\ 34160770513210828760*n^6+238962467554292372800*n^5-810510163898067295152*n^4+ 1577031591081028761216*n^3-1719401064817453297920*n^2+919027384666659072000*n-\ 165477351391672320000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 323/16384*(6370631*n^17-900130450*n^16+57948391844*n^15-2241904801000*n^ 14+57794630253962*n^13-1037506331630860*n^12+13088972034274708*n^11-\ 112665998878810520*n^10+583594048888633783*n^9-661742718950357330*n^8-\ 15919809690132905648*n^7+143994181464543613840*n^6-658601841137546207376*n^5+ 1848518263691414684640*n^4-3224294353866009803904*n^3+3303538247844521671680*n^ 2-1728010683496677888000*n+321220740936775680000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n -25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 17/16384*(116502938*n^17-16199207401* n^16+1019951994176*n^15-38261669857060*n^14+944446278003116*n^13-\ 15909586687969702*n^12+181251740015189272*n^11-1275071423401206860*n^10+ 3056275836629254954*n^9+40692650933142243607*n^8-548906528495968988072*n^7+ 3495742731195575169400*n^6-13929091190750605807008*n^5+36272468976032663792496* n^4-60589044259866187157376*n^3+60802616307263347595520*n^2-\ 31834565492950547712000*n+6103194077798737920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n -25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 17/16384*(218884384*n^18-33552868788* n^17+2332184985003*n^16-96718803162840*n^15+2642142079214988*n^14-\ 49218912966867336*n^13+614876355095476786*n^12-4522555218138019920*n^11+ 4208184495691719972*n^10+340039185059221140636*n^9-4614453046720028638581*n^8+ 34700574434886765929160*n^7-173525345676741186733544*n^6+ 598876952505205532623488*n^5-1417259831945341364432208*n^4+ 2219370676477218401385600*n^3-2139759970649350297516800*n^2+ 1101586091682095712000000*n-213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-\ 33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 17/16384*(198495064*n^18 -29515783428*n^17+1970254165683*n^16-77280237154440*n^15+1944617284440348*n^14-\ 31555929767331816*n^13+289768840537665346*n^12-106459120357452720*n^11-\ 40291938765812950788*n^10+671608046138229573516*n^9-6418472595279398121741*n^8+ 41687113156873846585560*n^7-191852949400218608610824*n^6+ 628006928776751521009728*n^5-1435619876662847417598288*n^4+ 2200843623967687425513600*n^3-2101223476731144644524800*n^2+ 1083660626558491027200000*n-213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-\ 33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 4199/65536*(5538161*n^19 -889620688*n^18+63922525179*n^17-2676140167332*n^16+70320700603362*n^15-\ 1110782260882896*n^14+6371846686829318*n^13+157911315676700456*n^12-\ 5114900517123446907*n^11+79824117007670033856*n^10-833991722578971731073*n^9+ 6289518310444605520284*n^8-35045610133668196203016*n^7+144576950757903124742528 *n^6-436322793093695392121424*n^5+938704080115823577590592*n^4-\ 1377001592770091939193600*n^3+1276510866216783029683200*n^2-\ 648405711043932441600000*n+127994110619422924800000)/(2*n-29)/(2*n-35)/(2*n-23) /(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/32768*( 28663555*n^19-4315636172*n^18+281869313025*n^17-10047510882618*n^16+ 184606419749910*n^15-5009865785904*n^14-99474759899665550*n^13+ 3092878731814639924*n^12-56216137846105450785*n^11+712616859872702136924*n^10-\ 6654021823804046543475*n^9+46714505306546636401206*n^8-247730751956205993967880 *n^7+986380673010556484958352*n^6-2901989095682600029014000*n^5+ 6134335872903294893323488*n^4-8900760555866085926284800*n^3+ 8211902315108878719244800*n^2-4177305506188229184000000*n+ 831961719026249011200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15) /(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7) /(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(79628281*n^20-12072802910*n^19+ 741897037275*n^18-19430614302090*n^17-200141598115074*n^16+35650988591542980*n^ 15-1483806285367299050*n^14+38072884640792348620*n^13-694219678479194021739*n^ 12+9476910152198730268170*n^11-99192407582145831032625*n^10+ 804388320468127453140030*n^9-5064096577714867558472924*n^8+ 24633562379626580789965360*n^7-91532524483158142521073200*n^6+ 254824889884117531387273440*n^5-515720971091713067619188544*n^4+ 723848048614969846494566400*n^3-652164332656046109311462400*n^2+ 327018346224147946368000000*n-64893014084047422873600000)/(2*n-29)/(2*n-35)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/( 2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(40111837*n^20-3971931890*n^19-19782420825*n^18+24153454861050*n^17-\ 1898467710512538*n^16+83434073908629660*n^15-2488076392975308050*n^14+ 54162491621232135700*n^13-892937368948346548983*n^12+11377144790787022871910*n^ 11-113234972632658980797525*n^10+883902959237177035086450*n^9-\ 5403212734093945663954268*n^8+25689770502307399236231760*n^7-\ 93798772251766264796982000*n^6+257743993775883001330816800*n^5-\ 516864000895799620580106048*n^4+721360147688417474724418560*n^3-\ 648401696638262178227481600*n^2+325449504125412733946880000*n-\ 64893014084047422873600000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -323/524288*(33283193*n^21-\ 71133303339*n^20+13963207468105*n^19-1319673430406625*n^18+77734744190932668*n^ 17-3180119960968144614*n^16+95760000772942175330*n^15-2197740976154377060050*n^ 14+39295153369940890043713*n^13-554906763047405678763399*n^12+ 6237829657509320150958405*n^11-56000406924859303606661325*n^10+ 401182059819037474115782298*n^9-2282438940982859394241380504*n^8+ 10218349922839971686847885280*n^7-35481780148469898897226885200*n^6+ 93527533079422805207036068128*n^5-181304027644152386044611558144*n^4+ 246321254251108875490203962880*n^3-216956250396583214805317836800*n^2+ 107410510843199297196503040000*n-21284908619567554702540800000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29), -19/262144*(6082394348*n^21-1839923853579*n^20+240227116886470*n^ 19-18552780946196385*n^18+964919428852156638*n^17-36227541347662793814*n^16+ 1023931931582940882860*n^15-22379643255541558177170*n^14+ 384927276512330946059848*n^13-5267938738965170107962759*n^12+ 57718608185851602747016350*n^11-507366247141373339043401805*n^10+ 3572408990861625672514797998*n^9-20040087880132657268657761224*n^8+ 88708904630014255383786997840*n^7-305309552016658994918905701840*n^6+ 799418385005003233065528051168*n^5-1542428952811503258949807578624*n^4+ 2089578721408800782934738516480*n^3-1838397715170724193383418572800*n^2+ 910686888109788882031779840000*n-180921723266324214971596800000)/(2*n-5)/(-1+2* n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29), -19/262144*(23248253137*n^22-6479504021111*n^21+829619570581802*n ^20-65170810630388155*n^19+3533223972116180727*n^18-140823161035406225376*n^17+ 4288137016271221247332*n^16-102277738129716670861670*n^15+ 1942592531438665798107647*n^14-29698188288615900006184231*n^13+ 367790605004039788039383642*n^12-3700195956976337887665131535*n^11+ 30230861812352498563848002377*n^10-199880297736539061478480516466*n^9+ 1062232856414893208576430411512*n^8-4488859164418697283071120756800*n^7+ 14849200782010588735282682237712*n^6-37598011052035238985667213542816*n^5+ 70536786078739318511515799795712*n^4-93390771904304780279473066851840*n^3+ 80692851981617215131476688998400*n^2-39450975706742693060985937920000*n+ 7779634100451941243778662400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/262144* (-631367201149*n^22+165592264107413*n^21-20187315475822418*n^20+ 1522933397734906465*n^19-79812151333920581499*n^18+3090892469982872821728*n^17-\ 91835280958410658027348*n^16+2144688802123766115585170*n^15-\ 40002515000958370933131059*n^14+602088315317252191923307813*n^13-\ 7357283262319557160377263298*n^12+73177407157352776654938887325*n^11-\ 592098683518433765524711700389*n^10+3883137222680363590118280714038*n^9-\ 20498081484148158352110555517688*n^8+86151814706651795984444698924480*n^7-\ 283772739001485139659285313508304*n^6+716195935918478901495378017719008*n^5-\ 1340614737058925611536126362869248*n^4+1772570022783147111280963218946560*n^3-\ 1530776720057818140915083745177600*n^2+748637642040022901685118279680000*n-\ 147813047908586883631794585600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), -115/ 2097152*(110105700037*n^23-30403097802722*n^22+3936214364120258*n^21-\ 317850608530694932*n^20+17961928014404376387*n^19-755426254208754397038*n^18+ 24546619726076068963540*n^17-631412396923271066306984*n^16+ 13068089034347680553411147*n^15-219970320293517730373620414*n^14+ 3031689433588297321741125042*n^13-34330311583065369647515186356*n^12+ 319607538186516108668958428797*n^11-2441242338031587229403251799570*n^10+ 15225348580335730098979006202264*n^9-76922610391709649310944886185808*n^8+ 311190933095200028104606066711632*n^7-991652972887832520156860099156256*n^6+ 2432448534309893474321733433168896*n^5-4443986328692533685475820195345920*n^4+ 5757693274473768321981433661952000*n^3-4891000867540337665559441055744000*n^2+ 2362159583292523356449243136000000*n-462719106496445896586487398400000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/ (2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23 )/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -115/1048576*(63761286365*n^23-\ 17274156751828*n^22+2198758827133528*n^21-174877069627412306*n^20+ 9749521927693537605*n^19-405119354455267749066*n^18+13023359230626264162104*n^ 17-331827300635013047888452*n^16+6810195901357201679084935*n^15-\ 113788804030151305997374664*n^14+1558153623563424582366562800*n^13-\ 17545345381332916125076301658*n^12+162553976206098415535086375655*n^11-\ 1236509784453633330056930150362*n^10+7684998653687375215254088950976*n^9-\ 38715100274245162608418350630704*n^8+156257550054212816235867667281840*n^7-\ 497025893784790250452735748434080*n^6+1217494548629745130125706702430592*n^5-\ 2222194948292250638403181975226880*n^4+2877474758267295992458942418073600*n^3-\ 2443829174780100661093533235200000*n^2+1180444188477997329465574195200000*n-\ 231359553248222948293243699200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45) , -115/2097152*(279850994447*n^24-81490096964268*n^23+11193889600653746*n^22-\ 964770124526021616*n^21+58534525086262081241*n^20-2658783214221458944620*n^19+ 93872484967753795514552*n^18-2640155770684477331401104*n^17+ 60137577204489296839835585*n^16-1121880513493746087055793412*n^15+ 17265970792393670346082657658*n^14-220141112024156013871139052912*n^13+ 2328971579682761377877404073639*n^12-20428049847348293844178569086244*n^11+ 148083098813814446627272125118844*n^10-882093503052241523067992209558800*n^9+ 4280594189541212876098231439325968*n^8-16717448470613172491627541520453056*n^7+ 51662360460576359543132401481275200*n^6-123406209693563331317745361865165568*n^ 5+220403811966337963616362885494789120*n^4-280175027891158584572537826246758400 *n^3+234344697559024935959036767457280000*n^2-\ 111850981810035641500300310937600000*n+21747798005332957139564907724800000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 1/2097152*(-33968885488469* n^24+9826796538292140*n^23-1341849533380223606*n^22+115028890327171629600*n^21-\ 6945168761150565413699*n^20+314088726699868517872860*n^19-\ 11045929981668885280249976*n^18+309577627247579974751521680*n^17-\ 7029590230402015296611870939*n^16+130776623494129819943839748580*n^15-\ 2007786048859002182338745050286*n^14+25544818552396807825811525235360*n^13-\ 269750920946006075325426394285949*n^12+2362298381816653351764521683103700*n^11-\ 17101146722415447956108663315155076*n^10+101751407595947724334560884743473840*n ^9-493312387999018650688318140261188144*n^8+ 1925123939065549529887808348982966720*n^7-5945731376157665808241351379118181056 *n^6+14196313560302921038989968737302539520*n^5-\ 25346886805994764367798997966283852800*n^4+ 32214868676796757892274677774255616000*n^3-\ 26943514278167268783963827352023040000*n^2+ 12860581670601091818988427550720000000*n-2500996770613290071049964388352000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37) /(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), ((19447785605808*n+ 19447785605808)*(2*n-51)!-(n-26)!*(n-24)!*binomial(2*n,n))/(n-26)!/(n-24)!/ binomial(2*n,n)] The limits, as n goes to infinity are 33554405 1048565 33552055 33543361 536231545 267482717 1063983983 [--------, -------, --------, --------, ---------, ---------, ----------, 33554432 1048576 33554432 33554432 536870912 268435456 1073741824 1051685663 2057713813 990274973 116282329 421802011 23254738039 ----------, ----------, ----------, ---------, ---------, -----------, 1073741824 2147483648 1073741824 134217728 536870912 34359738368 9258328265 25719934763 12956123351 -10750471339 -28891373153 -----------, -----------, -----------, -------------, ------------, 17179869184 68719476736 68719476736 1099511627776 137438953472 -441716809603 -631367201149 -12662155504255 -7332547931975 -------------, -------------, ---------------, --------------, 1099511627776 1099511627776 17592186044416 8796093022208 -32182864361405 -33968885488469 -139522001754965 ---------------, ---------------, ----------------] 35184372088832 35184372088832 140737488355328 and in Maple notation [33554405/33554432, 1048565/1048576, 33552055/33554432, 33543361/33554432, 536231545/536870912, 267482717/268435456, 1063983983/1073741824, 1051685663/ 1073741824, 2057713813/2147483648, 990274973/1073741824, 116282329/134217728, 421802011/536870912, 23254738039/34359738368, 9258328265/17179869184, 25719934763/68719476736, 12956123351/68719476736, -10750471339/1099511627776, -\ 28891373153/137438953472, -441716809603/1099511627776, -631367201149/ 1099511627776, -12662155504255/17592186044416, -7332547931975/8796093022208, -\ 32182864361405/35184372088832, -33968885488469/35184372088832, -139522001754965 /140737488355328] and in floating point [.9999991953, .9999895096, .9999291599, .9996700585, .9988090862, .9964507706, .9909123024, .9794585994, .9581976631, .9222654374, .8663708642, .7856674697, .\ 6768019532, .5389056323, .3742743104, .1885364087, -.9777496724e-2, -.210212406\ 5, -.4017390980, -.5742251243, -.7197602090, -.8336141868, -.9146920195, -.9654\ 537930, -.9913634483] The cut off is at j=, 17 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 27], vs. those in the, 2, -th row from j=1 to j=, 26, are as follws 13 12 11 10 [3 (5592403 n - 472557644 n + 17920790303 n - 402850616740 n 9 8 7 + 5971651729329 n - 61423921307892 n + 449029010470469 n 6 5 4 - 2347708286389180 n + 8689561298528968 n - 21938277731605664 n 3 2 + 33337326484766928 n - 10625153105062080 n - 77454872739417600 n + 145720693198080000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 13) (2 n - 25) (2 n - 15) 14 13 12 (2 n - 17) (2 n - 23)), 15 (4473899 n - 438438007 n + 19439532931 n 11 10 9 - 515772685883 n + 9125530931517 n - 113542033505901 n 8 7 6 + 1020236670772033 n - 6681841799401769 n + 31645784841394724 n 5 4 3 - 104412438811873192 n + 211390553411537136 n - 111445083994437648 n 2 - 652083660853266240 n + 1446707245714214400 n - 58288277279232000)/(4096 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) 14 13 12 (2 n - 23)), 5 (6710621 n - 657601273 n + 29153061574 n 11 10 9 - 773237470187 n + 13669142955468 n - 169697037999489 n 8 7 6 + 1515987213751582 n - 9778263182328641 n + 44464176284398871 n 5 4 3 - 130767619429585138 n + 171376830175613244 n + 292471863263071128 n 2 - 1396089188067627360 n + 1117633639918521600 n - 87432415918848000)/( 2048 (2 n - 27) (2 n - 25) (2 n - 23) (2 n - 21) (2 n - 19) (2 n - 17) (2 n - 15) (2 n - 13) (2 n - 11) (2 n - 9) (2 n - 7) (2 n - 5) (2 n - 3) 15 14 13 (-1 + 2 n)), 115 (583444 n - 65622225 n + 3361756615 n 12 11 10 - 103830281100 n + 2155978404853 n - 31744851708570 n 9 8 7 + 339977690776295 n - 2661315767424600 n + 14935082761786187 n 6 5 4 - 56239832613849885 n + 113224834838960390 n + 44199425469626700 n 3 2 - 830642745554718984 n + 1611325514525179680 n - 1008493533576748800 n + 110240872245504000)/(2048 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 15 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 115 (1166279 n 14 13 12 11 - 131102553 n + 6708494681 n - 206706967467 n + 4271569603847 n 10 9 8 - 62290420241049 n + 654433557257623 n - 4932270040487361 n 7 6 5 + 25642327790813734 n - 81258522121474698 n + 81025668924450796 n 4 3 2 + 446016233455484328 n - 1887243655085073360 n + 2833420941696628800 n - 1641351502465113600 n + 220481744491008000)/(4096 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 23 ( 16 15 14 13 46581937 n - 5950858472 n + 347867874820 n - 12314032141040 n 12 11 10 + 294105249113494 n - 4989899405813504 n + 61487466581847020 n 9 8 7 - 549965575175507920 n + 3470023044867094001 n - 14158088181885223576 n 6 5 + 26445039254091840200 n + 57843494674671392960 n 4 3 - 499651249689068520432 n + 1316237151140428263552 n 2 - 1646950341643312631040 n + 898365473749271808000 n - 136698681584424960000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 16 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 3 (355836021 n 15 14 13 - 45333803848 n + 2637629104500 n - 92627315555600 n 12 11 10 + 2183052855973262 n - 36235219389978336 n + 430742488840060540 n 9 8 - 3629128318738036400 n + 20581726565708664213 n 7 6 - 65916701032834836184 n + 4889475642746542120 n 5 4 + 991392026546212200000 n - 4557470111375099640496 n 3 2 + 10097769714734499274368 n - 11695890413551225340160 n + 6289707236803635456000 n - 1048023225480591360000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 15 (2 n - 23) (2 n - 29)), 969 (2186174 n - 313133223 n + 20554958608 n 14 13 12 - 817453836700 n + 21910653972228 n - 415831442488826 n 11 10 9 + 5695615420625416 n - 55989325702364340 n + 379357871100887822 n 8 7 - 1545814189981855079 n + 1311472615737191544 n 6 5 + 26035449055806324680 n - 175501654449470752224 n 4 3 + 576454886323531018128 n - 1092170781565963613568 n 2 + 1164203632963394415360 n - 609434094456988416000 n + 107073580312258560000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 323 ( 17 16 15 14 6459632 n - 917841649 n + 59537949704 n - 2326895415940 n 13 12 11 + 60807408623024 n - 1112075618160598 n + 14411300046745888 n 10 9 8 - 129638205845402540 n + 741084462462257056 n - 1706169436464063857 n 7 6 - 11087846074771262888 n + 129095097476911329400 n 5 4 - 630843876252502523712 n + 1825957048673850277104 n 3 2 - 3237481578642245080704 n + 3342060020428493518080 n - 1747572202976802048000 n + 321220740936775680000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 323 (12573149 n - 1983894543 n 16 15 14 + 143227944213 n - 6246378052020 n + 182746300969218 n 13 12 11 - 3757793499861786 n + 55063076598162386 n - 563966831640300660 n 10 9 + 3690964705660763817 n - 9364604896331912799 n 8 7 - 86853877370537005911 n + 1192770899742619622880 n 6 5 - 7394426741943780217384 n + 28625826496296900706128 n 4 3 - 72613365727404291656688 n + 118512793450356329692800 n 2 - 116496193317259907692800 n + 59829581616404624640000 n - 11242725932787148800000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 18 17 (-1 + 2 n) (2 n - 5)), 17 (228298712 n - 35416905732 n 16 15 14 + 2499298721331 n - 105694141048200 n + 2964209083222044 n 13 12 11 - 57374413851591144 n + 764987726014762562 n - 6561592181804602800 n 10 9 + 24755154989275358076 n + 186944426980765395084 n 8 7 - 3781485995903727173517 n + 31474691123747052144600 n 6 5 - 165062970624172599348232 n + 585426815313158372553792 n 4 3 - 1408782477964665730007376 n + 2227925142697668506294400 n 2 - 2157553364137509450873600 n + 1109862787924352689920000 n - 213611792722955827200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 19 18 (-1 + 2 n) (2 n - 5)), 323 (22364792 n - 3799347136 n 17 16 15 + 293889190038 n - 13634616107829 n + 419518473892464 n 14 13 12 - 8885589526396212 n + 128140688204806196 n - 1128485770371737518 n 11 10 + 2360717514108535296 n + 93148904194603303332 n 9 8 - 1614239291113545909306 n + 15082200911256954670923 n 7 6 - 95014317049561195227352 n + 424394635682317981426616 n 5 4 - 1351744604898145790192928 n + 3015442683972349118117424 n 3 2 - 4523176818305461245859200 n + 4235013005941289435270400 n - 2145700633206054355200000 n + 415980859513124505600000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 19 18 17 16 969 (6678794 n - 1099579662 n + 81622281966 n - 3579466681743 n 15 14 13 + 101410215854748 n - 1871409933544704 n + 19948367555551772 n 12 11 - 20267850601730706 n - 3414356626439007678 n 10 9 + 68535376821008562894 n - 789317363623448052042 n 8 7 + 6273585920698628154441 n - 36120310414495849715464 n 6 5 + 152307391739192628273672 n - 466582012835431121073696 n 4 3 + 1013896146778212427669008 n - 1496358188199035867318400 n 2 + 1390795875003386335276800 n - 705896151371420486400000 n + 138660286504374835200000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), 969 (45280917 n - 7996756850 n 18 17 16 + 633661064735 n - 29344692972150 n + 854666818631022 n 15 14 13 - 14902932787894500 n + 86544991252019070 n + 3092179664721795700 n 12 11 - 109372089609443523343 n + 1959768754878916045350 n 10 9 - 23973655703547975927645 n + 215425087859419908130050 n 8 7 - 1458289475380993825540548 n + 7480918593414467869565200 n 6 5 - 28912924503733521765565360 n + 82837878517734644447546400 n 4 3 - 171029768134962282249498048 n + 243026118758046114190540800 n 2 - 220102205918829835524940800 n + 110150613010942568064000000 n - 21631004694682474291200000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 20 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (105573421 n 19 18 17 - 17391556610 n + 1241989610775 n - 48046417287990 n 16 15 14 + 914921000933766 n + 4278255807597180 n - 824437022796384050 n 13 12 + 27509001269796528820 n - 563748467565103978599 n 11 10 + 8229281349085204821270 n - 89972541639889722601125 n 9 8 + 752181739458145404387330 n - 4841444555849816277096284 n 7 6 + 23940092400088669688881360 n - 90044584029021496582345200 n 5 4 + 252908306520837172333431840 n - 514970496977918866180202304 n 3 2 + 725481518920282009777996800 n - 654634750243480003457510400 n + 328048394066751873715200000 n - 64893014084047422873600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 21 20 19 (2 n - 5)), 323 (140230126 n - 22729862085 n + 1452127245380 n 18 17 16 - 33257968232895 n - 1281722392256334 n + 136040668892099370 n 15 14 - 5856114061243835000 n + 164553710615006403810 n 13 12 - 3362727275255814683854 n + 52254589319144641385775 n 11 10 - 631211405775707414346780 n + 5991160813875387807876765 n 9 8 - 44841070313024125234527794 n + 264098033221299741478173180 n 7 6 - 1214982593582146164891718960 n + 4308626187698499657335787120 n 5 4 - 11537438048238323465390682144 n + 22613622288787692202233533760 n 3 2 - 30931174011782018693137904640 n + 27317741740799431077126835200 n - 13506568118223869231262720000 n + 2660613577445944337817600000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 21 20 (2 n - 35) (2 n - 29)), 323 (245886631 n - 20351860788 n 19 18 17 - 1479799670335 n + 327774473782980 n - 25098174006900414 n 16 15 + 1173756529974733992 n - 38375098431153160430 n 14 13 + 931433061984501761160 n - 17342625238278398982469 n 12 11 + 252500526446812138874652 n - 2906290890206321819913075 n 10 9 + 26580488766624047514217140 n - 193241333175371088630577844 n 8 7 + 1112277389423273826364861872 n - 5025297458767230878397688720 n 6 5 + 17572747570937363451834640320 n - 46563034473336903521777555904 n 4 3 + 90591141821260194037627990272 n - 123348575014368722708278717440 n 2 + 108737526618878292411216998400 n - 53809481736955437016043520000 n + 10642454309783777351270400000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), - 19 ( 22 21 20 5857136657 n - 3652592399929 n + 669060900445399 n 19 18 - 65311517983893335 n + 4112524320572477502 n 17 16 - 183098998106206808934 n + 6073814823637698220094 n 15 14 - 155094421537353987727390 n + 3113669561983036280193277 n 13 12 - 49824770868162259943522549 n + 640863792866353927101275499 n 11 10 - 6653993108045444039989305195 n + 55807641211687620447871532612 n 9 8 - 377076632452590698539189205884 n + 2039797729029531487878760984624 n 7 6 - 8743942070756584756463857211600 n + 29250872887978944015890200969152 n 5 - 74689425936347802002851489072704 n 4 + 140951898719046022050081411024384 n 3 - 187287208479531600395067064212480 n 2 + 162044323635360607739224232140800 n - 79160099151358602875159101440000 n + 15559268200903882487557324800000)/( 524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 22 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), - 57 (4739094037 n 21 20 19 - 1485522170429 n + 206309005515074 n - 17204746467946345 n 18 17 + 976457262709946787 n - 40349745578307558624 n 16 15 + 1264923804641876061364 n - 30895567032776201887010 n 14 13 + 598431527882353349066267 n - 9299035178212441401969229 n 12 11 + 116735570441937053352350514 n - 1187784620979376807169307525 n 10 9 + 9795806174216923444908069757 n - 65271026774304743952713332454 n 8 7 + 349067186864247142713439062584 n - 1482580188414223735530160815040 n 6 5 + 4923734798078387456381301206352 n - 12503563589887788221131913079264 n 4 + 23505662001438843316261994110464 n 3 - 31159696085540382538062848494080 n 2 + 26935513638204293084825139916800 n - 13165101368725155884989194240000 n + 2593211366817313747926220800000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), - 69 ( 23 22 21 13719571252 n - 4102952575019 n + 567791200203029 n 20 19 - 48508401303037060 n + 2876533902227015877 n 18 17 - 126098804689331205039 n + 4246889198183336226544 n 16 15 - 112687806113039216384600 n + 2395916588333413546512782 n 14 13 - 41281944959125642025438569 n + 580556844930013887300688869 n 12 11 - 6689351229137164182354827340 n + 63209415286001612881497085897 n 10 9 - 488941932198569403011430146669 n + 3081868053109474250485003671014 n 8 - 15707368534215278021192619192280 n 7 + 63996268822453112284361198414592 n 6 - 205072046271337027184227000186704 n 5 + 505135388550004686487502769560544 n 4 - 925556815053082069621490862558720 n 3 + 1201258302608281390647090137049600 n 2 - 1021092800122453299316341493248000 n + 492931114192186265758005043200000 n - 96399813853426228455518208000000)/ (262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 23 ( 23 22 21 113925826103 n - 32361773104909 n + 4292382921339403 n 20 19 - 353889010511081045 n + 20360224567779019308 n 18 17 - 869676976639074275364 n + 28641473601382683185738 n 16 15 - 745380273571005598327450 n + 15583365858213227985286783 n 14 13 - 264608536462106629649377829 n + 3674437545780120532548405543 n 12 11 - 41877928928819971294693054305 n + 392022747814857485901211874798 n 10 - 3008300580514253700509069845114 n 9 + 18834802735838773839142385161588 n 8 - 95462376387352252766772803879360 n 7 + 387186327934331851408831635479808 n 6 - 1236301162792967009732687388170784 n 5 + 3037102679428193439831213496657728 n 4 - 5554433610123822329755979577507840 n 3 + 7200851152509422639035511312563200 n 2 - 6118300622787274327095033394176000 n + 2954429732184818256072293990400000 n - 578398883120557370733109248000000 )/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 115 ( 24 23 22 224726885587 n - 67268076878388 n + 9471481695214186 n 21 20 - 834595643935308816 n + 51650096574015027781 n 19 18 - 2388003802672135803540 n + 85655087797856972283592 n 17 16 - 2443153991872129978935504 n + 56348627561446488790988125 n 15 14 - 1062842996503604075428435932 n + 16516618082825338673053350898 n 13 12 - 212378245615572816919190290512 n + 2263431115209264489033212484379 n 11 - 19979301078102717258283615367964 n 10 + 145613800012873529425262491383484 n 9 - 871330773033024628844804552571600 n 8 + 4244272937218287707290596303965008 n 7 - 16626036936439824127971901959955776 n 6 + 51502475459938369587415925217387840 n 5 - 123244115628678250134820464230093568 n 4 + 220386950202344142201633974047749120 n 3 - 280359285858020009823597590675558400 n 2 + 234560301168394970407075997306880000 n - 111931448743122306528381729177600000 n + 21747798005332957139564907724800000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), - 115 (257801350903 n 23 22 21 - 75867437860548 n + 10520405288245810 n - 914395729381834656 n 20 19 + 55896273527861759377 n - 2555960287500462163860 n 18 17 + 90778729284084674575336 n - 2566579059340463977815984 n 16 15 + 58733023477504776345628441 n - 1100147619143758186655503212 n 14 13 + 16991904030248854388808096922 n - 217319004991824332011502828832 n 12 + 2305277884837224132574627211695 n 11 - 20266659114365583230796022935804 n 10 + 147199267223360291146131768001708 n 9 - 878258259287550792429204794198160 n 8 + 4267704606803198432431299035144464 n 7 - 16685135323143608376187528259578176 n 6 + 51606065112534576076698756928540224 n 5 - 123349426415252986410964471460122368 n 4 + 220398275795154030876643550502405120 n 3 - 280239960368769440542502915680358400 n 2 + 234420326461748378567992334192640000 n - 111879145236615974260128807321600000 n + 21747798005332957139564907724800000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), - 5 (12939294675194 n 24 23 22 - 4084045806010225 n + 609605244734273000 n - 57245064687490293550 n 21 20 + 3795203875070278547030 n - 188968729385652314667415 n 19 18 + 7338979228223252713845500 n - 227916968229675885628171000 n 17 16 + 5756813542716437131581115190 n - 119653221211088445756910522015 n 15 + 2062610930913443345262347769200 n 14 - 29633994371271126109405820714950 n 13 + 355717656734920219984409155544810 n 12 - 3568408408621735696599704406115465 n 11 + 29860610655916812633854420471129900 n 10 - 207607382478662824457707321390942900 n 9 + 1191629160240395569533048413652320720 n 8 - 5595223989366049661542932845258512240 n 7 + 21221951926289709212417312644696065600 n 6 - 63908611650970543650012591535661937600 n 5 + 149225737425522593220967664449320397056 n 4 - 261283766388787385430806558935557872640 n 3 + 326527340560133905711954086636186316800 n 2 - 269240417373302655273052297428234240000 n + 127052315184448175083563647213568000000 n - 24509768352010242696289651005849600000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 15 (4538188336243 n 24 23 22 - 1423923652910985 n + 211396378832316070 n - 19753900038267739990 n 21 20 + 1303810855222675364565 n - 64657542503108640207815 n 19 18 + 2501997772299307433226020 n - 77447974757182741274504040 n 17 16 + 1950502243474301024218507885 n - 40435011809177058575907013015 n 15 14 + 695420164899997878676439501670 n - 9970922140533773033212866514190 n 13 + 119474478908236453077069864447275 n 12 - 1196657497145470905178597826475465 n 11 + 10000291534982942989441951198439920 n 10 - 69448210898716103816313145011379540 n 9 + 398237658831495376589813027943352320 n 8 - 1868416580809297387607955226552073840 n 7 + 7082108579056951188561943344305470720 n 6 - 21316601696419900072982570462901282240 n 5 + 49755232063991445274654283368454291712 n 4 - 87095215973269611974589468926580218880 n 3 + 108826505095359440483397798520680345600 n 2 - 89728852518733869679090563108679680000 n + 42344193690481111972917273265766400000 n - 8169922784003414232096550335283200000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), ( (73469412288608 n + 73469412288608) (2 n - 53)! - (n - 27)! (n - 25)! binomial(2 n, n))/((n - 27)! (n - 25)! binomial(2 n, n))] and in Maple notation [3/2048*(5592403*n^13-472557644*n^12+17920790303*n^11-402850616740*n^10+ 5971651729329*n^9-61423921307892*n^8+449029010470469*n^7-2347708286389180*n^6+ 8689561298528968*n^5-21938277731605664*n^4+33337326484766928*n^3-\ 10625153105062080*n^2-77454872739417600*n+145720693198080000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-13)/(2*n-25)/(2*n-15)/( 2*n-17)/(2*n-23), 15/4096*(4473899*n^14-438438007*n^13+19439532931*n^12-\ 515772685883*n^11+9125530931517*n^10-113542033505901*n^9+1020236670772033*n^8-\ 6681841799401769*n^7+31645784841394724*n^6-104412438811873192*n^5+ 211390553411537136*n^4-111445083994437648*n^3-652083660853266240*n^2+ 1446707245714214400*n-58288277279232000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n -23), 5/2048*(6710621*n^14-657601273*n^13+29153061574*n^12-773237470187*n^11+ 13669142955468*n^10-169697037999489*n^9+1515987213751582*n^8-9778263182328641*n ^7+44464176284398871*n^6-130767619429585138*n^5+171376830175613244*n^4+ 292471863263071128*n^3-1396089188067627360*n^2+1117633639918521600*n-\ 87432415918848000)/(2*n-27)/(2*n-25)/(2*n-23)/(2*n-21)/(2*n-19)/(2*n-17)/(2*n-\ 15)/(2*n-13)/(2*n-11)/(2*n-9)/(2*n-7)/(2*n-5)/(2*n-3)/(-1+2*n), 115/2048*( 583444*n^15-65622225*n^14+3361756615*n^13-103830281100*n^12+2155978404853*n^11-\ 31744851708570*n^10+339977690776295*n^9-2661315767424600*n^8+14935082761786187* n^7-56239832613849885*n^6+113224834838960390*n^5+44199425469626700*n^4-\ 830642745554718984*n^3+1611325514525179680*n^2-1008493533576748800*n+ 110240872245504000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/ (2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 115/ 4096*(1166279*n^15-131102553*n^14+6708494681*n^13-206706967467*n^12+ 4271569603847*n^11-62290420241049*n^10+654433557257623*n^9-4932270040487361*n^8 +25642327790813734*n^7-81258522121474698*n^6+81025668924450796*n^5+ 446016233455484328*n^4-1887243655085073360*n^3+2833420941696628800*n^2-\ 1641351502465113600*n+220481744491008000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 23/16384*(46581937*n^16-5950858472*n^15+347867874820*n^14-\ 12314032141040*n^13+294105249113494*n^12-4989899405813504*n^11+ 61487466581847020*n^10-549965575175507920*n^9+3470023044867094001*n^8-\ 14158088181885223576*n^7+26445039254091840200*n^6+57843494674671392960*n^5-\ 499651249689068520432*n^4+1316237151140428263552*n^3-1646950341643312631040*n^2 +898365473749271808000*n-136698681584424960000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-17)/(2*n-23)/(2*n-29), 3/16384*(355836021*n^16-45333803848*n^15+ 2637629104500*n^14-92627315555600*n^13+2183052855973262*n^12-36235219389978336* n^11+430742488840060540*n^10-3629128318738036400*n^9+20581726565708664213*n^8-\ 65916701032834836184*n^7+4889475642746542120*n^6+991392026546212200000*n^5-\ 4557470111375099640496*n^4+10097769714734499274368*n^3-11695890413551225340160* n^2+6289707236803635456000*n-1048023225480591360000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/( 2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 969/16384*(2186174*n^17-313133223*n^16+ 20554958608*n^15-817453836700*n^14+21910653972228*n^13-415831442488826*n^12+ 5695615420625416*n^11-55989325702364340*n^10+379357871100887822*n^9-\ 1545814189981855079*n^8+1311472615737191544*n^7+26035449055806324680*n^6-\ 175501654449470752224*n^5+576454886323531018128*n^4-1092170781565963613568*n^3+ 1164203632963394415360*n^2-609434094456988416000*n+107073580312258560000)/(2*n-\ 5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 323/16384*( 6459632*n^17-917841649*n^16+59537949704*n^15-2326895415940*n^14+60807408623024* n^13-1112075618160598*n^12+14411300046745888*n^11-129638205845402540*n^10+ 741084462462257056*n^9-1706169436464063857*n^8-11087846074771262888*n^7+ 129095097476911329400*n^6-630843876252502523712*n^5+1825957048673850277104*n^4-\ 3237481578642245080704*n^3+3342060020428493518080*n^2-1747572202976802048000*n+ 321220740936775680000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 323/16384*(12573149*n^18-1983894543*n^17+143227944213*n^16-\ 6246378052020*n^15+182746300969218*n^14-3757793499861786*n^13+55063076598162386 *n^12-563966831640300660*n^11+3690964705660763817*n^10-9364604896331912799*n^9-\ 86853877370537005911*n^8+1192770899742619622880*n^7-7394426741943780217384*n^6+ 28625826496296900706128*n^5-72613365727404291656688*n^4+ 118512793450356329692800*n^3-116496193317259907692800*n^2+ 59829581616404624640000*n-11242725932787148800000)/(2*n-29)/(2*n-35)/(2*n-23)/( 2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33) /(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 17/16384*(228298712*n^18-\ 35416905732*n^17+2499298721331*n^16-105694141048200*n^15+2964209083222044*n^14-\ 57374413851591144*n^13+764987726014762562*n^12-6561592181804602800*n^11+ 24755154989275358076*n^10+186944426980765395084*n^9-3781485995903727173517*n^8+ 31474691123747052144600*n^7-165062970624172599348232*n^6+ 585426815313158372553792*n^5-1408782477964665730007376*n^4+ 2227925142697668506294400*n^3-2157553364137509450873600*n^2+ 1109862787924352689920000*n-213611792722955827200000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-\ 33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/16384*(22364792*n^19 -3799347136*n^18+293889190038*n^17-13634616107829*n^16+419518473892464*n^15-\ 8885589526396212*n^14+128140688204806196*n^13-1128485770371737518*n^12+ 2360717514108535296*n^11+93148904194603303332*n^10-1614239291113545909306*n^9+ 15082200911256954670923*n^8-95014317049561195227352*n^7+ 424394635682317981426616*n^6-1351744604898145790192928*n^5+ 3015442683972349118117424*n^4-4523176818305461245859200*n^3+ 4235013005941289435270400*n^2-2145700633206054355200000*n+ 415980859513124505600000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15) /(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7) /(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 969/16384*(6678794*n^19-1099579662*n^18+ 81622281966*n^17-3579466681743*n^16+101410215854748*n^15-1871409933544704*n^14+ 19948367555551772*n^13-20267850601730706*n^12-3414356626439007678*n^11+ 68535376821008562894*n^10-789317363623448052042*n^9+6273585920698628154441*n^8-\ 36120310414495849715464*n^7+152307391739192628273672*n^6-\ 466582012835431121073696*n^5+1013896146778212427669008*n^4-\ 1496358188199035867318400*n^3+1390795875003386335276800*n^2-\ 705896151371420486400000*n+138660286504374835200000)/(2*n-29)/(2*n-35)/(2*n-23) /(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 969/65536*( 45280917*n^20-7996756850*n^19+633661064735*n^18-29344692972150*n^17+ 854666818631022*n^16-14902932787894500*n^15+86544991252019070*n^14+ 3092179664721795700*n^13-109372089609443523343*n^12+1959768754878916045350*n^11 -23973655703547975927645*n^10+215425087859419908130050*n^9-\ 1458289475380993825540548*n^8+7480918593414467869565200*n^7-\ 28912924503733521765565360*n^6+82837878517734644447546400*n^5-\ 171029768134962282249498048*n^4+243026118758046114190540800*n^3-\ 220102205918829835524940800*n^2+110150613010942568064000000*n-\ 21631004694682474291200000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(105573421*n^20-\ 17391556610*n^19+1241989610775*n^18-48046417287990*n^17+914921000933766*n^16+ 4278255807597180*n^15-824437022796384050*n^14+27509001269796528820*n^13-\ 563748467565103978599*n^12+8229281349085204821270*n^11-89972541639889722601125* n^10+752181739458145404387330*n^9-4841444555849816277096284*n^8+ 23940092400088669688881360*n^7-90044584029021496582345200*n^6+ 252908306520837172333431840*n^5-514970496977918866180202304*n^4+ 725481518920282009777996800*n^3-654634750243480003457510400*n^2+ 328048394066751873715200000*n-64893014084047422873600000)/(2*n-29)/(2*n-35)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/( 2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(140230126*n^21-22729862085*n^20+1452127245380*n^19-33257968232895*n^ 18-1281722392256334*n^17+136040668892099370*n^16-5856114061243835000*n^15+ 164553710615006403810*n^14-3362727275255814683854*n^13+52254589319144641385775* n^12-631211405775707414346780*n^11+5991160813875387807876765*n^10-\ 44841070313024125234527794*n^9+264098033221299741478173180*n^8-\ 1214982593582146164891718960*n^7+4308626187698499657335787120*n^6-\ 11537438048238323465390682144*n^5+22613622288787692202233533760*n^4-\ 30931174011782018693137904640*n^3+27317741740799431077126835200*n^2-\ 13506568118223869231262720000*n+2660613577445944337817600000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35) /(2*n-29), 323/262144*(245886631*n^21-20351860788*n^20-1479799670335*n^19+ 327774473782980*n^18-25098174006900414*n^17+1173756529974733992*n^16-\ 38375098431153160430*n^15+931433061984501761160*n^14-17342625238278398982469*n^ 13+252500526446812138874652*n^12-2906290890206321819913075*n^11+ 26580488766624047514217140*n^10-193241333175371088630577844*n^9+ 1112277389423273826364861872*n^8-5025297458767230878397688720*n^7+ 17572747570937363451834640320*n^6-46563034473336903521777555904*n^5+ 90591141821260194037627990272*n^4-123348575014368722708278717440*n^3+ 108737526618878292411216998400*n^2-53809481736955437016043520000*n+ 10642454309783777351270400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), -19/524288*( 5857136657*n^22-3652592399929*n^21+669060900445399*n^20-65311517983893335*n^19+ 4112524320572477502*n^18-183098998106206808934*n^17+6073814823637698220094*n^16 -155094421537353987727390*n^15+3113669561983036280193277*n^14-\ 49824770868162259943522549*n^13+640863792866353927101275499*n^12-\ 6653993108045444039989305195*n^11+55807641211687620447871532612*n^10-\ 377076632452590698539189205884*n^9+2039797729029531487878760984624*n^8-\ 8743942070756584756463857211600*n^7+29250872887978944015890200969152*n^6-\ 74689425936347802002851489072704*n^5+140951898719046022050081411024384*n^4-\ 187287208479531600395067064212480*n^3+162044323635360607739224232140800*n^2-\ 79160099151358602875159101440000*n+15559268200903882487557324800000)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2* n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/( 2*n-39)/(2*n-35)/(2*n-29), -57/262144*(4739094037*n^22-1485522170429*n^21+ 206309005515074*n^20-17204746467946345*n^19+976457262709946787*n^18-\ 40349745578307558624*n^17+1264923804641876061364*n^16-30895567032776201887010*n ^15+598431527882353349066267*n^14-9299035178212441401969229*n^13+ 116735570441937053352350514*n^12-1187784620979376807169307525*n^11+ 9795806174216923444908069757*n^10-65271026774304743952713332454*n^9+ 349067186864247142713439062584*n^8-1482580188414223735530160815040*n^7+ 4923734798078387456381301206352*n^6-12503563589887788221131913079264*n^5+ 23505662001438843316261994110464*n^4-31159696085540382538062848494080*n^3+ 26935513638204293084825139916800*n^2-13165101368725155884989194240000*n+ 2593211366817313747926220800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), -69/ 262144*(13719571252*n^23-4102952575019*n^22+567791200203029*n^21-\ 48508401303037060*n^20+2876533902227015877*n^19-126098804689331205039*n^18+ 4246889198183336226544*n^17-112687806113039216384600*n^16+ 2395916588333413546512782*n^15-41281944959125642025438569*n^14+ 580556844930013887300688869*n^13-6689351229137164182354827340*n^12+ 63209415286001612881497085897*n^11-488941932198569403011430146669*n^10+ 3081868053109474250485003671014*n^9-15707368534215278021192619192280*n^8+ 63996268822453112284361198414592*n^7-205072046271337027184227000186704*n^6+ 505135388550004686487502769560544*n^5-925556815053082069621490862558720*n^4+ 1201258302608281390647090137049600*n^3-1021092800122453299316341493248000*n^2+ 492931114192186265758005043200000*n-96399813853426228455518208000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -23/524288*(113925826103*n^23-\ 32361773104909*n^22+4292382921339403*n^21-353889010511081045*n^20+ 20360224567779019308*n^19-869676976639074275364*n^18+28641473601382683185738*n^ 17-745380273571005598327450*n^16+15583365858213227985286783*n^15-\ 264608536462106629649377829*n^14+3674437545780120532548405543*n^13-\ 41877928928819971294693054305*n^12+392022747814857485901211874798*n^11-\ 3008300580514253700509069845114*n^10+18834802735838773839142385161588*n^9-\ 95462376387352252766772803879360*n^8+387186327934331851408831635479808*n^7-\ 1236301162792967009732687388170784*n^6+3037102679428193439831213496657728*n^5-\ 5554433610123822329755979577507840*n^4+7200851152509422639035511312563200*n^3-\ 6118300622787274327095033394176000*n^2+2954429732184818256072293990400000*n-\ 578398883120557370733109248000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45) , -115/2097152*(224726885587*n^24-67268076878388*n^23+9471481695214186*n^22-\ 834595643935308816*n^21+51650096574015027781*n^20-2388003802672135803540*n^19+ 85655087797856972283592*n^18-2443153991872129978935504*n^17+ 56348627561446488790988125*n^16-1062842996503604075428435932*n^15+ 16516618082825338673053350898*n^14-212378245615572816919190290512*n^13+ 2263431115209264489033212484379*n^12-19979301078102717258283615367964*n^11+ 145613800012873529425262491383484*n^10-871330773033024628844804552571600*n^9+ 4244272937218287707290596303965008*n^8-16626036936439824127971901959955776*n^7+ 51502475459938369587415925217387840*n^6-123244115628678250134820464230093568*n^ 5+220386950202344142201633974047749120*n^4-280359285858020009823597590675558400 *n^3+234560301168394970407075997306880000*n^2-\ 111931448743122306528381729177600000*n+21747798005332957139564907724800000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -115/2097152*(257801350903* n^24-75867437860548*n^23+10520405288245810*n^22-914395729381834656*n^21+ 55896273527861759377*n^20-2555960287500462163860*n^19+90778729284084674575336*n ^18-2566579059340463977815984*n^17+58733023477504776345628441*n^16-\ 1100147619143758186655503212*n^15+16991904030248854388808096922*n^14-\ 217319004991824332011502828832*n^13+2305277884837224132574627211695*n^12-\ 20266659114365583230796022935804*n^11+147199267223360291146131768001708*n^10-\ 878258259287550792429204794198160*n^9+4267704606803198432431299035144464*n^8-\ 16685135323143608376187528259578176*n^7+51606065112534576076698756928540224*n^6 -123349426415252986410964471460122368*n^5+220398275795154030876643550502405120* n^4-280239960368769440542502915680358400*n^3+ 234420326461748378567992334192640000*n^2-111879145236615974260128807321600000*n +21747798005332957139564907724800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13 )/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), -5/2097152*(12939294675194*n^25-4084045806010225*n^24+ 609605244734273000*n^23-57245064687490293550*n^22+3795203875070278547030*n^21-\ 188968729385652314667415*n^20+7338979228223252713845500*n^19-\ 227916968229675885628171000*n^18+5756813542716437131581115190*n^17-\ 119653221211088445756910522015*n^16+2062610930913443345262347769200*n^15-\ 29633994371271126109405820714950*n^14+355717656734920219984409155544810*n^13-\ 3568408408621735696599704406115465*n^12+29860610655916812633854420471129900*n^ 11-207607382478662824457707321390942900*n^10+ 1191629160240395569533048413652320720*n^9-5595223989366049661542932845258512240 *n^8+21221951926289709212417312644696065600*n^7-\ 63908611650970543650012591535661937600*n^6+ 149225737425522593220967664449320397056*n^5-\ 261283766388787385430806558935557872640*n^4+ 326527340560133905711954086636186316800*n^3-\ 269240417373302655273052297428234240000*n^2+ 127052315184448175083563647213568000000*n-\ 24509768352010242696289651005849600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-45), -15/2097152*(4538188336243*n^25-1423923652910985*n^24+ 211396378832316070*n^23-19753900038267739990*n^22+1303810855222675364565*n^21-\ 64657542503108640207815*n^20+2501997772299307433226020*n^19-\ 77447974757182741274504040*n^18+1950502243474301024218507885*n^17-\ 40435011809177058575907013015*n^16+695420164899997878676439501670*n^15-\ 9970922140533773033212866514190*n^14+119474478908236453077069864447275*n^13-\ 1196657497145470905178597826475465*n^12+10000291534982942989441951198439920*n^ 11-69448210898716103816313145011379540*n^10+ 398237658831495376589813027943352320*n^9-1868416580809297387607955226552073840* n^8+7082108579056951188561943344305470720*n^7-\ 21316601696419900072982570462901282240*n^6+ 49755232063991445274654283368454291712*n^5-\ 87095215973269611974589468926580218880*n^4+ 108826505095359440483397798520680345600*n^3-\ 89728852518733869679090563108679680000*n^2+ 42344193690481111972917273265766400000*n-8169922784003414232096550335283200000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), (( 73469412288608*n+73469412288608)*(2*n-53)!-(n-27)!*(n-25)!*binomial(2*n,n))/(n-\ 27)!/(n-25)!/binomial(2*n,n)] The limits, as n goes to infinity are 16777209 67108485 33553105 16774015 134122085 1071384551 1067508063 [--------, --------, --------, --------, ---------, ----------, ----------, 16777216 67108864 33554432 16777216 134217728 1073741824 1073741824 1059201303 130403821 4061127127 485134763 902978477 3235875693 ----------, ---------, ----------, ---------, ----------, ----------, 1073741824 134217728 4294967296 536870912 1073741824 4294967296 43877208573 34100214983 22647165349 79421381813 -111285596483 -----------, -----------, -----------, ------------, -------------, 68719476736 68719476736 68719476736 549755813888 2199023255552 -270128360109 -236662604097 -2620294000369 -25843591842505 -------------, -------------, --------------, ---------------, 1099511627776 549755813888 4398046511104 35184372088832 -29647155353845 -32348236687985 -68072825043645 -279179057576637 ---------------, ---------------, ---------------, ----------------] 35184372088832 35184372088832 70368744177664 281474976710656 and in Maple notation [16777209/16777216, 67108485/67108864, 33553105/33554432, 16774015/16777216, 134122085/134217728, 1071384551/1073741824, 1067508063/1073741824, 1059201303/ 1073741824, 130403821/134217728, 4061127127/4294967296, 485134763/536870912, 902978477/1073741824, 3235875693/4294967296, 43877208573/68719476736, 34100214983/68719476736, 22647165349/68719476736, 79421381813/549755813888, -\ 111285596483/2199023255552, -270128360109/1099511627776, -236662604097/ 549755813888, -2620294000369/4398046511104, -25843591842505/35184372088832, -\ 29647155353845/35184372088832, -32348236687985/35184372088832, -68072825043645/ 70368744177664, -279179057576637/281474976710656] and in floating point [.9999995828, .9999943525, .9999604523, .9998092055, .9992874041, .9978046184, .9941943577, .9864580845, .9715841785, .9455548429, .9036339130, .8409642400, .\ 7534110204, .6384974196, .4962234377, .3295596303, .1444666519, -.5060683019e-1 , -.2456803123, -.4304867691, -.5957858776, -.7345190580, -.8426228349, -.91939\ 21837, -.9673730267, -.9918432567] The cut off is at j=, 18 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 28], vs. those in the, 2, -th row from j=1 to j=, 27, are as follws 14 13 12 11 [21 (6391319 n - 626349001 n + 27771832231 n - 736895774069 n 10 9 8 + 13040295948537 n - 162337819286643 n + 1460989308174253 n 7 6 5 - 9613489925263367 n + 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2771404829735283823615771954381315200 n 5 + 6481028454743665532560987178051599872 n 4 - 11359788038770511481475853552511559680 n 3 + 14205228878459789300517253253246361600 n 2 - 11715589888704550829907649222778880000 n + 5527495866468956363883736252416000000 n - 1065642102261314899838680478515200000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 5 (11974622770144 n 24 23 22 - 3815867016406325 n + 574485399359022700 n - 54364156934186112950 n 21 20 + 3629162888502067766080 n - 181820303494690020444515 n 19 18 + 7100387672973724734084700 n - 221592745312715402491969400 n 17 16 + 5621537560278628644331960240 n - 117293486616170260084324069115 n 15 + 2028825839789800072738103860900 n 14 - 29235748585085136123358993818350 n 13 + 351852256749507163917551387263360 n 12 - 3537602576029339955262434301385565 n 11 + 29660233585871075013180944579430100 n 10 - 206554167882254985898804301900663300 n 9 + 1187223708448837626338204599112697520 n 8 - 5580901484991988945427314510404641840 n 7 + 21187132196459502993462287804381284800 n 6 - 63849763881986034265819563174173496000 n 5 + 149168653473448675216688827797831142656 n 4 - 261281078487328170440861062015296752640 n 3 + 326595662380213311800183645313536716800 n 2 - 269317359969161292752738880431370240000 n + 127080506774455088176151972093952000000 n - 24509768352010242696289651005849600000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 15 (34665564451359 n 25 24 - 11823557030067367 n + 1911461928040931500 n 23 22 - 194885332453145004450 n + 14065369279267870085405 n 21 20 - 764593227900841954319705 n + 32521011752767798798493670 n 19 18 - 1109911234837431281233950400 n + 30926158512062258094797841265 n 17 - 712081712417494650052992784185 n 16 + 13662403450700608355807201715920 n 15 - 219635075864419116583719047915450 n 14 + 2967721357719706919623894546737435 n 13 - 33742546619607280416663537041202295 n 12 + 322575312624099181130537396077685470 n 11 - 2586019952073567533858195164073842500 n 10 + 17304666335224853200162043259080952920 n 9 - 95991213625613617128870555195109617200 n 8 + 437194647655815152008174193849745369120 n 7 - 1613821471055342513936227386470688004800 n 6 + 4744230612095771199556457210137766111616 n 5 - 10844592516881840872375979700739643509248 n 4 + 18638047149456956050926555369357093304320 n 3 - 22921240045181261277811118338477959782400 n 2 + 18646495844546642788030958608016056320000 n - 8704677186299444458252099651721625600000 n + 1666664247936696503347696268397772800000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), ( 278134203664016 (n - 25)! (n + 1) (2 n - 55)! + 417201305496024 (n - 26)! ( (n + 1) (2 n - 54)! - 1/417201305496024 binomial(2 n, n) (n - 28)! (n - 25)!))/((n - 28)! (n - 26)! (n - 25)! binomial(2 n, n)), ( (278134203664016 n + 278134203664016) (2 n - 55)! - (n - 28)! (n - 26)! binomial(2 n, n))/((n - 28)! (n - 26)! binomial(2 n, n))] and in Maple notation [21/8192*(6391319*n^14-626349001*n^13+27771832231*n^12-736895774069*n^11+ 13040295948537*n^10-162337819286643*n^9+1460989308174253*n^8-9613489925263367*n ^7+46189037116369004*n^6-159523650266036056*n^5+379444916768991216*n^4-\ 540616627804007664*n^3+114140277753733440*n^2+1334434099899916800*n-\ 2331531091169280000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11) /(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23), 21/8192*( 6391301*n^14-626344105*n^13+27771219619*n^12-736849094345*n^11+13037879950953*n ^10-162248163051975*n^9+1458536551157257*n^8-9563451514663475*n^7+ 45429002906476046*n^6-151059886866708820*n^5+312691655889748824*n^4-\ 192976694712809280*n^3-900840550026384000*n^2+2144045820880512000*n-\ 83268967541760000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/( 2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23), 15/16384*( 35790607*n^15-4026316050*n^14+206354875720*n^13-6379618135800*n^12+ 132753191077234*n^11-1963885525061460*n^10+21249886586760260*n^9-\ 170102048356468800*n^8+1002301389553975511*n^7-4215686522304787530*n^6+ 11500538151411911420*n^5-13365206471101157400*n^4-27639776676778064352*n^3+ 115677271140041687040*n^2-89803344812973926400*n+6761440164390912000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2 *n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 5/4096*(26840599*n^15-3019129125*n^ 14+154694441965*n^13-4779598626075*n^12+99319246644163*n^11-1464546332794545*n^ 10+15730123148191595*n^9-123814673495681025*n^8+702113227516302602*n^7-\ 2698536382759327110*n^6+5727713833854570740*n^5+755968634044860600*n^4-\ 37402572852746946864*n^3+75511081122993017280*n^2-47732258132703244800*n+ 5071080123293184000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11) /(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 115/ 16384*(9332929*n^16-1194075512*n^15+69993069940*n^14-2489931674720*n^13+ 59995747609798*n^12-1033845676891664*n^11+13088439123453980*n^10-\ 122675740850222560*n^9+840649620017109617*n^8-4024151729524062136*n^7+ 11783077856206540520*n^6-9994848925856005120*n^5-64652087952410047344*n^4+ 255407298401012349312*n^3-371309737451422429440*n^2+210282027049604966400*n-\ 27339736316884992000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11 )/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-\ 29), 23/16384*(46621493*n^16-5959719016*n^15+348766587140*n^14-12368530024480*n ^13+296301801783806*n^12-5051756836056112*n^11+62732203978401100*n^10-\ 567943363673697440*n^9+3654363354406044709*n^8-15463046601118444328*n^7+ 32501850232825235080*n^6+41070273224455121920*n^5-477401947737265225008*n^4+ 1316034552487635939456*n^3-1675442041818236248320*n^2+916664291971948800000*n-\ 136698681584424960000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 1173/16384*(1824028*n^17-262810593*n^16+17410690064*n^15-702218829140*n^ 14+19227276843096*n^13-376766062609846*n^12+5414519597540768*n^11-\ 57300551343260940*n^10+438574204227426124*n^9-2293495620104303329*n^8+ 6827165738497885872*n^7+24875236936786600*n^6-96535636430854585248*n^5+ 425322921818395518768*n^4-914772664013843896704*n^3+1035122646818525748480*n^2-\ 545214597885796608000*n+88452088084039680000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25) /(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 51/16384*(41735086*n^17-5992647277*n^16+ 394798065792*n^15-15782555085620*n^14+426151532059652*n^13-8171634825899454*n^ 12+113567785422143624*n^11-1140456730868303420*n^10+8001777852597681438*n^9-\ 35233015876618249261*n^8+54962507542972314376*n^7+392696031441335458200*n^6-\ 3125853608731763758176*n^5+10759458060740574898992*n^4-20829306381864600177792* n^3+22420970783227873347840*n^2-11740707955471106304000*n+ 2034398025932912640000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 2907/32768*(2898417*n^18-464351829*n^17+34232810544*n^16-\ 1536236065060*n^15+46735483721094*n^14-1014394170239758*n^13+16061020712100468* n^12-185554479285869980*n^11+1522472903689756161*n^10-8108453676847234997*n^9+ 18214823988253642932*n^8+100240902303689241640*n^7-1156320212857495696272*n^6+ 5488387509143428337584*n^5-15463792201896734746944*n^4+26783485132088794982400* n^3-27085661757329636966400*n^2+13891968175446789120000*n-\ 2498383540619366400000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25)/( 2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/( 2*n-3)/(-1+2*n)/(2*n-5), 323/32768*(25591303*n^18-4063465161*n^17+295784973246* n^16-13038430135140*n^15+386835905938146*n^14-8104816522655622*n^13+ 121941730059783712*n^12-1302462134682328620*n^11+9288365311249223199*n^10-\ 34059962734305495273*n^9-84158917519917538662*n^8+2015287848702160283160*n^7-\ 13756548335684166518048*n^6+55506438207572067835056*n^5-\ 144002843675649583795296*n^4+238026340382187523161600*n^3-\ 235340046671932789977600*n^2+120789384358327534080000*n-22485451865574297600000 )/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 323/65536*(99072175*n^19-17350702112*n^18+1395685007445*n^17-68139869583588* n^16+2245221332741310*n^15-52420432521206064*n^14+882360015444231130*n^13-\ 10581273617461430936*n^12+84557244965959148955*n^11-329225419395962568336*n^10-\ 1541944260917627006895*n^9+34920648203864386654236*n^8-285773291609860818737240 *n^7+1459179487691115526707712*n^6-5041238485676050320133680*n^5+ 11845442996039873405268288*n^4-18337986145094074079443200*n^3+ 17420894879352388963660800*n^2-8802538069099981409280000*n+ 1663923438052498022400000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15 )/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7 )/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/16384*(23495734*n^19-4042499666*n^18+ 317700042906*n^17-15041349623949*n^16+475515418109028*n^15-10474535678909232*n^ 14+161312829751908052*n^13-1647074276532630998*n^12+8482557105200949342*n^11+ 38526822864226716162*n^10-1248150216945958189422*n^9+13266330440849474486043*n^ 8-88531675869654109038104*n^7+408595545050906028349336*n^6-\ 1328429622859902683293536*n^5+3002694745167809152313904*n^4-\ 4539740770106661587184000*n^3+4265431048107838500998400*n^2-\ 2159306548764925190400000*n+415980859513124505600000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 969/32768*( 28877271*n^20-5407898050*n^19+462890444305*n^18-23874476436450*n^17+ 821475024539136*n^16-19610910542266500*n^15+322595377380118410*n^14-\ 3326053898405722900*n^13+10532593939003839391*n^12+307298071378466432550*n^11-\ 6654482435587975948635*n^10+75490943086414300920150*n^9-\ 583240971030261426246774*n^8+3260083504738171979861600*n^7-\ 13369486264823886301443680*n^6+39934310171228273648239200*n^5-\ 84832916925668240669709024*n^4+122685604481852576324870400*n^3-\ 111971347698703027458470400*n^2+55900665353711610432000000*n-\ 10815502347341237145600000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 969/32768*(25550971*n^20-\ 4632870150*n^19+380195300005*n^18-18517313950350*n^17+585898741523136*n^16-\ 12161760064262700*n^15+147510187602205810*n^14-207031540295942700*n^13-\ 32029102618493549309*n^12+753925234704029495250*n^11-10251099531654200000535*n^ 10+97529228941259407911450*n^9-684281648295608951186174*n^8+ 3596484621722112196303200*n^7-14138595855922440566854880*n^6+ 40996374742748741551581600*n^5-85334000166228062333238624*n^4+ 121858909124586166017734400*n^3-110596474835665831523750400*n^2+ 55306406982978575424000000*n-10815502347341237145600000)/(2*n-29)/(2*n-35)/(2*n -39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2 *n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 969 /131072*(170313193*n^21-32900149569*n^20+2858399443865*n^19-145408122572195*n^ 18+4650441533338188*n^17-88051798913388954*n^16+463741447338657730*n^15+ 27657534424179202410*n^14-1042176788101264990447*n^13+21097191807724378251451*n ^12-297453001068417490469355*n^11+3129690487549671968423865*n^10-\ 25215422834847174797525942*n^9+156870182046979898351030736*n^8-\ 752192955507590752571345440*n^7+2752187359876477156031690320*n^6-\ 7541847504653373837287584992*n^5+15022806911434968909759006336*n^4-\ 20753856919327558152989836800*n^3+18405072618459436264484505600*n^2-\ 9083680984387052960716800000*n+1773742384963962891878400000)/(2*n-5)/(-1+2*n)/( 2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/ (2*n-29), 323/65536*(193417663*n^21-34643870373*n^20+2682620913875*n^19-\ 111056442353535*n^18+2090052902449848*n^17+29745602563756842*n^16-\ 3331979666174852330*n^15+118425959266718223330*n^14-2705684291043907888897*n^13 +44913443874819492641487*n^12-566811555485507438855385*n^11+ 5549524558750810513393245*n^10-42498191416886857182552902*n^9+ 254662860039546842960274492*n^8-1187063475430166933141062240*n^7+ 4251520257024944557434626160*n^6-11467663117783947808471295712*n^5+ 22590551059444542488298207552*n^4-30994674965364766444248913920*n^3+ 27407803097632581916929100800*n^2-13543169151564395449666560000*n+ 2660613577445944337817600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 323/1048576*( 3902301125*n^22-667622117239*n^21+42935460692971*n^20-625511807748905*n^19-\ 92021840908801650*n^18+7999221854427329526*n^17-359034674876631911674*n^16+ 10996046047514762190110*n^15-249866124925381759816175*n^14+ 4377237155008437577038181*n^13-60330682584752145618963729*n^12+ 661309721211334049569389675*n^11-5791498506494473828186532500*n^10+ 40513762061314399560937673516*n^9-225348837065523791961741457904*n^8+ 987617970514163833197221231440*n^7-3361311464123783809618885752000*n^6+ 8694650616056867027236420766016*n^5-16558183884052856422746272369664*n^4+ 22123788829841806309833604807680*n^3-19183814864304346260834755788800*n^2+ 9360346211046223348613898240000*n-1830502141282809704418508800000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29), 19/1048576*(23764541431*n^22-961379035727*n^21-\ 444304021020013*n^20+71039240387553515*n^19-5480989810779349944*n^18+ 273484534926695907738*n^17-9765771354939805064618*n^16+262607587752945766948870 *n^15-5477894029905797995208329*n^14+90268992848862195151156373*n^13-\ 1188149596070143236610964793*n^12+12565148959499604532365476175*n^11-\ 106952546113410509118979387334*n^10+731289720601522051800092886448*n^9-\ 3993786984671139396840292355008*n^8+17249751333367978909515116642480*n^7-\ 58043617932328575468012734944224*n^6+148857207353728718210940137655168*n^5-\ 281774825125883841768391952045568*n^4+375091467191803537480646943528960*n^3-\ 324766312966075978554592367001600*n^2+158586541206872111924241530880000*n-\ 31118536401807764975114649600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2* n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), -1311/ 2097152*(1193274029*n^23-581712091258*n^22+105445898377138*n^21-\ 10764184081532420*n^20+725820252654445179*n^19-35090470597172034198*n^18+ 1276844960846546372468*n^17-36069665493195382635400*n^16+ 807492086320689593919139*n^15-14524119800324917248577958*n^14+ 211754663605774020831171618*n^13-2515090932923390587498239780*n^12+ 24380729014467440321744955269*n^11-192676451535892312234246964458*n^10+ 1236331067257321329804698886808*n^9-6394415100028552138037329932560*n^8+ 26363632754350430405912325695184*n^7-85272607605931569420976173076128*n^6+ 211526301941459284058116203091968*n^5-389491683204074862904556041059840*n^4+ 507016046220723232157632533811200*n^3-431456489106966072050162835456000*n^2+ 208134877712533460863805030400000*n-40589395306705780402323456000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -69/524288*(17763102119*n^23-5835275255713 *n^22+864469424858743*n^21-77780495659280645*n^20+4803775614326401944*n^19-\ 217580012898125559528*n^18+7526389086484223730698*n^17-204175580524209973579450 *n^16+4422078178234353611532679*n^15-77385257252735127674322113*n^14+ 1102614768574302098455171923*n^13-12845491778479332589026769905*n^12+ 122510792990723284351087225034*n^11-955032133584025352214547344838*n^10+ 6058554141775501051102957402588*n^9-31041829359636499520053476529760*n^8+ 127010277951147088537294505505024*n^7-408347495043201142135599290051808*n^6+ 1008345160326081213513924861786048*n^5-1850779124043308043724501806090240*n^4+ 2404567216381950241961638287283200*n^3-2044713001657389905409116322816000*n^2+ 986828642558341716763684454400000*n-192799627706852456911036416000000)/(2*n-5)/ (-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/( 2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23) /(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -69/2097152*(233059596571*n^24-\ 74761365300444*n^23+11162914984268434*n^22-1034393975641456056*n^21+ 66858459388851880981*n^20-3210085683449806882644*n^19+118995560616376665696904* n^18-3493169497298442852133056*n^17+82617899595625769593589701*n^16-\ 1592955721187953139905232004*n^15+25233573342702442634557932634*n^14-\ 329914717332193592119112633496*n^13+3567103222075902178685971997851*n^12-\ 31878952358700295871860901868684*n^11+234802953994137375230458655810284*n^10-\ 1417543328569297287853959639989616*n^9+6955823360834106042131632968830416*n^8-\ 27410897741932375881861394450453824*n^7+85310167593771614253236749270599744*n^6 -204866989027863546659509875610987776*n^5+367248634944995749068215543968404480* n^4-467872138006013084130961611484262400*n^3+ 391650134680332462349836424507392000*n^2-186820905905269683191001214156800000*n +36246330008888261899274846208000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13 )/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), -23/2097152*(941724868697*n^24-289043898990060*n^23+ 41588328714396998*n^22-3734077749720651960*n^21+234896509623918115127*n^20-\ 11016258346804884035940*n^19+400095410815032498813368*n^18-\ 11536932088284812900834880*n^17+268628960268911862404418887*n^16-\ 5109039557997172765393309620*n^15+79969017703297356928615651358*n^14-\ 1034717051508480751588232491800*n^13+11086998343092544405688281421657*n^12-\ 98316036191067823442599835216700*n^11+719348930406690457661531435517188*n^10-\ 4318552690765229244509821433911920*n^9+21092490503374429548179116498338032*n^8-\ 82805143555328307274673565151855680*n^7+256942634210412712246024051675601088*n^ 6-615641368817230201155310281385309440*n^5+ 1101872460251266323295617199738137600*n^4-1402452719480978180164008665851392000 *n^3+1173571366728531107150340133662720000*n^2-\ 559944913001396360117299716096000000*n+108738990026664785697824538624000000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n -41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -115/2097152*(457722387503 *n^25-148102773596200*n^24+22606672227758750*n^23-2166093226312381600*n^22+ 146234474986041811985*n^21-7400867126362712020480*n^20+291660739237352522427500 *n^19-9177028160912624452561000*n^18+234519856780129009924761905*n^17-\ 4925303749149838651966400680*n^16+85688346108525043833570323150*n^15-\ 1241128219989588175109068548400*n^14+15004382260347667977872931480095*n^13-\ 151451189238467616605705074772080*n^12+1274129855599622003099916447657800*n^11-\ 8898891591588946571504219881913800*n^10+51274473377068955413170694326658640*n^9 -241523260345666848358210537846050880*n^8+918430613455698530108417176570771200* n^7-2771404829735283823615771954381315200*n^6+ 6481028454743665532560987178051599872*n^5-\ 11359788038770511481475853552511559680*n^4+ 14205228878459789300517253253246361600*n^3-\ 11715589888704550829907649222778880000*n^2+ 5527495866468956363883736252416000000*n-1065642102261314899838680478515200000)/ (2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -5/2097152*( 11974622770144*n^25-3815867016406325*n^24+574485399359022700*n^23-\ 54364156934186112950*n^22+3629162888502067766080*n^21-181820303494690020444515* n^20+7100387672973724734084700*n^19-221592745312715402491969400*n^18+ 5621537560278628644331960240*n^17-117293486616170260084324069115*n^16+ 2028825839789800072738103860900*n^15-29235748585085136123358993818350*n^14+ 351852256749507163917551387263360*n^13-3537602576029339955262434301385565*n^12+ 29660233585871075013180944579430100*n^11-206554167882254985898804301900663300*n ^10+1187223708448837626338204599112697520*n^9-\ 5580901484991988945427314510404641840*n^8+ 21187132196459502993462287804381284800*n^7-\ 63849763881986034265819563174173496000*n^6+ 149168653473448675216688827797831142656*n^5-\ 261281078487328170440861062015296752640*n^4+ 326595662380213311800183645313536716800*n^3-\ 269317359969161292752738880431370240000*n^2+ 127080506774455088176151972093952000000*n-\ 24509768352010242696289651005849600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-45), -15/8388608*(34665564451359*n^26-11823557030067367*n^ 25+1911461928040931500*n^24-194885332453145004450*n^23+14065369279267870085405* n^22-764593227900841954319705*n^21+32521011752767798798493670*n^20-\ 1109911234837431281233950400*n^19+30926158512062258094797841265*n^18-\ 712081712417494650052992784185*n^17+13662403450700608355807201715920*n^16-\ 219635075864419116583719047915450*n^15+2967721357719706919623894546737435*n^14-\ 33742546619607280416663537041202295*n^13+322575312624099181130537396077685470*n ^12-2586019952073567533858195164073842500*n^11+ 17304666335224853200162043259080952920*n^10-\ 95991213625613617128870555195109617200*n^9+ 437194647655815152008174193849745369120*n^8-\ 1613821471055342513936227386470688004800*n^7+ 4744230612095771199556457210137766111616*n^6-\ 10844592516881840872375979700739643509248*n^5+ 18638047149456956050926555369357093304320*n^4-\ 22921240045181261277811118338477959782400*n^3+ 18646495844546642788030958608016056320000*n^2-\ 8704677186299444458252099651721625600000*n+ 1666664247936696503347696268397772800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), (278134203664016*(n-25)!*(n+1)*(2*n-55)!+ 417201305496024*(n-26)!*((n+1)*(2*n-54)!-1/417201305496024*binomial(2*n,n)*(n-\ 28)!*(n-25)!))/(n-28)!/(n-26)!/(n-25)!/binomial(2*n,n), ((278134203664016*n+ 278134203664016)*(2*n-55)!-(n-28)!*(n-26)!*binomial(2*n,n))/(n-28)!/(n-26)!/ binomial(2*n,n)] The limits, as n goes to infinity are 134217699 134217321 536859105 134202995 1073286835 1072294339 534896211 [---------, ---------, ---------, ---------, ----------, ----------, ---------, 134217728 134217728 536870912 134217728 1073741824 1073741824 536870912 1064244693 8425698219 8265990869 32000312525 3794561041 27982075599 ----------, ----------, ----------, -----------, ----------, -----------, 1073741824 8589934592 8589934592 34359738368 4294967296 34359738368 24758890899 165033484017 62473905149 1260443263375 451526287189 -----------, ------------, ------------, -------------, -------------, 34359738368 274877906944 137438953472 4398046511104 4398046511104 -1564382252019 -1225654046211 -16081112163399 -21659671980031 --------------, --------------, ---------------, ---------------, 17592186044416 4398046511104 35184372088832 35184372088832 -52638074562845 -1871034807835 -519983466770385 -545566565692311 ---------------, --------------, ----------------, ----------------, 70368744177664 2199023255552 562949953421312 562949953421312 -2234416425956247 -----------------] 2251799813685248 and in Maple notation [134217699/134217728, 134217321/134217728, 536859105/536870912, 134202995/ 134217728, 1073286835/1073741824, 1072294339/1073741824, 534896211/536870912, 1064244693/1073741824, 8425698219/8589934592, 8265990869/8589934592, 32000312525/34359738368, 3794561041/4294967296, 27982075599/34359738368, 24758890899/34359738368, 165033484017/274877906944, 62473905149/137438953472, 1260443263375/4398046511104, 451526287189/4398046511104, -1564382252019/ 17592186044416, -1225654046211/4398046511104, -16081112163399/35184372088832, -\ 21659671980031/35184372088832, -52638074562845/70368744177664, -1871034807835/ 2199023255552, -519983466770385/562949953421312, -545566565692311/ 562949953421312, -2234416425956247/2251799813685248] and in floating point [.9999997839, .9999969676, .9999780077, .9998902306, .9995762585, .9986519245, .9963218328, .9911551075, .9808803698, .9622879872, .9313316703, .8834900896, .\ 8143855841, .7205785630, .6003883173, .4545574859, .2865916175, .1026651915, -.\ 8892483561e-1, -.2786814653, -.4570526972, -.6156049034, -.7480320301, -.850848\ 1223, -.9236761876, -.9691209003, -.9922802251] The cut off is at j=, 19 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 29], vs. those in the, 2, -th row from j=1 to j=, 28, are as follws 14 13 12 11 [203 (661171 n - 64794743 n + 2872950509 n - 76230776167 n 10 9 8 + 1349005389303 n - 16793911022889 n + 151146222520007 n 7 6 5 - 994690675834141 n + 4781088261884506 n - 16534874791503068 n 4 3 2 + 39508682044605384 n - 57257811815323392 n + 15696430372149120 n + 134942946269414400 n - 249806902625280000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) 15 (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23)), 3045 (88156 n 14 13 12 11 - 9917523 n + 508324351 n - 15718001637 n + 327219307831 n 10 9 8 - 4846100006319 n + 52584293709233 n - 423962078782671 n 7 6 5 + 2544538854814385 n - 11227756009796898 n + 34985333046850316 n 4 3 2 - 67597595471169192 n + 33169819823440128 n + 204157626700314240 n - 445462957684454400 n + 16653793508352000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 105 ( 15 14 13 12 2556497 n - 287600472 n + 14740397054 n - 455741435058 n 11 10 9 + 9484984568540 n - 140366574311106 n + 1520054708268382 n 8 7 6 - 12190174013574414 n + 72120856296427939 n - 306049399767453342 n 5 4 3 + 852299572017902164 n - 1069359707210283528 n - 1796865234816398976 n 2 + 8382333377102497920 n - 6623291826692121600 n + 482960011742208000)/( 8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 16 15 (2 n - 17) (2 n - 23) (2 n - 29)), 15 (71578289 n - 9161295592 n 14 13 12 + 537413129540 n - 19147128009520 n + 462772765300118 n 11 10 9 - 8023279184799424 n + 102800880257827180 n - 986303522298482960 n 8 7 + 7071411099518662897 n - 37018435863149116376 n 6 5 + 132263915457063437320 n - 258354087370612425920 n 4 3 - 76494617036717336304 n + 1715839027013616331392 n 2 - 3308435252618010359040 n + 2042385206346064742400 n - 209604645096118272000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 16 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 5 (214694567 n 15 14 13 - 27472925176 n + 1610874024620 n - 57338366750560 n 12 11 10 + 1383071967667754 n - 23879719144717072 n + 303370299528121540 n 9 8 - 2860710436604614880 n + 19808117765289644791 n 7 6 - 96570758784658486328 n + 293781082975687903960 n 5 4 - 308368178616873589760 n - 1354117412581969892112 n 3 2 + 5840748845565892488576 n - 8696250379864505445120 n + 4952646831861669427200 n - 628813935288354816000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 15 (2 n - 23) (2 n - 29)), 1955 (1097554 n - 158473793 n + 10536665008 n 14 13 12 - 427578027380 n + 11827276095628 n - 235669224130886 n 11 10 9 + 3479892531977576 n - 38471610318778780 n + 316261880387392082 n 8 7 - 1874336599186900849 n + 7371078170183897624 n 6 5 - 14285294551453143400 n - 20238952667216957664 n 4 3 + 207533490052816380528 n - 550689444410510510208 n 2 + 683536410945422818560 n - 366313469477279385600 n + 53071252850423808000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 2737 ( 17 16 15 14 782798 n - 112889251 n + 7489546016 n - 302766089260 n 13 12 11 + 8319452811236 n - 163904591434402 n + 2374528917770152 n 10 9 8 - 25431546388622660 n + 198233421736055134 n - 1069219223324409443 n 7 6 + 3428698146265000648 n - 1920984016641817400 n 5 4 - 36911657136267195168 n + 177468292311738582096 n 3 2 - 393082200293882898816 n + 450455765353743640320 n - 237600481832347392000 n + 37908037750302720000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 3519 (1213518 n - 195802206 n 16 15 14 + 14587414911 n - 664763118640 n + 20677719582576 n 13 12 11 - 463350330763012 n + 7680506475754242 n - 94918363599322120 n 10 9 + 865528463942908194 n - 5605381267918933358 n 8 7 + 23088539589585525783 n - 34066964417867402840 n 6 5 - 232806733149599315688 n + 1785323221614662212576 n 4 3 - 5962588452070486401936 n + 11261818922184419529600 n 2 - 11872792553176321401600 n + 6098276610418164480000 n - 1031941027647129600000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 18 17 (-1 + 2 n) (2 n - 5)), 153 (55427973 n - 8906468451 n 16 15 14 + 659337771186 n - 29758884665740 n + 912438066514086 n 13 12 11 - 20015535827643202 n + 321530310196513392 n - 3791676201490168420 n 10 9 + 32123250937730547309 n - 182241258739535810243 n 8 7 + 525734724800895313158 n + 1098524926964988719560 n 6 5 - 19544463533282045460768 n + 99854191531280122250896 n 4 3 - 290315396972929050104736 n + 511076753438612768985600 n 2 - 520732714493563543641600 n + 267016530286405025280000 n - 47469287271767961600000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 19 18 (-1 + 2 n) (2 n - 5)), 969 (17271152 n - 3075491411 n 17 16 15 + 252935127033 n - 12717181310394 n + 435745553978724 n 14 13 12 - 10724933715086682 n + 194354996965598246 n - 2605478504184036728 n 11 10 + 25383555861271832796 n - 168831056943456339003 n 9 8 + 601881732871799984409 n + 1064005208024621910138 n 7 6 - 28814512277120922338872 n + 192672912031985528509696 n 5 4 - 757736429104159123328688 n + 1919577076538438329773984 n 3 2 - 3106429163380984057804800 n + 3013540478845592984870400 n - 1519146104299012062720000 n + 277320573008749670400000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 19 18 17 16 323 (50614841 n - 8925621868 n + 724314035859 n - 35760263856042 n 15 14 13 + 1195349377327842 n - 28440600205191216 n + 491439356155431158 n 12 11 - 6137018239631485804 n + 53009906923078426893 n 10 9 - 267063558144482005764 n - 39544918658573136393 n 8 7 + 13614242181098010913734 n - 128397940855062332249176 n 6 5 + 692452459124346511205648 n - 2462862324780637608012624 n 4 3 + 5887675635151828275014112 n - 9206940152177353871078400 n 2 + 8786579968648300269427200 n - 4436309786755283458560000 n + 831961719026249011200000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (194810021 n - 37661615890 n 18 17 16 + 3356057289135 n - 182295778112310 n + 6719307350902806 n 15 14 - 176742262033393380 n + 3384907175828529470 n 13 12 - 46899963776047633420 n + 446587302073115700081 n 11 10 - 2345909572941413506170 n - 4854856278235271059245 n 9 8 + 230035237870378129192770 n - 2441862826381490944463164 n 7 6 + 15924852843080098696297040 n - 71645729241694451374446960 n 5 4 + 227369844449955852983292960 n - 502755577631714211215079744 n 3 2 + 745213582728750783829478400 n - 687734238342762020766182400 n + 342402828168802617216000000 n - 64893014084047422873600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), 323 (91701553 n - 17404943570 n + 1514710139055 n 17 16 15 - 79788483857130 n + 2823475822214208 n - 70186264424308740 n 14 13 + 1234640111043242710 n - 14731981978956902660 n 12 11 + 96467677949473055433 n + 241172813480575595190 n 10 9 - 14481706767242303601285 n + 192883441852892774173710 n 8 7 - 1595722984293254602798202 n + 9267528811776924299353120 n 6 5 - 38936229693555931368773280 n + 118184197615022590554106080 n 4 3 - 253735030803461821404712992 n + 369316811196977568132384000 n 2 - 338009538977014049972467200 n + 168607727784803801894400000 n - 32446507042023711436800000)/(32768 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 969 (55804642 n 20 19 18 - 11465491611 n + 1080400850060 n - 61599520474205 n 17 16 15 + 2354950511797422 n - 62873599594492176 n + 1167565462531330120 n 14 13 - 13842043877566254210 n + 57519348571953613982 n 12 11 + 1467524076675370346269 n - 38751000400142843341620 n 10 9 + 523177492772988201355935 n - 4850520151856541862526798 n 8 7 + 33058707892154104110401934 n - 168938480416550125617619360 n 6 5 + 647125297983392446943546080 n - 1832792461955808911155489248 n 4 3 + 3735079464902194976116735584 n - 5233377323888316156134419200 n 2 + 4669233869877027099708326400 n - 2299275591631309350259200000 n + 443435596240990722969600000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) 21 (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 969 (48750592 n 20 19 18 - 9687871011 n + 874246238810 n - 47000811296705 n 17 16 15 + 1648579487866122 n - 38109431916726576 n + 516557892058421620 n 14 13 - 738765362501229210 n - 146908344018816539068 n 12 11 + 3954570879991479980869 n - 62364040663369504929870 n 10 9 + 697377314119549700253435 n - 5838830128677036240535598 n 8 7 + 37292056091743947190503534 n - 182202014523239120077447360 n 6 5 + 675788297143168146098066080 n - 1870054860831391829427342048 n 4 3 + 3750097547590961479065113184 n - 5202003064078564351102483200 n 2 + 4621142739832687893687206400 n - 2279111790362298783091200000 n + 443435596240990722969600000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) 22 (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 969 (319229101 n 21 20 19 - 67132526637 n + 6360004811237 n - 353013694907335 n 18 17 16 + 12281361205466276 n - 247829650991764842 n + 925436531629602322 n 15 14 + 121624568209110430530 n - 4808394586442696184859 n 13 12 + 108522357174696038196623 n - 1740409825352396000756863 n 11 10 + 21133365932899983715397445 n - 199297045856380970133427814 n 9 8 + 1473998389112239719838118328 n - 8557096939669486654214379688 n 7 6 + 38762645265361357847666057360 n - 135308502587734481404001583904 n 5 4 + 356661156766709575713531126528 n - 688276872876887490298765627008 n 3 2 + 927138905314211074402380672000 n - 806594449039271658182988748800 n + 392924680900055371497830400000 n - 76270922553450404350771200000)/( 131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 22 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 323 (5637315263 n 21 20 19 - 1090965566911 n + 90787150619011 n - 3953876179381205 n 18 17 + 67589571941625888 n + 2395864725215395434 n 16 15 - 209792153038130929354 n + 7912119827177669631110 n 14 13 - 199733808999782998003217 n + 3731069456643723368763589 n 12 11 - 53707284786377105198425929 n + 607399100252744601400373775 n 10 9 - 5445222865306204541297791582 n + 38779514562037583352680677664 n 8 7 - 218712041425260299374346666624 n + 968861870670422395306592315440 n 6 5 - 3324580431055083937649005491552 n + 8651936166370642187979452740224 n 4 - 16546540974144365283052952047104 n 3 + 22165068635175954294864377210880 n 2 - 19239178552242505516116198604800 n + 9382325762619508317355376640000 n - 1830502141282809704418508800000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 437 ( 23 22 21 4938829684 n - 876904452263 n + 53678825498903 n 20 19 18 + 268937639467805 n - 263195302751479791 n + 21000644744698059672 n 17 16 - 988778834070687402842 n + 32883709940430057882250 n 15 14 - 825453149046096526340506 n + 16169434343625529011613337 n 13 12 - 251849565997203768486991917 n + 3152569627793951218343125545 n 11 10 - 31882942407611104220798444651 n + 260802062279704362562405737862 n 9 8 - 1721145465274706392711525454752 n + 9107060969252993731608481605440 n 7 - 38239158609128334446541145821936 n 6 + 125464048318094913180061012327392 n 5 - 314593812954129351833828732299392 n 4 + 583675158872221046271656347368960 n 3 - 763300245913374199804033094092800 n 2 + 650705362685726152929173489664000 n - 313566175922834797048383897600000 n + 60884092960058670603485184000000)/ (1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 437 ( 23 22 21 1268607469 n + 95704434712 n - 66223664042932 n 20 19 18 + 9407038559263730 n - 745801101823146831 n + 39764371339961675022 n 17 16 - 1545724205704430752352 n + 45798949647911442535300 n 15 14 - 1062930119128807759968721 n + 19663132109381410857249812 n 13 12 - 293158862514048835969098252 n + 3545351150694844495442498970 n 11 10 - 34876251409240646205587879741 n + 278945507725127995704919483462 n 9 8 - 1807466778004706774480910063712 n + 9422612673502376586179264948240 n 7 - 39094569912035806868608246175376 n 6 + 127073768924637858470646164272992 n 5 - 316380315276656839728171962842752 n 4 + 584062124413848457708409539253760 n 3 - 761474621855943152113650610636800 n 2 + 648374585099706310491812726784000 n - 312660530185768339473484185600000 n + 60884092960058670603485184000000)/ (1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 69 ( 24 23 22 63695524177 n - 28694337609276 n + 5297159700974638 n 21 20 - 569566149480648144 n + 41151626641404335287 n 19 18 - 2155529431806438006396 n + 85702874513570299640248 n 17 16 - 2664914322758119882656624 n + 66126923330633661107619007 n 15 14 - 1327568129962686362826072756 n + 21762254416769119200634622278 n 13 12 - 292935505019467130310298418064 n + 3246707398455761683321384460137 n 11 - 29632056661469459934220669475796 n 10 + 222162336535814049965417098188148 n 9 - 1361307886773635250590046014213424 n 8 + 6762396817692159804220803306466672 n 7 - 26915241933462025489195098342346176 n 6 + 84427403072246714581644855211466688 n 5 - 203951100811051616827400106441663744 n 4 + 367126442453095587507351085390494720 n 3 - 468885480940524972146801719201689600 n 2 + 392861482738024633883081033822208000 n - 187277626578962182420533170995200000 n + 36246330008888261899274846208000000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), - 69 (157786675507 n 23 22 21 - 54663495376356 n + 8649251058257218 n - 838598720290885224 n 20 19 + 56207073764153182717 n - 2779947350067076865676 n 18 17 + 105616989557993130679528 n - 3165003553591361503859904 n 16 15 + 76170404652925390882801837 n - 1490488426363821411936317436 n 14 13 + 23908929236608985350955288458 n - 315957610587121186012972932744 n 12 + 3447406736152331805352410313267 n 11 - 31047482978193315787614206472276 n 10 + 230166381034878582550065579507628 n 9 - 1397084558034261548939080928765904 n 8 + 6885989437115174911233082681705552 n 7 - 27233240743112354617661241477886656 n 6 + 84996057839913654757484189450799168 n 5 - 204544105731216786294302086267156224 n 4 + 367209359126150518557224678089221120 n 3 - 468233284586480313443371456912281600 n 2 + 392078111777304756111647536877568000 n - 186981603920087414401392087859200000 n + 36246330008888261899274846208000000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), - 345 (98242033366 n 24 23 22 - 33774279703655 n + 5431058010471160 n - 544418476284087770 n 21 20 + 38230134925952133010 n - 2002717590656868935465 n 19 18 + 81354087237301260785260 n - 2629005354664393435624520 n 17 16 + 68782062657398205440084650 n - 1474730674313410483632439865 n 15 14 + 26127431508221311955166410560 n - 384512550220938162199824077570 n 13 + 4713578499781332625743986408830 n 12 - 48155632332541421591706300855815 n 11 + 409360152886524570756985710685660 n 10 - 2884596336939512964417021312940220 n 9 + 16745741542975134660336631468569520 n 8 - 79371395983683699344183168677498640 n 7 + 303352151754848529439640369021060160 n 6 - 919034055445819690055996307797209920 n 5 + 2155653316941363489450894501490170624 n 4 - 3786289702232893512235973147660866560 n 3 + 4740598729028403440752032126459187200 n 2 - 3911490910511601995935521753845760000 n + 1844818463026793611410911649792000000 n - 355214034087104966612893492838400000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 115 (387866835758 n 24 23 22 - 128333652452365 n + 19974095904696680 n - 1946727094567489510 n 21 20 + 133404200870443586930 n - 6840849766925239552795 n 19 18 + 272727396522853784304980 n - 8669113103794863896826760 n 17 16 + 223533194846683286390713250 n - 4731645109839026254561724995 n 15 14 + 82888641590896985544177874880 n - 1207827553249779508412841510910 n 13 + 14678449478757491405560395324590 n 12 - 148833419825970420651784226003845 n 11 + 1256980004913012467974536696050180 n 10 - 8808149323106432755783870141799860 n 9 + 50892571684422323945260691518404560 n 8 - 240274532902276779169801537986614320 n 7 + 915378223049842960406480285517151680 n 6 - 2766217453276514552866822591872192960 n 5 + 6475959536809799492736829530337734912 n 4 - 11359500623810202208590232779044551680 n 3 + 14211244138946223057634441229645721600 n 2 - 11722406801099748308155644821729280000 n + 5530001294609069173209375783321600000 n - 1065642102261314899838680478515200000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), - 5 (21404507583863 n 25 24 - 7466294717380369 n + 1231835483412233375 n 23 22 - 127926052554635855650 n + 9387702937995285576335 n 21 20 - 518044733667976708127935 n + 22335082284370726401560315 n 19 18 - 771628920205181214625177300 n + 21737019093868700498775575105 n 17 16 - 505425656453566628120405931295 n + 9782426694357310207885977403565 n 15 - 158484756090515130669071351777650 n 14 + 2156166319950381749254192830695045 n 13 - 24663096802725305907616683029162065 n 12 + 237015498787924959064964924493846665 n 11 - 1908730257098946152460969693031147000 n 10 + 12822074399205280457728541058281509940 n 9 - 71358655816044401094386004901490898400 n 8 + 325888280054523091000722130435097577840 n 7 - 1205608028187144448331237512050273705600 n 6 + 3550333299784352033839208788358051819712 n 5 - 8126090638345700584361515527304428999936 n 4 + 13978515112399406971395217224097482378240 n 3 - 17200082633327725592226367018329221836800 n 2 + 13994900909506534195220457330543882240000 n - 6532133303896377533850163845714739200000 n + 1249998185952522377510772201298329600000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), - 5 ( 26 25 24 96645228146627 n - 33250528597552201 n + 5417721420311885750 n 23 22 - 556283752537802483350 n + 40404511327140866220965 n 21 20 - 2208984169377942890548615 n + 94439909291332662282342260 n 19 18 - 3237998873865356821301996200 n + 90593815650759913287222758045 n 17 - 2093589617419745215952997156055 n 16 + 40299461332652398035794375471510 n 15 - 649711187919475946773441621766350 n 14 + 8801120517607388922461502307670555 n 13 - 100288482345108759339780580846533385 n 12 + 960586131779728908104752292469880160 n 11 - 7713531010468021016728537183654772500 n 10 + 51688591215713660174732097757439917760 n 9 - 287062183409307310172527356702584479600 n 8 + 1308710545987074418298638645566891087360 n 7 - 4834676245194646176689138300227265534400 n 6 + 14221543584842729220402161955284705826048 n 5 - 32523366627520498077727304270135812230144 n 4 + 55914175075295183497243384521948641832960 n 3 - 68776744805904621833336587130307248947200 n 2 + 55953708205470114259667862369115176960000 n - 26119162914341488228522763537272012800000 n + 4999992743810089510043088805193318400000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), - 15 ( 27 26 25 69625187511016 n - 25588779782002551 n + 4466945655312287589 n 24 23 - 492884365336603806150 n + 38591474198000796131670 n 22 21 - 2281841032778469359417025 n + 105870156113327166441967515 n 20 19 - 3953726512847411336003913660 n + 120959476706896955022369576960 n 18 - 3069587525724703473812512426665 n 17 + 65183077322351816362204790317275 n 16 - 1165203324124154517451378134567510 n 15 + 17599674842584112865954062792892990 n 14 - 225028223836677222342991470695560095 n 13 + 2435782202260158703405361915357015445 n 12 - 22285449015970744437836857231625618760 n 11 + 171771807594942906631712213909029196580 n 10 - 1109644065831562902297491323358801748960 n 9 + 5963970843179537960729381905311012319280 n 8 - 26405322396186129938889342745627843954560 n 7 + 95035836036281807506917087271698912955584 n 6 - 273157458204827075184746701443336831784704 n 5 + 612061567505599557747530831985376408912896 n 4 - 1033658386735687211204470458930866922639360 n 3 + 1252090532955661949304433337610029757235200 n 2 - 1005643281055073236389375375469467893760000 n + 464670340896204461004071769163974574080000 n - 88333205140644914677427902225081958400000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 45)), ( 1054991807001440 (n - 26)! (n + 1) (2 n - 57)! + 1582487710502160 (n - 27)! ((n + 1) (2 n - 56)! - 1/1582487710502160 binomial(2 n, n) (n - 29)! (n - 26)!))/((n - 29)! (n - 27)! (n - 26)! binomial(2 n, n)), ( (1054991807001440 n + 1054991807001440) (2 n - 57)! - (n - 29)! (n - 27)! binomial(2 n, n))/((n - 29)! (n - 27)! binomial(2 n, n))] and in Maple notation [203/8192*(661171*n^14-64794743*n^13+2872950509*n^12-76230776167*n^11+ 1349005389303*n^10-16793911022889*n^9+151146222520007*n^8-994690675834141*n^7+ 4781088261884506*n^6-16534874791503068*n^5+39508682044605384*n^4-\ 57257811815323392*n^3+15696430372149120*n^2+134942946269414400*n-\ 249806902625280000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/ (2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23), 3045/8192*( 88156*n^15-9917523*n^14+508324351*n^13-15718001637*n^12+327219307831*n^11-\ 4846100006319*n^10+52584293709233*n^9-423962078782671*n^8+2544538854814385*n^7-\ 11227756009796898*n^6+34985333046850316*n^5-67597595471169192*n^4+ 33169819823440128*n^3+204157626700314240*n^2-445462957684454400*n+ 16653793508352000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/( 2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 105/ 8192*(2556497*n^15-287600472*n^14+14740397054*n^13-455741435058*n^12+ 9484984568540*n^11-140366574311106*n^10+1520054708268382*n^9-12190174013574414* n^8+72120856296427939*n^7-306049399767453342*n^6+852299572017902164*n^5-\ 1069359707210283528*n^4-1796865234816398976*n^3+8382333377102497920*n^2-\ 6623291826692121600*n+482960011742208000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2* n-23)/(2*n-29), 15/16384*(71578289*n^16-9161295592*n^15+537413129540*n^14-\ 19147128009520*n^13+462772765300118*n^12-8023279184799424*n^11+ 102800880257827180*n^10-986303522298482960*n^9+7071411099518662897*n^8-\ 37018435863149116376*n^7+132263915457063437320*n^6-258354087370612425920*n^5-\ 76494617036717336304*n^4+1715839027013616331392*n^3-3308435252618010359040*n^2+ 2042385206346064742400*n-209604645096118272000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-17)/(2*n-23)/(2*n-29), 5/16384*(214694567*n^16-27472925176*n^15+ 1610874024620*n^14-57338366750560*n^13+1383071967667754*n^12-23879719144717072* n^11+303370299528121540*n^10-2860710436604614880*n^9+19808117765289644791*n^8-\ 96570758784658486328*n^7+293781082975687903960*n^6-308368178616873589760*n^5-\ 1354117412581969892112*n^4+5840748845565892488576*n^3-8696250379864505445120*n^ 2+4952646831861669427200*n-628813935288354816000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-17)/(2*n-23)/(2*n-29), 1955/16384*(1097554*n^17-158473793*n^16+ 10536665008*n^15-427578027380*n^14+11827276095628*n^13-235669224130886*n^12+ 3479892531977576*n^11-38471610318778780*n^10+316261880387392082*n^9-\ 1874336599186900849*n^8+7371078170183897624*n^7-14285294551453143400*n^6-\ 20238952667216957664*n^5+207533490052816380528*n^4-550689444410510510208*n^3+ 683536410945422818560*n^2-366313469477279385600*n+53071252850423808000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/ (2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 2737/16384*( 782798*n^17-112889251*n^16+7489546016*n^15-302766089260*n^14+8319452811236*n^13 -163904591434402*n^12+2374528917770152*n^11-25431546388622660*n^10+ 198233421736055134*n^9-1069219223324409443*n^8+3428698146265000648*n^7-\ 1920984016641817400*n^6-36911657136267195168*n^5+177468292311738582096*n^4-\ 393082200293882898816*n^3+450455765353743640320*n^2-237600481832347392000*n+ 37908037750302720000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33 )/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-\ 23)/(2*n-29), 3519/16384*(1213518*n^18-195802206*n^17+14587414911*n^16-\ 664763118640*n^15+20677719582576*n^14-463350330763012*n^13+7680506475754242*n^ 12-94918363599322120*n^11+865528463942908194*n^10-5605381267918933358*n^9+ 23088539589585525783*n^8-34066964417867402840*n^7-232806733149599315688*n^6+ 1785323221614662212576*n^5-5962588452070486401936*n^4+11261818922184419529600*n ^3-11872792553176321401600*n^2+6098276610418164480000*n-1031941027647129600000) /(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 153/32768*(55427973*n^18-8906468451*n^17+659337771186*n^16-29758884665740*n^ 15+912438066514086*n^14-20015535827643202*n^13+321530310196513392*n^12-\ 3791676201490168420*n^11+32123250937730547309*n^10-182241258739535810243*n^9+ 525734724800895313158*n^8+1098524926964988719560*n^7-19544463533282045460768*n^ 6+99854191531280122250896*n^5-290315396972929050104736*n^4+ 511076753438612768985600*n^3-520732714493563543641600*n^2+ 267016530286405025280000*n-47469287271767961600000)/(2*n-29)/(2*n-35)/(2*n-23)/ (2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33 )/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 969/32768*(17271152*n^19-\ 3075491411*n^18+252935127033*n^17-12717181310394*n^16+435745553978724*n^15-\ 10724933715086682*n^14+194354996965598246*n^13-2605478504184036728*n^12+ 25383555861271832796*n^11-168831056943456339003*n^10+601881732871799984409*n^9+ 1064005208024621910138*n^8-28814512277120922338872*n^7+192672912031985528509696 *n^6-757736429104159123328688*n^5+1919577076538438329773984*n^4-\ 3106429163380984057804800*n^3+3013540478845592984870400*n^2-\ 1519146104299012062720000*n+277320573008749670400000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/32768*( 50614841*n^19-8925621868*n^18+724314035859*n^17-35760263856042*n^16+ 1195349377327842*n^15-28440600205191216*n^14+491439356155431158*n^13-\ 6137018239631485804*n^12+53009906923078426893*n^11-267063558144482005764*n^10-\ 39544918658573136393*n^9+13614242181098010913734*n^8-128397940855062332249176*n ^7+692452459124346511205648*n^6-2462862324780637608012624*n^5+ 5887675635151828275014112*n^4-9206940152177353871078400*n^3+ 8786579968648300269427200*n^2-4436309786755283458560000*n+ 831961719026249011200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15) /(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7) /(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(194810021*n^20-37661615890*n^19+ 3356057289135*n^18-182295778112310*n^17+6719307350902806*n^16-\ 176742262033393380*n^15+3384907175828529470*n^14-46899963776047633420*n^13+ 446587302073115700081*n^12-2345909572941413506170*n^11-4854856278235271059245*n ^10+230035237870378129192770*n^9-2441862826381490944463164*n^8+ 15924852843080098696297040*n^7-71645729241694451374446960*n^6+ 227369844449955852983292960*n^5-502755577631714211215079744*n^4+ 745213582728750783829478400*n^3-687734238342762020766182400*n^2+ 342402828168802617216000000*n-64893014084047422873600000)/(2*n-29)/(2*n-35)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/( 2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/32768*(91701553*n^20-17404943570*n^19+1514710139055*n^18-79788483857130*n^ 17+2823475822214208*n^16-70186264424308740*n^15+1234640111043242710*n^14-\ 14731981978956902660*n^13+96467677949473055433*n^12+241172813480575595190*n^11-\ 14481706767242303601285*n^10+192883441852892774173710*n^9-\ 1595722984293254602798202*n^8+9267528811776924299353120*n^7-\ 38936229693555931368773280*n^6+118184197615022590554106080*n^5-\ 253735030803461821404712992*n^4+369316811196977568132384000*n^3-\ 338009538977014049972467200*n^2+168607727784803801894400000*n-\ 32446507042023711436800000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 969/32768*(55804642*n^21-\ 11465491611*n^20+1080400850060*n^19-61599520474205*n^18+2354950511797422*n^17-\ 62873599594492176*n^16+1167565462531330120*n^15-13842043877566254210*n^14+ 57519348571953613982*n^13+1467524076675370346269*n^12-38751000400142843341620*n ^11+523177492772988201355935*n^10-4850520151856541862526798*n^9+ 33058707892154104110401934*n^8-168938480416550125617619360*n^7+ 647125297983392446943546080*n^6-1832792461955808911155489248*n^5+ 3735079464902194976116735584*n^4-5233377323888316156134419200*n^3+ 4669233869877027099708326400*n^2-2299275591631309350259200000*n+ 443435596240990722969600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/ (2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 969/32768*(48750592*n^ 21-9687871011*n^20+874246238810*n^19-47000811296705*n^18+1648579487866122*n^17-\ 38109431916726576*n^16+516557892058421620*n^15-738765362501229210*n^14-\ 146908344018816539068*n^13+3954570879991479980869*n^12-62364040663369504929870* n^11+697377314119549700253435*n^10-5838830128677036240535598*n^9+ 37292056091743947190503534*n^8-182202014523239120077447360*n^7+ 675788297143168146098066080*n^6-1870054860831391829427342048*n^5+ 3750097547590961479065113184*n^4-5202003064078564351102483200*n^3+ 4621142739832687893687206400*n^2-2279111790362298783091200000*n+ 443435596240990722969600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/ (2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 969/131072*(319229101* n^22-67132526637*n^21+6360004811237*n^20-353013694907335*n^19+12281361205466276 *n^18-247829650991764842*n^17+925436531629602322*n^16+121624568209110430530*n^ 15-4808394586442696184859*n^14+108522357174696038196623*n^13-\ 1740409825352396000756863*n^12+21133365932899983715397445*n^11-\ 199297045856380970133427814*n^10+1473998389112239719838118328*n^9-\ 8557096939669486654214379688*n^8+38762645265361357847666057360*n^7-\ 135308502587734481404001583904*n^6+356661156766709575713531126528*n^5-\ 688276872876887490298765627008*n^4+927138905314211074402380672000*n^3-\ 806594449039271658182988748800*n^2+392924680900055371497830400000*n-\ 76270922553450404350771200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 323/1048576 *(5637315263*n^22-1090965566911*n^21+90787150619011*n^20-3953876179381205*n^19+ 67589571941625888*n^18+2395864725215395434*n^17-209792153038130929354*n^16+ 7912119827177669631110*n^15-199733808999782998003217*n^14+ 3731069456643723368763589*n^13-53707284786377105198425929*n^12+ 607399100252744601400373775*n^11-5445222865306204541297791582*n^10+ 38779514562037583352680677664*n^9-218712041425260299374346666624*n^8+ 968861870670422395306592315440*n^7-3324580431055083937649005491552*n^6+ 8651936166370642187979452740224*n^5-16546540974144365283052952047104*n^4+ 22165068635175954294864377210880*n^3-19239178552242505516116198604800*n^2+ 9382325762619508317355376640000*n-1830502141282809704418508800000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29), 437/1048576*(4938829684*n^23-876904452263*n^22+ 53678825498903*n^21+268937639467805*n^20-263195302751479791*n^19+ 21000644744698059672*n^18-988778834070687402842*n^17+32883709940430057882250*n^ 16-825453149046096526340506*n^15+16169434343625529011613337*n^14-\ 251849565997203768486991917*n^13+3152569627793951218343125545*n^12-\ 31882942407611104220798444651*n^11+260802062279704362562405737862*n^10-\ 1721145465274706392711525454752*n^9+9107060969252993731608481605440*n^8-\ 38239158609128334446541145821936*n^7+125464048318094913180061012327392*n^6-\ 314593812954129351833828732299392*n^5+583675158872221046271656347368960*n^4-\ 763300245913374199804033094092800*n^3+650705362685726152929173489664000*n^2-\ 313566175922834797048383897600000*n+60884092960058670603485184000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/1048576*(1268607469*n^23+95704434712*n ^22-66223664042932*n^21+9407038559263730*n^20-745801101823146831*n^19+ 39764371339961675022*n^18-1545724205704430752352*n^17+45798949647911442535300*n ^16-1062930119128807759968721*n^15+19663132109381410857249812*n^14-\ 293158862514048835969098252*n^13+3545351150694844495442498970*n^12-\ 34876251409240646205587879741*n^11+278945507725127995704919483462*n^10-\ 1807466778004706774480910063712*n^9+9422612673502376586179264948240*n^8-\ 39094569912035806868608246175376*n^7+127073768924637858470646164272992*n^6-\ 316380315276656839728171962842752*n^5+584062124413848457708409539253760*n^4-\ 761474621855943152113650610636800*n^3+648374585099706310491812726784000*n^2-\ 312660530185768339473484185600000*n+60884092960058670603485184000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -69/2097152*(63695524177*n^24-\ 28694337609276*n^23+5297159700974638*n^22-569566149480648144*n^21+ 41151626641404335287*n^20-2155529431806438006396*n^19+85702874513570299640248*n ^18-2664914322758119882656624*n^17+66126923330633661107619007*n^16-\ 1327568129962686362826072756*n^15+21762254416769119200634622278*n^14-\ 292935505019467130310298418064*n^13+3246707398455761683321384460137*n^12-\ 29632056661469459934220669475796*n^11+222162336535814049965417098188148*n^10-\ 1361307886773635250590046014213424*n^9+6762396817692159804220803306466672*n^8-\ 26915241933462025489195098342346176*n^7+84427403072246714581644855211466688*n^6 -203951100811051616827400106441663744*n^5+367126442453095587507351085390494720* n^4-468885480940524972146801719201689600*n^3+ 392861482738024633883081033822208000*n^2-187277626578962182420533170995200000*n +36246330008888261899274846208000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13 )/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), -69/2097152*(157786675507*n^24-54663495376356*n^23+ 8649251058257218*n^22-838598720290885224*n^21+56207073764153182717*n^20-\ 2779947350067076865676*n^19+105616989557993130679528*n^18-\ 3165003553591361503859904*n^17+76170404652925390882801837*n^16-\ 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2002717590656868935465*n^20+81354087237301260785260*n^19-\ 2629005354664393435624520*n^18+68782062657398205440084650*n^17-\ 1474730674313410483632439865*n^16+26127431508221311955166410560*n^15-\ 384512550220938162199824077570*n^14+4713578499781332625743986408830*n^13-\ 48155632332541421591706300855815*n^12+409360152886524570756985710685660*n^11-\ 2884596336939512964417021312940220*n^10+16745741542975134660336631468569520*n^9 -79371395983683699344183168677498640*n^8+303352151754848529439640369021060160*n ^7-919034055445819690055996307797209920*n^6+ 2155653316941363489450894501490170624*n^5-3786289702232893512235973147660866560 *n^4+4740598729028403440752032126459187200*n^3-\ 3911490910511601995935521753845760000*n^2+1844818463026793611410911649792000000 *n-355214034087104966612893492838400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-45), -115/2097152*(387866835758*n^25-128333652452365*n^24+ 19974095904696680*n^23-1946727094567489510*n^22+133404200870443586930*n^21-\ 6840849766925239552795*n^20+272727396522853784304980*n^19-\ 8669113103794863896826760*n^18+223533194846683286390713250*n^17-\ 4731645109839026254561724995*n^16+82888641590896985544177874880*n^15-\ 1207827553249779508412841510910*n^14+14678449478757491405560395324590*n^13-\ 148833419825970420651784226003845*n^12+1256980004913012467974536696050180*n^11-\ 8808149323106432755783870141799860*n^10+50892571684422323945260691518404560*n^9 -240274532902276779169801537986614320*n^8+915378223049842960406480285517151680* n^7-2766217453276514552866822591872192960*n^6+ 6475959536809799492736829530337734912*n^5-\ 11359500623810202208590232779044551680*n^4+ 14211244138946223057634441229645721600*n^3-\ 11722406801099748308155644821729280000*n^2+ 5530001294609069173209375783321600000*n-1065642102261314899838680478515200000)/ (2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -5/2097152*( 21404507583863*n^26-7466294717380369*n^25+1231835483412233375*n^24-\ 127926052554635855650*n^23+9387702937995285576335*n^22-518044733667976708127935 *n^21+22335082284370726401560315*n^20-771628920205181214625177300*n^19+ 21737019093868700498775575105*n^18-505425656453566628120405931295*n^17+ 9782426694357310207885977403565*n^16-158484756090515130669071351777650*n^15+ 2156166319950381749254192830695045*n^14-24663096802725305907616683029162065*n^ 13+237015498787924959064964924493846665*n^12-\ 1908730257098946152460969693031147000*n^11+ 12822074399205280457728541058281509940*n^10-\ 71358655816044401094386004901490898400*n^9+ 325888280054523091000722130435097577840*n^8-\ 1205608028187144448331237512050273705600*n^7+ 3550333299784352033839208788358051819712*n^6-\ 8126090638345700584361515527304428999936*n^5+ 13978515112399406971395217224097482378240*n^4-\ 17200082633327725592226367018329221836800*n^3+ 13994900909506534195220457330543882240000*n^2-\ 6532133303896377533850163845714739200000*n+ 1249998185952522377510772201298329600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), -5/8388608*(96645228146627*n^26-\ 33250528597552201*n^25+5417721420311885750*n^24-556283752537802483350*n^23+ 40404511327140866220965*n^22-2208984169377942890548615*n^21+ 94439909291332662282342260*n^20-3237998873865356821301996200*n^19+ 90593815650759913287222758045*n^18-2093589617419745215952997156055*n^17+ 40299461332652398035794375471510*n^16-649711187919475946773441621766350*n^15+ 8801120517607388922461502307670555*n^14-100288482345108759339780580846533385*n^ 13+960586131779728908104752292469880160*n^12-\ 7713531010468021016728537183654772500*n^11+ 51688591215713660174732097757439917760*n^10-\ 287062183409307310172527356702584479600*n^9+ 1308710545987074418298638645566891087360*n^8-\ 4834676245194646176689138300227265534400*n^7+ 14221543584842729220402161955284705826048*n^6-\ 32523366627520498077727304270135812230144*n^5+ 55914175075295183497243384521948641832960*n^4-\ 68776744805904621833336587130307248947200*n^3+ 55953708205470114259667862369115176960000*n^2-\ 26119162914341488228522763537272012800000*n+ 4999992743810089510043088805193318400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), -15/8388608*(69625187511016*n^27-\ 25588779782002551*n^26+4466945655312287589*n^25-492884365336603806150*n^24+ 38591474198000796131670*n^23-2281841032778469359417025*n^22+ 105870156113327166441967515*n^21-3953726512847411336003913660*n^20+ 120959476706896955022369576960*n^19-3069587525724703473812512426665*n^18+ 65183077322351816362204790317275*n^17-1165203324124154517451378134567510*n^16+ 17599674842584112865954062792892990*n^15-225028223836677222342991470695560095*n ^14+2435782202260158703405361915357015445*n^13-\ 22285449015970744437836857231625618760*n^12+ 171771807594942906631712213909029196580*n^11-\ 1109644065831562902297491323358801748960*n^10+ 5963970843179537960729381905311012319280*n^9-\ 26405322396186129938889342745627843954560*n^8+ 95035836036281807506917087271698912955584*n^7-\ 273157458204827075184746701443336831784704*n^6+ 612061567505599557747530831985376408912896*n^5-\ 1033658386735687211204470458930866922639360*n^4+ 1252090532955661949304433337610029757235200*n^3-\ 1005643281055073236389375375469467893760000*n^2+ 464670340896204461004071769163974574080000*n-\ 88333205140644914677427902225081958400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), (1054991807001440*(n-26)!*(n+1)*( 2*n-57)!+1582487710502160*(n-27)!*((n+1)*(2*n-56)!-1/1582487710502160*binomial( 2*n,n)*(n-29)!*(n-26)!))/(n-29)!/(n-27)!/(n-26)!/binomial(2*n,n), (( 1054991807001440*n+1054991807001440)*(2*n-57)!-(n-29)!*(n-27)!*binomial(2*n,n)) /(n-29)!/(n-27)!/binomial(2*n,n)] The limits, as n goes to infinity are 134217713 67108755 268432185 1073674335 1073472835 1072859035 1071259063 [---------, --------, ---------, ----------, ----------, ----------, ----------, 134217728 67108864 268435456 1073741824 1073741824 1073741824 1073741824 2135184921 8480479869 1045984143 16348593643 62923636783 29619601619 ----------, ----------, ----------, -----------, -----------, -----------, 2147483648 8589934592 1073741824 17179869184 68719476736 34359738368 27037349049 5766519 309332998869 1820852829949 539567142977 -----------, -------, ------------, -------------, -------------, 34359738368 8388608 549755813888 4398046511104 2199023255552 554381463953 -4394991168213 -10887280609983 -16946750755635 -------------, --------------, ---------------, ---------------, 8796093022208 35184372088832 35184372088832 35184372088832 -22302343056085 -107022537919315 -483226140733135 -130547226583155 ---------------, ----------------, ----------------, ----------------, 35184372088832 140737488355328 562949953421312 140737488355328 -1092931412873829 -4470631133401701 -----------------, -----------------] 1125899906842624 4503599627370496 and in Maple notation [134217713/134217728, 67108755/67108864, 268432185/268435456, 1073674335/ 1073741824, 1073472835/1073741824, 1072859035/1073741824, 1071259063/1073741824 , 2135184921/2147483648, 8480479869/8589934592, 1045984143/1073741824, 16348593643/17179869184, 62923636783/68719476736, 29619601619/34359738368, 27037349049/34359738368, 5766519/8388608, 309332998869/549755813888, 1820852829949/4398046511104, 539567142977/2199023255552, 554381463953/ 8796093022208, -4394991168213/35184372088832, -10887280609983/35184372088832, -\ 16946750755635/35184372088832, -22302343056085/35184372088832, -107022537919315 /140737488355328, -483226140733135/562949953421312, -130547226583155/ 140737488355328, -1092931412873829/1125899906842624, -4470631133401701/ 4503599627370496] and in floating point [.9999998882, .9999983758, .9999878146, .9999371460, .9997494845, .9991778387, .9976877486, .9942729590, .9872577932, .9741486451, .9516133952, .9156594283, .\ 8620438637, .7868904227, .6874226332, .5626734471, .4140140004, .2453667289, .\ 6302587553e-1, -.1249131619, -.3094351260, -.4816556258, -.6338707140, -.760440\ 8688, -.8583820601, -.9275938352, -.9707180951, -.9926795238] The cut off is at j=, 20 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 30], vs. those in the, 2, -th row from j=1 to j=, 29, are as follws 15 14 13 12 [29 (18512789 n - 2082688530 n + 106749321560 n - 3300883185600 n 11 10 9 + 68722052859278 n - 1017934435818300 n + 11050697835385780 n 8 7 6 - 89223300577983000 n + 537834947371799077 n - 2405750813002547010 n 5 4 + 7836276631269218860 n - 17766895376367173400 n 3 2 + 24283332700460009856 n - 4218185729928044160 n - 60646936899788467200 n + 104918899102617600000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) 15 14 13 (2 n - 29)), 203 (1322341 n - 148763130 n + 7624902640 n 12 11 10 - 235773010200 n + 4908451667782 n - 72697808460420 n 9 8 7 + 788945509686620 n - 6363309932522400 n + 38229963210852413 n 6 5 4 - 169140618832755570 n + 530790184085745740 n - 1046962841751005400 n 3 2 + 599724705118126464 n + 2971188022623477120 n - 6915097474383744000 n + 249806902625280000)/(8192 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) 16 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 609 (1763111 n 15 14 13 12 - 225675976 n + 13240503260 n - 471903283120 n + 11414675216282 n 11 10 9 - 198257998224832 n + 2550586905698260 n - 24694935839047760 n 8 7 6 + 180679926552914503 n - 989579028586556408 n + 3924145575188339800 n 5 4 - 10223572407941285120 n + 11492358520945093104 n 3 2 + 23128592556531321216 n - 95537954877599128320 n + 73376875308724992000 n - 5162675987589120000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 16 15 (2 n - 23) (2 n - 29)), 21 (51128731 n - 6544158392 n 14 13 12 + 383913333580 n - 13680052027760 n + 330731849325202 n 11 10 9 - 5737432048902944 n + 73601737598288900 n - 707869377537459280 n 8 7 + 5099217603653891003 n - 26938353004127844136 n 6 5 + 98006019500033966360 n - 200539573841693344960 n 4 3 - 16080147204335769936 n + 1201831358797438745472 n 2 - 2405307079115362563840 n + 1499115448664669952000 n - 149717603640084480000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 17 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 255 (8420264 n 16 15 14 - 1216528663 n + 80978287328 n - 3293203941580 n 13 12 11 + 91459234434848 n - 1835828556005626 n + 27469299940269616 n 10 9 - 310926816216996980 n + 2664518969707535512 n 8 7 - 17007501740567096759 n + 77166822483058073584 n 6 5 - 218917858362373843400 n + 200884362006769970976 n 4 3 + 1017520129656906756048 n - 4101990367769790150528 n 2 + 5926066639461012360960 n - 3305310829797928089600 n + 406879605186582528000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 595 ( 17 16 15 14 3607418 n - 521005921 n + 34657114016 n - 1407546020260 n 13 12 11 + 38989651571276 n - 778772305089142 n + 11545041210140152 n 10 9 8 - 128454894886492460 n + 1066814099813513194 n - 6428127173423193953 n 7 6 + 26055614534470589848 n - 55220194189170756200 n 5 4 - 44044153532600635488 n + 652665667899526940016 n 3 2 - 1809675577348159142016 n + 2283689878427947633920 n - 1227653396076770150400 n + 174376973651392512000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 41055 (104464 n - 16902708 n 16 15 14 + 1264983543 n - 58064530680 n + 1826746231548 n 13 12 11 - 41660613579816 n + 709412681700346 n - 9133951749402480 n 10 9 8 + 88719859851063012 n - 636835276956122244 n + 3199674007152096279 n 7 6 - 9614783585460526200 n + 4528350196458065176 n 5 4 + 98783585859443976768 n - 455157184771073649168 n 3 2 + 980295356035423359360 n - 1099308210745431571200 n + 568809083451071232000 n - 88452088084039680000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 18 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 1173 (3648056 n 17 16 15 14 - 589319628 n + 43984784061 n - 2009912571000 n + 62771007019812 n 13 12 11 - 1414785837764376 n + 23646719192751902 n - 295699253161369200 n 10 9 + 2743081742374309908 n - 18256263414894611964 n 8 7 + 79481582508185889213 n - 152974327352400099000 n 6 5 - 530127745955455529176 n + 5018268267948640723968 n 4 3 - 17582311486160284432176 n + 33918159947160316291200 n 2 - 36102487049101292601600 n + 18552052570165551360000 n - 3095823082941388800000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 19 18 17 (-1 + 2 n) (2 n - 5)), 66861 (509573 n - 91492172 n + 7613232807 n 16 15 14 - 389133803168 n + 13642159100746 n - 346577991259304 n 13 12 11 + 6563081004773214 n - 93651785752812976 n + 1002217856382774889 n 10 9 - 7838398251171907276 n + 41687020086228269051 n 8 7 - 115208697421037291344 n - 209722443914391799208 n 6 5 + 3774411047135765801952 n - 18775261255718963485072 n 4 3 + 53244708280179461839488 n - 91729057692451975776000 n 2 + 91703529748296613324800 n - 46183690624951756800000 n + 8038277478514483200000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 19 18 (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (52255051 n - 9337314578 n 17 16 15 + 771512718819 n - 39032541853602 n + 1348458293107662 n 14 13 12 - 33550559386338036 n + 616821803696244478 n - 8435168192049182324 n 11 10 + 84676210985139682623 n - 594704080767274757994 n 9 8 + 2485452083926863723087 n - 647752054829766705546 n 7 6 - 70986651138420821930936 n + 535858943401286612911408 n 5 4 - 2203337115613681492818384 n + 5711225395996599568277472 n 3 2 - 9360678614656593653222400 n + 9134215661548323733363200 n - 4602064612001892456960000 n + 831961719026249011200000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 20 19 18 323 (205575763 n - 40460708810 n + 3690172092105 n 17 16 15 - 206561975895150 n + 7917560983822578 n - 219326737156350420 n 14 13 + 4509731192951795410 n - 69390867501191104700 n 12 11 + 790191392287828596663 n - 6367037765022599022930 n 10 9 + 31072925102397390000165 n - 12733765412712999480150 n 8 7 - 1222541101649910313373612 n + 11505588489163864327057360 n 6 5 - 60691894094540037426390480 n + 210880827342347162412420000 n 4 3 - 493341264139708165636291392 n + 756577087269331847877964800 n 2 - 709357981218147513179827200 n + 352194417754734119685120000 n - 64893014084047422873600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 20 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (199909583 n 19 18 17 - 38936506390 n + 3502083247005 n - 192452422760850 n 16 15 14 + 7198713431444298 n - 193001672163057900 n + 3794246284560266810 n 13 12 - 54694881341801580100 n + 559984642715685634083 n 11 10 - 3610306015250795370510 n + 5923086965735020393065 n 9 8 + 160414031496901871217750 n - 2106755068689835721838892 n 7 6 + 14757937764750657563455280 n - 68862014588449020412120080 n 5 4 + 223346962924739447468623200 n - 500647925972208818267019072 n 3 2 + 748155212054894720387819520 n - 692971604163607773603916800 n + 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(n + 1) (2 n - 59)! + 6013453299908208 (n - 28)! ((n + 1) (2 n - 58)! - 1/6013453299908208 binomial(2 n, n) (n - 30)! (n - 27)!))/((n - 30)! (n - 28)! (n - 27)! binomial(2 n, n)), ( (4008968866605472 n + 4008968866605472) (2 n - 59)! - (n - 30)! (n - 28)! binomial(2 n, n))/((n - 30)! (n - 28)! binomial(2 n, n))] and in Maple notation [29/16384*(18512789*n^15-2082688530*n^14+106749321560*n^13-3300883185600*n^12+ 68722052859278*n^11-1017934435818300*n^10+11050697835385780*n^9-\ 89223300577983000*n^8+537834947371799077*n^7-2405750813002547010*n^6+ 7836276631269218860*n^5-17766895376367173400*n^4+24283332700460009856*n^3-\ 4218185729928044160*n^2-60646936899788467200*n+104918899102617600000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2 *n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 203/8192*(1322341*n^15-148763130*n^ 14+7624902640*n^13-235773010200*n^12+4908451667782*n^11-72697808460420*n^10+ 788945509686620*n^9-6363309932522400*n^8+38229963210852413*n^7-\ 169140618832755570*n^6+530790184085745740*n^5-1046962841751005400*n^4+ 599724705118126464*n^3+2971188022623477120*n^2-6915097474383744000*n+ 249806902625280000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11)/ (2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 609/ 16384*(1763111*n^16-225675976*n^15+13240503260*n^14-471903283120*n^13+ 11414675216282*n^12-198257998224832*n^11+2550586905698260*n^10-\ 24694935839047760*n^9+180679926552914503*n^8-989579028586556408*n^7+ 3924145575188339800*n^6-10223572407941285120*n^5+11492358520945093104*n^4+ 23128592556531321216*n^3-95537954877599128320*n^2+73376875308724992000*n-\ 5162675987589120000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11) /(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-\ 29), 21/16384*(51128731*n^16-6544158392*n^15+383913333580*n^14-13680052027760*n ^13+330731849325202*n^12-5737432048902944*n^11+73601737598288900*n^10-\ 707869377537459280*n^9+5099217603653891003*n^8-26938353004127844136*n^7+ 98006019500033966360*n^6-200539573841693344960*n^5-16080147204335769936*n^4+ 1201831358797438745472*n^3-2405307079115362563840*n^2+1499115448664669952000*n-\ 149717603640084480000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 255/16384*(8420264*n^17-1216528663*n^16+80978287328*n^15-3293203941580*n ^14+91459234434848*n^13-1835828556005626*n^12+27469299940269616*n^11-\ 310926816216996980*n^10+2664518969707535512*n^9-17007501740567096759*n^8+ 77166822483058073584*n^7-218917858362373843400*n^6+200884362006769970976*n^5+ 1017520129656906756048*n^4-4101990367769790150528*n^3+5926066639461012360960*n^ 2-3305310829797928089600*n+406879605186582528000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n -25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 595/16384*(3607418*n^17-521005921*n^ 16+34657114016*n^15-1407546020260*n^14+38989651571276*n^13-778772305089142*n^12 +11545041210140152*n^11-128454894886492460*n^10+1066814099813513194*n^9-\ 6428127173423193953*n^8+26055614534470589848*n^7-55220194189170756200*n^6-\ 44044153532600635488*n^5+652665667899526940016*n^4-1809675577348159142016*n^3+ 2283689878427947633920*n^2-1227653396076770150400*n+174376973651392512000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 41055/16384 *(104464*n^18-16902708*n^17+1264983543*n^16-58064530680*n^15+1826746231548*n^14 -41660613579816*n^13+709412681700346*n^12-9133951749402480*n^11+ 88719859851063012*n^10-636835276956122244*n^9+3199674007152096279*n^8-\ 9614783585460526200*n^7+4528350196458065176*n^6+98783585859443976768*n^5-\ 455157184771073649168*n^4+980295356035423359360*n^3-1099308210745431571200*n^2+ 568809083451071232000*n-88452088084039680000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-\ 17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 1173/16384*(3648056*n^18-\ 589319628*n^17+43984784061*n^16-2009912571000*n^15+62771007019812*n^14-\ 1414785837764376*n^13+23646719192751902*n^12-295699253161369200*n^11+ 2743081742374309908*n^10-18256263414894611964*n^9+79481582508185889213*n^8-\ 152974327352400099000*n^7-530127745955455529176*n^6+5018268267948640723968*n^5-\ 17582311486160284432176*n^4+33918159947160316291200*n^3-36102487049101292601600 *n^2+18552052570165551360000*n-3095823082941388800000)/(2*n-29)/(2*n-35)/(2*n-\ 23)/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2* n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 66861/65536*(509573*n^ 19-91492172*n^18+7613232807*n^17-389133803168*n^16+13642159100746*n^15-\ 346577991259304*n^14+6563081004773214*n^13-93651785752812976*n^12+ 1002217856382774889*n^11-7838398251171907276*n^10+41687020086228269051*n^9-\ 115208697421037291344*n^8-209722443914391799208*n^7+3774411047135765801952*n^6-\ 18775261255718963485072*n^5+53244708280179461839488*n^4-91729057692451975776000 *n^3+91703529748296613324800*n^2-46183690624951756800000*n+ 8038277478514483200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/( 2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/( 2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/32768*(52255051*n^19-9337314578*n^18+ 771512718819*n^17-39032541853602*n^16+1348458293107662*n^15-33550559386338036*n ^14+616821803696244478*n^13-8435168192049182324*n^12+84676210985139682623*n^11-\ 594704080767274757994*n^10+2485452083926863723087*n^9-647752054829766705546*n^8 -70986651138420821930936*n^7+535858943401286612911408*n^6-\ 2203337115613681492818384*n^5+5711225395996599568277472*n^4-\ 9360678614656593653222400*n^3+9134215661548323733363200*n^2-\ 4602064612001892456960000*n+831961719026249011200000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*( 205575763*n^20-40460708810*n^19+3690172092105*n^18-206561975895150*n^17+ 7917560983822578*n^16-219326737156350420*n^15+4509731192951795410*n^14-\ 69390867501191104700*n^13+790191392287828596663*n^12-6367037765022599022930*n^ 11+31072925102397390000165*n^10-12733765412712999480150*n^9-\ 1222541101649910313373612*n^8+11505588489163864327057360*n^7-\ 60691894094540037426390480*n^6+210880827342347162412420000*n^5-\ 493341264139708165636291392*n^4+756577087269331847877964800*n^3-\ 709357981218147513179827200*n^2+352194417754734119685120000*n-\ 64893014084047422873600000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(199909583*n^20-\ 38936506390*n^19+3502083247005*n^18-192452422760850*n^17+7198713431444298*n^16-\ 193001672163057900*n^15+3794246284560266810*n^14-54694881341801580100*n^13+ 559984642715685634083*n^12-3610306015250795370510*n^11+5923086965735020393065*n ^10+160414031496901871217750*n^9-2106755068689835721838892*n^8+ 14757937764750657563455280*n^7-68862014588449020412120080*n^6+ 223346962924739447468623200*n^5-500647925972208818267019072*n^4+ 748155212054894720387819520*n^3-692971604163607773603916800*n^2+ 344721888860207446748160000*n-64893014084047422873600000)/(2*n-29)/(2*n-35)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/( 2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/131072*(764791325*n^21-162416296413*n^20+15950641665805*n^19-\ 958639301548215*n^18+39287302427281020*n^17-1156192915850650818*n^16+ 24983898571916424170*n^15-395525083954823613630*n^14+4405344182219337121765*n^ 13-29248349403906018815793*n^12-4473066037934195753415*n^11+ 2767536991196202998268405*n^10-37667476264742152137193390*n^9+ 306166773709383350469072192*n^8-1735864676549478031252522400*n^7+ 7119202192158892470386359440*n^6-21128645972304474510130650720*n^5+ 44443315948070703782172200832*n^4-63501892260582495427181084160*n^3+ 57148840855301134200033024000*n^2-28067206415859809939220480000*n+ 5321227154891888675635200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 323/32768*(178606352 *n^21-37216966185*n^20+3568301200030*n^19-207961846650915*n^18+8185626893363262 *n^17-227913278165531280*n^16+4535628399411005180*n^15-62316666876601935630*n^ 14+496916651293216151452*n^13+456457968764550418575*n^12-\ 78786977316108893638170*n^11+1293135428449087025817105*n^10-\ 12983441959014441174473098*n^9+92459211199779761310777930*n^8-\ 485770633800370505643264320*n^7+1895897268616666564838799840*n^6-\ 5439254379651501836475127968*n^5+11181409703507075410338780960*n^4-\ 15749912463896421332387262720*n^3+14084006202634766172678489600*n^2-\ 6929820006240758150684160000*n+1330306788722972168908800000)/(2*n-5)/(-1+2*n)/( 2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/ (2*n-29), 969/32768*(107611989*n^22-24151380803*n^21+2493682204398*n^20-\ 156360374939615*n^19+6602681610132839*n^18-195765135562413648*n^17+ 4066371198609620538*n^16-54601327018252985630*n^15+275121004927401963099*n^14+ 6852264129430963480237*n^13-214587299353957591235902*n^12+ 3386732315929126503143805*n^11-37002383793294405054032871*n^10+ 301170499395766465732328182*n^9-1870069987319523001228233402*n^8+ 8898035734468635023303428240*n^7-32211509842753524563969313856*n^6+ 87196705493362844592147736032*n^5-171423333506195100717612505632*n^4+ 233581954895895571047173203200*n^3-204192360898508612679866611200*n^2+ 99264397008373392687091200000*n-19067730638362601087692800000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-35)/(2*n-29), 1615/32768*(55669685*n^22-12070706197*n^21+1191896740342*n ^20-70330717691465*n^19+2715908400010575*n^18-69265895144986848*n^17+ 1031133286656907214*n^16-960661653860432722*n^15-396904038254257160165*n^14+ 11945534789854334760091*n^13-215148208503168626343294*n^12+ 2784626744307516817960251*n^11-27348042398637679142723695*n^10+ 208009169046282765373541450*n^9-1232250252976605230158304822*n^8+ 5666095311720550573447574224*n^7-19999443199893032272319593440*n^6+ 53145064268969465699896185504*n^5-103134238527305948483728009440*n^4+ 139402624613920551509560019712*n^3-121447198989352342113123432960*n^2+ 59122804361861461873107456000*n-11440638383017560652615680000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-35)/(2*n-29), 37145/2097152*(248885309*n^23-56721340006*n^22+ 5828731098322*n^21-350652949449572*n^20+13139363454845859*n^19-\ 276115399873638402*n^18+145055545005336284*n^17+218350123224256886456*n^16-\ 8985549087934708884221*n^15+223702319404958129640982*n^14-\ 4035817943559903392170734*n^13+55873668471209572553526684*n^12-\ 608274825094531511883049171*n^11+5262462930798169524204590066*n^10-\ 36276690753168465102378959072*n^9+198638946858749558423788508752*n^8-\ 856775945079402094460983614576*n^7+2870181231702658684781570883360*n^6-\ 7309857848834830476645678844800*n^5+13711760316185513729668569607680*n^4-\ 18052626827872486682858778163200*n^3+15430938631310866383722225664000*n^2-\ 7425089767435381863620198400000*n+1432566893177851073023180800000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/1048576*(7541350891*n^23-1566572572118*n^ 22+138700590810386*n^21-6210806649114760*n^20+79016082044793201*n^19+ 7695456795329314242*n^18-593853934184942118644*n^17+23725630875125076037360*n^ 16-657060388441992197040499*n^15+13692085018816812793798382*n^14-\ 222557519376168175181498334*n^13+2874051820646045076399933480*n^12-\ 29760414206455610813402299769*n^11+247936710054767604515895990982*n^10-\ 1659935807157069758365961822944*n^9+8883306124421613162003744326000*n^8-\ 37632594230703035819984474662224*n^7+124322610069819006519464268220512*n^6-\ 313327020398155314963294441550464*n^5+583400765124521609071049538577920*n^4-\ 764594779335916215439031582361600*n^3+652358095883085677566574757888000*n^2-\ 314208361081845557874221875200000*n+60884092960058670603485184000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/2097152*(16574459483*n^24-\ 2979439712412*n^23+153611227575962*n^22+8833656498801552*n^21-\ 1868545019111024947*n^20+144662976702179043108*n^19-7176171061014016055368*n^18 +258359499382063193198832*n^17-7124775468741236350305547*n^16+ 154971433363192724705135148*n^15-2705436438262670113369705918*n^14+ 38307213712879889033406997392*n^13-442466816477317820614779567757*n^12+ 4177856608153338405908611650828*n^11-32213830530900625305917434409908*n^10+ 202005550561719189866018264470512*n^9-1022615194761906086266003249335472*n^8+ 4132575947461211353272485442654528*n^7-13119113381770145707272649410360768*n^6+ 31979624617917406687607959916051712*n^5-57932846106521458133277244226165760*n^4 +74276660806086170853928512316108800*n^3-62324843562840265051689102127104000*n^ 2+29681866938896865233077778841600000*n-5723104738245515036727607296000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 23/2097152*(38914019609*n^ 24+22602849397188*n^23-7697478007344274*n^22+1051063275350253792*n^21-\ 86652675846382489921*n^20+4940233384115530140948*n^19-208429675654343978602504* n^18+6772302626237546796139632*n^17-173830037870743421650187881*n^16+ 3584454776463062301764286828*n^15-60039335912921018123342238394*n^14+ 822530257004135921435468662752*n^13-9249523814331224751666090183871*n^12+ 85436238987972287716588917991548*n^11-646921567498617736911555006894604*n^10+ 3996469575017152578028052695956432*n^9-19985075161153553595521282557704976*n^8+ 79968395376796383042445834029050688*n^7-251892164229109845537327304160476224*n^ 6+610403734850528880674217701973787392*n^5-\ 1101176642158485819955695585130152960*n^4+1408250700575906304382123576534732800 *n^3-1180499355006944713979413871775744000*n^2+ 562556490680802646863944382873600000*n-108738990026664785697824538624000000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n -41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -1725/2097152*(6475645487* n^25-2868891390802*n^24+547840852414238*n^23-62302572333012700*n^22+ 4822637749650357329*n^21-273077520228372412630*n^20+11821889678667756445868*n^ 19-402817032598060129591960*n^18+11020087792342376207090609*n^17-\ 245421633049227188649067390*n^16+4491583838439661899355137998*n^15-\ 67969260243942020681737411420*n^14+853378991868786919895339453279*n^13-\ 8899106543894067974263636222330*n^12+76986430344950423019519546341768*n^11-\ 550619775043903254439374414240880*n^10+3236679396322538863477174308470864*n^9-\ 15501002995720473606198710558388640*n^8+59744941849826458781590626270167808*n^7 -182211168718346753418100355307248640*n^6+429541661966405656761237330984402432* n^5-757128689585818838073011109674078208*n^4+ 949964260041940263189531612553912320*n^3-784423392030077741581797204280934400*n ^2+369751112877965605213097611100160000*n-71042806817420993322578698567680000)/ (2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -1725/2097152*( 13793846146*n^25-5057033387843*n^24+854839370059288*n^23-89179761983289050*n^22 +6469731230799231934*n^21-348202564162796871605*n^20+14469008973629109177388*n^ 19-476640685573915718613560*n^18+12676086385083210397873774*n^17-\ 275620512257614994781868205*n^16+4942230200338184982659195968*n^15-\ 73490001881578402497410175170*n^14+908916535535989789082438441314*n^13-\ 9356659317297954520379439840875*n^12+80055487669945906650029283828988*n^11-\ 567216619186949264381342158400780*n^10+3307955065106649105742990475201584*n^9-\ 15738486085516839911637934584098000*n^8+60335919233036728213852925444248128*n^7 -183233850195851879098701145254949440*n^6+430564975126313844928889721529237248* n^5-757218245085283054500306362693993472*n^4+ 948783017402988459069481891846410240*n^3-783056612015381167622305064128512000*n ^2+369243664257841169546507763253248000*n-71042806817420993322578698567680000)/ (2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -345/2097152*( 205798140298*n^26-75735002194982*n^25+13090451621198185*n^24-\ 1415940779227471640*n^23+107695697246799114700*n^22-6133882122104954587370*n^21 +271960039753466464480795*n^20-9631471049947164540233540*n^19+ 277350793993117929412827070*n^18-6575745587903901142434979850*n^17+ 129484772856233717042143586095*n^16-2129935467657648289655733101840*n^15+ 29367946961965382487760027702360*n^14-339882761731774239678497465005910*n^13+ 3299816775782570913232343870942845*n^12-26809384188213106077781850267512340*n^ 11+181458668568083381201283741243553780*n^10-\ 1016333915137256126296938810722204240*n^9+4666195434707346436441363621405951920 *n^8-17337080394787836527450086612631797440*n^7+ 51229431383607285035960067012475641792*n^6-\ 117556859608713593178014941936849467648*n^5+ 202584137804389237805676327282565340160*n^4-\ 249538494962919438652689148649496883200*n^3+ 203113691707479363683211402586091520000*n^2-\ 94773453800736994643963149378191360000*n+18115915738442353297257568134758400000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), -5/ 2097152*(18316892196734*n^26-6518396793531766*n^25+1094529227146606745*n^24-\ 115443780705243318640*n^23+8588667056950562879960*n^22-479739267950286611877610 *n^21+20907199841292794532252035*n^20-729213626152684265040816940*n^19+ 20716212096788869336981846970*n^18-485297298804032591148421370650*n^17+ 9454895418894826190695846963535*n^16-154068550294438811921867611674040*n^15+ 2106758853364740410129668154925740*n^14-24204990165147195909008805384794230*n^ 13+233508740532076818948394998035388485*n^12-\ 1886719462118719958533265337802695340*n^11+ 12709992185915369721459519671473624100*n^10-\ 70902938975408083471790670514437050320*n^9+ 324444200839208115797912095763677360560*n^8-\ 1202179671102245300989264801482945906240*n^7+ 3544674391786862130026517283839352226496*n^6-\ 8120769497587145777082767722247362055424*n^5+ 13978483017204091400129712848941042928640*n^4-\ 17206686637001484142185653682371920588800*n^3+ 14002151630622220501160368077400780800000*n^2-\ 6534756988244703667775973562479083520000*n+ 1249998185952522377510772201298329600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), -15/16777216*(115872715662341*n^27-\ 43462931975307072*n^26+7729217146160338257*n^25-867349197186692851800*n^24+ 68959406813175905760045*n^23-4134508576695354137670000*n^22+ 194258579056226530540722885*n^21-7337653128138488775342414120*n^20+ 226803665657333219599253029635*n^19-5809008382182890069661919666080*n^18+ 124381065276947397870134424527715*n^17-2239932429393907190537532378865320*n^16+ 34056213885960222370987958871204015*n^15-437983968593420641497552499260509040*n ^14+4765237236124936333313408266316505975*n^13-\ 43793567967965650149921275723255743320*n^12+ 338861969991991384774834464600301175580*n^11-\ 2196319387298250658665055533427222528320*n^10+ 11837648111052309511788386031278608592080*n^9-\ 52533211151091029937585830359180028161920*n^8+ 189431802156798039940700262554674445073984*n^7-\ 545289169365490779293287034647199470399488*n^6+ 1223196253349766253302223763078201975513088*n^5-\ 2067361186064462925519682683558671465963520*n^4+ 2505386004296672905268178919941697429094400*n^3-\ 2012569088440215457355724910282802872320000*n^2+ 929797577681067430187509544718768537600000*n-\ 176666410281289829354855804450163916800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -3/8388608*(324748278195389*n^27 -120299404299393798*n^26+21151304404455160143*n^25-2349042557796232084800*n^24+ 185007000863445187443705*n^23-10997326670460033156637410*n^22+ 512689602986202468210776535*n^21-19229122865262156355091711700*n^20+ 590573660108994379919443971915*n^19-15039044585690184590602695611130*n^18+ 320346409340806426904050667084805*n^17-5742262643255942788877341717546200*n^16+ 86945214550057321260332595676276635*n^15-1114068691819767790739261303978306670* n^14+12081799883804695833350956737739217445*n^13-\ 110720063416783375158744300526830215700*n^12+ 854615376144299345073792188612539613220*n^11-\ 5527470969464683160120130171457826534640*n^10+ 29738543487548466378576095052319421953840*n^9-\ 131777522197716058399076782696183845259200*n^8+ 474606327021687908871811139062750461875136*n^7-\ 1364871701314068676987219278232939604036352*n^6+ 3059483948978444892001747603848053548127232*n^5-\ 5168336177235064078374461453058421422182400*n^4+ 6261528929368826224332334032587945754624000*n^3-\ 5029358308909866425472826607489998356480000*n^2+ 2323757834159829867846461962611712327680000*n-\ 441666025703224573387139511125409792000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -3/16777216*(1397898594687941*n^ 28-552130279357741042*n^27+103782866080737228777*n^26-12356153782196749425810*n ^25+1046215014875609264148645*n^24-67058088891483605199355890*n^23+ 3381488265999134695185718365*n^22-137639087149870184111977994250*n^21+ 4603869137019963776725721596635*n^20-128171277587441695780255534722270*n^19+ 2997140399741535381930731320216395*n^18-59244770023761333767839369581900150*n^ 17+994161393216412368528180670830577815*n^16-\ 14196307842190444220779511001236927430*n^15+ 172644112070566824511523116934664057855*n^14-\ 1786784951641430418621827403873983215950*n^13+ 15702388184521063944542739161110210326180*n^12-\ 116726025669318925930936839862518139069560*n^11+ 729865774342828381882117197253805591667760*n^10-\ 3809379914713027372488392609090901859661600*n^9+ 16426937759682589997275508064156974672846784*n^8-\ 57740341102550319196050662931435678041543808*n^7+ 162489644437213779738463807622046965432590848*n^6-\ 357317476699717065900182770945789320392202240*n^5+ 593553113442155422600737533535693732258816000*n^4-\ 708748734949151019955283115422662303088640000*n^3+ 562382624283696756705756352289647276523520000*n^2-\ 257330196872878948163764414602329692569600000*n+ 48583262827354703072585346223795077120000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), (4008968866605472*(n -27)!*(n+1)*(2*n-59)!+6013453299908208*(n-28)!*((n+1)*(2*n-58)!-1/ 6013453299908208*binomial(2*n,n)*(n-30)!*(n-27)!))/(n-30)!/(n-28)!/(n-27)!/ binomial(2*n,n), ((4008968866605472*n+4008968866605472)*(2*n-59)!-(n-30)!*(n-28 )!*binomial(2*n,n))/(n-30)!/(n-28)!/binomial(2*n,n)] The limits, as n goes to infinity are 536870881 268435223 1073734599 1073703351 268395915 1073206855 268048095 [---------, ---------, ----------, ----------, ---------, ----------, ---------, 536870912 268435456 1073741824 1073741824 268435456 1073741824 268435456 534896211 34070560353 16878381473 66400971449 64570795309 247027597975 ---------, -----------, -----------, -----------, -----------, ------------, 536870912 34359738368 17179869184 68719476736 68719476736 274877906944 3605615731 104276017341 89906541275 9244844802805 3295570339367 ----------, ------------, ------------, --------------, -------------, 4294967296 137438953472 137438953472 17592186044416 8796093022208 7243038794071 895022451007 -11170488465075 -11897192300925 --------------, --------------, ---------------, ---------------, 35184372088832 35184372088832 70368744177664 35184372088832 -35500179201405 -45792230491835 -1738090734935115 -974244834586167 ---------------, ---------------, -----------------, ----------------, 70368744177664 70368744177664 2251799813685248 1125899906842624 -4193695784063823 -4378319350289075 -17889118232400563 -----------------, -----------------, ------------------] 4503599627370496 4503599627370496 18014398509481984 and in Maple notation [536870881/536870912, 268435223/268435456, 1073734599/1073741824, 1073703351/ 1073741824, 268395915/268435456, 1073206855/1073741824, 268048095/268435456, 534896211/536870912, 34070560353/34359738368, 16878381473/17179869184, 66400971449/68719476736, 64570795309/68719476736, 247027597975/274877906944, 3605615731/4294967296, 104276017341/137438953472, 89906541275/137438953472, 9244844802805/17592186044416, 3295570339367/8796093022208, 7243038794071/ 35184372088832, 895022451007/35184372088832, -11170488465075/70368744177664, -\ 11897192300925/35184372088832, -35500179201405/70368744177664, -45792230491835/ 70368744177664, -1738090734935115/2251799813685248, -974244834586167/ 1125899906842624, -4193695784063823/4503599627370496, -4378319350289075/ 4503599627370496, -17889118232400563/18014398509481984] and in floating point [.9999999423, .9999991320, .9999932712, .9999641692, .9998526983, .9995017713, .9985569678, .9963218328, .9915838121, .9824511055, .9662613076, .9396287396, .\ 8986811662, .8394978314, .7587078823, .6541561836, .5255085854, .3746629704, .2\ 058595440, .2543806804e-1, -.1587421887, -.3381385426, -.5044878890, -.65074673\ 46, -.7718673411, -.8653032376, -.9311875235, -.9721821904, -.9930455476] The cut off is at j=, 21 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 31], vs. those in the, 2, -th row from j=1 to j=, 30, are as follws 15 14 13 12 [31 (1082401 n - 121770105 n + 6241393165 n - 192995309325 n 11 10 9 + 4018030771027 n - 59516660798595 n + 646120358721695 n 8 7 6 - 5216961044927775 n + 31451268016536368 n - 140734991383906920 n 5 4 3 + 458986363630798640 n - 1045059236908268400 n + 1451822248270456704 n 2 - 338996379464858880 n - 3472977303129600000 n + 6338850154116480000)/( 1024 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 25) (2 n - 15) 16 15 (2 n - 17) (2 n - 23) (2 n - 29)), 899 (1194373 n - 152879624 n 14 13 12 + 8969714740 n - 319706873360 n + 7734382665646 n 11 10 9 - 134386186389728 n + 1730541414284540 n - 16797160994751280 n 8 7 6 + 123700998162530789 n - 689210435811970072 n + 2860463476835018600 n 5 4 - 8477010163035319360 n + 15711656512646778192 n 3 2 - 7190389530144072576 n - 46675932366443546880 n + 100308661278126336000 n - 3497296636753920000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 16 15 14 (2 n - 23) (2 n - 29)), 203 (5289349 n - 677032712 n + 39722168500 n 13 12 11 - 1415765152400 n + 34247184056878 n - 594903563487584 n 10 9 8 + 7655688487707260 n - 74173880224615600 n + 543550535211223397 n 7 6 - 2987655782801414296 n + 11942266856537362280 n 5 4 - 31702373417249704000 n + 38233619050058327376 n 3 2 + 63675001598002238592 n - 290541752937474935040 n + 226892066090565888000 n - 15488027962767360000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 15 (2 n - 23) (2 n - 29)), 10353 (207422 n - 29971687 n + 1995639056 n 14 13 12 - 81206591260 n + 2258071091204 n - 45441111161914 n 11 10 9 + 683475125204872 n - 7817833656063860 n + 68408368888070926 n 8 7 6 - 455016161329326311 n + 2242468099628426488 n - 7649164544150487800 n 5 4 + 14527000758419084448 n + 3296440433631264912 n 3 2 - 88700154723192940416 n + 170021235230704177920 n - 103669391816964864000 n + 10021665152378880000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 17 16 (2 n - 17) (2 n - 23) (2 n - 29)), 357 (6014842 n - 869054579 n 15 14 13 + 57855530464 n - 2353399622540 n + 65388168747244 n 12 11 10 - 1313631990656258 n + 19686981534056408 n - 223483973150435140 n 9 8 + 1925037623034654986 n - 12398125828836637747 n 7 6 + 57156027540299180792 n - 167576857087715584600 n 5 4 + 181270670773207256928 n + 663920398709155909584 n 3 2 - 2914989334383261665664 n + 4307242700649509441280 n - 2415780677743716096000 n + 290628289418987520000)/(16384 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 1785 (2405422 n - 389562534 n 16 15 14 + 29201706489 n - 1344209372640 n + 42498197146104 n 13 12 11 - 977387692241268 n + 16880055413412958 n - 222477487582676040 n 10 9 + 2245346772917560026 n - 17168628196743599862 n 8 7 + 96301443150489474417 n - 365884711855929138600 n 6 5 + 721959444034732555048 n + 671542347806905709664 n 4 3 - 8475390968273274330864 n + 22653908268491049137280 n 2 - 27918034813151752377600 n + 14723252488675374336000 n - 2034398025932912640000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 18 17 16 (-1 + 2 n) (2 n - 5)), 1785 (2403992 n - 389129244 n + 29141337609 n 15 14 13 - 1339088399640 n + 42203084468244 n - 965183731292088 n 12 11 10 + 16507253722133798 n - 213950093150405040 n + 2098962155995884036 n 9 8 - 15297690608469790092 n + 78835567929359098377 n 7 6 - 250743747971836992600 n + 215219796910710952328 n 5 4 + 2020273642046549927424 n - 10202194278900386396784 n 3 2 + 22610047626569495537280 n - 25666879222465702329600 n + 13301361395281403136000 n - 2034398025932912640000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 19 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 111435 (76904 n 18 17 16 15 - 13856752 n + 1159330386 n - 59747446143 n + 2120625493008 n 14 13 12 - 54864903308124 n + 1066973606135132 n - 15825568976101066 n 11 10 + 179273981777116512 n - 1530666279038892996 n 9 8 + 9496051328588522178 n - 38932914696986458239 n 7 6 + 70906329750602779736 n + 240599094862143689672 n 5 4 - 2186225110836305553696 n + 7442324595964138034448 n 3 2 - 14029663557440203148160 n + 14647481967821383699200 n - 7392949486234405632000 n + 1205741621777172480000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 22287 (383299 n 18 17 16 15 - 68923565 n + 5747843676 n - 294713962455 n + 10377944610858 n 14 13 12 - 265279720884570 n + 5066221076846692 n - 73136976718681070 n 11 10 + 795565090790485587 n - 6377808223468832865 n 9 8 + 35479906758214432488 n - 112388892024873804795 n 7 6 - 43742574425929170344 n + 2496229022626628839000 n 5 4 - 13482142772096207009856 n + 39471784461291220186320 n 3 2 - 69103887408767981462400 n + 69600457604771712864000 n - 35047784944936431360000 n + 6028708108885862400000)/(16384 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 20 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 37145 (1828195 n 19 18 17 16 - 363194558 n + 33551552961 n - 1911012892962 n + 74984131706154 n 15 14 13 - 2143464021432156 n + 45983352720376642 n - 750215131180941764 n 12 11 + 9303536922157215543 n - 86191342872642684534 n 10 9 + 567743153595212881533 n - 2275201108728180468186 n 8 7 + 1022224532543234527372 n + 54983197262294601968368 n 6 5 - 408982804758982615147536 n + 1643177884014594813858912 n 4 3 - 4167128891728016296611264 n + 6697835459408827172090880 n 2 - 6421851211402245115929600 n + 3181485173187246621696000 n - 564287078991716720640000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 20 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 323 (207797983 n 19 18 17 - 41058485990 n + 3763938685005 n - 212095603694850 n 16 15 14 + 8199485915230698 n - 229651168418575500 n + 4790337328500934810 n 13 12 - 75154488558608728100 n + 880476212578431354483 n 11 10 - 7448200811197958350110 n + 40936443964144431391065 n 9 8 - 80640626339913398684250 n - 875761090807426833992492 n 7 6 + 10230049129808093125685680 n - 57487653289628364982072080 n 5 4 + 205991731723446283213807200 n - 490475663319151882516260672 n 3 2 + 759880063796555916554065920 n - 715784556939028124117068800 n + 355125070003691100072960000 n - 64893014084047422873600000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 21 20 19 (2 n - 5)), 323 (407313146 n - 88160126103 n + 8869484728300 n 18 17 16 - 549638907197205 n + 23425320274553526 n - 725474955054044538 n 15 14 + 16796017833976959320 n - 293853352123977236010 n 13 12 + 3861292680374512392646 n - 36870652351690590912843 n 11 10 + 229591410266739639731340 n - 483647319031196004568065 n 9 8 - 7331021900314614656082454 n + 99896896352706259071883212 n 7 6 - 688978892265841016108549840 n + 3145492621995863999206638480 n 5 4 - 9983982985926045833363616864 n + 21937377502920888236449150272 n 3 2 - 32193365299260329779675269120 n + 29327885621881136411790412800 n - 14362213692299694869007360000 n + 2660613577445944337817600000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 21 20 (2 n - 35) (2 n - 29)), 323 (394198681 n - 84383160183 n 19 18 17 + 8368420364045 n - 509032746591285 n + 21177393461169456 n 16 15 - 635403223947683898 n + 14095196049378784810 n 14 13 - 232037521923103192170 n + 2769243591612969046121 n 12 11 - 21906264622165687264683 n + 70568672177376204365625 n 10 9 + 820166819618750606406495 n - 15490499026000149785396834 n 8 7 + 138162601694193841639481292 n - 819371796446423089766515360 n 6 5 + 3450774189124967937398774160 n - 10417067035548367605696107424 n 4 3 + 22155496922904871178355377472 n - 31868644302999228827551749120 n 2 + 28764900088994239935691852800 n - 14115138140141958102543360000 n + 2660613577445944337817600000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 323 ( 22 21 20 1498713217 n - 348087356705 n + 37500010879793 n 19 18 17 - 2481236443009555 n + 112443188474927772 n - 3679373589101500290 n 16 15 + 89046178534565634778 n - 1595411953711716203030 n 14 13 + 20481847613133549484457 n - 165163530103008429517525 n 12 11 + 239364243059274471198333 n + 15389236358973717816320745 n 10 9 - 266466410835415071568905398 n + 2647044432108157748089003240 n 8 7 - 18441919570818638147690647672 n + 94626402936672274571704890320 n 6 5 - 361006958537285851358718937248 n + 1014152076053025640085396321280 n 4 3 - 2045199117894412287652200335232 n + 2831144353762976243690064491520 n 2 - 2491899688642002275689316812800 n + 1208362786276830860095580160000 n - 228812767660351213052313600000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 323 ( 22 21 20 19 347232361 n - 79089961577 n + 8315452080782 n - 533476870565485 n 18 17 + 23223675414781611 n - 719438013989151072 n 16 15 + 16061649136578975622 n - 250997898388719133850 n 14 13 + 2366266708424443267751 n - 924013629018055272697 n 12 11 - 406870109167759996735158 n + 8096650975085654416970535 n 10 9 - 96895472979038438730619979 n + 828637824001817290603362418 n 8 7 - 5308026021645157019787409678 n + 25796743363459246600408059920 n 6 5 - 94790475656954832066594056544 n + 259322423489736936670261232928 n 4 3 - 513501604841002867168365161568 n + 702792852026635979112269018880 n 2 - 615506066299686811170493555200 n + 298988219304759493152483840000 n - 57203191915087803263078400000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 37145 ( 23 22 21 20 5402396 n - 1319149969 n + 148604454823 n - 10198829177768 n 19 18 17 + 473141748997071 n - 15479189975158833 n + 356674501071717356 n 16 15 - 5352696150035664166 n + 30932827855287995626 n 14 13 + 824293229066070813973 n - 29705282724087225663681 n 12 11 + 539246572011268120799436 n - 6836473152892355772528949 n 10 9 + 65323860962546439327813629 n - 483042927139180153581252098 n 8 7 + 2785593946513022722178368498 n - 12493298346771236941588632144 n 6 5 + 43104210251574024816817831200 n - 112199215851170077643550266400 n 4 3 + 213705848442295411578184644000 n - 284031953586783268048982304000 n 2 + 243726572285930608364548800000 n - 117054988759798386436339200000 n + 22383857705903923015987200000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) 23 22 21 (2 n - 45)), 37145 (36761299 n - 8661044135 n + 931483101065 n 20 19 18 - 60007692569209 n + 2534616830341134 n - 70675146987850482 n 17 16 + 1134792231303710914 n + 409614091442486062 n 15 14 - 611082186479244808801 n + 20174818875794532425861 n 13 12 - 409498477234927847692107 n + 6054765176854723412475483 n 11 10 - 68760034017825083727863696 n + 612617973987776029724331364 n 9 8 - 4314654985310927880635727088 n + 24008867954878266275071455584 n 7 6 - 104827235259934355842251975936 n + 354417986985556878247359811392 n 5 4 - 908785465925678214594630902784 n + 1712758590974856817523455822080 n 3 2 - 2261494967966687017582781184000 n + 1935300571992567002199062016000 n - 930663430670411885551104000000 n + 179070861647231384127897600000)/( 262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 2185 ( 24 23 22 7874434427 n - 1936393446420 n + 214718024786954 n 21 20 - 13905310047560448 n + 555052989467880365 n 19 18 - 11764503911949752964 n - 67338186095118138904 n 17 16 + 15562192034185069900176 n - 664639739806321195474987 n 15 14 + 18098482335427791622168932 n - 363901357410990251568221422 n 13 12 + 5685241583449095138560019840 n - 70624053601528781611097950189 n 11 10 + 705191945556945683039586856308 n - 5681876422499751156061635745060 n 9 + 36899208628821609867117526815984 n 8 - 192092005459090637257064546833456 n 7 + 793697603854924864680067649310144 n 6 - 2563610046754313222753425479269568 n 5 + 6331082050811729895403834684424448 n 4 - 11574896037377945659043784419036160 n 3 + 14923999280063423832692484939264000 n 2 - 12550064496528394205792157560832000 n + 5969000997334962067613471539200000 n - 1144620947649103007345521459200000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 23 (2 n - 35) (2 n - 29) (2 n - 45)), 437 (27096480707 n - 5999259803700 n 22 21 20 + 558330251935898 n - 24837632135653920 n + 81467198726362757 n 19 18 + 61110839611083241980 n - 4428160704896685392152 n 17 16 + 187318039576437094578000 n - 5658584995015197925299763 n 15 14 + 130572700069024962474145860 n - 2376203105074711997237354782 n 13 12 + 34697209155775975873848157920 n - 410342165696061456744373059973 n 11 + 3946945978740537367620550489140 n 10 - 30884903151576355132050546461572 n 9 + 195968208959143402131536816601840 n 8 - 1001442103885643935432942045041328 n 7 + 4077318232283986205784239290038720 n 6 - 13018887195021752738683468912627392 n 5 + 31873227924570556256740909985276160 n 4 - 57915592852124051973439815244262400 n 3 + 74391253508773860718892547644928000 n 2 - 62464857477594705479943729377280000 n + 29735200531439654978939756544000000 n - 5723104738245515036727607296000000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 115 (102817445378 n 24 23 22 - 17952028081285 n + 506298544595720 n + 156238924638050810 n 21 20 - 23765560779357191410 n + 1841387724178314729725 n 19 18 - 96527689487530514668180 n + 3752261974826315720281160 n 17 16 - 113138720756617340802283090 n + 2716148632154718935309719925 n 15 14 - 52771494605147570825697747280 n + 838320452365686825633429321410 n 13 - 10955383721540416613808044841550 n 12 + 118110072188969548392840960157475 n 11 - 1050557818558209709815116137615780 n 10 + 7689997952139585332702274803242460 n 9 - 46082715219468046058284060269048880 n 8 + 224222552635223074731878833749102800 n 7 - 875368182425681996784738057288012480 n 6 + 2696862514438613254481808983074532160 n 5 - 6406403254968248977044619518785480448 n 4 + 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25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), ( 15259946008369216 (n - 28)! (n + 1) (2 n - 61)! + 22889919012553824 (n - 29)! ((n + 1) (2 n - 60)! - 1/22889919012553824 binomial(2 n, n) (n - 31)! (n - 28)!))/((n - 31)! (n - 29)! (n - 28)! binomial(2 n, n)), ( (15259946008369216 n + 15259946008369216) (2 n - 61)! - (n - 31)! (n - 29)! binomial(2 n, n))/((n - 31)! (n - 29)! binomial(2 n, n))] and in Maple notation [31/1024*(1082401*n^15-121770105*n^14+6241393165*n^13-192995309325*n^12+ 4018030771027*n^11-59516660798595*n^10+646120358721695*n^9-5216961044927775*n^8 +31451268016536368*n^7-140734991383906920*n^6+458986363630798640*n^5-\ 1045059236908268400*n^4+1451822248270456704*n^3-338996379464858880*n^2-\ 3472977303129600000*n+6338850154116480000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-25)/(2*n-15)/(2*n-17)/(2 *n-23)/(2*n-29), 899/16384*(1194373*n^16-152879624*n^15+8969714740*n^14-\ 319706873360*n^13+7734382665646*n^12-134386186389728*n^11+1730541414284540*n^10 -16797160994751280*n^9+123700998162530789*n^8-689210435811970072*n^7+ 2860463476835018600*n^6-8477010163035319360*n^5+15711656512646778192*n^4-\ 7190389530144072576*n^3-46675932366443546880*n^2+100308661278126336000*n-\ 3497296636753920000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11) /(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-\ 29), 203/16384*(5289349*n^16-677032712*n^15+39722168500*n^14-1415765152400*n^13 +34247184056878*n^12-594903563487584*n^11+7655688487707260*n^10-\ 74173880224615600*n^9+543550535211223397*n^8-2987655782801414296*n^7+ 11942266856537362280*n^6-31702373417249704000*n^5+38233619050058327376*n^4+ 63675001598002238592*n^3-290541752937474935040*n^2+226892066090565888000*n-\ 15488027962767360000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-11 )/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-\ 29), 10353/16384*(207422*n^17-29971687*n^16+1995639056*n^15-81206591260*n^14+ 2258071091204*n^13-45441111161914*n^12+683475125204872*n^11-7817833656063860*n^ 10+68408368888070926*n^9-455016161329326311*n^8+2242468099628426488*n^7-\ 7649164544150487800*n^6+14527000758419084448*n^5+3296440433631264912*n^4-\ 88700154723192940416*n^3+170021235230704177920*n^2-103669391816964864000*n+ 10021665152378880000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33 )/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-\ 23)/(2*n-29), 357/16384*(6014842*n^17-869054579*n^16+57855530464*n^15-\ 2353399622540*n^14+65388168747244*n^13-1313631990656258*n^12+19686981534056408* n^11-223483973150435140*n^10+1925037623034654986*n^9-12398125828836637747*n^8+ 57156027540299180792*n^7-167576857087715584600*n^6+181270670773207256928*n^5+ 663920398709155909584*n^4-2914989334383261665664*n^3+4307242700649509441280*n^2 -2415780677743716096000*n+290628289418987520000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 1785/16384*(2405422*n^18-389562534*n^ 17+29201706489*n^16-1344209372640*n^15+42498197146104*n^14-977387692241268*n^13 +16880055413412958*n^12-222477487582676040*n^11+2245346772917560026*n^10-\ 17168628196743599862*n^9+96301443150489474417*n^8-365884711855929138600*n^7+ 721959444034732555048*n^6+671542347806905709664*n^5-8475390968273274330864*n^4+ 22653908268491049137280*n^3-27918034813151752377600*n^2+14723252488675374336000 *n-2034398025932912640000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25 )/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19 )/(2*n-3)/(-1+2*n)/(2*n-5), 1785/16384*(2403992*n^18-389129244*n^17+29141337609 *n^16-1339088399640*n^15+42203084468244*n^14-965183731292088*n^13+ 16507253722133798*n^12-213950093150405040*n^11+2098962155995884036*n^10-\ 15297690608469790092*n^9+78835567929359098377*n^8-250743747971836992600*n^7+ 215219796910710952328*n^6+2020273642046549927424*n^5-10202194278900386396784*n^ 4+22610047626569495537280*n^3-25666879222465702329600*n^2+ 13301361395281403136000*n-2034398025932912640000)/(2*n-29)/(2*n-35)/(2*n-23)/(2 *n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/ (2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 111435/16384*(76904*n^19-\ 13856752*n^18+1159330386*n^17-59747446143*n^16+2120625493008*n^15-\ 54864903308124*n^14+1066973606135132*n^13-15825568976101066*n^12+ 179273981777116512*n^11-1530666279038892996*n^10+9496051328588522178*n^9-\ 38932914696986458239*n^8+70906329750602779736*n^7+240599094862143689672*n^6-\ 2186225110836305553696*n^5+7442324595964138034448*n^4-14029663557440203148160*n ^3+14647481967821383699200*n^2-7392949486234405632000*n+1205741621777172480000) /(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-\ 13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2* n)/(2*n-5), 22287/16384*(383299*n^19-68923565*n^18+5747843676*n^17-294713962455 *n^16+10377944610858*n^15-265279720884570*n^14+5066221076846692*n^13-\ 73136976718681070*n^12+795565090790485587*n^11-6377808223468832865*n^10+ 35479906758214432488*n^9-112388892024873804795*n^8-43742574425929170344*n^7+ 2496229022626628839000*n^6-13482142772096207009856*n^5+39471784461291220186320* n^4-69103887408767981462400*n^3+69600457604771712864000*n^2-\ 35047784944936431360000*n+6028708108885862400000)/(2*n-29)/(2*n-35)/(2*n-23)/(2 *n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/ (2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 37145/65536*( 1828195*n^20-363194558*n^19+33551552961*n^18-1911012892962*n^17+74984131706154* n^16-2143464021432156*n^15+45983352720376642*n^14-750215131180941764*n^13+ 9303536922157215543*n^12-86191342872642684534*n^11+567743153595212881533*n^10-\ 2275201108728180468186*n^9+1022224532543234527372*n^8+54983197262294601968368*n ^7-408982804758982615147536*n^6+1643177884014594813858912*n^5-\ 4167128891728016296611264*n^4+6697835459408827172090880*n^3-\ 6421851211402245115929600*n^2+3181485173187246621696000*n-\ 564287078991716720640000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37) /(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9 )/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(207797983*n^20-\ 41058485990*n^19+3763938685005*n^18-212095603694850*n^17+8199485915230698*n^16-\ 229651168418575500*n^15+4790337328500934810*n^14-75154488558608728100*n^13+ 880476212578431354483*n^12-7448200811197958350110*n^11+40936443964144431391065* n^10-80640626339913398684250*n^9-875761090807426833992492*n^8+ 10230049129808093125685680*n^7-57487653289628364982072080*n^6+ 205991731723446283213807200*n^5-490475663319151882516260672*n^4+ 759880063796555916554065920*n^3-715784556939028124117068800*n^2+ 355125070003691100072960000*n-64893014084047422873600000)/(2*n-29)/(2*n-35)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/( 2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 323/65536*(407313146*n^21-88160126103*n^20+8869484728300*n^19-549638907197205*n ^18+23425320274553526*n^17-725474955054044538*n^16+16796017833976959320*n^15-\ 293853352123977236010*n^14+3861292680374512392646*n^13-36870652351690590912843* n^12+229591410266739639731340*n^11-483647319031196004568065*n^10-\ 7331021900314614656082454*n^9+99896896352706259071883212*n^8-\ 688978892265841016108549840*n^7+3145492621995863999206638480*n^6-\ 9983982985926045833363616864*n^5+21937377502920888236449150272*n^4-\ 32193365299260329779675269120*n^3+29327885621881136411790412800*n^2-\ 14362213692299694869007360000*n+2660613577445944337817600000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35) /(2*n-29), 323/65536*(394198681*n^21-84383160183*n^20+8368420364045*n^19-\ 509032746591285*n^18+21177393461169456*n^17-635403223947683898*n^16+ 14095196049378784810*n^15-232037521923103192170*n^14+2769243591612969046121*n^ 13-21906264622165687264683*n^12+70568672177376204365625*n^11+ 820166819618750606406495*n^10-15490499026000149785396834*n^9+ 138162601694193841639481292*n^8-819371796446423089766515360*n^7+ 3450774189124967937398774160*n^6-10417067035548367605696107424*n^5+ 22155496922904871178355377472*n^4-31868644302999228827551749120*n^3+ 28764900088994239935691852800*n^2-14115138140141958102543360000*n+ 2660613577445944337817600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 323/131072*( 1498713217*n^22-348087356705*n^21+37500010879793*n^20-2481236443009555*n^19+ 112443188474927772*n^18-3679373589101500290*n^17+89046178534565634778*n^16-\ 1595411953711716203030*n^15+20481847613133549484457*n^14-\ 165163530103008429517525*n^13+239364243059274471198333*n^12+ 15389236358973717816320745*n^11-266466410835415071568905398*n^10+ 2647044432108157748089003240*n^9-18441919570818638147690647672*n^8+ 94626402936672274571704890320*n^7-361006958537285851358718937248*n^6+ 1014152076053025640085396321280*n^5-2045199117894412287652200335232*n^4+ 2831144353762976243690064491520*n^3-2491899688642002275689316812800*n^2+ 1208362786276830860095580160000*n-228812767660351213052313600000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29), 323/32768*(347232361*n^22-79089961577*n^21+ 8315452080782*n^20-533476870565485*n^19+23223675414781611*n^18-\ 719438013989151072*n^17+16061649136578975622*n^16-250997898388719133850*n^15+ 2366266708424443267751*n^14-924013629018055272697*n^13-406870109167759996735158 *n^12+8096650975085654416970535*n^11-96895472979038438730619979*n^10+ 828637824001817290603362418*n^9-5308026021645157019787409678*n^8+ 25796743363459246600408059920*n^7-94790475656954832066594056544*n^6+ 259322423489736936670261232928*n^5-513501604841002867168365161568*n^4+ 702792852026635979112269018880*n^3-615506066299686811170493555200*n^2+ 298988219304759493152483840000*n-57203191915087803263078400000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-35)/(2*n-29), 37145/32768*(5402396*n^23-1319149969*n^22+148604454823*n ^21-10198829177768*n^20+473141748997071*n^19-15479189975158833*n^18+ 356674501071717356*n^17-5352696150035664166*n^16+30932827855287995626*n^15+ 824293229066070813973*n^14-29705282724087225663681*n^13+ 539246572011268120799436*n^12-6836473152892355772528949*n^11+ 65323860962546439327813629*n^10-483042927139180153581252098*n^9+ 2785593946513022722178368498*n^8-12493298346771236941588632144*n^7+ 43104210251574024816817831200*n^6-112199215851170077643550266400*n^5+ 213705848442295411578184644000*n^4-284031953586783268048982304000*n^3+ 243726572285930608364548800000*n^2-117054988759798386436339200000*n+ 22383857705903923015987200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 37145/262144*(36761299*n^23-8661044135*n^22+931483101065*n^21-60007692569209*n^ 20+2534616830341134*n^19-70675146987850482*n^18+1134792231303710914*n^17+ 409614091442486062*n^16-611082186479244808801*n^15+20174818875794532425861*n^14 -409498477234927847692107*n^13+6054765176854723412475483*n^12-\ 68760034017825083727863696*n^11+612617973987776029724331364*n^10-\ 4314654985310927880635727088*n^9+24008867954878266275071455584*n^8-\ 104827235259934355842251975936*n^7+354417986985556878247359811392*n^6-\ 908785465925678214594630902784*n^5+1712758590974856817523455822080*n^4-\ 2261494967966687017582781184000*n^3+1935300571992567002199062016000*n^2-\ 930663430670411885551104000000*n+179070861647231384127897600000)/(2*n-5)/(-1+2* n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-35)/(2*n-29)/(2*n-45), 2185/2097152*(7874434427*n^24-1936393446420*n^ 23+214718024786954*n^22-13905310047560448*n^21+555052989467880365*n^20-\ 11764503911949752964*n^19-67338186095118138904*n^18+15562192034185069900176*n^ 17-664639739806321195474987*n^16+18098482335427791622168932*n^15-\ 363901357410990251568221422*n^14+5685241583449095138560019840*n^13-\ 70624053601528781611097950189*n^12+705191945556945683039586856308*n^11-\ 5681876422499751156061635745060*n^10+36899208628821609867117526815984*n^9-\ 192092005459090637257064546833456*n^8+793697603854924864680067649310144*n^7-\ 2563610046754313222753425479269568*n^6+6331082050811729895403834684424448*n^5-\ 11574896037377945659043784419036160*n^4+14923999280063423832692484939264000*n^3 -12550064496528394205792157560832000*n^2+5969000997334962067613471539200000*n-\ 1144620947649103007345521459200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/ (2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/ (2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29 )/(2*n-45), 437/2097152*(27096480707*n^24-5999259803700*n^23+558330251935898*n^ 22-24837632135653920*n^21+81467198726362757*n^20+61110839611083241980*n^19-\ 4428160704896685392152*n^18+187318039576437094578000*n^17-\ 5658584995015197925299763*n^16+130572700069024962474145860*n^15-\ 2376203105074711997237354782*n^14+34697209155775975873848157920*n^13-\ 410342165696061456744373059973*n^12+3946945978740537367620550489140*n^11-\ 30884903151576355132050546461572*n^10+195968208959143402131536816601840*n^9-\ 1001442103885643935432942045041328*n^8+4077318232283986205784239290038720*n^7-\ 13018887195021752738683468912627392*n^6+31873227924570556256740909985276160*n^5 -57915592852124051973439815244262400*n^4+74391253508773860718892547644928000*n^ 3-62464857477594705479943729377280000*n^2+29735200531439654978939756544000000*n -5723104738245515036727607296000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47) /(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13) /(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), 115/2097152*(102817445378*n^25-17952028081285*n^24+ 506298544595720*n^23+156238924638050810*n^22-23765560779357191410*n^21+ 1841387724178314729725*n^20-96527689487530514668180*n^19+ 3752261974826315720281160*n^18-113138720756617340802283090*n^17+ 2716148632154718935309719925*n^16-52771494605147570825697747280*n^15+ 838320452365686825633429321410*n^14-10955383721540416613808044841550*n^13+ 118110072188969548392840960157475*n^12-1050557818558209709815116137615780*n^11+ 7689997952139585332702274803242460*n^10-46082715219468046058284060269048880*n^9 +224222552635223074731878833749102800*n^8-875368182425681996784738057288012480* n^7+2696862514438613254481808983074532160*n^6-\ 6406403254968248977044619518785480448*n^5+ 11353206364038677869233633535040271360*n^4-\ 14291321929208638291009042981028352000*n^3+ 11815257226222350712789156766748672000*n^2-\ 5564509858757149486074291039436800000*n+1065642102261314899838680478515200000)/ (2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), -575/2097152*( 1249470308*n^25-3171611792659*n^24+880977852124364*n^23-120148245932222650*n^22 +10376745569968383452*n^21-632631645106062936565*n^20+28890566271099264165764*n ^19-1025082669434797151528680*n^18+28946976549896669502229772*n^17-\ 661254779823621531359277565*n^16+12355404100280718684697270004*n^15-\ 190194970857696984632089563010*n^14+2422188883271695568479353131492*n^13-\ 25560817581614485727600041614475*n^12+223336148646959261331163484621564*n^11-\ 1610634905808659996688074329809340*n^10+9532998624568375020932560188694352*n^9-\ 45913131635086575479279349417758800*n^8+177766913984731287754959071184561984*n^ 7-544093297323817851305712007019392320*n^6+ 1286083208050412760964059280954746624*n^5-2271163624452335718577041894101799936 *n^4+2852826834422572044963557047161016320*n^3-\ 2356665081288027940709291444834304000*n^2+1110513775529044607456306326339584000 *n-213128420452262979967736095703040000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-45), -1725/2097152*(15548071853*n^26-6976914009091*n^25+ 1390528435257299*n^24-167637778285987438*n^23+13893549645911544713*n^22-\ 848573918147080394269*n^21+39869554240330473727211*n^20-\ 1482650248718102623781908*n^19+44507426629108533911749643*n^18-\ 1093551707437588913811051469*n^17+22206080033899243102291612721*n^16-\ 375123514539889754672715148558*n^15+5292804741309366628741561383563*n^14-\ 62487949495829932141293983661619*n^13+617196543991547689252503022034081*n^12-\ 5088976737838212677263845947269648*n^11+34880608027009112227309794785411828*n^ 10-197446783080524167161699579681653824*n^9+ 914548415984334957321453877125828336*n^8-3422472043011182179987185828653822208* n^7+10170723275460088175802406313498270400*n^6-\ 23439432584400345523961166063387241728*n^5+ 40514693203998414028437858528691940352*n^4-\ 49995162378437379085671567717889290240*n^3+ 40720180847578084844529866222727168000*n^2-\ 18990209187317868226417751542530048000*n+3623183147688470659451513626951680000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), -1725 /2097152*(29776714190*n^26-11572765483942*n^25+2089183231288673*n^24-\ 234126772605270064*n^23+18338225310406576952*n^22-1070551351079853372730*n^21+ 48469545927734006767835*n^20-1747560227391285686577484*n^19+ 51104239965636307889486762*n^18-1227856435814890609167896170*n^17+ 24457933119224532215466672455*n^16-406347453828134966864910599224*n^15+ 5651366775833272759835250950252*n^14-65894304588784045987701851620390*n^13+ 643867365286248488815971250851245*n^12-5259926736816771522008947926283084*n^11+ 35768201765088704576005297025664452*n^10-201121440637762531921225847191616080*n ^9+926391024698885291850139989110683440*n^8-\ 3451044307910881829189900755121699904*n^7+ 10218668876763965405684347061540659392*n^6-\ 23485529936712558687769614944223122688*n^5+ 40516336753408997814947119097681636352*n^4-\ 49939411938500819872979942838959370240*n^3+ 40657834462769206173456710105346048000*n^2-\ 18967440964772695599734981243240448000*n+3623183147688470659451513626951680000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), -15/ 2097152*(9865116439144*n^27-3880891562647446*n^26+719623696345709220*n^25-\ 83788807441536170415*n^24+6883074744005026824660*n^23-424845205855374117212100* n^22+20484447817356782168112060*n^21-791808813200569817572295805*n^20+ 24983258430443136153117893820*n^19-651728938218187745859574663890*n^18+ 14184427098538109112157447093260*n^17-259175741807120738348799607911705*n^16+ 3991526218554471142275254155331340*n^15-51919236011530830279434854372634520*n^ 14+570534587472784197074379265983750660*n^13-\ 5289134637075004424137924971228310155*n^12+ 41235179134794042033854396412099594300*n^11-\ 268995715828965915198046174911213213660*n^10+ 1457775324860216120014896134823498444560*n^9-\ 6498874100515123089492966542275371539280*n^8+ 23521777089182421603205303217841090543936*n^7-\ 67907685145015308263994898150283559768384*n^6+ 152668636262516131467408818135613866730240*n^5-\ 258428635246592398717336681332139212272640*n^4+ 313470599299422370652537171869806151372800*n^3-\ 251889619504127327174683255439999324160000*n^2+ 116338520677486819398066865439312117760000*n-\ 22083301285161228669356975556270489600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -75/4194304*(5001811837063*n^27-\ 1910261048675409*n^26+345177532232596908*n^25-39287824524474966180*n^24+ 3163262641260285735405*n^23-191798375282988562731123*n^22+ 9102277057909865259252966*n^21-346899010231403339034379626*n^20+ 10808081761061405120449168305*n^19-278786982297432516660078966783*n^18+ 6006936415334255294793810773616*n^17-108780424732926896018055407999136*n^16+ 1662059859310600288452498300050955*n^15-21467710067240343981433559186285973*n^ 14+234452353566096865051244437497645726*n^13-\ 2161765480727041383342554823160090386*n^12+ 16774636699114904135654812940394069360*n^11-\ 108987880627708031016470714202882444168*n^10+ 588621236296409513674142173115993967296*n^9-\ 2616623589038682513376757816063400040416*n^8+ 9448403812975870825614292551177430497792*n^7-\ 27227227999557145089573990953958081596544*n^6+ 61126092637719910048261777273046097919488*n^5-\ 103369567417683316108155801245249059424256*n^4+ 125313018181317491301688922999872241541120*n^3-\ 100675070181154984240093986409362923520000*n^2+ 46506500723730374584718003991733469184000*n-\ 8833320514064491467742790222508195840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -3/16777216*(1174563635544491*n^28 -472623033902672842*n^27+90359318361420166527*n^26-10926031051341177072210*n^25 +938296850214415426948395*n^24-60920642180950858851616890*n^23+ 3108208445958473977189757115*n^22-127869030529749213041339252250*n^21+ 4318515308040594936107979030885*n^20-121279672287815557268338951635270*n^19+ 2858360075332914225337949080677645*n^18-56901769713199401648422916250704150*n^ 17+960896390053120227569342317808731065*n^16-\ 13798754190308033241972684605581154430*n^15+ 168648791087916795680038011830582064105*n^14-\ 1753124637031603762419338783129648041950*n^13+ 15465924336558526177607293534993727415180*n^12-\ 115352105376057359051461275667307710573560*n^11+ 723340433613591097597539943289557560927760*n^10-\ 3784473367602439352894519977524059780333600*n^9+ 16352455479382750137524307457937539534833984*n^8-\ 57572886820499150942690042398787425869307008*n^7+ 162228112025192128153612557232621235247886848*n^6-\ 357089643402286388395199280309696349144995840*n^5+ 593575652925458169834440093863948845290496000*n^4-\ 709057819987672088652752322565824259031040000*n^3+ 562702865326017752503654080255097516523520000*n^2-\ 257442713941461738212236903410788440473600000*n+ 48583262827354703072585346223795077120000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -3/16777216*( 1308564611030561*n^28-520595383126685902*n^27+98501351632929265797*n^26-\ 11797751617199972825310*n^25+1004381782322485750546545*n^24-\ 64695259072146026669116590*n^23+3276961105818995143356644265*n^22-\ 133925026786459095954976959750*n^21+4496020729402516998168266116335*n^20-\ 125580852591086061768920229278370*n^19+2945244116879783286491841145960095*n^18-\ 58372862575579408026433890733210650*n^17+981838797394570111709834204073185115*n ^16-14049669032816141610582097981963146330*n^15+ 171176333821163176276388375476139401155*n^14-\ 1774465406954301618409753665374119596450*n^13+ 15616146750477388681681739381216446767780*n^12-\ 116226579225611828869900151760294518196360*n^11+ 727500935542316517082557920578263478861360*n^10-\ 3800379374990751536192730763736863879109600*n^9+ 16400094488680497273714833175913843279817664*n^8-\ 57680148174790049556893305526989704797336448*n^7+ 162395890629180986467323109464281402474147328*n^6-\ 357236125885446490641616419214530592882698240*n^5+ 593561619060745853684241499781310435265536000*n^4-\ 708859606477054849433185346685210531594240000*n^3+ 562497168304528616618345039556096533790720000*n^2-\ 257370381540229944609647446319636388249600000*n+ 48583262827354703072585346223795077120000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -29/16777216*( 290247226299314*n^29-122897631503606647*n^28+24809349727549073316*n^27-\ 3178285366292891017443*n^26+290163670275699298735020*n^25-\ 20097715250031701895427815*n^24+1097779274128711442628335820*n^23-\ 48527346639642510979967076735*n^22+1767793632443386565908209359040*n^21-\ 53765582162020023470046035309145*n^20+1378160265578841954529991444379060*n^19-\ 29974354954786361539459724580764505*n^18+555741303683370102074566752087331860*n ^17-8809050145409963819285414296353980205*n^16+ 119541679663191260040042442896731344740*n^15-\ 1388787833639651324818609267343104926645*n^14+ 13793702563156815777353057080019806500270*n^13-\ 116802400102198196061510367617598075221660*n^12+ 839642165432452141077325964351064304543480*n^11-\ 5093491690276516146438939422651706211609040*n^10+ 25867087930543005487700206364274499935656736*n^9-\ 108826844482045447919693832134628444535230528*n^8+ 374127969381247470638729295301546370700883584*n^7-\ 1032118706836295586020101256711013756897925632*n^6+ 2229798276778258445658181783692741831113717760*n^5-\ 3646547456702561974162480718093748342580224000*n^4+ 4295454644194363962271388927400088668733440000*n^3-\ 3369283455257396439603877356109495091527680000*n^2+ 1527386197708508945585084461093528299110400000*n-\ 286473722188884628462486007043757178880000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), ( 15259946008369216*(n-28)!*(n+1)*(2*n-61)!+22889919012553824*(n-29)!*((n+1)*(2*n -60)!-1/22889919012553824*binomial(2*n,n)*(n-31)!*(n-28)!))/(n-31)!/(n-29)!/(n-\ 28)!/binomial(2*n,n), ((15259946008369216*n+15259946008369216)*(2*n-61)!-(n-31) !*(n-29)!*binomial(2*n,n))/(n-31)!/(n-29)!/binomial(2*n,n)] The limits, as n goes to infinity are 33554431 1073741327 1073737847 1073719983 1073649297 2146839135 [--------, ----------, ----------, ----------, ----------, ----------, 33554432 1073741824 1073741824 1073741824 1073741824 2147483648 536390715 1071224655 8542584813 67908303275 67118748509 65781073079 ---------, ----------, ----------, -----------, -----------, -----------, 536870912 1073741824 8589934592 68719476736 68719476736 68719476736 127326173963 484084369091 112156052603 50167999855 1365498451355 ------------, ------------, ------------, -----------, -------------, 137438953472 549755813888 137438953472 68719476736 2199023255552 17205639222995 11841162068959 5912003109235 -179611356775 --------------, --------------, --------------, --------------, 35184372088832 35184372088832 35184372088832 17592186044416 -26820423946425 -25682415988875 -18497093323395 -375135887779725 ---------------, ---------------, ---------------, ----------------, 140737488355328 70368744177664 35184372088832 562949953421312 -3523690906633473 -3925693833091683 -4208584781340053 -8768762598360223 -----------------, -----------------, -----------------, -----------------, 4503599627370496 4503599627370496 4503599627370496 9007199254740992 -35790360362583199 ------------------] 36028797018963968 and in Maple notation [33554431/33554432, 1073741327/1073741824, 1073737847/1073741824, 1073719983/ 1073741824, 1073649297/1073741824, 2146839135/2147483648, 536390715/536870912, 1071224655/1073741824, 8542584813/8589934592, 67908303275/68719476736, 67118748509/68719476736, 65781073079/68719476736, 127326173963/137438953472, 484084369091/549755813888, 112156052603/137438953472, 50167999855/68719476736, 1365498451355/2199023255552, 17205639222995/35184372088832, 11841162068959/ 35184372088832, 5912003109235/35184372088832, -179611356775/17592186044416, -\ 26820423946425/140737488355328, -25682415988875/70368744177664, -18497093323395 /35184372088832, -375135887779725/562949953421312, -3523690906633473/ 4503599627370496, -3925693833091683/4503599627370496, -4208584781340053/ 4503599627370496, -8768762598360223/9007199254740992, -35790360362583199/ 36028797018963968] and in floating point [.9999999702, .9999995371, .9999962961, .9999796590, .9999138275, .9996998752, .9991055634, .9976557037, .9944877602, .9881958726, .9767063385, .9572405991, .\ 9264198449, .8805443378, .8160426849, .7300404811, .6209568034, .4890136786, .3\ 365460676, .1680292345, -.1020972359e-1, -.1905705741, -.3649690824, -.52571901\ 17, -.6663752000, -.7824165552, -.8716791362, -.9344935451, -.9735282134, -.993\ 3820534] The cut off is at j=, 21 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 32], vs. those in the, 2, -th row from j=1 to j=, 31, are as follws 16 15 14 13 [31 (69273665 n - 8867028856 n + 520245161780 n - 18543167255200 n 12 11 10 + 448605686915750 n - 7794971282814352 n + 100392924751791100 n 9 8 7 - 974856954895023200 n + 7188554796833343505 n - 40214447055770115128 n 6 5 + 169086817294165967560 n - 523215986364809192000 n 4 3 + 1136893363874604760080 n - 1496732133430766945664 n 2 + 208054458971597986560 n + 3785312479249240934400 n - 6490982557815275520000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 16 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 31 (69273649 n 15 14 13 - 8867023112 n + 520244202100 n - 18543068280800 n 12 11 10 + 448598641387078 n - 7794604664812784 n + 100378522582112060 n 9 8 7 - 974422191353696800 n + 7178392542996772097 n - 40030696905914410696 n 6 5 + 166541694129696583880 n - 496775427097716894400 n 4 3 + 938704330051218364176 n - 500620978575916449408 n 2 - 2633849281353403895040 n + 6008067858754821888000 n - 202843204931727360000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) 17 (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 15283 (562057 n 16 15 14 13 - 81216989 n + 5408054044 n - 220091944100 n + 6121742598454 n 12 11 10 - 123275023826318 n + 1857023186778188 n - 21316877806844620 n 9 8 7 + 188054848018787321 n - 1274263539960007597 n + 6550614653972344592 n 6 5 - 24667035940416174760 n + 61684629360533817168 n 4 3 - 67220650799923206096 n - 132689268774255325824 n 2 + 540837360157804980480 n - 411368235837140736000 n + 27155479767736320000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 3451 ( 17 16 15 14 1244543 n - 179833576 n + 11974334396 n - 487283756560 n 13 12 11 + 13551090238826 n - 272762229000352 n + 4104567337253812 n 10 9 8 - 46996813127366960 n + 412083687626186719 n - 2752245380468008568 n 7 6 + 13673573801907369688 n - 47401901561920942400 n 5 4 + 93829014306651900912 n + 3991875209180666496 n 3 2 - 522681904799972464896 n + 1037443331413139281920 n - 638422786594796544000 n + 60129990914273280000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 10353 (3318653 n - 537590601 n 16 15 14 + 40316708706 n - 1857528995940 n + 58829083952646 n 13 12 11 - 1357438563013302 n + 23589261815299232 n - 314479348237789020 n 10 9 + 3240535532067268149 n - 25720721753705857593 n 8 7 + 154338377994038922918 n - 667804432626109691640 n 6 5 + 1839516993130930473152 n - 1772628185702501919504 n 4 3 - 7400708737511688557856 n + 30399032220768393177600 n 2 - 43662483403323563865600 n + 24019749107890905600000 n - 2806066242666086400000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 18 17 (-1 + 2 n) (2 n - 5)), 357 (96228463 n - 15586011009 n 16 15 14 + 1168557247488 n - 53809854786900 n + 1702318298534826 n 13 12 11 - 39194359773339078 n + 678221910235879516 n - 8968171398208436700 n 10 9 + 90999586313767976559 n - 701899722334761853617 n 8 7 + 3993531315310735022604 n - 15568030281698311926600 n 6 5 + 32982968903093877183952 n + 15936970428935110222704 n 4 3 - 325028531914851365500608 n + 906511601939206549651200 n 2 - 1134955752810627100492800 n + 600447417403506140160000 n - 81375921037316505600000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 19 18 17 (-1 + 2 n) (2 n - 5)), 33915 (4050221 n - 730707748 n + 61265485239 n 16 15 14 - 3168569619732 n + 113114332111242 n - 2953851400358376 n 13 12 + 58297188721561118 n - 884786217776658184 n 11 10 + 10385375975523376113 n - 93698137803627218004 n 9 8 + 635534285918939275947 n - 3075797759952086025636 n 7 6 + 9229081838502272276264 n - 7083489975160879072672 n 5 4 - 71882563529084276756304 n + 347594876463252288399552 n 3 2 - 750115358834393656131840 n + 834472260090260461900800 n - 424791137314494010368000 n + 63387559544857067520000)/(262144 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 4845 (7081273 n 18 17 16 - 1276622159 n + 106902436782 n - 5516881386456 n 15 14 13 + 196223459627646 n - 5092795082627658 n + 99504806533576984 n 12 11 - 1485906686610419372 n + 16997230413970203969 n 10 9 - 147198913530206316207 n + 933638471653784932386 n 8 7 - 3993206559423046589988 n + 8514753452950171080832 n 6 5 + 15668948985737667496624 n - 188377624107366714733152 n 4 3 + 673393055421348049098816 n - 1295951199176663861022720 n 2 + 1365753102211243328486400 n - 689736165164743302144000 n + 110928229203499868160000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), 111435 (4915751 n - 980769250 n 18 17 16 + 91176091905 n - 5241046924050 n + 208374392757366 n 15 14 13 - 6068759792148180 n + 133650740199112610 n - 2262078978121342900 n 12 11 + 29551601533878371571 n - 295662015729158488410 n 10 9 + 2209290033360064723965 n - 11591331963863321785050 n 8 7 + 35091153645271557592256 n + 11036883803898377441360 n 6 5 - 705121705492654802528880 n + 3713466877366129267356000 n 4 3 - 10621428900434005449826944 n + 18224610280659824373204480 n 2 - 18034415764833545734809600 n + 8929760737798335891456000 n - 1504765543977911255040000)/(524288 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 20 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 111435 (4894587 n 19 18 17 - 974144918 n + 90220727781 n - 5156914373262 n 16 15 14 + 203320297494006 n - 5849611946407356 n + 126572656733245482 n 13 12 - 2088829647650761964 n + 26312848724718038535 n 11 10 - 249397577971142002974 n + 1707267605613464906193 n 9 8 - 7503770426162666606886 n + 10609399642733536554264 n 7 6 + 115614070122910891762288 n - 1007647552830736463884656 n 5 4 + 4246678523548698236798112 n - 11023011641668122755995392 n 3 2 + 17943204193625596333800960 n - 17305320153995114469580800 n + 8569243992886961319936000 n - 1504765543977911255040000)/(524288 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 21 20 19 (2 n - 5)), 7429 (291288979 n - 63703053729 n + 6497749463555 n 18 17 16 - 410038674025155 n + 17896485726998004 n - 571792969167240114 n 15 14 + 13793656830441034870 n - 255095853344167510710 n 13 12 + 3626591114776227973739 n - 39190988218592728445109 n 11 10 + 310861525655152558712535 n - 1636468587271139924062215 n 9 8 + 3418903331951939276988094 n + 27760902514570359509945976 n 7 6 - 331102570768142579936034400 n + 1827205128800762054633521680 n 5 4 - 6411030641395810133328678816 n + 14965395667149391368623582976 n 3 2 - 22772426546772283542053506560 n + 21101405925022185992605286400 n - 10305269939979577235312640000 n + 1850861619092830843699200000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 21 20 (2 n - 35) (2 n - 29)), 323 (3301989973 n - 717804632664 n 19 18 17 + 72617301771035 n - 4531752737481480 n + 194856214261859598 n 16 15 - 6102458538311341584 n + 143323499115483305830 n 14 13 - 2555795096078926559760 n + 34511346316258058658593 n 12 11 - 344583978127212565715064 n + 2364017203167069561431775 n 10 9 - 8192351748825706799249640 n - 31593066838401721819880372 n 8 7 + 672294147847243877745661536 n - 5079475719001640200423565680 n 6 5 + 24151691569170409461752868480 n - 78435848354450141842859097792 n 4 3 + 174775781946578109821062027776 n - 258623628855369446657601192960 n 2 + 236489826478831958557176422400 n - 115716960053446896651386880000 n + 21284908619567554702540800000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 323 ( 22 21 20 25792637579 n - 6111552313273 n + 675022404253813 n 19 18 - 46076110553115335 n + 2171649807766568514 n 17 16 - 74742253555525113558 n + 1935146648863285358138 n 15 14 - 38180960569776332625790 n + 572712218531179798014559 n 13 12 - 6372488009231830252179413 n + 48575548273359969901909233 n 11 10 - 175893579538686227596973595 n - 1152379877473507663129249276 n 9 8 + 24652475158071387803060427092 n - 218844961397493148119216831632 n 7 6 + 1273502188950309329329767156400 n - 5251936779125225409650086848576 n 5 + 15537608153449796116037352169152 n 4 - 32436349518495856795799977239552 n 3 + 45870009575768207521422148408320 n 2 - 40760787769223444184227335372800 n + 19709576085962484277257461760000 n - 3661004282565619408837017600000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 323 ( 22 21 20 24838308049 n - 5817618818033 n + 633157876431773 n 19 18 - 42417121428119515 n + 1952029177157760624 n 17 16 - 65141487067287014538 n + 1618759115637592179658 n 15 14 - 30153585841790969589350 n + 413899670261016689574209 n 13 12 - 3905862025945929914677813 n + 18435142148014969358731113 n 11 10 + 113248376861105150928328065 n - 3315912044911450269453932786 n 9 8 + 37130451331461005257356323872 n - 273241622509783818546341785792 n 7 6 + 1447048571764232471183728982480 n - 5634075597535341125932537147296 n 5 + 16045415382643254936823913776512 n 4 - 32654281704224908892092404106752 n 3 + 45454568587990296341573470648320 n 2 - 40104126272269266947733572812800 n + 19430893868940261645975797760000 n - 3661004282565619408837017600000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 7429 ( 23 22 21 4079641487 n - 1032132610354 n + 121460592173734 n 20 19 18 - 8807662373015660 n + 439172044756073937 n - 15889275467353428294 n 17 16 + 427855431560988262964 n - 8604010116712646794840 n 15 14 + 125793400775740905725617 n - 1199004212475455389266494 n 13 12 + 3462176276626924858801254 n + 111275165279321333069887380 n 11 10 - 2436882349512746199683248993 n + 29102558347653673481582203286 n 9 8 - 244163422139189481158499506336 n + 1528726633509934583540421852400 n 7 6 - 7262204230656101502002018532048 n + 26118167178204517813190649557856 n 5 - 70053118917351525813538914431616 n 4 + 136236118486861706929170930270720 n 3 - 183421084691743584833809762560000 n 2 + 158256151941309536333270790144000 n - 75829389440195043364032921600000 n + 14325668931778510730231808000000)/( 4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 7429 ( 23 22 21 20 468786271 n - 116254212683 n + 13346687431781 n - 938209665721405 n 19 18 + 44937603835699806 n - 1539565977288754338 n 17 16 + 38272714823600880106 n - 673192397322643996490 n 15 14 + 7339713481245149368331 n - 11012555586554808395263 n 13 12 - 1402570730421007812926079 n + 33275234356568838469501575 n 11 10 - 467197775728166449032400664 n + 4715755388041756227314763772 n 9 8 - 36091668618015321258581142784 n + 213077522885322005620950603440 n 7 6 - 971805726322367013573014031744 n + 3394025758612455466972795912512 n 5 4 - 8912517413101139848622058273024 n + 17078834322576836285166225722880 n 3 - 22783533364398943162503355392000 n 2 + 19580360485212089902562098176000 n - 9397132484095730415231590400000 n + 1790708616472313841278976000000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) 24 23 22 (2 n - 45)), 37145 (2656242979 n - 703079929980 n + 86070667820578 n 21 20 19 - 6437274667469976 n + 326457326664107485 n - 11715331241239225908 n 18 17 + 297173154238317164632 n - 4923993491715038371488 n 16 15 + 31104270730579724138461 n + 966699692910441984194844 n 14 13 - 38899354510943041059380534 n + 800414796957326414481825480 n 12 11 - 11618542526241028401721297613 n + 128473253042264782148197923156 n 10 9 - 1112881612399612477275967776740 n + 7627305521450234513041489076208 n 8 - 41373378532086052046223700617392 n 7 + 176451452481823626929847835389888 n 6 - 583923975501113924528576682783936 n 5 + 1468393459736091772022029768939776 n 4 - 2719036528407520906587527220433920 n 3 + 3533415998538224106627356763648000 n 2 - 2980756978809790861287538089984000 n + 1415250752856344804640470630400000 n - 269322575917436001728357990400000 )/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 2185 ( 24 23 22 37871654411 n - 9657748796580 n + 1126410878839538 n 21 20 - 78825101512511112 n + 3618914635035984053 n 19 18 - 109331105986500246156 n + 1855064253876634306616 n 17 16 + 5389323060801488887584 n - 1445954536310727424323595 n 15 14 + 51583010369001125761577028 n - 1166551572210623699278680166 n 13 12 + 19484869536867082383311279640 n - 252784844452513706932856236453 n 11 + 2602178767308420391477373625132 n 10 - 21442138996631789576226467424868 n 9 + 141640133034402698593650339416016 n 8 - 747088866683763506201609698995376 n 7 + 3118282420203300019790168132352576 n 6 - 10150159457137348297307139975933120 n 5 + 25211230566634967984452129368367872 n 4 - 46278237188717037649406597002583040 n 3 + 59814767467785771830835068226048000 n 2 - 50348480200272733935158256734208000 n + 23933017841096563553077832908800000 n - 4578483790596412029382085836800000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 2185 (116784182261 n 24 23 22 - 30855911206825 n + 3672770225595050 n - 254201937604883350 n 21 20 + 10669063318416051395 n - 216865747822951959655 n 19 18 - 3941878724381484333700 n + 507883465260356467799000 n 17 16 - 22510540004441937120996565 n + 666000158042234081939611145 n 15 14 - 14807499630980169555880268950 n + 258775038056417783140301260850 n 13 - 3631975277118814843263273620035 n 12 + 41386344568976614948817209937495 n 11 - 384673150920252904763842141025200 n 10 + 2917262418228426501823535542318700 n 9 - 17990300520988713541134950202037120 n 8 + 89586941776088859082613320204914320 n 7 - 356297359508231247664936945683283200 n 6 + 1113776572719506750196270995970100800 n 5 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(n + 1) (2 n - 63)! + 87267816235361454 ( (n + 1) (2 n - 62)! - 1/87267816235361454 binomial(2 n, n) (n - 32)! (n - 29)!) (n - 30)!)/( (n - 32)! (n - 30)! (n - 29)! binomial(2 n, n)), ( (58178544156907636 n + 58178544156907636) (2 n - 63)! - (n - 32)! (n - 30)! binomial(2 n, n))/((n - 32)! (n - 30)! binomial(2 n, n))] and in Maple notation [31/32768*(69273665*n^16-8867028856*n^15+520245161780*n^14-18543167255200*n^13+ 448605686915750*n^12-7794971282814352*n^11+100392924751791100*n^10-\ 974856954895023200*n^9+7188554796833343505*n^8-40214447055770115128*n^7+ 169086817294165967560*n^6-523215986364809192000*n^5+1136893363874604760080*n^4-\ 1496732133430766945664*n^3+208054458971597986560*n^2+3785312479249240934400*n-\ 6490982557815275520000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 31/32768*(69273649*n^16-8867023112*n^15+520244202100*n^14-18543068280800 *n^13+448598641387078*n^12-7794604664812784*n^11+100378522582112060*n^10-\ 974422191353696800*n^9+7178392542996772097*n^8-40030696905914410696*n^7+ 166541694129696583880*n^6-496775427097716894400*n^5+938704330051218364176*n^4-\ 500620978575916449408*n^3-2633849281353403895040*n^2+6008067858754821888000*n-\ 202843204931727360000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-\ 11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2* n-29), 15283/65536*(562057*n^17-81216989*n^16+5408054044*n^15-220091944100*n^14 +6121742598454*n^13-123275023826318*n^12+1857023186778188*n^11-\ 21316877806844620*n^10+188054848018787321*n^9-1274263539960007597*n^8+ 6550614653972344592*n^7-24667035940416174760*n^6+61684629360533817168*n^5-\ 67220650799923206096*n^4-132689268774255325824*n^3+540837360157804980480*n^2-\ 411368235837140736000*n+27155479767736320000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25) /(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 3451/32768*(1244543*n^17-179833576*n^16+ 11974334396*n^15-487283756560*n^14+13551090238826*n^13-272762229000352*n^12+ 4104567337253812*n^11-46996813127366960*n^10+412083687626186719*n^9-\ 2752245380468008568*n^8+13673573801907369688*n^7-47401901561920942400*n^6+ 93829014306651900912*n^5+3991875209180666496*n^4-522681904799972464896*n^3+ 1037443331413139281920*n^2-638422786594796544000*n+60129990914273280000)/(2*n-5 )/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27) /(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 10353/131072*( 3318653*n^18-537590601*n^17+40316708706*n^16-1857528995940*n^15+58829083952646* n^14-1357438563013302*n^13+23589261815299232*n^12-314479348237789020*n^11+ 3240535532067268149*n^10-25720721753705857593*n^9+154338377994038922918*n^8-\ 667804432626109691640*n^7+1839516993130930473152*n^6-1772628185702501919504*n^5 -7400708737511688557856*n^4+30399032220768393177600*n^3-43662483403323563865600 *n^2+24019749107890905600000*n-2806066242666086400000)/(2*n-29)/(2*n-35)/(2*n-\ 23)/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2* n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 357/131072*(96228463*n ^18-15586011009*n^17+1168557247488*n^16-53809854786900*n^15+1702318298534826*n^ 14-39194359773339078*n^13+678221910235879516*n^12-8968171398208436700*n^11+ 90999586313767976559*n^10-701899722334761853617*n^9+3993531315310735022604*n^8-\ 15568030281698311926600*n^7+32982968903093877183952*n^6+15936970428935110222704 *n^5-325028531914851365500608*n^4+906511601939206549651200*n^3-\ 1134955752810627100492800*n^2+600447417403506140160000*n-\ 81375921037316505600000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25)/ (2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/ (2*n-3)/(-1+2*n)/(2*n-5), 33915/262144*(4050221*n^19-730707748*n^18+61265485239 *n^17-3168569619732*n^16+113114332111242*n^15-2953851400358376*n^14+ 58297188721561118*n^13-884786217776658184*n^12+10385375975523376113*n^11-\ 93698137803627218004*n^10+635534285918939275947*n^9-3075797759952086025636*n^8+ 9229081838502272276264*n^7-7083489975160879072672*n^6-71882563529084276756304*n ^5+347594876463252288399552*n^4-750115358834393656131840*n^3+ 834472260090260461900800*n^2-424791137314494010368000*n+63387559544857067520000 )/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-\ 13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2* n)/(2*n-5), 4845/65536*(7081273*n^19-1276622159*n^18+106902436782*n^17-\ 5516881386456*n^16+196223459627646*n^15-5092795082627658*n^14+99504806533576984 *n^13-1485906686610419372*n^12+16997230413970203969*n^11-147198913530206316207* n^10+933638471653784932386*n^9-3993206559423046589988*n^8+ 8514753452950171080832*n^7+15668948985737667496624*n^6-188377624107366714733152 *n^5+673393055421348049098816*n^4-1295951199176663861022720*n^3+ 1365753102211243328486400*n^2-689736165164743302144000*n+ 110928229203499868160000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15) /(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7) /(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 111435/524288*(4915751*n^20-980769250*n^19+ 91176091905*n^18-5241046924050*n^17+208374392757366*n^16-6068759792148180*n^15+ 133650740199112610*n^14-2262078978121342900*n^13+29551601533878371571*n^12-\ 295662015729158488410*n^11+2209290033360064723965*n^10-11591331963863321785050* n^9+35091153645271557592256*n^8+11036883803898377441360*n^7-\ 705121705492654802528880*n^6+3713466877366129267356000*n^5-\ 10621428900434005449826944*n^4+18224610280659824373204480*n^3-\ 18034415764833545734809600*n^2+8929760737798335891456000*n-\ 1504765543977911255040000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37 )/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-\ 9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 111435/524288*(4894587*n^20-\ 974144918*n^19+90220727781*n^18-5156914373262*n^17+203320297494006*n^16-\ 5849611946407356*n^15+126572656733245482*n^14-2088829647650761964*n^13+ 26312848724718038535*n^12-249397577971142002974*n^11+1707267605613464906193*n^ 10-7503770426162666606886*n^9+10609399642733536554264*n^8+ 115614070122910891762288*n^7-1007647552830736463884656*n^6+ 4246678523548698236798112*n^5-11023011641668122755995392*n^4+ 17943204193625596333800960*n^3-17305320153995114469580800*n^2+ 8569243992886961319936000*n-1504765543977911255040000)/(2*n-29)/(2*n-35)/(2*n-\ 39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2* n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 7429 /1048576*(291288979*n^21-63703053729*n^20+6497749463555*n^19-410038674025155*n^ 18+17896485726998004*n^17-571792969167240114*n^16+13793656830441034870*n^15-\ 255095853344167510710*n^14+3626591114776227973739*n^13-39190988218592728445109* n^12+310861525655152558712535*n^11-1636468587271139924062215*n^10+ 3418903331951939276988094*n^9+27760902514570359509945976*n^8-\ 331102570768142579936034400*n^7+1827205128800762054633521680*n^6-\ 6411030641395810133328678816*n^5+14965395667149391368623582976*n^4-\ 22772426546772283542053506560*n^3+21101405925022185992605286400*n^2-\ 10305269939979577235312640000*n+1850861619092830843699200000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35) /(2*n-29), 323/524288*(3301989973*n^21-717804632664*n^20+72617301771035*n^19-\ 4531752737481480*n^18+194856214261859598*n^17-6102458538311341584*n^16+ 143323499115483305830*n^15-2555795096078926559760*n^14+34511346316258058658593* n^13-344583978127212565715064*n^12+2364017203167069561431775*n^11-\ 8192351748825706799249640*n^10-31593066838401721819880372*n^9+ 672294147847243877745661536*n^8-5079475719001640200423565680*n^7+ 24151691569170409461752868480*n^6-78435848354450141842859097792*n^5+ 174775781946578109821062027776*n^4-258623628855369446657601192960*n^3+ 236489826478831958557176422400*n^2-115716960053446896651386880000*n+ 21284908619567554702540800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 323/2097152*( 25792637579*n^22-6111552313273*n^21+675022404253813*n^20-46076110553115335*n^19 +2171649807766568514*n^18-74742253555525113558*n^17+1935146648863285358138*n^16 -38180960569776332625790*n^15+572712218531179798014559*n^14-\ 6372488009231830252179413*n^13+48575548273359969901909233*n^12-\ 175893579538686227596973595*n^11-1152379877473507663129249276*n^10+ 24652475158071387803060427092*n^9-218844961397493148119216831632*n^8+ 1273502188950309329329767156400*n^7-5251936779125225409650086848576*n^6+ 15537608153449796116037352169152*n^5-32436349518495856795799977239552*n^4+ 45870009575768207521422148408320*n^3-40760787769223444184227335372800*n^2+ 19709576085962484277257461760000*n-3661004282565619408837017600000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2 *n-39)/(2*n-35)/(2*n-29), 323/2097152*(24838308049*n^22-5817618818033*n^21+ 633157876431773*n^20-42417121428119515*n^19+1952029177157760624*n^18-\ 65141487067287014538*n^17+1618759115637592179658*n^16-30153585841790969589350*n ^15+413899670261016689574209*n^14-3905862025945929914677813*n^13+ 18435142148014969358731113*n^12+113248376861105150928328065*n^11-\ 3315912044911450269453932786*n^10+37130451331461005257356323872*n^9-\ 273241622509783818546341785792*n^8+1447048571764232471183728982480*n^7-\ 5634075597535341125932537147296*n^6+16045415382643254936823913776512*n^5-\ 32654281704224908892092404106752*n^4+45454568587990296341573470648320*n^3-\ 40104126272269266947733572812800*n^2+19430893868940261645975797760000*n-\ 3661004282565619408837017600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 7429/ 4194304*(4079641487*n^23-1032132610354*n^22+121460592173734*n^21-\ 8807662373015660*n^20+439172044756073937*n^19-15889275467353428294*n^18+ 427855431560988262964*n^17-8604010116712646794840*n^16+125793400775740905725617 *n^15-1199004212475455389266494*n^14+3462176276626924858801254*n^13+ 111275165279321333069887380*n^12-2436882349512746199683248993*n^11+ 29102558347653673481582203286*n^10-244163422139189481158499506336*n^9+ 1528726633509934583540421852400*n^8-7262204230656101502002018532048*n^7+ 26118167178204517813190649557856*n^6-70053118917351525813538914431616*n^5+ 136236118486861706929170930270720*n^4-183421084691743584833809762560000*n^3+ 158256151941309536333270790144000*n^2-75829389440195043364032921600000*n+ 14325668931778510730231808000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2* n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 7429/524288*(468786271*n^23-116254212683*n^22+13346687431781*n^21-\ 938209665721405*n^20+44937603835699806*n^19-1539565977288754338*n^18+ 38272714823600880106*n^17-673192397322643996490*n^16+7339713481245149368331*n^ 15-11012555586554808395263*n^14-1402570730421007812926079*n^13+ 33275234356568838469501575*n^12-467197775728166449032400664*n^11+ 4715755388041756227314763772*n^10-36091668618015321258581142784*n^9+ 213077522885322005620950603440*n^8-971805726322367013573014031744*n^7+ 3394025758612455466972795912512*n^6-8912517413101139848622058273024*n^5+ 17078834322576836285166225722880*n^4-22783533364398943162503355392000*n^3+ 19580360485212089902562098176000*n^2-9397132484095730415231590400000*n+ 1790708616472313841278976000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 37145/8388608*(2656242979*n^24-703079929980*n^23+86070667820578*n^22-\ 6437274667469976*n^21+326457326664107485*n^20-11715331241239225908*n^19+ 297173154238317164632*n^18-4923993491715038371488*n^17+31104270730579724138461* n^16+966699692910441984194844*n^15-38899354510943041059380534*n^14+ 800414796957326414481825480*n^13-11618542526241028401721297613*n^12+ 128473253042264782148197923156*n^11-1112881612399612477275967776740*n^10+ 7627305521450234513041489076208*n^9-41373378532086052046223700617392*n^8+ 176451452481823626929847835389888*n^7-583923975501113924528576682783936*n^6+ 1468393459736091772022029768939776*n^5-2719036528407520906587527220433920*n^4+ 3533415998538224106627356763648000*n^3-2980756978809790861287538089984000*n^2+ 1415250752856344804640470630400000*n-269322575917436001728357990400000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/ (2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17 )/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 2185/8388608*(37871654411*n^24-\ 9657748796580*n^23+1126410878839538*n^22-78825101512511112*n^21+ 3618914635035984053*n^20-109331105986500246156*n^19+1855064253876634306616*n^18 +5389323060801488887584*n^17-1445954536310727424323595*n^16+ 51583010369001125761577028*n^15-1166551572210623699278680166*n^14+ 19484869536867082383311279640*n^13-252784844452513706932856236453*n^12+ 2602178767308420391477373625132*n^11-21442138996631789576226467424868*n^10+ 141640133034402698593650339416016*n^9-747088866683763506201609698995376*n^8+ 3118282420203300019790168132352576*n^7-10150159457137348297307139975933120*n^6+ 25211230566634967984452129368367872*n^5-46278237188717037649406597002583040*n^4 +59814767467785771830835068226048000*n^3-50348480200272733935158256734208000*n^ 2+23933017841096563553077832908800000*n-4578483790596412029382085836800000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 2185/16777216*(116784182261 *n^25-30855911206825*n^24+3672770225595050*n^23-254201937604883350*n^22+ 10669063318416051395*n^21-216865747822951959655*n^20-3941878724381484333700*n^ 19+507883465260356467799000*n^18-22510540004441937120996565*n^17+ 666000158042234081939611145*n^16-14807499630980169555880268950*n^15+ 258775038056417783140301260850*n^14-3631975277118814843263273620035*n^13+ 41386344568976614948817209937495*n^12-384673150920252904763842141025200*n^11+ 2917262418228426501823535542318700*n^10-17990300520988713541134950202037120*n^9 +89586941776088859082613320204914320*n^8-356297359508231247664936945683283200*n ^7+1113776572719506750196270995970100800*n^6-\ 2675018134406483566456443235556679936*n^5+4777482568411873431009433666730603520 *n^4-6042388349711316252708422841705984000*n^3+ 5004556106319576546306618710421504000*n^2-2354127012387798819174069844377600000 *n+448691411478448378879444412006400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-45), 115/8388608*(733506681497*n^25-171701551320490*n^24+ 16535521784707430*n^23-688346566288663060*n^22-11985840409178439265*n^21+ 3460319869991440875650*n^20-244513747885677631379920*n^19+ 10952009753732601407099840*n^18-359202586229365252478392585*n^17+ 9121042603499772700482815450*n^16-184478742818015671082051935570*n^15+ 3020439258327207468931162645340*n^14-40407378287835576461674595137375*n^13+ 443798878676710592847029699063150*n^12-4006868087688378439503847116054820*n^11+ 29686970104070093499082903986730040*n^10-179655948890631325687017234538127920*n ^9+881103762353908791055076045583687200*n^8-\ 3461685675731439863077419301367421120*n^7+ 10717711049039410284607289642231859840*n^6-\ 25554666154261724826113978184722084352*n^5+ 45405108897768572975236696149003479040*n^4-\ 57245615180981170177783069807727616000*n^3+ 47355372917882299014828239351881728000*n^2-\ 22293319606495653361184862407884800000*n+4262568409045259599354721914060800000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 115/33554432*( 2581237072129*n^26-402443388061481*n^25-12280356247800680*n^24+ 9103829716107038770*n^23-1218109507517375502965*n^22+96229785412315225856185*n^ 21-5341138789870447013334170*n^20+223651646265757523003338720*n^19-\ 7342859568556910194415695505*n^18+193547894302247863938226383145*n^17-\ 4159227555527000425693515698420*n^16+73605379520609455339131642082570*n^15-\ 1079497846973418748674769077145235*n^14+13165194899330361524308971377665015*n^ 13-133639034321342279883376788931692170*n^12+ 1127615511692128455938097667636574020*n^11-\ 7880363057584390400871287086823265800*n^10+ 45337872614392852021173834829794353200*n^9-\ 212836155793280236334547806885718100320*n^8+ 805217570596143755052772018540984355520*n^7-\ 2413625515823064973981623828294312618624*n^6+ 5598904874453362977195257931735893223936*n^5-\ 9722227772220039749067188587249700874240*n^4+ 12030586627167529267770152599486827110400*n^3-\ 9808651197442636493709221400414265344000*n^2+ 4570782156779592439656659519827476480000*n-\ 869563955445232958268363270468403200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-51)/(2*n-45), -575/33554432*(172418874685*n^26-\ 149525862950711*n^25+38566288119084934*n^24-5364890586849664442*n^23+ 487621797869121179251*n^22-31778458276104682126985*n^21+ 1566872272143920239600960*n^20-60482509465317209075443352*n^19+ 1870285006957256077328585731*n^18-47073524579119766150204400905*n^17+ 975067100519043674678603334670*n^16-16746563146622151332538443953922*n^15+ 239592625523718717558816133887901*n^14-2862031920384020817643660541420615*n^13+ 28549710986041356835060259828676940*n^12-237375900124099001770612158982067852*n ^11+1638469571193893828846818226009578576*n^10-\ 9329234399723212645127353701817572560*n^9+ 43420672082427881125439446251983937600*n^8-\ 163126243380248858788208751074512398912*n^7+ 486260911302827850833277226210207289856*n^6-\ 1123233504174623987543195199515540404224*n^5+ 1944638983965696993975810366427962064896*n^4-\ 2402016395242435417690797453908859371520*n^3+ 1957083318204372845604050875425275904000*n^2-\ 912448359300579637477590476112789504000*n+ 173912791089046591653672654093680640000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-51)/(2*n-45), -1725/67108864*(1151591664299*n^27-\ 531741691150344*n^26+111191250182068719*n^25-14225848720097975064*n^24+ 1261155365308960682067*n^23-82905188635540231624872*n^22+ 4214845588519844177111355*n^21-170436184828720324816827096*n^20+ 5590057590357119459651022237*n^19-150795199026319774876180720632*n^18+ 3378922858566247425008172623325*n^17-63325710348961401487635554464776*n^16+ 997096723088262414314465353474257*n^15-13222337181143733986126610625792632*n^14 +147759425213820574191448727237239305*n^13-\ 1389889388634885205754829365164613256*n^12+ 10972690841625567205509354571243292052*n^11-\ 72351993713391025387510451804361169632*n^10+ 395673436779296445109434896466279300400*n^9-\ 1777330860056627288307819536117566659456*n^8+ 6472623906627278165523505319820010746048*n^7-\ 18778129717017526429517460129547757541888*n^6+ 42373110863640421202610088153127251408896*n^5-\ 71912955000668370649908879436659209060352*n^4+ 87364698651358880672967868266087439319040*n^3-\ 70240201712689202298684228722741821440000*n^2+ 32426004629399330992853348403770032128000*n-\ 6144918618479646238429767111310049280000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -75/16777216*(11711948865781*n^27-\ 4828938009143967*n^26+930275421332212332*n^25-111797500221853094610*n^24+ 9431364722011839326655*n^23-595434056596791340703697*n^22+ 29270910202611835099805034*n^21-1150502944880663690380322544*n^20+ 36830496845870151016638017115*n^19-972969731252778922726930071777*n^18+ 21410016136259072501147277880584*n^17-394970693138232864455156479725834*n^16+ 6133994372401736763119733902273385*n^15-80370717205455568378408671166891167*n^ 14+888792482588090997036106291315154274*n^13-\ 8284756290859997140112195591671483284*n^12+ 64894107155623787583159538810793028840*n^11-\ 425033281612733328130329509224123529712*n^10+ 2311183395274528253017982327426446951904*n^9-\ 10332341901041182872524686128049481824704*n^8+ 37481741858809251854132967415816195503744*n^7-\ 108404845314613409198484286221338587399680*n^6+ 244043642365273902997132287515894346319872*n^5-\ 413493093592511890980171479383619723649024*n^4+ 501843099098817127030030980019883544084480*n^3-\ 403334962028516660480070569518169456640000*n^2+ 186253133623427130424297847412580417536000*n-\ 35333282056257965870971160890032783360000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -75/134217728*(262048868194735*n^ 28-110039609593875542*n^27+21847629459492910143*n^26-2731936653240134779530*n^ 25+241741617844485572250255*n^24-16121430734255512655549742*n^23+ 842490215430936636892970499*n^22-35412642721743081118294844874*n^21+ 1219274690863334621966235770985*n^20-34838648719006645486951345890522*n^19+ 833894222881478793272292001500189*n^18-16831643779803737889510694608179214*n^17 +287761351139195674859558104680643125*n^16-\ 4177864479806150902755734089997413962*n^15+ 51559482011614694513301191898885949809*n^14-\ 540561267618640221701119247924576105214*n^13+ 4804521710577340785334044710435507378340*n^12-\ 36067242482154285229784709265890091747032*n^11+ 227429481393764668706759362559661607461104*n^10-\ 1195526668858672916585473904187736000986784*n^9+ 5186173701713596240105981493095821885328320*n^8-\ 18317993219735712477515011918952280960912000*n^7+ 51747611374673366883637893980693651582432256*n^6-\ 114123501886688700613153250927292739240464384*n^5+ 189956995654748144800580500458595311329034240*n^4-\ 227093714505981073800523703348422923600691200*n^3+ 180268313170178428147256183600864772980736000*n^2-\ 82453341833954993488792784462440523366400000*n+ 15546644104753504983227310791614424678400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -3/134217728*( 8177100207432691*n^28-3344435893282863890*n^27+648782044120628528859*n^26-\ 79475591261414375224470*n^25+6905001919531883470843995*n^24-\ 453016123737730843673687850*n^23+23330018362937049205399383855*n^22-\ 967842678301933769816610879750*n^21+32932823129931268724115219769485*n^20-\ 931096637542762867191421986529950*n^19+22076235824586805346203174851350865*n^18 -441827212657937155147483462214824050*n^17+ 7496595213617767874876262377463316665*n^16-\ 108106708455640479781435815119573109150*n^15+ 1326189689830792302013515785470084745685*n^14-\ 13830796089956279609841934836612493189650*n^13+ 122360370725806560633780891079323073212780*n^12-\ 914859130977513197063896432892281735221000*n^11+ 5748862901357327463466655952989236634226320*n^10-\ 30131042273585873955142441665655960943807200*n^9+ 130386128850452502160860675629915952199320384*n^8-\ 459607016239946030152270644723658668709388160*n^7+ 1296298110905238414574134432552862360224124416*n^6-\ 2855384156621534176719198278443579175146874880*n^5+ 4748732931572547435642327957063600293551104000*n^4-\ 5674266302847993586120078061040178991923200000*n^3+ 4503494765503693156585544912401689075056640000*n^2-\ 2060199936382903227481459286815791199027200000*n+ 388666102618837624580682769790360616960000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -87/268435456*( 1312242750056033*n^29-565164673703297359*n^28+115882689469143290127*n^27-\ 15059174124369127883571*n^26+1392939484170742173708315*n^25-\ 97640771733128188605095055*n^24+5391929873069090020824900915*n^23-\ 240736455938164194736825535295*n^22+8849602963048643762332360499505*n^21-\ 271375928854009332118978795648065*n^20+7008164246657156924187726951425445*n^19-\ 153453903821825138837275267562522985*n^18+2862409652133709912154674648334731545 *n^17-45619098414886218705604385494732938885*n^16+ 622073415414517025174658930352286101905*n^15-\ 7258159425075983913191443361544273068565*n^14+ 72363534820378955518766161820813444289690*n^13-\ 614800703345705692329379328893621119949020*n^12+ 4432311492011582147414830354996959062975560*n^11-\ 26954321185500446694151237647644415638284880*n^10+ 137174423158627003096042756897183285020420192*n^9-\ 578125902501653781186481760692593595067543616*n^8+ 1990336490703323070285533049173281970764626048*n^7-\ 5496996250897766306765129114567033549428744704*n^6+ 11885839648736682584233046578562573471038494720*n^5-\ 19449219041528485872988425378998593529238528000*n^4+ 22918171033939522818328443290260893196615680000*n^3-\ 17978685872560812828255030530431877135400960000*n^2+ 8149241007277334928419523803452264336588800000*n-\ 1527859851674051351799925370900038287360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -87/ 134217728*(726844112090109*n^29-309810590620554292*n^28+62921569155747136821*n^ 27-8105499637151763381948*n^26+743740939412713034100945*n^25-\ 51751414867070227285316340*n^24+2838620471488558715855857545*n^23-\ 125958939474636009418147502460*n^22+4604367123959425342031189292615*n^21-\ 140473970212255490290140384944220*n^20+3610871843379210984804433626612735*n^19-\ 78733668489353864621725793458572180*n^18+1463079935759766326031227453239123035* n^17-23238280632390411003174762904050652380*n^16+ 315919431174430044712630924159527153315*n^15-\ 3676077986229942634576098304941297479220*n^14+ 36562675023380182004409797989144976679120*n^13-\ 309984433353181605290703984918889133369760*n^12+ 2230703898318673426539958795393004186870880*n^11-\ 13544309054210839167466231217171468279805440*n^10+ 68836920750797397215966007395578866201666816*n^9-\ 289791991678272441935144840615505389336299008*n^8+ 996770469798813629946392333946531839820328704*n^7-\ 2750942589919185439705555051859371600872778752*n^6+ 5944978428684818140214297394313981997845647360*n^5-\ 9724306008802438609700688469329614989092864000*n^4+ 11456180250623367679978800919576089969623040000*n^3-\ 8986403163372305333600779116556001046036480000*n^2+ 4073600298075316735475921139592785454694400000*n-\ 763929925837025675899962685450019143680000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -29/ 536870912*(18636441972066461*n^30-8440047135115580475*n^29+ 1825349667077528071945*n^28-250971464021096428700475*n^27+ 24637747869669378101061789*n^26-1838729606979601823865327495*n^25+ 108456285912772049131895514525*n^24-5189525146761488892599926008375*n^23+ 205161032795408732955462293830515*n^22-6790646647950332720142974718359025*n^21+ 190014302682109974534937871906490075*n^20-4526737225624318901540162631345248625 *n^19+92274194235328561596593250520758991455*n^18-\ 1614776921792660662586660231516355461325*n^17+ 24304984661756526671083533944574570326175*n^16-\ 314829113497148300680956099025391137984125*n^15+ 3507187100818124864950991051386789457602740*n^14-\ 33536506916393509414757840873407159342804800*n^13+ 274383678740581988873184872714847174412381600*n^12-\ 1911909955846148886766099816652592828316140000*n^11+ 11275096502971307238860113741899867115071680384*n^10-\ 55813281193064691116432950685288168917435200000*n^9+ 229441758314269401537908705400551591302050343680*n^8-\ 772479553548671685789784359603926033999814278400*n^7+ 2091452911994680595607314997287482182453152326656*n^6-\ 4443325212238889492676301235046822387827987066880*n^5+ 7159574980800274483609519807912908175905871872000*n^4-\ 8325326400454580804196691799636838757260656640000*n^3+ 6458834188601609849119828048440386558839357440000*n^2-\ 2901996187733361431259205062964620030325555200000*n+ 540862387492614178537173581298613553725440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2 *n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), (58178544156907636*(n-29)!*(n+1)*(2*n-63)!+87267816235361454*((n+1)*(2*n-62)!-1 /87267816235361454*binomial(2*n,n)*(n-32)!*(n-29)!)*(n-30)!)/(n-32)!/(n-30)!/(n -29)!/binomial(2*n,n), ((58178544156907636*n+58178544156907636)*(2*n-63)!-(n-32 )!*(n-30)!*binomial(2*n,n))/(n-32)!/(n-30)!/binomial(2*n,n)] The limits, as n goes to infinity are 2147483615 2147483119 8589917131 4294917893 34358014509 34353561291 [----------, ----------, ----------, ----------, -----------, -----------, 2147483648 2147483648 8589934592 4294967296 34359738368 34359738368 137363245215 34308767685 547786712685 545428302345 2163985824991 ------------, -----------, ------------, ------------, -------------, 137438953472 34359738368 549755813888 549755813888 2199023255552 1066542761279 8331021938017 8022773499827 30307656606923 3482613207259 -------------, -------------, -------------, --------------, -------------, 1099511627776 8796093022208 8796093022208 35184372088832 4398046511104 98666145454955 82749564888035 255173438240285 84353268372155 ---------------, ---------------, ---------------, ---------------, 140737488355328 140737488355328 562949953421312 281474976710656 296842263294835 -99140852943875 -1986495620915775 -878396164933575 ----------------, ----------------, -----------------, ----------------, 2251799813685248 2251799813685248 9007199254740992 2251799813685248 -19653665114605125 -24531300622298073 -114165119254874871 ------------------, ------------------, -------------------, 36028797018963968 36028797018963968 144115188075855872 -63235437751839483 -540456817189927369 -561916116264196579 ------------------, -------------------, -------------------, 72057594037927936 576460752303423488 576460752303423488 -2291298373174467043 --------------------] 2305843009213693952 and in Maple notation [2147483615/2147483648, 2147483119/2147483648, 8589917131/8589934592, 4294917893/4294967296, 34358014509/34359738368, 34353561291/34359738368, 137363245215/137438953472, 34308767685/34359738368, 547786712685/549755813888, 545428302345/549755813888, 2163985824991/2199023255552, 1066542761279/ 1099511627776, 8331021938017/8796093022208, 8022773499827/8796093022208, 30307656606923/35184372088832, 3482613207259/4398046511104, 98666145454955/ 140737488355328, 82749564888035/140737488355328, 255173438240285/ 562949953421312, 84353268372155/281474976710656, 296842263294835/ 2251799813685248, -99140852943875/2251799813685248, -1986495620915775/ 9007199254740992, -878396164933575/2251799813685248, -19653665114605125/ 36028797018963968, -24531300622298073/36028797018963968, -114165119254874871/ 144115188075855872, -63235437751839483/72057594037927936, -540456817189927369/ 576460752303423488, -561916116264196579/576460752303423488, -\ 2291298373174467043/2305843009213693952] and in floating point [.9999999846, .9999997537, .9999979673, .9999884975, .9999498291, .9998202234, .9994491499, .9985165579, .9964182258, .9921283024, .9840668213, .9700149906, .\ 9471275391, .9120837490, .8613954096, .7918545651, .7010651292, .5879710222, .4\ 532790823, .2996830104, .1318244462, -.4402738305e-1, -.2205453177, -.390086258\ 8, -.5454987882, -.6808803694, -.7921796500, -.8775679871, -.9375431285, -.9747\ 690784, -.9936922696] The cut off is at j=, 22 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 33], vs. those in the, 2, -th row from j=1 to j=, 32, are as follws 16 15 14 13 [3 (715827877 n - 91625966760 n + 5375866991620 n - 191612761295200 n 12 11 10 + 4635594446638974 n - 80548158795082160 n + 1037398356491734380 n 9 8 - 10073666788429015200 n + 74285120318556740021 n 7 6 - 415610536292909757800 n + 1748078819761204792680 n 5 4 - 5415378712192059083200 n + 11813961104645377986128 n 3 2 - 15798269097069541937280 n + 3097197322814846488320 n + 38373977159073629337600 n - 69169532881719029760000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) 17 16 15 (2 n - 23) (2 n - 29)), 1581 (2716614 n - 392550635 n + 26139237776 n 14 13 12 - 1063809690300 n + 29590697677828 n - 595947668890210 n 11 10 9 + 8980262770053432 n - 103171968459968980 n + 912213326558864902 n 8 7 - 6218276429030941275 n + 32483166849392904008 n 6 5 - 127762104308676732440 n + 362140917414715306656 n 4 3 - 646560705954840968880 n + 276366617071047782784 n 2 + 1896282482797380222720 n - 4018824805914825984000 n + 131251485544058880000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 3689 ( 17 16 15 14 1164262 n - 168235559 n + 11202456064 n - 455910440540 n 13 12 11 + 12681134065684 n - 255373942423418 n + 3847364757472808 n 10 9 8 - 44175012003977140 n + 389935228440317846 n - 2645849374412016487 n 7 6 + 13644708413400113192 n - 51752181540771220600 n 5 4 + 131662429064244866208 n - 153137104109750983536 n 3 2 - 254049983141491880064 n + 1134862788411998657280 n - 876889373575792896000 n + 56250636661739520000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) 18 17 (2 n - 17) (2 n - 23) (2 n - 29)), 962829 (17843 n - 2890541 n 16 15 14 13 + 216799526 n - 9990946740 n + 316571299226 n - 7312098719982 n 12 11 10 + 127341268945072 n - 1705279329173420 n + 17735310815927419 n 9 8 7 - 143441801598508413 n + 893939504734131778 n - 4177761937707616440 n 6 5 + 13673012269880810112 n - 25296285517347562064 n 4 3 - 4431093973449953376 n + 143749829923757337600 n 2 - 273949809942333657600 n + 165182878116748800000 n - 15086377648742400000)/(65536 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 18 17 16 (2 n - 5)), 31059 (1106241 n - 179204567 n + 13440076312 n 15 14 13 - 619285733580 n + 19616662731862 n - 452800057020234 n 12 11 10 + 7874061238110964 n - 105110268530979140 n + 1085745185055312353 n 9 8 - 8655880705262049831 n + 52348261513382210636 n 7 6 - 229706412567212151480 n + 651262536613608149744 n 5 4 - 716845361965008432368 n - 2260378448368070526912 n 3 2 + 10086359659940150611200 n - 14799683517317649331200 n + 8186390942296089600000 n - 935355414222028800000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 19 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 196707 (349312 n 18 17 16 15 - 63043823 n + 5289507633 n - 273910556112 n + 9800435922324 n 14 13 12 - 256957046372586 n + 5107044364180966 n - 78451879920890684 n 11 10 + 939816086560066716 n - 8772169664397526959 n 9 8 + 62968114588556348769 n - 336330993458499723996 n 7 6 + 1237475716268644466248 n - 2459758591451209930832 n 5 4 - 1477151020078808032368 n + 24166884159663265593792 n 3 2 - 65056544942835667449600 n + 79656441040000892851200 n - 41398210022190658560000 n + 5464444788349747200000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 6783 (10127717 n 18 17 16 - 1827462010 n + 153265875333 n - 7930625468700 n 15 14 13 + 283350906427314 n - 7409646821332140 n + 146566152297080786 n 12 11 - 2232410008095776080 n + 26348698043854307121 n 10 9 - 239734633550350657290 n + 1647269124191501979129 n 8 7 - 8144220061132503631140 n + 25571804882796017004248 n 6 5 - 26893652500826665058560 n - 159224654590593489211248 n 4 3 + 847681074994643124649920 n - 1880770437415137788870400 n 2 + 2117131929371847966336000 n - 1079535476966983787520000 n + 158468898862142668800000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), 4845 (56681449 n - 11327848250 n 18 17 16 + 1055908256595 n - 60953594680950 n + 2439380840697834 n 15 14 - 71763569368832820 n + 1604573124497794390 n 13 12 - 27776654540978677100 n + 375115979234785339629 n 11 10 - 3941838019187948541090 n + 31751914803213623696535 n 9 8 - 189225742418956763209950 n + 766565923452155392580944 n 7 6 - 1552209814183396802369360 n - 2859502418049974191527120 n 5 4 + 32691213426086124600564000 n - 113649581198618012009609856 n 3 2 + 214016596083221223543851520 n - 221511427723219778110310400 n + 110007929184992846751744000 n - 17304803755745979432960000)/(262144 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), 4845 (113206573 n - 22602858650 n + 2103563960115 n 17 16 15 - 121117710281250 n + 4827070773851418 n - 141075665076731940 n 14 13 + 3122226684574138630 n - 53209061931817574900 n 12 11 + 701769240796204816833 n - 7115665710120757634130 n 10 9 + 54236550956462851045095 n - 294470372925906322725450 n 8 7 + 974017583858550513517288 n - 459177942867240825582320 n 6 5 - 14155122994480503676193040 n + 81774204228176184721173600 n 4 3 - 241554800565204052803962112 n + 421084714321318424488727040 n 2 - 419762669028706310527180800 n + 207842565684666552127488000 n - 34609607511491958865920000)/(524288 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 37145 (29446406 n 20 19 18 - 6472225473 n + 664902619780 n - 42380707939815 n 17 16 15 + 1875568924007646 n - 61073595990366498 n + 1511854747580627000 n 14 13 - 28958217748284514830 n + 432040834061163243826 n 12 11 - 5001879758192922133893 n + 44170203644334892144020 n 10 9 - 285221390715305965151595 n + 1199753987408034099335786 n 8 7 - 1731060821856212735384328 n - 16020518046329962759810160 n 6 5 + 138412524192165855144799440 n - 571083318033909398613965664 n 4 3 + 1452564067127261120007720192 n - 2321954592793701079402160640 n 2 + 2203053892616572830625996800 n - 1073795558237230151651328000 n + 185086161909283084369920000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 7429 ( 21 20 19 18 146403748 n - 32086137741 n + 3282316272410 n - 207935083263495 n 17 16 15 + 9122105904746898 n - 293406970410793386 n + 7139741977674304660 n 14 13 - 133544866630480577790 n + 1927493200303488003368 n 12 11 - 21279817702734519959121 n + 174651244444515721638210 n 10 9 - 986917420695766824371835 n + 2835014446227541440164578 n 8 7 + 8276581909194945733047864 n - 145364044825849059314697040 n 6 5 + 863779824908926331547437520 n - 3130769352786820537940858592 n 4 3 + 7440257646680090778250032384 n - 11437947819933948343661898240 n 2 + 10648740131700349827378585600 n - 5196978529613887689953280000 n + 925430809546415421849600000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 7429 ( 22 21 20 579678971 n - 138911090131 n + 15568525514773 n 19 18 17 - 1082857232717645 n + 52282081213298706 n - 1855913273937668946 n 16 15 + 50013837968988476858 n - 1040539552938735314410 n 14 13 + 16801688053596170725231 n - 209145164006238894785351 n 12 11 + 1957012941286086455466513 n - 12840266310739212510990825 n 10 9 + 45213650076446390828877236 n + 114702497551037193994661564 n 8 7 - 2853055962944292103439402192 n + 21396430937612604663080866960 n 6 5 - 99857263473391000503697345344 n + 318039082177681533463944012864 n 4 3 - 695827241924300085416453885952 n + 1012548957709317340922214865920 n 2 - 911705762195912243247232204800 n + 439547924413569035117813760000 n - 79587049620991726279065600000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 323 ( 22 21 20 26208159049 n - 6239532926033 n + 693250500099773 n 19 18 - 47669258928113515 n + 2267274101476623624 n 17 16 - 78922491595858448538 n + 2072903900172079995658 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4164166487622402721683252661432484113104568320000 n 2 + 3230900796823293479924625016755976271086878720000 n - 1451506209234920981174314131843528800442777600000 n + 270431193746307089268586790649306776862720000000)/(268435456 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 30 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), - 29 (17552638988517511 n 29 28 - 7997855517827608875 n + 1739419263723866022245 n 27 26 - 240385806380750170099575 n + 23709564300357206364397689 n 25 24 - 1777068388837550980786882995 n + 105230828249819721117925073025 n 23 - 5053215319306722451529116657875 n 22 + 200423045871450810129765513556515 n 21 - 6653453218172603111145253512660525 n 20 + 186674603406131794921421139203365575 n 19 - 4457953003815383003326636854680647125 n 18 + 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540862387492614178537173581298613553725440000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), - 31 (34973065913887148 n 30 29 - 16903726144617503539 n + 3907723753178734676935 n 28 27 - 575253893903822546928905 n + 60570310398395593315054227 n 26 25 - 4857591433350719858129424621 n + 308520568825695056915902285455 n 24 - 15930671500856280194673802984725 n 23 + 681250598256479883536100987191895 n 22 - 24453742080406930163567774382523935 n 21 + 744148676492379876201498452455023825 n 20 - 19338825102057866316308021530170030675 n 19 + 431480641823033089450423310708841589065 n 18 - 8295619144050874184957033006027895812895 n 17 + 137748244120614547069214677589245523288325 n 16 - 1977570599606341997749589665645472176837575 n 15 + 24544138299413909055237032109804417270096945 n 14 - 263032241786861192422311889697594399013793410 n 13 + 2428256047408401406786894841436748474676108500 n 12 - 19242068124257667651007305177136056300925147400 n 11 + 130237053881504889513166195473873233995327307312 n 10 - 748006969712660882904725536279550125668492344416 n 9 + 3614857398456148025726021379885581677357923483840 n 8 - 14540012824520803388712093367565576408298239022720 n 7 + 47997910709524935375773809173689929773070600593408 n 6 - 127666495944005367113262015345628714931444937437184 n 5 + 266955852906782614431250033818264120314882835333120 n 4 - 424133092045209032249722937567734583451165954048000 n 3 + 487158477017806698142020857646166749848574689280000 n 2 - 373993439367449180446064290149958726129684316160000 n + 166610784686902214737565116891488330084306124800000 n - 30864050434659176833298711784427334727106560000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), ( 222136259508192792 (n - 30)! (n + 1) (2 n - 65)! + 333204389262289188 (n - 31)! ((n + 1) (2 n - 64)! - 1/333204389262289188 binomial(2 n, n) (n - 33)! (n - 30)!))/((n - 33)! (n - 31)! (n - 30)! binomial(2 n, n)), ( (222136259508192792 n + 222136259508192792) (2 n - 65)! - (n - 33)! (n - 31)! binomial(2 n, n))/((n - 33)! (n - 31)! binomial(2 n, n))] and in Maple notation [3/32768*(715827877*n^16-91625966760*n^15+5375866991620*n^14-191612761295200*n^ 13+4635594446638974*n^12-80548158795082160*n^11+1037398356491734380*n^10-\ 10073666788429015200*n^9+74285120318556740021*n^8-415610536292909757800*n^7+ 1748078819761204792680*n^6-5415378712192059083200*n^5+11813961104645377986128*n ^4-15798269097069541937280*n^3+3097197322814846488320*n^2+ 38373977159073629337600*n-69169532881719029760000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-7)/(2*n-9)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2* n-15)/(2*n-17)/(2*n-23)/(2*n-29), 1581/32768*(2716614*n^17-392550635*n^16+ 26139237776*n^15-1063809690300*n^14+29590697677828*n^13-595947668890210*n^12+ 8980262770053432*n^11-103171968459968980*n^10+912213326558864902*n^9-\ 6218276429030941275*n^8+32483166849392904008*n^7-127762104308676732440*n^6+ 362140917414715306656*n^5-646560705954840968880*n^4+276366617071047782784*n^3+ 1896282482797380222720*n^2-4018824805914825984000*n+131251485544058880000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 3689/32768* (1164262*n^17-168235559*n^16+11202456064*n^15-455910440540*n^14+12681134065684* n^13-255373942423418*n^12+3847364757472808*n^11-44175012003977140*n^10+ 389935228440317846*n^9-2645849374412016487*n^8+13644708413400113192*n^7-\ 51752181540771220600*n^6+131662429064244866208*n^5-153137104109750983536*n^4-\ 254049983141491880064*n^3+1134862788411998657280*n^2-876889373575792896000*n+ 56250636661739520000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33 )/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-\ 23)/(2*n-29), 962829/65536*(17843*n^18-2890541*n^17+216799526*n^16-9990946740*n ^15+316571299226*n^14-7312098719982*n^13+127341268945072*n^12-1705279329173420* n^11+17735310815927419*n^10-143441801598508413*n^9+893939504734131778*n^8-\ 4177761937707616440*n^7+13673012269880810112*n^6-25296285517347562064*n^5-\ 4431093973449953376*n^4+143749829923757337600*n^3-273949809942333657600*n^2+ 165182878116748800000*n-15086377648742400000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-\ 17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 31059/131072*(1106241*n^18-\ 179204567*n^17+13440076312*n^16-619285733580*n^15+19616662731862*n^14-\ 452800057020234*n^13+7874061238110964*n^12-105110268530979140*n^11+ 1085745185055312353*n^10-8655880705262049831*n^9+52348261513382210636*n^8-\ 229706412567212151480*n^7+651262536613608149744*n^6-716845361965008432368*n^5-\ 2260378448368070526912*n^4+10086359659940150611200*n^3-14799683517317649331200* n^2+8186390942296089600000*n-935355414222028800000)/(2*n-29)/(2*n-35)/(2*n-23)/ (2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33 )/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 196707/131072*(349312*n^19 -63043823*n^18+5289507633*n^17-273910556112*n^16+9800435922324*n^15-\ 256957046372586*n^14+5107044364180966*n^13-78451879920890684*n^12+ 939816086560066716*n^11-8772169664397526959*n^10+62968114588556348769*n^9-\ 336330993458499723996*n^8+1237475716268644466248*n^7-2459758591451209930832*n^6 -1477151020078808032368*n^5+24166884159663265593792*n^4-65056544942835667449600 *n^3+79656441040000892851200*n^2-41398210022190658560000*n+ 5464444788349747200000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/( 2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/( 2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 6783/131072*(10127717*n^19-1827462010*n^18+ 153265875333*n^17-7930625468700*n^16+283350906427314*n^15-7409646821332140*n^14 +146566152297080786*n^13-2232410008095776080*n^12+26348698043854307121*n^11-\ 239734633550350657290*n^10+1647269124191501979129*n^9-8144220061132503631140*n^ 8+25571804882796017004248*n^7-26893652500826665058560*n^6-\ 159224654590593489211248*n^5+847681074994643124649920*n^4-\ 1880770437415137788870400*n^3+2117131929371847966336000*n^2-\ 1079535476966983787520000*n+158468898862142668800000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 4845/262144*( 56681449*n^20-11327848250*n^19+1055908256595*n^18-60953594680950*n^17+ 2439380840697834*n^16-71763569368832820*n^15+1604573124497794390*n^14-\ 27776654540978677100*n^13+375115979234785339629*n^12-3941838019187948541090*n^ 11+31751914803213623696535*n^10-189225742418956763209950*n^9+ 766565923452155392580944*n^8-1552209814183396802369360*n^7-\ 2859502418049974191527120*n^6+32691213426086124600564000*n^5-\ 113649581198618012009609856*n^4+214016596083221223543851520*n^3-\ 221511427723219778110310400*n^2+110007929184992846751744000*n-\ 17304803755745979432960000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 4845/524288*(113206573*n^20-\ 22602858650*n^19+2103563960115*n^18-121117710281250*n^17+4827070773851418*n^16-\ 141075665076731940*n^15+3122226684574138630*n^14-53209061931817574900*n^13+ 701769240796204816833*n^12-7115665710120757634130*n^11+54236550956462851045095* n^10-294470372925906322725450*n^9+974017583858550513517288*n^8-\ 459177942867240825582320*n^7-14155122994480503676193040*n^6+ 81774204228176184721173600*n^5-241554800565204052803962112*n^4+ 421084714321318424488727040*n^3-419762669028706310527180800*n^2+ 207842565684666552127488000*n-34609607511491958865920000)/(2*n-29)/(2*n-35)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/( 2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 37145/524288*(29446406*n^21-6472225473*n^20+664902619780*n^19-42380707939815*n^ 18+1875568924007646*n^17-61073595990366498*n^16+1511854747580627000*n^15-\ 28958217748284514830*n^14+432040834061163243826*n^13-5001879758192922133893*n^ 12+44170203644334892144020*n^11-285221390715305965151595*n^10+ 1199753987408034099335786*n^9-1731060821856212735384328*n^8-\ 16020518046329962759810160*n^7+138412524192165855144799440*n^6-\ 571083318033909398613965664*n^5+1452564067127261120007720192*n^4-\ 2321954592793701079402160640*n^3+2203053892616572830625996800*n^2-\ 1073795558237230151651328000*n+185086161909283084369920000)/(2*n-5)/(-1+2*n)/(2 *n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/( 2*n-29), 7429/524288*(146403748*n^21-32086137741*n^20+3282316272410*n^19-\ 207935083263495*n^18+9122105904746898*n^17-293406970410793386*n^16+ 7139741977674304660*n^15-133544866630480577790*n^14+1927493200303488003368*n^13 -21279817702734519959121*n^12+174651244444515721638210*n^11-\ 986917420695766824371835*n^10+2835014446227541440164578*n^9+ 8276581909194945733047864*n^8-145364044825849059314697040*n^7+ 863779824908926331547437520*n^6-3130769352786820537940858592*n^5+ 7440257646680090778250032384*n^4-11437947819933948343661898240*n^3+ 10648740131700349827378585600*n^2-5196978529613887689953280000*n+ 925430809546415421849600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/ (2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 7429/1048576*( 579678971*n^22-138911090131*n^21+15568525514773*n^20-1082857232717645*n^19+ 52282081213298706*n^18-1855913273937668946*n^17+50013837968988476858*n^16-\ 1040539552938735314410*n^15+16801688053596170725231*n^14-\ 209145164006238894785351*n^13+1957012941286086455466513*n^12-\ 12840266310739212510990825*n^11+45213650076446390828877236*n^10+ 114702497551037193994661564*n^9-2853055962944292103439402192*n^8+ 21396430937612604663080866960*n^7-99857263473391000503697345344*n^6+ 318039082177681533463944012864*n^5-695827241924300085416453885952*n^4+ 1012548957709317340922214865920*n^3-911705762195912243247232204800*n^2+ 439547924413569035117813760000*n-79587049620991726279065600000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-35)/(2*n-29), 323/2097152*(26208159049*n^22-6239532926033*n^21+ 693250500099773*n^20-47669258928113515*n^19+2267274101476623624*n^18-\ 78922491595858448538*n^17+2072903900172079995658*n^16-41676133298229289737350*n ^15+641860265864121629919209*n^14-7446473485207987815397813*n^13+ 61698882997792482100135113*n^12-301787924191226971356793935*n^11-\ 210363479211054183820415786*n^10+19219480747648157236835897872*n^9-\ 195160290769653669607884913792*n^8+1197938931361471980484262246480*n^7-\ 5085550977807902298732847723296*n^6+15316505484279438447656600464512*n^5-\ 32341460384996604447653513866752*n^4+46050895364705001384418462648320*n^3-\ 41046702583686746234566724812800*n^2+19830916190024791738724597760000*n-\ 3661004282565619408837017600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 7429/ 2097152*(2216433676*n^23-572582443577*n^22+69130713608297*n^21-5173953644803405 *n^20+268355907138423951*n^19-10209131893306572072*n^18+293801136792894686362*n ^17-6490217500634050108970*n^16+110114361314303197986266*n^15-\ 1408675690861389660252697*n^14+12786852169846647045674157*n^13-\ 64807642375027933943874585*n^12-175926786050463906751321589*n^11+ 7437316042623067462801005418*n^10-84303966309857511510793533088*n^9+ 611442992010810026647388547200*n^8-3174973418918635085883445772304*n^7+ 12116460329827757923731590960928*n^6-33858500110120951858371156985728*n^5+ 67710963533251208345713239989760*n^4-92754670076282439655530336000000*n^3+ 80637950484573352321650941952000*n^2-38533586396861158954083532800000*n+ 7162834465889255365115904000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 7429/2097152*(2123479501*n^23-542000520002*n^22+64460974718822*n^21-\ 4734549313481680*n^20+239823362850483951*n^19-8852153847738538722*n^18+ 244834140198561554212*n^17-5119531098790794007520*n^16+79934643778108614691691* n^15-881683544578333597306822*n^14+5460661671467269379694582*n^13+ 16284541437904541948895840*n^12-888049500024566649855753239*n^11+ 12360821815846527243441071818*n^10-110760472977432592326671637088*n^9+ 719684473949082428573466587600*n^8-3500987275508079393714113801904*n^7+ 12796426651626224247113471341728*n^6-34709802943145090894292004230528*n^5+ 68017837775766325607753665205760*n^4-92003370390717898715593056000000*n^3+ 79537863312000269839666225152000*n^2-38080393407011355917329612800000*n+ 7162834465889255365115904000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 7429/4194304*(7969656457*n^24-2188311704028*n^23+280212353782318*n^22-\ 22176903193465152*n^21+1211200411512723247*n^20-48206649174704177388*n^19+ 1435771531388581786168*n^18-32171602077135492464592*n^17+ 530262164511484108333927*n^16-5855578098665321128073268*n^15+ 24762121299899075094303238*n^14+546352172679809980609140288*n^13-\ 15044995919239442146186143983*n^12+212901350215256900347431994812*n^11-\ 2115165283497171457108229474252*n^10+15840495382673454665284262969808*n^9-\ 91429273260124664266147324051088*n^8+408016856524345766711598968023872*n^7-\ 1396359795717965137798787371025472*n^6+3598941122203818981966087576699648*n^5-\ 6779952921719530588742774470978560*n^4+8905178485376579231709816992256000*n^3-\ 7545666325000259491891532070912000*n^2+3574980208106836919034649804800000*n-\ 673306439793590004320894976000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/( 2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29) /(2*n-45), 7429/8388608*(14532084551*n^24-3909423591540*n^23+488186046778394*n^ 22-37442390036199720*n^21+1963848437465136761*n^20-74001928195566150780*n^19+ 2034824854615534079384*n^18-39919488783214296912480*n^17+ 494615912828857272101561*n^16-1202208252840733558970220*n^15-\ 107734517262613548513300766*n^14+2992751703917207155933933560*n^13-\ 48601969544244780293847892009*n^12+570565569099923662111836566940*n^11-\ 5131139607864199589782917078196*n^10+36080053204365030122550218845200*n^9-\ 199355737732641264634464812132464*n^8+861890278811046698753210552481600*n^7-\ 2881249353752913088648310502138816*n^6+7299288412124257510399849894813440*n^5-\ 13585804101649563138633426179558400*n^4+17710576726278884631026783261184000*n^3 -14959487959416561195472116142080000*n^2+7097932191320532792280866816000000*n-\ 1346612879587180008641789952000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/ (2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/ (2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29 )/(2*n-45), 37145/8388608*(5091744254*n^25-1455965785675*n^24+192979332066200*n ^23-15664396874333650*n^22+864381119848083530*n^21-33837930243292297045*n^20+ 938139172667401228700*n^19-16962674527152458467000*n^18+ 111348145364030187461090*n^17+4406585434789815850134155*n^16-\ 192669091229384072397404800*n^15+4435251986747757034791830150*n^14-\ 72859916253313789222824354490*n^13+920992814700773715860561356805*n^12-\ 9218276605680636040888708117300*n^11+73895215107324789731882998331300*n^10-\ 475592680674505943811514093829680*n^9+2448742025105228313559726321250480*n^8-\ 9996854896739495608543222389796800*n^7+31888385463221419642787441399083200*n^6-\ 77760054462996816441685488142504704*n^5+140373042785296666322458186169441280*n^ 4-178712949119315848649848899058176000*n^3+148402100179736286577352888979456000 *n^2-69698477407989597486454982246400000*n+13196806219954364084689541529600000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 6555/8388608*( 23845651006*n^25-6557915331025*n^24+826065379772200*n^23-62490204530391150*n^22 +3099178873979958770*n^21-100486301493912558455*n^20+1756375690505807741700*n^ 19+13235626100977651824200*n^18-2057672096460988939409390*n^17+ 78435444730832149763169345*n^16-1957339724933710540185296800*n^15+ 36578166053794123191235517050*n^14-536522532722520009417148217010*n^13+ 6307647110313902598307924212695*n^12-60004113102723085010205079724300*n^11+ 463215331582359641542158996741900*n^10-2896424283320513480093176926227920*n^9+ 14581182423275128847448962416005520*n^8-58487080193059019013380885849588800*n^7 +184034631488168419231514686035624000*n^6-444181415438375105542533459458355456* n^5+796025363991072432200324302088401920*n^4-\ 1008895425899162664763139788168704000*n^3+836282778700741524327147854966784000* n^2-393180691006223292950559439257600000*n+74781901913074729813240735334400000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 345/16777216*( 1365344172281*n^26-385955953825403*n^25+49049213928612300*n^24-\ 3599311930638617050*n^23+156219268539351823195*n^22-2768533132469363424165*n^21 -127947575232878421416110*n^20+12763228822474869548180800*n^19-\ 587917870459264770796064265*n^18+18799941686225485959296266715*n^17-\ 459135347140278318394332766760*n^16+8907990653551567172526160743150*n^15-\ 140064966375594598767375388192435*n^14+1803934705167014728821968852951125*n^13-\ 19129568002220542857520239024560310*n^12+167246602615148669318902159307415500*n ^11-1203295303328529912023972058694318520*n^10+ 7089932150923216546144613726397371280*n^9-\ 33937039459549915205180347072376576160*n^8+ 130421196554020775371299868909716801600*n^7-\ 395791125689612816091436613183043728256*n^6+ 926745852445075635292857693381176360448*n^5-\ 1619910205435611248383703443948842792960*n^4+ 2012598029061879869652165340624382976000*n^3-\ 1643365018525097154242247164717064192000*n^2+ 764970923010104150644058474771251200000*n-\ 144927325907538826378060545078067200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-51)/(2*n-45), 115/33554432*(5170849977463*n^26-\ 1267374098443037*n^25+123505996443387610*n^24-4223250555387645470*n^23-\ 300589236059780744375*n^22+49103437948621467055645*n^21-\ 3466070575430174031474440*n^20+164418241335712931796537880*n^19-\ 5832285540235648314310916855*n^18+162096693459314492202983395165*n^17-\ 3620667033351729076597196783390*n^16+65988877906248972525544761342730*n^15-\ 990404123512998001899088458147065*n^14+12304105742837249348145755253872755*n^13 -126788072541121867564656702739427540*n^12+ 1083045073737737081431803084314981980*n^11-\ 7645729862104703081482731979935323120*n^10+ 44353944978324155991371132641234771600*n^9-\ 209626935790667142073133776984459055040*n^8+ 797385820069167316715311456130459023680*n^7-\ 2400329708361182664090604245229338498048*n^6+ 5585922223013711048208602196268086607872*n^5-\ 9721500413274845409448349287373668147200*n^4+ 12046007480365356113371566194732862259200*n^3-\ 9826125157643382577822411475779534848000*n^2+ 4577205072359585637643869021257072640000*n-\ 869563955445232958268363270468403200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-51)/(2*n-45), 345/33554432*(1268670080488*n^27-\ 138913603521507*n^26-32789840559535215*n^25+9869685891836541030*n^24-\ 1256437659905902357650*n^23+102392238246737897013075*n^22-\ 6024317321724477541573665*n^21+271155378405345922915558560*n^20-\ 9658313866014733787664962880*n^19+278226065528282763444110095395*n^18-\ 6577247294061810015937774505265*n^17+128863627164338049293857485006510*n^16-\ 2106065847278709538597578685052490*n^15+28822744618287933025460507285114685*n^ 14-330842490829666302632806130583460815*n^13+ 3183880670255399335888595536322814060*n^12-\ 25628203863643969843589533491719675820*n^11+ 171788080791386343882387316537706089680*n^10-\ 952525428491965207109016123168339061840*n^9+ 4327982601274271675058485388277387152960*n^8-\ 15909390352021979321277844138468693674048*n^7+ 46498626488162319348715329905420686428672*n^6-\ 105515739340278553578002722615284193843200*n^5+ 179783786820663693248480874112180706426880*n^4-\ 218933709450055707140525363891885416857600*n^3+ 176170534324920355678541174603811962880000*n^2-\ 81270756616527779923152224818955550720000*n+ 15362296546199115596074417778275123200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -1725/33554432*(198736810219*n^27 -134654305943358*n^26+34045570810640493*n^25-4890785835509006940*n^24+ 469116840322234755495*n^23-32657837136849004044978*n^22+ 1734419233621201267710141*n^21-72588841887284471839260456*n^20+ 2447794298784729053999982285*n^19-67553352338979130609283624898*n^18+ 1542705155245853356404065178591*n^17-29377828609169233640021329709916*n^16+ 468867753051966178948969112307765*n^15-6289518337691933810975849011519758*n^14+ 70977802413678976322740948085210151*n^13-673248967931560587542457146139717216*n ^12+5352928027014503887830177283685046060*n^11-\ 35508639985529962163966750987436377088*n^10+ 195165523293827948951848435437188466896*n^9-\ 880318062938270980098004301734822327296*n^8+ 3216744961039387052956359191254365838656*n^7-\ 9357168603563383049665478918370991249920*n^6+ 21157055068906026817443986251718244169728*n^5-\ 35956951401173536455088112116638547378176*n^4+ 43719676595790287509782768393052551659520*n^3-\ 35160559954081360611369058482674319360000*n^2+ 16227549944004778295286593158114443264000*n-\ 3072459309239823119214883555655024640000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -75/67108864*(59760278952503*n^28-\ 28652067393337906*n^27+6305256098962931691*n^26-856379019784133360970*n^25+ 81115057736411131156815*n^24-5727152077516189059056226*n^23+ 314178834396480100577588487*n^22-13768444206118433868501747282*n^21+ 491497601921652591865554379425*n^20-14492960607577611752748651061566*n^19+ 356593795897451712439063600731537*n^18-7373739281611031521473930851031102*n^17+ 128769900106013566880831609864733285*n^16-1904728371998345892673377903141244086 *n^15+23893858227811867349437087112504850957*n^14-\ 254113998938932619758187607786721946502*n^13+ 2286834137548298106783538418323449520500*n^12-\ 17352741269013606510106261866006007954296*n^11+ 110435014953465943539576959165889614353072*n^10-\ 585078852840897697145441463753074350855712*n^9+ 2554680336956567089275316451996965222556352*n^8-\ 9071608663806839447624055450135845555080320*n^7+ 25735605490683011980211468794707842368736256*n^6-\ 56939178001605811496877611111219663397138432*n^5+ 94987523161115044515139539736607909274501120*n^4-\ 113709710811583686985161496966254529489305600*n^3+ 90305135015417326235720650139539321356288000*n^2-\ 41287159959628155762055935251222219980800000*n+ 7773322052376752491613655395807212339200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -75/134217728*( 198702952510411*n^28-86918350369097282*n^27+17850565525727750067*n^26-\ 2296462305302043646950*n^25+208178183815144890513075*n^24-\ 14174144160956433670422042*n^23+754135813723262612877056679*n^22-\ 32197320244144766500402386774*n^21+1123782904174455277487301011445*n^20-\ 32495905099489889771978423581422*n^19+786015771483433253750374800960729*n^18-\ 16012054760472164854067210309074914*n^17+275973243276681025376935646481400065*n ^16-4035263319007967504864469302868670062*n^15+ 50110028523026709869886153405459871869*n^14-\ 528219797508629543753822352918156992514*n^13+ 4716963478316053473628988607119842881420*n^12-\ 35553811225388464464098526100499334284232*n^11+ 224970032049094120169187730520790317291024*n^10-\ 1186063985009878720202139070643740121020384*n^9+ 5157662546871547039136716364426195713921344*n^8-\ 18253427541298574105525582824802216882213760*n^7+ 51646007743306967825851384252036986491653632*n^6-\ 114034023596260778740801289683347517391038464*n^5+ 189964474607845138996899379201502421316362240*n^4-\ 227213580472274577393758035400516200149811200*n^3+ 180393497472156384527614352858784133349376000*n^2-\ 82497508436525315946131498413553064345600000*n+ 15546644104753504983227310791614424678400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -87/134217728*( 467098786880242*n^29-209052550062164399*n^28+44358370572860717268*n^27-\ 5943891502997347335471*n^26+565130652624970820944920*n^25-\ 40606074425240277867004455*n^24+2292861990433137519378767460*n^23-\ 104446253679337548323959380195*n^22+3909589882656362228819778503220*n^21-\ 121858596505381526175908920462065*n^20+3193437566642479785843030495658980*n^19-\ 70852485580670184703651730539851885*n^18+1337334231618310704243136292876719080* n^17-21539821121599724508326283379351253085*n^16+ 296499138685623821736362089403757370620*n^15-\ 3488449913110332116719442740553904228465*n^14+ 35036656761556987133576000608450592975610*n^13-\ 299597769487932388390357580794682346137820*n^12+ 2172047235297253317691481632384119736945640*n^11-\ 13272789776017703688882076271559386435139280*n^10+ 67824461280994894720804649745070918447552608*n^9-\ 286828451602058547097153000469220397433818176*n^8+ 990241166826656607878243946903729488670940032*n^7-\ 2740954741385936948862239399286024703143304704*n^6+ 5936534119188891982119538206188128025245624320*n^5-\ 9725502805836738097288604323383961207004160000*n^4+ 11468043310995413722232623475101905238425600000*n^3-\ 8998439850223959066597083459406438451445760000*n^2+ 4077783511868895941731480570071191833804800000*n-\ 763929925837025675899962685450019143680000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -87/ 134217728*(575111694393256*n^29-251393609807265887*n^28+52236724033952445414*n^ 27-6869905385403953673963*n^26+642322734237688492735110*n^25-\ 45463025375075251067739615*n^24+2532559505587220220540911430*n^23-\ 113964322062720343565989344135*n^22+4219117200189567007811715346110*n^21-\ 130205812345717358304380085304545*n^20+3381759353088763026989028575590890*n^19-\ 74428349284697994913282105423614705*n^18+1394693336782247386857426812685171090* n^17-22318456847958018371466339658800950805*n^16+ 305443769917589303061984157613357624610*n^15-\ 3575245326309030270909108456723824906445*n^14+ 35745455053261943805988841347488357860130*n^13-\ 304440403566294449347618715542429648630860*n^12+ 2199491498811034924099228851546569471685320*n^11-\ 13400245082349881589173644627803964018746640*n^10+ 68301181073795584692012076200224459009086944*n^9-\ 288227868579389259056150889492476046923604288*n^8+ 993332970075832772004067870228947824844222336*n^7-\ 2745698102510173261776229990769355792023834112*n^6+ 5940561786696740538840891132607472162309007360*n^5-\ 9724957179920491529003158614420932252949504000*n^4+ 11462413367511994226231158112583618736619520000*n^3-\ 8992710312520594924549049399135503365242880000*n^2+ 4075789102738505571668524956533420222054400000*n-\ 763929925837025675899962685450019143680000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -29/ 268435456*(7963467256597043*n^30-3663219784759517175*n^29+803713345833940911535 *n^28-111973230002383288595175*n^27+11126519865901942677553707*n^26-\ 839674956824215070030088435*n^25+50035651057339525664412570075*n^24-\ 2416617907975829564779009813875*n^23+96356320179135923206520557067445*n^22-\ 3214210325005930211183392517672325*n^21+90577824601586778433137035581339725*n^ 20-2171741434667449481023257928516147125*n^19+ 44527053382855606230498977675889796665*n^18-\ 783300288058402786202015473796995122225*n^17+ 11845429119698951281300108303334947934025*n^16-\ 154082415207513689210265497919678780428625*n^15+ 1722889361765652078424042912342404499237620*n^14-\ 16529041413464956947539365596886333474682400*n^13+ 135626150317016846606583981467830896863360800*n^12-\ 947422956860042669795982362528307067230540000*n^11+ 5599321439473272284076832027031098575570584192*n^10-\ 27768307072416896375164723894409763660326784000*n^9+ 114327087190224489940417145620665555314362947840*n^8-\ 385394959986777677855340741250648729515731155200*n^7+ 1044469551402497492200658767839750726355065443328*n^6-\ 2220643128257195061370239075922715314805993533440*n^5+ 3579994307953333692234372234004505673910695936000*n^4-\ 4164166487622402721683252661432484113104568320000*n^3+ 3230900796823293479924625016755976271086878720000*n^2-\ 1451506209234920981174314131843528800442777600000*n+ 270431193746307089268586790649306776862720000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2 *n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -29/536870912*(17552638988517511*n^30-7997855517827608875*n^29+ 1739419263723866022245*n^28-240385806380750170099575*n^27+ 23709564300357206364397689*n^26-1777068388837550980786882995*n^25+ 105230828249819721117925073025*n^24-5053215319306722451529116657875*n^23+ 200423045871450810129765513556515*n^22-6653453218172603111145253512660525*n^21+ 186674603406131794921421139203365575*n^20-4457953003815383003326636854680647125 *n^19+91070538853823963542229520590525369955*n^18-\ 1596839100295589197480091228029440519825*n^17+ 24077164204425986006914880847658604445675*n^16-\ 312365448622743011330225823409086199400625*n^15+ 3484559438848649301819146114785982798486490*n^14-\ 33360756664994674453330168651196173200339300*n^13+ 273236654873720887839473890929450088562930600*n^12-\ 1905676171450834905779592607960911814852518000*n^11+ 11247236930259335290692091138278659639773487584*n^10-\ 55712702415664053371402571006690100927218801600*n^9+ 229156083941866662872072498812944109493970890880*n^8-\ 771868024805446759836825733471517866982953036800*n^7+ 2090545208004297764859363723908146291403579744256*n^6-\ 4442591610411970066115737617780150534804132986880*n^5+ 7159727767825821245463228891376474046654902272000*n^4-\ 8326412678753311133532971034059028679056752640000*n^3+ 6459903809392116916357205860667113562933821440000*n^2-\ 2902362030798494422451397415224697555727155200000*n+ 540862387492614178537173581298613553725440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2 *n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -31/536870912*(34973065913887148*n^31-16903726144617503539*n^30+ 3907723753178734676935*n^29-575253893903822546928905*n^28+ 60570310398395593315054227*n^27-4857591433350719858129424621*n^26+ 308520568825695056915902285455*n^25-15930671500856280194673802984725*n^24+ 681250598256479883536100987191895*n^23-24453742080406930163567774382523935*n^22 +744148676492379876201498452455023825*n^21-\ 19338825102057866316308021530170030675*n^20+ 431480641823033089450423310708841589065*n^19-\ 8295619144050874184957033006027895812895*n^18+ 137748244120614547069214677589245523288325*n^17-\ 1977570599606341997749589665645472176837575*n^16+ 24544138299413909055237032109804417270096945*n^15-\ 263032241786861192422311889697594399013793410*n^14+ 2428256047408401406786894841436748474676108500*n^13-\ 19242068124257667651007305177136056300925147400*n^12+ 130237053881504889513166195473873233995327307312*n^11-\ 748006969712660882904725536279550125668492344416*n^10+ 3614857398456148025726021379885581677357923483840*n^9-\ 14540012824520803388712093367565576408298239022720*n^8+ 47997910709524935375773809173689929773070600593408*n^7-\ 127666495944005367113262015345628714931444937437184*n^6+ 266955852906782614431250033818264120314882835333120*n^5-\ 424133092045209032249722937567734583451165954048000*n^4+ 487158477017806698142020857646166749848574689280000*n^3-\ 373993439367449180446064290149958726129684316160000*n^2+ 166610784686902214737565116891488330084306124800000*n-\ 30864050434659176833298711784427334727106560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2 *n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/ (2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57 )/(2*n-45), (222136259508192792*(n-30)!*(n+1)*(2*n-65)!+333204389262289188*(n-\ 31)!*((n+1)*(2*n-64)!-1/333204389262289188*binomial(2*n,n)*(n-33)!*(n-30)!))/(n -33)!/(n-31)!/(n-30)!/binomial(2*n,n), ((222136259508192792*n+ 222136259508192792)*(2*n-65)!-(n-33)!*(n-31)!*binomial(2*n,n))/(n-33)!/(n-31)!/ binomial(2*n,n)] The limits, as n goes to infinity are 2147483631 2147483367 2147481259 17179757847 34358739219 536813403 [----------, ----------, ----------, -----------, -----------, ---------, 2147483648 2147483648 2147483648 17179869184 34359738368 536870912 68696304411 274621620405 548485846185 546893375435 271908360973 -----------, ------------, ------------, ------------, ------------, 68719476736 274877906944 549755813888 549755813888 274877906944 4306435075559 8465235372827 4116471444751 15775329212929 59206577819053 -------------, -------------, -------------, --------------, --------------, 4398046511104 8796093022208 4398046511104 17592186044416 70368744177664 107958856129379 94566420157415 78154121172165 471043739436945 ---------------, ---------------, ---------------, ----------------, 140737488355328 140737488355328 140737488355328 1125899906842624 594647747408245 54711397221045 -342820997627775 -4482020921437725 ----------------, ---------------, ----------------, -----------------, 2251799813685248 562949953421312 4503599627370496 18014398509481984 -14902721438280825 -20318797229290527 -6254339676526659 ------------------, ------------------, -----------------, 36028797018963968 36028797018963968 9007199254740992 -230940550441314247 -509026530667007819 -271041260832625397 -------------------, -------------------, -------------------, 288230376151711744 576460752303423488 288230376151711744 -1125154472168322877 -4583918985988863805 --------------------, --------------------] 1152921504606846976 4611686018427387904 and in Maple notation [2147483631/2147483648, 2147483367/2147483648, 2147481259/2147483648, 17179757847/17179869184, 34358739219/34359738368, 536813403/536870912, 68696304411/68719476736, 274621620405/274877906944, 548485846185/549755813888, 546893375435/549755813888, 271908360973/274877906944, 4306435075559/ 4398046511104, 8465235372827/8796093022208, 4116471444751/4398046511104, 15775329212929/17592186044416, 59206577819053/70368744177664, 107958856129379/ 140737488355328, 94566420157415/140737488355328, 78154121172165/140737488355328 , 471043739436945/1125899906842624, 594647747408245/2251799813685248, 54711397221045/562949953421312, -342820997627775/4503599627370496, -\ 4482020921437725/18014398509481984, -14902721438280825/36028797018963968, -\ 20318797229290527/36028797018963968, -6254339676526659/9007199254740992, -\ 230940550441314247/288230376151711744, -509026530667007819/576460752303423488, -271041260832625397/288230376151711744, -1125154472168322877/ 1152921504606846976, -4583918985988863805/4611686018427387904] and in floating point [.9999999921, .9999998691, .9999988875, .9999935193, .9999709209, .9998928811, .9996627983, .9990676350, .9976899422, .9947932548, .9891968547, .9791699712, .\ 9623858401, .9359772422, .8967236461, .8413760756, .7670938098, .6719348289, .5\ 553184307, .4183708841, .2640766483, .9718696465e-1, -.7612155298e-1, -.2488021\ 412, -.4136336118, -.5639599129, -.6943711913, -.8012359888, -.8830202726, -.94\ 03632762, -.9759159385, -.9939789846] The cut off is at j=, 23 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 34], vs. those in the, 2, -th row from j=1 to j=, 33, are as follws 17 16 15 [3 (2863311519 n - 413748511223 n + 27550782558788 n 14 13 12 - 1121258355785580 n + 31188827053276858 n - 628142275710007786 n 11 10 + 9465789873530048916 n - 108763574429838130660 n 9 8 + 962019040657963786927 n - 6565609984008814240359 n 7 6 + 34428402949849261384304 n - 137108887547122205045240 n 5 4 + 405388786687887020130096 n - 847959494457950325259632 n 3 2 + 1078942803926129303524992 n - 114969141280045479179520 n - 2764783101481720313702400 n + 4703528235956894023680000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) 17 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 21 (204522238 n 16 15 14 - 29553459651 n + 1967912005376 n - 80089760378060 n 13 12 11 + 2227763339503716 n - 44866700144559202 n + 676100009272073672 n 10 9 - 7767832344753769860 n + 68687778611865695854 n 8 7 - 468357771203943120643 n + 2448561555404031075528 n 6 5 - 9651694347745655168600 n + 27522986673470531004192 n 4 3 - 50040348331180016691504 n + 24903879180240233303424 n 2 + 139219107951258432587520 n - 311860201363852356864000 n + 9881361840245575680000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 651 ( 18 17 16 15 13194976 n - 2137584132 n + 160328324487 n - 7388852535480 n 14 13 12 + 234145265585532 n - 5409520907393064 n + 94260516179967514 n 11 10 - 1263960859910445840 n + 13186796804734432308 n 9 8 - 107440927423001791476 n + 681125509044073409511 n 7 6 - 3311856545948937593880 n + 11911195860760286827384 n 5 4 - 28709025725894657843328 n + 30388935941413580869488 n 3 2 + 59196156755319010627200 n - 238071746545757117971200 n + 179485773643626912000000 n - 11156376271245004800000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 18 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 1953 (4398312 n 17 16 15 14 - 712522844 n + 53441830309 n - 2462844853560 n + 78040204169284 n 13 12 11 - 1802707366284888 n + 31400171513201998 n - 420659879580134480 n 10 9 + 4378676158892202596 n - 35477159590955059692 n 8 7 + 221894081807030463077 n - 1044475731430257486360 n 6 5 + 3468629636585120731208 n - 6661296550081444256576 n 4 3 - 99439353448326798384 n + 34847397973872585878400 n 2 - 68621230500338765318400 n + 41745053206249747200000 n - 3718792090415001600000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) 19 18 17 (-1 + 2 n) (2 n - 5)), 1076103 (127717 n - 23052456 n + 1934500543 n 16 15 14 - 100212005824 n + 3588131641754 n - 94210037463472 n 13 12 11 + 1877589640296366 n - 28995314021353888 n + 350832112218772041 n 10 9 - 3335809730091269288 n + 24769023732277509139 n 8 7 - 140747423263352358752 n + 583666624196985713288 n 6 5 - 1564257529940303653184 n + 1555303914481742299952 n 4 3 + 5544292238211720670464 n - 23167354690205791660800 n 2 + 33095530098467715686400 n - 17979937429847500800000 n + 1997754008643993600000)/(262144 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 19 18 (2 n - 3) (-1 + 2 n) (2 n - 5)), 34713 (1979521 n - 357278288 n 17 16 15 + 29978685699 n - 1552633175692 n + 55567719682802 n 14 13 12 - 1457649467351056 n + 28996885999402038 n - 446141931018752904 n 11 10 + 5359178878151030933 n - 50251860939629601824 n 9 8 + 363441767812064944327 n - 1965482249937572079116 n 7 6 + 7394399091781380587144 n - 15584533746047908976032 n 5 4 - 4059882631990724914064 n + 131507105261632267163712 n 3 2 - 368871138599294580998400 n + 458531123700307945651200 n - 239071805447100595200000 n + 30965187133981900800000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 20 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 57855 (4750177 n 19 18 17 - 949853582 n + 88624247379 n - 5124392056698 n 16 15 14 + 205654971823818 n - 6078548397076284 n + 136964746514949238 n 13 12 - 2400797383122390356 n + 33072726464001171909 n 11 10 - 358542522210277039206 n + 3032649877233278594487 n 9 8 - 19556681398850211612594 n + 91408496785169647796320 n 7 6 - 272286478533770837289008 n + 262334365698311774264496 n 5 4 + 1657599107034004825671648 n - 8438016005222229494308224 n 3 2 + 18251961290595042341230080 n - 20159043383226932852889600 n + 10109223360329465788416000 n - 1449170757870352957440000)/(262144 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), 1995 (137703185 n - 27526792858 n + 2566895125251 n 17 16 15 - 148273769198622 n + 5940160950308394 n - 175039408911110676 n 14 13 + 3923641339373262422 n - 68180957291562776284 n 12 11 + 925950159595117290933 n - 9810021233603746629474 n 10 9 + 79971410505507814136103 n - 485462403264600076364166 n 8 7 + 2034186786695069122946912 n - 4585898411685254234750992 n 6 5 - 4341360840678223117540176 n + 74334912730185088212327072 n 4 3 - 271607243133385037426903424 n + 521898692222889812722752000 n 2 - 545133753422434291224729600 n + 270918444049309677164544000 n - 42025951978240235765760000)/(262144 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 285 (7704456289 n 20 19 18 - 1697015568087 n + 174904253384945 n - 11203109584564485 n 17 16 + 499424808459419724 n - 16437427594530130062 n 15 14 + 413238166749973179250 n - 8091660257804631487770 n 13 12 + 124553511063793743973769 n - 1507528813388837753672067 n 11 10 + 14207700853805575284901005 n - 101732671953202921405534305 n 9 8 + 523751159464717087396188634 n - 1657925314613406653890109832 n 7 6 + 782610896964233875517204960 n + 22199658557950421960200197360 n 5 4 - 125130045860711658417557896416 n + 361554671999187363139293570048 n 3 2 - 618483834042547570412470940160 n + 606457233594561448729992499200 n - 295579724875893220443512832000 n + 48245792871019790659092480000)/( 1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 21 20 (2 n - 39) (2 n - 35) (2 n - 29)), 95 (11534251796 n - 2537514900543 n 19 18 17 + 261031844996230 n - 16670068215357165 n + 739740917169272886 n 16 15 - 24178801270759278318 n + 601623500309745915500 n 14 13 - 11603818692352795207530 n + 174745740367973033730016 n 12 11 - 2048974512659184822110763 n + 18425543969363911305359070 n 10 9 - 122501302091703707408927145 n + 548197696346857059253151126 n 8 7 - 1100141931807679414784509848 n - 4632486191260034301194616560 n 6 5 + 49826366860434422966030573040 n - 216409758285826736236284817824 n 4 3 + 563516617470664468467732809472 n - 912179874383201048379967434240 n 2 + 870557305297642834918549708800 n - 424154044810063566003253248000 n + 72368689306529685988638720000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 2185 ( 22 21 20 999107099 n - 240804474493 n + 27202832442478 n 19 18 17 - 1912530585076265 n + 93683685065211549 n - 3390148711375634208 n 16 15 + 93713222160465448508 n - 2016530004421157588770 n 14 13 + 34067099230143445482709 n - 451545379481444173818893 n 12 11 + 4641731625718780813705758 n - 35921496635082737017664325 n 10 9 + 194166208673048224612936739 n - 553442802429836512517310118 n 8 7 - 1262623536269996349628438952 n + 23861923893147503384834979520 n 6 5 - 139744369328589777026920633296 n + 496893099024081017486235719712 n 4 3 - 1159415737302134229449673537792 n + 1752865113368987796234877539840 n 2 - 1607109543205229616654715084800 n + 772747561957312585883086848000 n - 135297984355685934674411520000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 437 ( 22 21 20 4959919369 n - 1191485496113 n + 133968300702698 n 19 18 17 - 9357298890735565 n + 454233521336975919 n - 16235942757418351968 n 16 15 + 441384240336501334708 n - 9286635209761574081450 n 14 13 + 152172733069946181410279 n - 1933034183987244987441793 n 12 11 + 18657732308172270217988538 n - 129775429791481649496906585 n 10 9 + 547901124416050619327511409 n - 20969189409072873408338558 n 8 7 - 19685281651153894818544031992 n + 166605155238900556351720057280 n 6 5 - 813635535378686627455515304176 n + 2654281648217001040992440118432 n 4 3 - 5890716752322350689004384093952 n + 8644263046952311419124785016320 n 2 - 7813196298474198135801899212800 n + 3763892238443864218265333760000 n - 676489921778429673372057600000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 437 ( 23 22 21 78379266439 n - 20494780376318 n + 2512718757330518 n 20 19 - 191735522636816620 n + 10189806722829083889 n 18 17 - 399741281217862435698 n + 11963015580392965353628 n 16 15 - 278119068159757659928280 n + 5059572346517442994636649 n 14 13 - 71786921382715170710422498 n + 779975066914076220622520358 n 12 11 - 6171304438931643382763896140 n + 30151185263906155087715179279 n 10 9 - 5752451483942983952314314638 n - 1430367433654333856629761463672 n 8 + 14730356203695870714921369563600 n 7 - 89162423229915810082307468186256 n 6 + 371648507720283470401050081061152 n 5 - 1099067350306663060408985957240832 n 4 + 2281414988388447846774115271677440 n 3 - 3197813980725268077938641159680000 n 2 + 2809080550227102706269590335488000 n - 1338367512469003847783195443200000 n + 243536371840234682413940736000000 )/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 437 ( 23 22 21 38398544267 n - 9970509054784 n + 1211351163801724 n 20 19 - 91356813893462810 n + 4782802211370427167 n 18 17 - 184053967065080193774 n + 5373469246498523901104 n 16 15 - 120938481777671412216340 n + 2105439851702133510203497 n 14 13 - 28024741771149373763929724 n + 274058066006433448034302644 n 12 11 - 1729127342507110348426775970 n + 2518825634731224281877609437 n 10 9 + 88341985952810642248980788906 n - 1228477612523707333581696310496 n 8 7 + 9557083609187662922314369937200 n - 51452230389056358920486567584368 n 6 + 200719033538420592622535129705376 n 5 - 569008106269185211156501008494976 n 4 + 1148712142504758845165549158417920 n 3 - 1582642078337748185626688352000000 n 2 + 1379356358993970311828684611584000 n - 658577251352319757293253017600000 n + 121768185920117341206970368000000) /(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 437 ( 24 23 22 148765452293 n - 41743501011684 n + 5487626394952742 n 21 20 - 448463204951022096 n + 25483304104747387283 n 19 18 - 1066395548182922894244 n + 33925025807964096152072 n 17 16 - 833714166560845936653936 n + 15872502588506449515882923 n 15 14 - 230898480078662776245627564 n + 2446578452698729411076502782 n 13 12 - 15906423566944813737100755216 n - 3392971352957418528135755347 n 11 + 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(2 n - 51) (2 n - 57) (2 n - 45)), - 29 (3513148220705299 n 29 28 - 1638084119043757695 n + 363811755089069127770 n 27 26 - 51248215163408138106105 n + 5143438145544425871393186 n 25 24 - 391671564975243292687934805 n + 23530667995599805069465716150 n 23 - 1144903097138514177821365174725 n 22 + 45955749088710495590792657523960 n 21 - 1542242834033103977649667109492175 n 20 + 43698154912881897071261421708354450 n 19 - 1052876384454055810966182323759430675 n 18 + 21682319795442363097718037680894645070 n 17 - 382934334118846112932025656067103069975 n 16 + 5811367546400204530229945046911905024050 n 15 - 75830441179179985707632475130917606163575 n 14 + 850266709997238320955657019612628624909685 n 13 - 8177279423996165136391327296009332582648550 n 12 + 67241114230598584852180282076882351684773100 n 11 - 470589815907628812879204320897620672796561400 n 10 + 2785654199064125874953810559829627746944211856 n 9 - 13833398681675020126214226431465716496506354080 n 8 + 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306744902939185240430054448 n + 19669189253428384987398329040 n 24 - 1024574744314511731536379555800 n 23 + 44167921656871141129580375136510 n 22 - 1597127497979345226774343929816780 n 21 + 48929506944906537338500539883236600 n 20 - 1279376728554424368687522100391849400 n 19 + 28704092036259770090754397484009597220 n 18 - 554648468271164067656171089480028105760 n 17 + 9251848096252086002733311772289083327600 n 16 - 133366993228186172122427291049715288806600 n 15 + 1661312980005334832594095322300525499757785 n 14 - 17861807945241211891800164189085390838246830 n 13 + 165371446745158246067639877164679067285425500 n 12 - 1313756972888829908700398449461271360513046200 n 11 + 8911542351487745018535237364389558762993263056 n 10 - 51279805257286916137086379364801230070518373408 n 9 + 248216555318463494994718970750387554284343321920 n 8 - 999742569904746753182767334410776554135174559360 n 7 + 3303866691680277449161884071377194794841902114304 n 6 - 8795374020982776988337566645085293512699951163392 n 5 + 18403534121575330568417122378879949783445914050560 n 4 - 29252406465846514542718991456094444725494554624000 n 3 + 33608219825301793248972519148711018623409520640000 n 2 - 25803351884418619481625555623701189231265710080000 n + 11494020270816479306480958728883938228856422400000 n - 2128555202390288057468876674788092050145280000000)/(1073741824 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), - 31 (33028595855166973 n 30 29 - 16053992728956787064 n + 3730553363778445931810 n 28 27 - 551793269581995725900530 n + 58354668669507297255045402 n 26 25 - 4698715860053950709693624346 n + 299529129921297131312752774830 n 24 - 15518516987483256937908301082850 n 23 + 665668512145390636654118773498020 n 22 - 23961493182571425690612829912581810 n 21 + 731031656380074527941999862481969450 n 20 - 19041994461470013974532602399080787550 n 19 + 425749808336540357523871232862059153190 n 18 - 8200949798755166853996899607315288804270 n 17 + 136408391968665008032989565456958945811450 n 16 - 1961326133878176546751496006396713154536950 n 15 + 24375685685918820421015663383181702473583695 n 14 - 261542741172294087481316837352351932697596910 n 13 + 2417079040912866380533748314814114334040023500 n 12 - 19171373739111003280285730265910963642030077400 n 11 + 129863673312538549789435806454347892025261532112 n 10 - 746381324366572203236239540358346933244581082016 n 9 + 3609127778651119448257950734353654142370264075840 n 8 - 14524098535125512802208417966756027472949009966720 n 7 + 47964565926612453375523119934873361288363564740608 n 6 - 127618105940588050343309772598514775582378967363584 n 5 + 266918086189886177061575106539804359984249844613120 n 4 - 424142969740693164838466954183786797033119387648000 n 3 + 487216731784559110018855922263715825651620577280000 n 2 - 374049563810348937521683297677022609462398812160000 n + 166629755767137422296009581032974854968888524800000 n - 30864050434659176833298711784427334727106560000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 32 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), - 31 (140281157667292627 n 31 30 - 72215957889491812304 n + 17806905225854126930264 n 29 28 - 2800313240286619759403680 n + 315501964437696691330218228 n 27 26 - 27121917044044070550257802336 n + 1849938915909389806178837629656 n 25 - 102792000708332716464051684120480 n 24 + 4740576566224528781084944891545330 n 23 - 183947327766511011385123786702805760 n 22 + 6066591948235981131426665400333992760 n 21 - 171342674227803947523045657189891458400 n 20 + 4167500762920128999548184652257871171860 n 19 - 87640179816894615899048052034037603567520 n 18 + 1597701792813984155758787572366733129382120 n 17 - 25286692187882998203775463183543387322348000 n 16 + 347588819538928630346816680162406070002212755 n 15 - 4147110795926548848428501861050228871786694960 n 14 + 42876089763820768399866044432518470078153696560 n 13 - 383083503629362322064111931947264741716683945600 n 12 + 2946414224549126921095423155899977265793968838688 n 11 - 19406926392158453226730542875656487933392821118976 n 10 + 108729936832941586428216040768479188275200543119616 n 9 - 513707459570980282989725492549709793708205669406720 n 8 + 2024240447730684126270815448905985976820202212512512 n 7 - 6558832555441544062357697150225957561909797270278144 n 6 + 17154419167958244192771347312891602150485951279489024 n 5 - 35333443719529243831676551291757600030929486064517120 n 4 + 55389267395851114303039631941833456807947986526208000 n 3 - 62877196626763780346615639608330314253654227025920000 n 2 + 47788905536849621614489872638791450577708329205760000 n - 21116176036677352149029999327282508715985259724800000 n + 3888870354767056280995637684837844175615426560000000)/(1073741824 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45)), ( 849344521648972440 (n - 31)! (2 n - 67)! (n + 1) + 1274016782473458660 ( (n + 1) (2 n - 66)! - 1/1274016782473458660 binomial(2 n, n) (n - 34)! (n - 31)!) (n - 32)!)/( binomial(2 n, n) (n - 34)! (n - 32)! (n - 31)!), ( (849344521648972440 n + 849344521648972440) (2 n - 67)! - (n - 34)! (n - 32)! binomial(2 n, n))/((n - 34)! (n - 32)! binomial(2 n, n))] and in Maple notation [3/65536*(2863311519*n^17-413748511223*n^16+27550782558788*n^15-\ 1121258355785580*n^14+31188827053276858*n^13-628142275710007786*n^12+ 9465789873530048916*n^11-108763574429838130660*n^10+962019040657963786927*n^9-\ 6565609984008814240359*n^8+34428402949849261384304*n^7-137108887547122205045240 *n^6+405388786687887020130096*n^5-847959494457950325259632*n^4+ 1078942803926129303524992*n^3-114969141280045479179520*n^2-\ 2764783101481720313702400*n+4703528235956894023680000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31) /(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 21/32768*(204522238*n^17-\ 29553459651*n^16+1967912005376*n^15-80089760378060*n^14+2227763339503716*n^13-\ 44866700144559202*n^12+676100009272073672*n^11-7767832344753769860*n^10+ 68687778611865695854*n^9-468357771203943120643*n^8+2448561555404031075528*n^7-\ 9651694347745655168600*n^6+27522986673470531004192*n^5-50040348331180016691504* n^4+24903879180240233303424*n^3+139219107951258432587520*n^2-\ 311860201363852356864000*n+9881361840245575680000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-29), 651/32768*(13194976*n^18-2137584132* n^17+160328324487*n^16-7388852535480*n^15+234145265585532*n^14-5409520907393064 *n^13+94260516179967514*n^12-1263960859910445840*n^11+13186796804734432308*n^10 -107440927423001791476*n^9+681125509044073409511*n^8-3311856545948937593880*n^7 +11911195860760286827384*n^6-28709025725894657843328*n^5+ 30388935941413580869488*n^4+59196156755319010627200*n^3-\ 238071746545757117971200*n^2+179485773643626912000000*n-11156376271245004800000 )/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 1953/32768*(4398312*n^18-712522844*n^17+53441830309*n^16-2462844853560*n^15+ 78040204169284*n^14-1802707366284888*n^13+31400171513201998*n^12-\ 420659879580134480*n^11+4378676158892202596*n^10-35477159590955059692*n^9+ 221894081807030463077*n^8-1044475731430257486360*n^7+3468629636585120731208*n^6 -6661296550081444256576*n^5-99439353448326798384*n^4+34847397973872585878400*n^ 3-68621230500338765318400*n^2+41745053206249747200000*n-3718792090415001600000) /(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 1076103/262144*(127717*n^19-23052456*n^18+1934500543*n^17-100212005824*n^16+ 3588131641754*n^15-94210037463472*n^14+1877589640296366*n^13-28995314021353888* n^12+350832112218772041*n^11-3335809730091269288*n^10+24769023732277509139*n^9-\ 140747423263352358752*n^8+583666624196985713288*n^7-1564257529940303653184*n^6+ 1555303914481742299952*n^5+5544292238211720670464*n^4-23167354690205791660800*n ^3+33095530098467715686400*n^2-17979937429847500800000*n+1997754008643993600000 )/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-\ 13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2* n)/(2*n-5), 34713/131072*(1979521*n^19-357278288*n^18+29978685699*n^17-\ 1552633175692*n^16+55567719682802*n^15-1457649467351056*n^14+28996885999402038* n^13-446141931018752904*n^12+5359178878151030933*n^11-50251860939629601824*n^10 +363441767812064944327*n^9-1965482249937572079116*n^8+7394399091781380587144*n^ 7-15584533746047908976032*n^6-4059882631990724914064*n^5+ 131507105261632267163712*n^4-368871138599294580998400*n^3+ 458531123700307945651200*n^2-239071805447100595200000*n+30965187133981900800000 )/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-\ 13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2* n)/(2*n-5), 57855/262144*(4750177*n^20-949853582*n^19+88624247379*n^18-\ 5124392056698*n^17+205654971823818*n^16-6078548397076284*n^15+ 136964746514949238*n^14-2400797383122390356*n^13+33072726464001171909*n^12-\ 358542522210277039206*n^11+3032649877233278594487*n^10-19556681398850211612594* n^9+91408496785169647796320*n^8-272286478533770837289008*n^7+ 262334365698311774264496*n^6+1657599107034004825671648*n^5-\ 8438016005222229494308224*n^4+18251961290595042341230080*n^3-\ 20159043383226932852889600*n^2+10109223360329465788416000*n-\ 1449170757870352957440000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37 )/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-\ 9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 1995/262144*(137703185*n^20-\ 27526792858*n^19+2566895125251*n^18-148273769198622*n^17+5940160950308394*n^16-\ 175039408911110676*n^15+3923641339373262422*n^14-68180957291562776284*n^13+ 925950159595117290933*n^12-9810021233603746629474*n^11+79971410505507814136103* n^10-485462403264600076364166*n^9+2034186786695069122946912*n^8-\ 4585898411685254234750992*n^7-4341360840678223117540176*n^6+ 74334912730185088212327072*n^5-271607243133385037426903424*n^4+ 521898692222889812722752000*n^3-545133753422434291224729600*n^2+ 270918444049309677164544000*n-42025951978240235765760000)/(2*n-29)/(2*n-35)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/( 2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 285/1048576*(7704456289*n^21-1697015568087*n^20+174904253384945*n^19-\ 11203109584564485*n^18+499424808459419724*n^17-16437427594530130062*n^16+ 413238166749973179250*n^15-8091660257804631487770*n^14+124553511063793743973769 *n^13-1507528813388837753672067*n^12+14207700853805575284901005*n^11-\ 101732671953202921405534305*n^10+523751159464717087396188634*n^9-\ 1657925314613406653890109832*n^8+782610896964233875517204960*n^7+ 22199658557950421960200197360*n^6-125130045860711658417557896416*n^5+ 361554671999187363139293570048*n^4-618483834042547570412470940160*n^3+ 606457233594561448729992499200*n^2-295579724875893220443512832000*n+ 48245792871019790659092480000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 95/524288*( 11534251796*n^21-2537514900543*n^20+261031844996230*n^19-16670068215357165*n^18 +739740917169272886*n^17-24178801270759278318*n^16+601623500309745915500*n^15-\ 11603818692352795207530*n^14+174745740367973033730016*n^13-\ 2048974512659184822110763*n^12+18425543969363911305359070*n^11-\ 122501302091703707408927145*n^10+548197696346857059253151126*n^9-\ 1100141931807679414784509848*n^8-4632486191260034301194616560*n^7+ 49826366860434422966030573040*n^6-216409758285826736236284817824*n^5+ 563516617470664468467732809472*n^4-912179874383201048379967434240*n^3+ 870557305297642834918549708800*n^2-424154044810063566003253248000*n+ 72368689306529685988638720000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 2185/524288*( 999107099*n^22-240804474493*n^21+27202832442478*n^20-1912530585076265*n^19+ 93683685065211549*n^18-3390148711375634208*n^17+93713222160465448508*n^16-\ 2016530004421157588770*n^15+34067099230143445482709*n^14-\ 451545379481444173818893*n^13+4641731625718780813705758*n^12-\ 35921496635082737017664325*n^11+194166208673048224612936739*n^10-\ 553442802429836512517310118*n^9-1262623536269996349628438952*n^8+ 23861923893147503384834979520*n^7-139744369328589777026920633296*n^6+ 496893099024081017486235719712*n^5-1159415737302134229449673537792*n^4+ 1752865113368987796234877539840*n^3-1607109543205229616654715084800*n^2+ 772747561957312585883086848000*n-135297984355685934674411520000)/(2*n-5)/(-1+2* n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-35)/(2*n-29), 437/524288*(4959919369*n^22-1191485496113*n^21+ 133968300702698*n^20-9357298890735565*n^19+454233521336975919*n^18-\ 16235942757418351968*n^17+441384240336501334708*n^16-9286635209761574081450*n^ 15+152172733069946181410279*n^14-1933034183987244987441793*n^13+ 18657732308172270217988538*n^12-129775429791481649496906585*n^11+ 547901124416050619327511409*n^10-20969189409072873408338558*n^9-\ 19685281651153894818544031992*n^8+166605155238900556351720057280*n^7-\ 813635535378686627455515304176*n^6+2654281648217001040992440118432*n^5-\ 5890716752322350689004384093952*n^4+8644263046952311419124785016320*n^3-\ 7813196298474198135801899212800*n^2+3763892238443864218265333760000*n-\ 676489921778429673372057600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-\ 9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n -25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 437/ 4194304*(78379266439*n^23-20494780376318*n^22+2512718757330518*n^21-\ 191735522636816620*n^20+10189806722829083889*n^19-399741281217862435698*n^18+ 11963015580392965353628*n^17-278119068159757659928280*n^16+ 5059572346517442994636649*n^15-71786921382715170710422498*n^14+ 779975066914076220622520358*n^13-6171304438931643382763896140*n^12+ 30151185263906155087715179279*n^11-5752451483942983952314314638*n^10-\ 1430367433654333856629761463672*n^9+14730356203695870714921369563600*n^8-\ 89162423229915810082307468186256*n^7+371648507720283470401050081061152*n^6-\ 1099067350306663060408985957240832*n^5+2281414988388447846774115271677440*n^4-\ 3197813980725268077938641159680000*n^3+2809080550227102706269590335488000*n^2-\ 1338367512469003847783195443200000*n+243536371840234682413940736000000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/ (2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23 )/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/2097152*(38398544267*n^23-\ 9970509054784*n^22+1211351163801724*n^21-91356813893462810*n^20+ 4782802211370427167*n^19-184053967065080193774*n^18+5373469246498523901104*n^17 -120938481777671412216340*n^16+2105439851702133510203497*n^15-\ 28024741771149373763929724*n^14+274058066006433448034302644*n^13-\ 1729127342507110348426775970*n^12+2518825634731224281877609437*n^11+ 88341985952810642248980788906*n^10-1228477612523707333581696310496*n^9+ 9557083609187662922314369937200*n^8-51452230389056358920486567584368*n^7+ 200719033538420592622535129705376*n^6-569008106269185211156501008494976*n^5+ 1148712142504758845165549158417920*n^4-1582642078337748185626688352000000*n^3+ 1379356358993970311828684611584000*n^2-658577251352319757293253017600000*n+ 121768185920117341206970368000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45) , 437/4194304*(148765452293*n^24-41743501011684*n^23+5487626394952742*n^22-\ 448463204951022096*n^21+25483304104747387283*n^20-1066395548182922894244*n^19+ 33925025807964096152072*n^18-833714166560845936653936*n^17+ 15872502588506449515882923*n^16-230898480078662776245627564*n^15+ 2446578452698729411076502782*n^14-15906423566944813737100755216*n^13-\ 3392971352957418528135755347*n^12+1593127234937141973296595700596*n^11-\ 23032587583372604088915152628748*n^10+204681504821487710411989814580624*n^9-\ 1306664727381194799588084265438672*n^8+6233581795071815730675383362696896*n^7-\ 22353485733661575069424574758582848*n^6+59556908951522005928142334342650624*n^5 -114796544875288202210921037159828480*n^4+152940708794574089887569966567936000* n^3-130384425657218297545385137668096000*n^2+ 61616442324882845776177247846400000*n-11446209476491030073455214592000000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/4194304*(141775298333*n ^24-39289956971724*n^23+5086559321343782*n^22-407905450915703136*n^21+ 22640405895868268123*n^20-919705043716559692284*n^19+28148738611269645745112*n^ 18-656105609911321506560976*n^17+11544400304386549358561363*n^16-\ 146526006723265345117563204*n^15+1123995223836124811976387422*n^14+ 793657567887556329273642144*n^13-173005500883646510831508806107*n^12+ 2972147776078476160748113566156*n^11-31930663519092555515366323541308*n^10+ 249622024719250071651561372791184*n^9-1480545496472602401714802599086032*n^8+ 6731280737505702000857126053525056*n^7-23342336413317719472470213674099008*n^6+ 60728653857084741191568001724630784*n^5-115143419220761540652907244238535680*n^ 4+151834110638463306425058456807936000*n^3-128865750847615046180487342145536000 *n^2+61007260307926740534172628582400000*n-11446209476491030073455214592000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37) /(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 37145/16777216*( 12436893403*n^25-3693863069915*n^24+512856907389430*n^23-44131238744212610*n^22 +2629054981860943405*n^21-114577597118123667605*n^20+3754433550007804584580*n^ 19-93132863308572086546360*n^18+1715537571487475472177925*n^17-\ 21607781260573213738224005*n^16+118902208798037611048917430*n^15+ 2050650735972822203996443990*n^14-70438077077792431642109181485*n^13+ 1165525362129030543795755875045*n^12-13516933882124696093403844578320*n^11+ 119109100000557011641921235953540*n^10-819104535790661441508228111741440*n^9+ 4427797538332565250117305950751920*n^8-18750489721947513950516336108901120*n^7+ 61487938327714188129718449922549440*n^6-153042901477495921682391011592791808*n^ 5+280283798387407727296666696384834560*n^4-360033704432331213588455371110912000 *n^3+300054371294079530202162640269312000*n^2-\ 140646652374686994602141928652800000*n+26393612439908728169379083059200000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 2185/8388608*( 95573271898*n^25-27798021774515*n^24+3762120148610680*n^23-313589226827171210*n ^22+17932644486703946230*n^21-739516359928507874405*n^20+ 22355909253642859485580*n^19-485208874120306643585960*n^18+ 6732479327531401189878550*n^17-21324676689214878450055805*n^16-\ 1717387776558567796875876320*n^15+54798155142709569602113849390*n^14-\ 1016049163189599049909955651510*n^13+13697239761296020394453005322245*n^12-\ 142729720089611747293803717286820*n^11+1176162254070986925931331360475940*n^10-\ 7722557204369637407505464884957040*n^9+40359654720966978203416423589203120*n^8-\ 166637708843808046328290054473213120*n^7+536101747920919459667091971190939840*n ^6-1315546933604438217237285050451088128*n^5+ 2385301108576138946010278920049679360*n^4-3044980449426898286702238435678720000 *n^3+2531228123227492251115132441300992000*n^2-\ 1188073341686115124320568526438400000*n+224345705739224189439722206003200000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2* n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/( 2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 6555/8388608*( 55202651662*n^26-16998487989106*n^25+2430970909419975*n^24-213342656364788600*n ^23+12753834519564876140*n^22-541797690919477822830*n^21+ 16304352421176321478405*n^20-317913610988644646017900*n^19+ 2031221190962362250678970*n^18+113054530015782365474536930*n^17-\ 5253212573879968094775792895*n^16+133741571582708613627498376800*n^15-\ 2461207700971228246006506263120*n^14+35191844963141630666804132396750*n^13-\ 402285870736913267051359742341245*n^12+3721862847843731109160732840106500*n^11-\ 27980599465752160238026605936457540*n^10+170686860007171207712121953115832560*n ^9-839880181140036499098636828266902320*n^8+ 3299034669683328376855158819400171200*n^7-\ 10183805973038450321312003155016402112*n^6+ 24154137509252163046999212986264445696*n^5-\ 42606059244711481350608763140817361920*n^4+ 53230413749047917279434303996017152000*n^3-\ 43558341736311825505879909827907584000*n^2+ 20247101409939340266658492106342400000*n-3813876997566811220475277502054400000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 345/ 8388608*(853555290678*n^26-252274448887214*n^25+34184634516811025*n^24-\ 2782270276882886400*n^23+148126876763787432160*n^22-5101157784754910648770*n^21 +88649805371847186519195*n^20+1413352623858265401496900*n^19-\ 163598170666280593003545070*n^18+6611676536440708112724728670*n^17-\ 180593977274321333052867730505*n^16+3744666481473771737259245706200*n^15-\ 61547510039639470803663026166780*n^14+818228805534007309251824803389250*n^13-\ 8885447772162386224035274003679155*n^12+79122992594517487062257067862390500*n^ 11-577555002222323575393269987052509260*n^10+ 3442344351887804160261025738603066640*n^9-\ 16628887431507556536863267735837632080*n^8+ 64370177420390327477080237471139044800*n^7-\ 196448563171531194949452211222117105728*n^6+ 461933610139774399295102147822882951424*n^5-\ 809839764445114565416618202535472788480*n^4+ 1007972130707745540286570557699829248000*n^3-\ 823609260026530497236003192795344896000*n^2+ 383199288750585970470246535313817600000*n-\ 72463662953769413189030272539033600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 345/67108864*(10034160714653*n^27-\ 3021663529688796*n^26+407642122690682841*n^25-31421030784373795500*n^24+ 1377910112373899430885*n^23-16642163622415168372860*n^22-\ 2254669481142786685938675*n^21+197128740051112593538978020*n^20-\ 9456276398369918910120945045*n^19+325464679135522354375058096700*n^18-\ 8681787546259095075799914004485*n^17+185786279516849395379996481490620*n^16-\ 3248626639693344204916134290097705*n^15+46899149557251870612741761641807020*n^ 14-562081593470542683636852897789433905*n^13+ 5604067907397877210273084231151395020*n^12-\ 46448209039456319568803027312345848260*n^11+ 318991628661816899603375789358677255280*n^10-\ 1804610566640052128456669874675631328240*n^9+ 8335994787713668273267530193513550003520*n^8-\ 31054767061354826609029457298958271022528*n^7+ 91727963019343540598485258259774699878656*n^6-\ 209826677633268516272412311057422224577536*n^5+ 359544834808678890264622273785108255928320*n^4-\ 439346572387018978750689193791801311232000*n^3+ 353978426899171236466077316132575412224000*n^2-\ 163136523496533249044716193703120076800000*n+ 30724593092398231192148835556550246400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1725/33554432*(599741118575*n^27-\ 151999270493688*n^26+14461963363595007*n^25-256594147682226120*n^24-\ 84712178543903666145*n^23+11164796864787666440808*n^22-799909052545367574659421 *n^21+40192783876316317857821496*n^20-1536973959522760295528823975*n^19+ 46538988986056934279057261688*n^18-1141665690951456850727870571171*n^17+ 23015881827460485790756919047656*n^16-384771429058629229720521979430235*n^15+ 5363101724690144624122176856525368*n^14-62490794917657124917717909317237231*n^ 13+608885143436594610428213548124738856*n^12-\ 4951832872392662854102640956612497420*n^11+ 33477421466848738538985655081739037408*n^10-\ 186941221654825170957651004607446135376*n^9+ 854338869967535724063344494535978391936*n^8-\ 3155215154036461425837317373206200361280*n^7+ 9255798063685290379214161970217719548416*n^6-\ 21061924714464292520073206730551058167808*n^5+ 35956608982866246210278811009222536626176*n^4-\ 43838027115003559130351540837742901739520*n^3+ 35290348620380261536673487491065036800000*n^2-\ 16274493554535562452952213991661502464000*n+ 3072459309239823119214883555655024640000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1725/67108864*(668928961913*n^28+ 4205315837702*n^27-51027614884073331*n^26+12003752762981154630*n^25-\ 1511537793915126982095*n^24+128030714338930306736742*n^23-\ 7989288303758138914101279*n^22+385770068255942147323399854*n^21-\ 14858864280670991466169389105*n^20+465877866346960227903911290122*n^19-\ 12057397371310552973517387689289*n^18+260105293038587167239014243218194*n^17-\ 4707920614132909217082723163216005*n^16+71797387255570595540429718166658562*n^ 15-924531673457553013228565234566112949*n^14+ 10055804003726650841868499285004234394*n^13-\ 92255750220666781500258800086899405300*n^12+ 711690578843192153206211014069028996232*n^11-\ 4593332692276208764625935534809546828464*n^10+ 24625215680526203382921867704887060356064*n^9-\ 108588963191822018348886597378937457803968*n^8+ 388715876569196968150617568978328838253440*n^7-\ 1109839595232479828708756494401641212658688*n^6+ 2467440716771453626508265539227028367396864*n^5-\ 4130347153359310483644708063248866312765440*n^4+ 4954601106231182677609869949469106086707200*n^3-\ 3937654721663449570345016960744511799296000*n^2+ 1799127660082920095028511183706770636800000*n-\ 337970524016380543113637191122052710400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -75/67108864*( 25603167554093*n^28-15877307730332566*n^27+4044840937950353121*n^26-\ 604585022800935471750*n^25+61294322373056271933525*n^24-\ 4553887761235223918844246*n^23+259921595406937883756385477*n^22-\ 11758061847256203029388499662*n^21+430765652759195688676608701835*n^20-\ 12978849912374422382894049648186*n^19+325177718384752570572603851928627*n^18-\ 6828235823973732700991074589182482*n^17+120818218784006005791890563030565895*n^ 16-1807321614150344601901030904613871506*n^15+ 22892076551677250609000547073879287447*n^14-\ 245490085372720566949877077951747668282*n^13+ 2225020520473238112247027458511591393260*n^12-\ 16986795560740125495096613785482066694216*n^11+ 108666386578415568481131000966708360453712*n^10-\ 578217320581724497732742735340780013114592*n^9+ 2533844234896075267061711822006576283070272*n^8-\ 9024067003001590533386714619946206986743680*n^7+ 25660202965924173450026648665733947494383616*n^6-\ 56872027275192450194230287892788679218143232*n^5+ 94992083508496787666793528089111727688581120*n^4-\ 113798505033537442371130694764369096042905600*n^3+ 90398644746611873191881294124764403826688000*n^2-\ 41320294324727800913341585956234185932800000*n+ 7773322052376752491613655395807212339200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), -2175/268435456*( 18252341335091*n^29-9135326043687793*n^28+2119227063796217949*n^27-\ 305503001399240952237*n^26+30877047178936798755825*n^25-\ 2336759280414466846555617*n^24+137962803667995439820877801*n^23-\ 6532237771146349644020237313*n^22+252902472751538956843808089779*n^21-\ 8119616751550087387750771191567*n^20+218404343669525192519360665239231*n^19-\ 4958538949355136809248432437602103*n^18+95515166367807728025082927365250539*n^ 17-1566322175515914096533779664703500427*n^16+ 21905513274080132337817839215457626211*n^15-\ 261355167969358431870929196275856161643*n^14+ 2657335513585749747105186160778216825694*n^13-\ 22967382632256850740789069430706701244052*n^12+ 168063669985613380068684042308967434034136*n^11-\ 1035218077812861502441491583396409158425968*n^10+ 5325977113952664067119632192272876657298208*n^9-\ 22651475357032135727806337991926122322829504*n^8+ 78565156381358983788363664281493464324569472*n^7-\ 218268245442933584818896857928898615699124736*n^6+ 474060941939390177726140579444759998221604864*n^5-\ 778148616075614826647853008097944525043671040*n^4+ 918638723261744184019585487300197042426675200*n^3-\ 721097354469910082598692833690706225135616000*n^2+ 326649174448214271491266873971262095360000000*n-\ 61114394066962054071997014836001531494400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -435/ 134217728*(72180244470128*n^29-33484620953312083*n^28+7322485282727215398*n^27-\ 1006688175823716624063*n^26+97847183384342117803350*n^25-\ 7166150580626724708718515*n^24+411438600989101997932945350*n^23-\ 19017630566361985222241446155*n^22+721052940982726295945373161790*n^21-\ 22730334001380015556137874367085*n^20+601656170447462109749981266607370*n^19-\ 13467344085257980143206390038390125*n^18+256187829005711566284136517705446530*n ^17-4154854341994028293283094348650730225*n^16+ 57540967478066005834503036400880499810*n^15-\ 680622649270389643444408062746275435065*n^14+ 6867954209236641240711330598215465668250*n^13-\ 58967304843522005187315965292572713947420*n^12+ 429012843981950751017781327078493495347880*n^11-\ 2629495315134860377551014417873867913444560*n^10+ 13471150717110439907805856175479789999259232*n^9-\ 57090510847558443370350342924565237486251072*n^8+ 197440265331459706108890135550408846755576192*n^7-\ 547258220123345963535579141699978491218540032*n^6+ 1186514828714386327287103788479536085660190720*n^5-\ 1945207852313903667339060609128345834046873600*n^4+ 2294715726715990443660258220087002090897408000*n^3-\ 1800814618635201232848189776001790892113920000*n^2+ 815948880303620656072961335499816986214400000*n-\ 152785985167405135179992537090003828736000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -87/ 134217728*(962789225749223*n^30-458558763597184365*n^29+103797555830991323590*n ^28-14873089535366981833635*n^27+1515846259987737105619422*n^26-\ 117046079528438182479382935*n^25+7120785859918660699971082050*n^24-\ 350434364676502653570541573575*n^23+14212092266787025317119199921420*n^22-\ 481426641621992959934565918118725*n^21+13756746082668201264021515864778150*n^20 -334005482944400403034842211923516225*n^19+ 6925994982299521052026499503532761890*n^18-\ 123084285625717281250422571936468205325*n^17+ 1878375878009865018089209265785538431350*n^16-\ 24633018358759586009166963746071464528525*n^15+ 277435021603501608269082301201862940253245*n^14-\ 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n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -29/134217728*(3513148220705299*n^30-1638084119043757695*n^29+ 363811755089069127770*n^28-51248215163408138106105*n^27+ 5143438145544425871393186*n^26-391671564975243292687934805*n^25+ 23530667995599805069465716150*n^24-1144903097138514177821365174725*n^23+ 45955749088710495590792657523960*n^22-1542242834033103977649667109492175*n^21+ 43698154912881897071261421708354450*n^20-1052876384454055810966182323759430675* n^19+21682319795442363097718037680894645070*n^18-\ 382934334118846112932025656067103069975*n^17+ 5811367546400204530229945046911905024050*n^16-\ 75830441179179985707632475130917606163575*n^15+ 850266709997238320955657019612628624909685*n^14-\ 8177279423996165136391327296009332582648550*n^13+ 67241114230598584852180282076882351684773100*n^12-\ 470589815907628812879204320897620672796561400*n^11+ 2785654199064125874953810559829627746944211856*n^10-\ 13833398681675020126214226431465716496506354080*n^9+ 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18403534121575330568417122378879949783445914050560*n^5-\ 29252406465846514542718991456094444725494554624000*n^4+ 33608219825301793248972519148711018623409520640000*n^3-\ 25803351884418619481625555623701189231265710080000*n^2+ 11494020270816479306480958728883938228856422400000*n-\ 2128555202390288057468876674788092050145280000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-45), -31/536870912*(33028595855166973*n^31-16053992728956787064*n^30+ 3730553363778445931810*n^29-551793269581995725900530*n^28+ 58354668669507297255045402*n^27-4698715860053950709693624346*n^26+ 299529129921297131312752774830*n^25-15518516987483256937908301082850*n^24+ 665668512145390636654118773498020*n^23-23961493182571425690612829912581810*n^22 +731031656380074527941999862481969450*n^21-\ 19041994461470013974532602399080787550*n^20+ 425749808336540357523871232862059153190*n^19-\ 8200949798755166853996899607315288804270*n^18+ 136408391968665008032989565456958945811450*n^17-\ 1961326133878176546751496006396713154536950*n^16+ 24375685685918820421015663383181702473583695*n^15-\ 261542741172294087481316837352351932697596910*n^14+ 2417079040912866380533748314814114334040023500*n^13-\ 19171373739111003280285730265910963642030077400*n^12+ 129863673312538549789435806454347892025261532112*n^11-\ 746381324366572203236239540358346933244581082016*n^10+ 3609127778651119448257950734353654142370264075840*n^9-\ 14524098535125512802208417966756027472949009966720*n^8+ 47964565926612453375523119934873361288363564740608*n^7-\ 127618105940588050343309772598514775582378967363584*n^6+ 266918086189886177061575106539804359984249844613120*n^5-\ 424142969740693164838466954183786797033119387648000*n^4+ 487216731784559110018855922263715825651620577280000*n^3-\ 374049563810348937521683297677022609462398812160000*n^2+ 166629755767137422296009581032974854968888524800000*n-\ 30864050434659176833298711784427334727106560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2 *n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/ (2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57 )/(2*n-45), -31/1073741824*(140281157667292627*n^32-72215957889491812304*n^31+ 17806905225854126930264*n^30-2800313240286619759403680*n^29+ 315501964437696691330218228*n^28-27121917044044070550257802336*n^27+ 1849938915909389806178837629656*n^26-102792000708332716464051684120480*n^25+ 4740576566224528781084944891545330*n^24-183947327766511011385123786702805760*n^ 23+6066591948235981131426665400333992760*n^22-\ 171342674227803947523045657189891458400*n^21+ 4167500762920128999548184652257871171860*n^20-\ 87640179816894615899048052034037603567520*n^19+ 1597701792813984155758787572366733129382120*n^18-\ 25286692187882998203775463183543387322348000*n^17+ 347588819538928630346816680162406070002212755*n^16-\ 4147110795926548848428501861050228871786694960*n^15+ 42876089763820768399866044432518470078153696560*n^14-\ 383083503629362322064111931947264741716683945600*n^13+ 2946414224549126921095423155899977265793968838688*n^12-\ 19406926392158453226730542875656487933392821118976*n^11+ 108729936832941586428216040768479188275200543119616*n^10-\ 513707459570980282989725492549709793708205669406720*n^9+ 2024240447730684126270815448905985976820202212512512*n^8-\ 6558832555441544062357697150225957561909797270278144*n^7+ 17154419167958244192771347312891602150485951279489024*n^6-\ 35333443719529243831676551291757600030929486064517120*n^5+ 55389267395851114303039631941833456807947986526208000*n^4-\ 62877196626763780346615639608330314253654227025920000*n^3+ 47788905536849621614489872638791450577708329205760000*n^2-\ 21116176036677352149029999327282508715985259724800000*n+ 3888870354767056280995637684837844175615426560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/ (2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37 )/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 57)/(2*n-63)/(2*n-45), (849344521648972440*(n-31)!*(2*n-67)!*(n+1)+ 1274016782473458660*((n+1)*(2*n-66)!-1/1274016782473458660*binomial(2*n,n)*(n-\ 34)!*(n-31)!)*(n-32)!)/binomial(2*n,n)/(n-34)!/(n-32)!/(n-31)!, (( 849344521648972440*n+849344521648972440)*(2*n-67)!-(n-34)!*(n-32)!*binomial(2*n ,n))/(n-34)!/(n-32)!/binomial(2*n,n)] The limits, as n goes to infinity are 8589934557 2147483499 268435293 1073737917 137436646851 68715112473 [----------, ----------, ---------, ----------, ------------, -----------, 8589934592 2147483648 268435456 1073741824 137438953472 68719476736 274821490335 274717854075 2195770042365 273938480155 2183049011315 ------------, ------------, -------------, ------------, -------------, 274877906944 274877906944 2199023255552 274877906944 2199023255552 2167484764253 34251739433843 16780163844679 65010502652041 -------------, --------------, --------------, --------------, 2199023255552 35184372088832 17592186044416 70368744177664 61955805371521 461968405454435 104413799548565 180926690822205 --------------, ---------------, ---------------, ---------------, 70368744177664 562949953421312 140737488355328 281474976710656 147238287641955 3461785446555285 1034553429541875 1153902459299925 ---------------, ----------------, ----------------, -----------------, 281474976710656 9007199254740992 4503599627370496 18014398509481984 -1920237566556975 -39698842403822925 -1962400396531605 -----------------, ------------------, -----------------, 18014398509481984 144115188075855872 4503599627370496 -83762662640182401 -101881298400453671 -1866937227559376051 ------------------, -------------------, --------------------, 144115188075855872 144115188075855872 2305843009213693952 -1023886471510176163 -4348715887686071437 -4505517953221266349 --------------------, --------------------, --------------------, 1152921504606846976 4611686018427387904 4611686018427387904 -18340576008503430061 ---------------------] 18446744073709551616 and in Maple notation [8589934557/8589934592, 2147483499/2147483648, 268435293/268435456, 1073737917/ 1073741824, 137436646851/137438953472, 68715112473/68719476736, 274821490335/ 274877906944, 274717854075/274877906944, 2195770042365/2199023255552, 273938480155/274877906944, 2183049011315/2199023255552, 2167484764253/ 2199023255552, 34251739433843/35184372088832, 16780163844679/17592186044416, 65010502652041/70368744177664, 61955805371521/70368744177664, 461968405454435/ 562949953421312, 104413799548565/140737488355328, 180926690822205/ 281474976710656, 147238287641955/281474976710656, 3461785446555285/ 9007199254740992, 1034553429541875/4503599627370496, 1153902459299925/ 18014398509481984, -1920237566556975/18014398509481984, -39698842403822925/ 144115188075855872, -1962400396531605/4503599627370496, -83762662640182401/ 144115188075855872, -101881298400453671/144115188075855872, -\ 1866937227559376051/2305843009213693952, -1023886471510176163/ 1152921504606846976, -4348715887686071437/4611686018427387904, -\ 4505517953221266349/4611686018427387904, -18340576008503430061/ 18446744073709551616] and in floating point [.9999999959, .9999999306, .9999993928, .9999963613, .9999832171, .9999364916, .9997947576, .9994177311, .9985206099, .9965823852, .9927357548, .9856579546, .\ 9734929857, .9538418820, .9238548082, .8804449489, .8206207366, .7419046678, .6\ 427807293, .5230954785, .3843353909, .2297170075, .6405445392e-1, -.1065945980, -.2754660555, -.4357404208, -.5812202292, -.7069435204, -.8096549592, -.8880799\ 494, -.9429774426, -.9769784706, -.9942446177] The cut off is at j=, 24 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 35], vs. those in the, 2, -th row from j=1 to j=, 34, are as follws 17 16 15 14 [(4294967287 n - 620622770294 n + 41326174496524 n - 1681887611428040 n 13 12 + 46783246957198234 n - 942213798732565268 n 11 10 + 14198702525918755988 n - 163145994444906530200 n 9 8 + 1443046294627484046071 n - 9848805939065658203782 n 7 6 + 51649356296328111454952 n - 205753566169714354508080 n 5 4 + 608995315179980511380208 n - 1278638997042445037506656 n 3 2 + 1651609137498410593345536 n - 266273559765416561287680 n - 4074390119603843530444800 n + 7262800952580498124800000)/(32768 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) 18 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 29)), 3 (5726622851 n 17 16 15 - 927712842957 n + 69582745863042 n - 3206791883712660 n 14 13 + 101621496391578282 n - 2347880114090657454 n 12 11 + 40915873074173089304 n - 548802231799707007980 n 10 9 + 5729885790153430760283 n - 46779630300678208897821 n 8 7 + 298203572615437825556766 n - 1472027770563536604414360 n 6 5 + 5520040830511845486802784 n - 15034787841095030750994768 n 4 3 + 25947367794727700748609888 n - 10368802428280141741632000 n 2 - 75348188816692285277683200 n + 157653365970007986946560000 n - 4841867301720332083200000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) 18 17 (2 n - 35) (2 n - 29)), 3 (2863310584 n - 463856065524 n 16 15 14 + 34791302518167 n - 1603387296100200 n + 50810010762998508 n 13 12 - 1173893733313355208 n + 20455720043532108634 n 11 10 - 274318734779733586800 n + 2862540119522916736332 n 9 8 - 23334668334064093675812 n + 148110012082298973865431 n 7 6 - 722205771633751127541000 n + 2614534579414879749290776 n 5 4 - 6402616276817017188459456 n + 7202006728521566561718768 n 3 2 + 11947714480011510115344000 n - 52299288490315162875667200 n + 40022575038359617777920000 n - 2420933650860166041600000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 19 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 1767 (9722600 n 18 17 16 - 1754924416 n + 147274122210 n - 7629798997359 n 15 14 13 + 273237166585680 n - 7176929688012252 n + 143156607455941340 n 12 11 - 2214799856266277098 n + 26903270147799027840 n 10 9 - 257920703660295824148 n + 1948132864678918402290 n 8 7 - 11464664199369686322927 n + 51106321784203285468280 n 6 5 - 161144433292881303002584 n + 290940455794794243488160 n 4 3 + 39532792540808478666384 n - 1553986219157004631478400 n 2 + 2944743267993261524294400 n - 1757631826137982798080000 n + 152079023908024012800000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 19 18 (2 n - 3) (-1 + 2 n) (2 n - 5)), 5301 (3240842 n - 584964766 n 17 16 15 + 49089469458 n - 2543042876429 n + 91060920600604 n 14 13 12 - 2391231228968792 n + 47670275478818196 n - 736577452369912998 n 11 10 + 8922044562861324466 n - 85009469370262226218 n 9 8 + 633633502216100906234 n - 3625491567968663473717 n 7 6 + 15223648984816486966888 n - 41858399314241904611624 n 5 4 + 46507610156588643502112 n + 129293771652284762820144 n 3 2 - 585427340038695760828800 n + 853411474455103697414400 n - 466145218461264249600000 n + 50693007969341337600000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) 20 (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 153729 (1788001 n 19 18 17 16 - 357585770 n + 33373356435 n - 1930730829550 n + 77562206801306 n 15 14 13 - 2296704213429940 n + 51922638453609270 n - 915552440141979900 n 12 11 + 12744629947925620901 n - 140669212670945511410 n 10 9 + 1226581386473659939255 n - 8326507919744448646550 n 8 7 + 42563202010291758825776 n - 151998046738689023499280 n 6 5 + 302430978852646153979440 n + 110657008489929276996000 n 4 3 - 2552746581936992060025984 n + 6918768812731587524006400 n 2 - 8421538302928647425894400 n + 4318462768907027220480000 n - 545386844359810252800000)/(262144 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 20 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 34713 (7917603 n 19 18 17 - 1583323390 n + 147746684505 n - 8544771951850 n 16 15 14 + 343052523812318 n - 10146204051967900 n + 228871322087387810 n 13 12 - 4019133500500114100 n + 55530580006186109903 n 11 10 - 604826572949156639510 n + 5152878634083331794565 n 9 8 - 33602880816627588001250 n + 159960576371268556983328 n 7 6 - 494680050220409420748720 n + 584743170558041408603920 n 5 4 + 2438411615103601274047200 n - 13731202243494575304193152 n 3 2 + 30511394560922186805419520 n - 34085794585515835761100800 n + 17123385025243490273280000 n - 2415284596450588262400000)/(262144 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 21 20 19 (2 n - 5)), 1653 (332460442 n - 73285163847 n + 7562751756140 n 18 17 16 - 485418108513765 n + 21711839778176142 n - 718397615222541702 n 15 14 + 18211134617522484760 n - 361176272556562052730 n 13 12 + 5667947349821347194422 n - 70610111897769202454787 n 11 10 + 694722386513553999963180 n - 5312146041142665666304545 n 9 8 + 30496837312852779870224962 n - 121637792495219425962318432 n 7 6 + 261418656751355609225190800 n + 255758016624601737414945840 n 5 4 - 4073066701615450250514175968 n + 14481878010890741538816048768 n 3 2 - 27259483390759601275768634880 n + 27995005164416449196514355200 n - 13695967105713670054717440000 n + 2079560037543956493926400000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 21 20 (2 n - 35) (2 n - 29)), 57 (19271523829 n - 4246245689238 n 19 18 17 + 437873703985655 n - 28070072756157810 n + 1252911163646311554 n 16 15 - 41314509270919092348 n + 1041535038217974788830 n 14 13 - 20476164400184071942020 n + 316977186942027926566889 n 12 11 - 3867095263765521906924078 n + 36855061232907655820146155 n 10 9 - 268262618013785330088974730 n + 1419273627066751028885509744 n 8 7 - 4786678596528413068154434848 n + 4644062130828968656062180080 n 6 5 + 47849562389709129231448193760 n - 299534315514204865258718822016 n 4 3 + 894081232185268080279878560512 n - 1553405105169970325381679390720 n 2 + 1534254010572861601209504460800 n - 747843363291480922180915200000 n + 120614482177549476647731200000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 285 ( 22 21 20 15397375793 n - 3720708042601 n + 421921885540771 n 19 18 - 29828590034405855 n + 1472774450530816818 n 17 16 - 53896840383929148606 n + 1513352327197437695606 n 15 14 - 33274875604276673713390 n + 579031745683785780583813 n 13 12 - 7994425458695472178282901 n + 87059631793318839437832231 n 11 10 - 735095670218147931490072275 n + 4634311581774478180645892648 n 9 8 - 19873393117978421643132107476 n + 39202962302083919863061289136 n 7 6 + 147651030998835938674967814640 n - 1585093089001904888406418505472 n 5 4 + 6749601105707192039712376555584 n - 17244306048364054740357652767744 n 3 + 27443810120661648256911667706880 n 2 - 25793638389518549864491557273600 n + 12381481758321938664720654336000 n - 2074569093453850998340976640000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 475 ( 22 21 20 4607086295 n - 1111640743807 n + 125776014281302 n 19 18 17 - 8862180530810915 n + 435403063982255985 n - 15819335752158146208 n 16 15 + 439623695513114919884 n - 9526214104216934979382 n 14 13 + 162416953426902177964345 n - 2179155026975207048086319 n 12 11 + 22782256450438905238930566 n - 180900387013473515752843479 n 10 9 + 1026085391565767420108334575 n - 3408942406895568145607064610 n 8 7 - 1605388162046467671878813192 n + 94900538665484417464440998464 n 6 5 - 606708368165995398165833955600 n + 2232337464537556809317430690144 n 4 3 - 5303883285191861028059464546560 n + 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(2 n - 69)! (n + 1) + 4877664252898384584 (n - 33)! ((n + 1) (2 n - 68)! - 1/4877664252898384584 binomial(2 n, n) (n - 35)! (n - 32)!))/((n - 35)! (n - 33)! (n - 32)! binomial(2 n, n)), ( (3251776168598923056 n + 3251776168598923056) (2 n - 69)! - (n - 35)! (n - 33)! binomial(2 n, n))/((n - 35)! (n - 33)! binomial(2 n, n))] and in Maple notation [1/32768*(4294967287*n^17-620622770294*n^16+41326174496524*n^15-\ 1681887611428040*n^14+46783246957198234*n^13-942213798732565268*n^12+ 14198702525918755988*n^11-163145994444906530200*n^10+1443046294627484046071*n^9 -9848805939065658203782*n^8+51649356296328111454952*n^7-\ 205753566169714354508080*n^6+608995315179980511380208*n^5-\ 1278638997042445037506656*n^4+1651609137498410593345536*n^3-\ 266273559765416561287680*n^2-4074390119603843530444800*n+ 7262800952580498124800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2 *n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17)/ (2*n-23)/(2*n-29), 3/65536*(5726622851*n^18-927712842957*n^17+69582745863042*n^ 16-3206791883712660*n^15+101621496391578282*n^14-2347880114090657454*n^13+ 40915873074173089304*n^12-548802231799707007980*n^11+5729885790153430760283*n^ 10-46779630300678208897821*n^9+298203572615437825556766*n^8-\ 1472027770563536604414360*n^7+5520040830511845486802784*n^6-\ 15034787841095030750994768*n^5+25947367794727700748609888*n^4-\ 10368802428280141741632000*n^3-75348188816692285277683200*n^2+ 157653365970007986946560000*n-4841867301720332083200000)/(2*n-5)/(-1+2*n)/(2*n-\ 3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-35)/(2*n-29), 3/32768*(2863310584* n^18-463856065524*n^17+34791302518167*n^16-1603387296100200*n^15+ 50810010762998508*n^14-1173893733313355208*n^13+20455720043532108634*n^12-\ 274318734779733586800*n^11+2862540119522916736332*n^10-23334668334064093675812* n^9+148110012082298973865431*n^8-722205771633751127541000*n^7+ 2614534579414879749290776*n^6-6402616276817017188459456*n^5+ 7202006728521566561718768*n^4+11947714480011510115344000*n^3-\ 52299288490315162875667200*n^2+40022575038359617777920000*n-\ 2420933650860166041600000)/(2*n-29)/(2*n-35)/(2*n-23)/(2*n-17)/(2*n-15)/(2*n-25 )/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19 )/(2*n-3)/(-1+2*n)/(2*n-5), 1767/32768*(9722600*n^19-1754924416*n^18+ 147274122210*n^17-7629798997359*n^16+273237166585680*n^15-7176929688012252*n^14 +143156607455941340*n^13-2214799856266277098*n^12+26903270147799027840*n^11-\ 257920703660295824148*n^10+1948132864678918402290*n^9-11464664199369686322927*n ^8+51106321784203285468280*n^7-161144433292881303002584*n^6+ 290940455794794243488160*n^5+39532792540808478666384*n^4-\ 1553986219157004631478400*n^3+2944743267993261524294400*n^2-\ 1757631826137982798080000*n+152079023908024012800000)/(2*n-29)/(2*n-35)/(2*n-23 )/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-\ 11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 5301/32768*( 3240842*n^19-584964766*n^18+49089469458*n^17-2543042876429*n^16+91060920600604* n^15-2391231228968792*n^14+47670275478818196*n^13-736577452369912998*n^12+ 8922044562861324466*n^11-85009469370262226218*n^10+633633502216100906234*n^9-\ 3625491567968663473717*n^8+15223648984816486966888*n^7-41858399314241904611624* n^6+46507610156588643502112*n^5+129293771652284762820144*n^4-\ 585427340038695760828800*n^3+853411474455103697414400*n^2-\ 466145218461264249600000*n+50693007969341337600000)/(2*n-29)/(2*n-35)/(2*n-23)/ (2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11 )/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 153729/262144*( 1788001*n^20-357585770*n^19+33373356435*n^18-1930730829550*n^17+77562206801306* n^16-2296704213429940*n^15+51922638453609270*n^14-915552440141979900*n^13+ 12744629947925620901*n^12-140669212670945511410*n^11+1226581386473659939255*n^ 10-8326507919744448646550*n^9+42563202010291758825776*n^8-\ 151998046738689023499280*n^7+302430978852646153979440*n^6+ 110657008489929276996000*n^5-2552746581936992060025984*n^4+ 6918768812731587524006400*n^3-8421538302928647425894400*n^2+ 4318462768907027220480000*n-545386844359810252800000)/(2*n-29)/(2*n-35)/(2*n-39 )/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-\ 21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 34713/ 262144*(7917603*n^20-1583323390*n^19+147746684505*n^18-8544771951850*n^17+ 343052523812318*n^16-10146204051967900*n^15+228871322087387810*n^14-\ 4019133500500114100*n^13+55530580006186109903*n^12-604826572949156639510*n^11+ 5152878634083331794565*n^10-33602880816627588001250*n^9+ 159960576371268556983328*n^8-494680050220409420748720*n^7+ 584743170558041408603920*n^6+2438411615103601274047200*n^5-\ 13731202243494575304193152*n^4+30511394560922186805419520*n^3-\ 34085794585515835761100800*n^2+17123385025243490273280000*n-\ 2415284596450588262400000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37 )/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-\ 9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 1653/262144*(332460442*n^21-\ 73285163847*n^20+7562751756140*n^19-485418108513765*n^18+21711839778176142*n^17 -718397615222541702*n^16+18211134617522484760*n^15-361176272556562052730*n^14+ 5667947349821347194422*n^13-70610111897769202454787*n^12+ 694722386513553999963180*n^11-5312146041142665666304545*n^10+ 30496837312852779870224962*n^9-121637792495219425962318432*n^8+ 261418656751355609225190800*n^7+255758016624601737414945840*n^6-\ 4073066701615450250514175968*n^5+14481878010890741538816048768*n^4-\ 27259483390759601275768634880*n^3+27995005164416449196514355200*n^2-\ 13695967105713670054717440000*n+2079560037543956493926400000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35) /(2*n-29), 57/524288*(19271523829*n^21-4246245689238*n^20+437873703985655*n^19-\ 28070072756157810*n^18+1252911163646311554*n^17-41314509270919092348*n^16+ 1041535038217974788830*n^15-20476164400184071942020*n^14+ 316977186942027926566889*n^13-3867095263765521906924078*n^12+ 36855061232907655820146155*n^11-268262618013785330088974730*n^10+ 1419273627066751028885509744*n^9-4786678596528413068154434848*n^8+ 4644062130828968656062180080*n^7+47849562389709129231448193760*n^6-\ 299534315514204865258718822016*n^5+894081232185268080279878560512*n^4-\ 1553405105169970325381679390720*n^3+1534254010572861601209504460800*n^2-\ 747843363291480922180915200000*n+120614482177549476647731200000)/(2*n-5)/(-1+2* n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29), 285/1048576*(15397375793*n^22-3720708042601*n^21+421921885540771* n^20-29828590034405855*n^19+1472774450530816818*n^18-53896840383929148606*n^17+ 1513352327197437695606*n^16-33274875604276673713390*n^15+ 579031745683785780583813*n^14-7994425458695472178282901*n^13+ 87059631793318839437832231*n^12-735095670218147931490072275*n^11+ 4634311581774478180645892648*n^10-19873393117978421643132107476*n^9+ 39202962302083919863061289136*n^8+147651030998835938674967814640*n^7-\ 1585093089001904888406418505472*n^6+6749601105707192039712376555584*n^5-\ 17244306048364054740357652767744*n^4+27443810120661648256911667706880*n^3-\ 25793638389518549864491557273600*n^2+12381481758321938664720654336000*n-\ 2074569093453850998340976640000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 475/ 524288*(4607086295*n^22-1111640743807*n^21+125776014281302*n^20-\ 8862180530810915*n^19+435403063982255985*n^18-15819335752158146208*n^17+ 439623695513114919884*n^16-9526214104216934979382*n^15+162416953426902177964345 *n^14-2179155026975207048086319*n^13+22782256450438905238930566*n^12-\ 180900387013473515752843479*n^11+1026085391565767420108334575*n^10-\ 3408942406895568145607064610*n^9-1605388162046467671878813192*n^8+ 94900538665484417464440998464*n^7-606708368165995398165833955600*n^6+ 2232337464537556809317430690144*n^5-5303883285191861028059464546560*n^4+ 8100913456320741096467454669312*n^3-7462585772831811934388501913600*n^2+ 3586018821711343204280802508800*n-622370728036155299502292992000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29), 10925/524288*(398625236*n^23-104909765059*n^22+ 12973801220413*n^21-1001340074510468*n^20+54016939704681261*n^19-\ 2160552036452343303*n^18+66300972796809236816*n^17-1592339601689612651656*n^16+ 30233226027954634497406*n^15-454592986233735165963377*n^14+ 5373112553497065646283469*n^13-48869219991133391877111084*n^12+ 324806458292788719994129361*n^11-1346641822587834844884755701*n^10+ 562987778985991527563804662*n^9+39452464652670086431893594808*n^8-\ 334746857013422713176010049664*n^7+1609054624887077381893988779440*n^6-\ 5156814613427825633499213315360*n^5+11252849122256316092597331014400*n^4-\ 16257910842953833720212818265600*n^3+14484351789024835974320655360000*n^2-\ 6880474373917475018195988480000*n+1217681859201173412069703680000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/1048576*(19755725179*n^23-5180013093317*n ^22+637334421471539*n^21-48851413502781505*n^20+2610933479253658944*n^19-\ 103152453222568640352*n^18+3114429762149730912514*n^17-73215613732701331925330* n^16+1351205705775904835834219*n^15-19547238479789170409437717*n^14+ 218573978135138642924888799*n^13-1818745115318390227693420605*n^12+ 10091859807248194639501593514*n^11-19992021236114901180810718742*n^10-\ 253187181669865381725618693796*n^9+3236752488546703364518606430720*n^8-\ 20893031240502464282535698873856*n^7+89882508478502582995377474916128*n^6-\ 270805711791876478976633971669056*n^5+568725590740071185086883263326720*n^4-\ 802761800416310808701971198848000*n^3+707392580359329915581361567744000*n^2-\ 336749255640905701954577817600000*n+60884092960058670603485184000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/4194304*(155718815969*n^24-\ 44260618662396*n^23+5912435195374286*n^22-492859931380097424*n^21+ 28702345260023936039*n^20-1238336916425476863996*n^19+40937716486256550392696*n ^18-1057124218958655317491824*n^17+21513665300287053589387439*n^16-\ 344819519693609563273304436*n^15+4295508987152394952489449446*n^14-\ 40057446938016069088537103184*n^13+250056020133115572779043158009*n^12-\ 533074382201371687691325977556*n^11-8899663730025056977129943178124*n^10+ 131275669098675370524873665011536*n^9-1015079204332880968727718667192336*n^8+ 5377939548812372435485695455736384*n^7-20611281349060025203100988072074304*n^6+ 57432479266754638521914912551600896*n^5-114095214235542777858013202640645120*n^ 4+154877612481990263852030973399552000*n^3-133133934432037132291170425831424000 *n^2+62735166871001426091048888729600000*n-11446209476491030073455214592000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37) /(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 437/4194304*( 152076577853*n^24-42905706083244*n^23+5677605535083302*n^22-467674772651962656* n^21+26829940098426970043*n^20-1135880523982779147804*n^19+ 36661161848503572660632*n^18-917844535500094350908496*n^17+ 17922656302036928537772083*n^16-270864388510166822569447524*n^15+ 3073065245317857905387083742*n^14-23816988315023304821172838176*n^13+ 76949805793158467299777795013*n^12+939906978607036305556403027436*n^11-\ 18817709508557890255333019038588*n^10+183393890133073960351140129112464*n^9-\ 1224300152548422777528059791605712*n^8+5997829664445238023747189456515136*n^7-\ 21885082780140243510087166851233088*n^6+59001871890992289224414386635396864*n^5 -114632235974800831369980202227809280*n^4+153464886868521303106654365927936000* n^3-131103797935451416612968303968256000*n^2+ 61905002227651527206600488550400000*n-11446209476491030073455214592000000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 2185/4194304*(58667362442*n ^25-17819774447065*n^24+2541581980107560*n^23-225938815773886750*n^22+ 14008633671625801670*n^21-641975497495520748175*n^20+22464992065512733148060*n^ 19-610673324027880216947800*n^18+12955689280505278317736550*n^17-\ 212335979165555611093485175*n^16+2586616355057137524712718960*n^15-\ 20558239141333474272716423350*n^14+35975781798379180145646876410*n^13+ 1833306151811251456132384364975*n^12-34002091829930472167261764476340*n^11+ 365305509135411401972445893545100*n^10-2813660587492869078216620851381360*n^9+ 16380889841713978381160292071123600*n^8-73075627748238412785422710549794240*n^7 +248841975155537895373137203280244800*n^6-636507810641196622942144560477395712* n^5+1188089252668303953645431667245091840*n^4-\ 1544283678314629602992984314581504000*n^3+1293411038648190603600605656301568000 *n^2-604837628314746401812931953459200000*n+ 112172852869612094719861103001600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29)/(2*n-45), 2185/8388608*(111218340169*n^25-33352021010720*n^24+ 4683458219089870*n^23-408515522110781000*n^22+24743314040765658415*n^21-\ 1101151606816488966200*n^20+37100649246696227831320*n^19-\ 957971875724250328475600*n^18+18843275373990131974913575*n^17-\ 271794364043086354722678800*n^16+2491613759797487188916992870*n^15-\ 2854750282855898023022724200*n^14-372175522231552690726827340655*n^13+ 7849388164360018060218424157800*n^12-99783084862646065951145543799980*n^11+ 923542687797340478598472898675200*n^10-6549725572711156883133510585003920*n^9+ 36157875154179281826826610948579200*n^8-155438808981187314944271063971273280*n^ 7+515343728054952646224743458909209600*n^6-\ 1292894286848180054991687381786307584*n^5+2380844068482100195856420404562718720 *n^4-3068595479866541702564528144628940800*n^3+ 2560903488076004331188718157111296000*n^2-1199526284227004090294066949980160000 *n+224345705739224189439722206003200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-45), 2185/16777216*(411844883215*n^26-131904277007789*n^25 +19792038258820360*n^24-1845243270799385990*n^23+119450160720541800565*n^22-\ 5676117545780107821395*n^21+203634902064550143225610*n^20-\ 5559753829839193667780240*n^19+113585119051071449698124065*n^18-\ 1611136177015350198519403955*n^17+10825652702544088226027990260*n^16+ 158260861543872204160574541010*n^15-6695003791532483932372065321485*n^14+ 127920739525134223626427633259875*n^13-1710180843430084489660866409126790*n^12+ 17487967420048050935486458267489060*n^11-141008705425538549120699800007373560*n ^10+905757180791071201597978847680446640*n^9-\ 4635095023599329978954885215460161440*n^8+ 18763788060379401368045832315996269760*n^7-\ 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91044541912377020275250757799010*n^13-1086204607189382198005448019673315*n^12+ 10348850404941066860056750255712420*n^11-79480647481019034858062697452704780*n^ 10+492700590308505790945382735348422160*n^9-\ 2454441755834985657765534164920533840*n^8+9733076478541867979787489381954179520 *n^7-30264239378403288904753663341016675392*n^6+ 72170498488398111290506263424932798720*n^5-\ 127786641393990791847049838287486202880*n^4+ 160022083640919769531934801758431129600*n^3-\ 131063522125379483291280624323112960000*n^2+ 60886594782106280275041201176248320000*n-11441630992700433661425832506163200000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 345/ 8388608*(2002843147576*n^27-662165703343962*n^26+101844010408846212*n^25-\ 9628536687718411125*n^24+621004079283595869420*n^23-28484236501787230568820*n^ 22+924065648260473926021100*n^21-19186233665321456139598335*n^20+ 107238349618748698067005860*n^19+9623669980237711888273319250*n^18-\ 467612275598849705699840336820*n^17+13036342783061529034938225418365*n^16-\ 266368230053369786093526834598860*n^15+4268978140156611220570488630935040*n^14-\ 55179029129910800252334422007925260*n^13+582589397073743839240341619732577415*n ^12-5049920028784790059117698387476847420*n^11+ 35941569283652893565552878910926735260*n^10-\ 209253506122415487860117415720941554480*n^9+ 989220476218581636142262981351454689040*n^8-\ 3754015727634784142955079356852036864576*n^7+ 11250511723686790004282935871117489255232*n^6-\ 26019761625246652871055437248667228650752*n^5+ 44933760300057470322479128136565795824640*n^4-\ 55168627912338237346933125434102195712000*n^3+ 44528961846424600122163926140019437568000*n^2-\ 20495256449360065714010745009001267200000*n+ 3840574136549778899018604444568780800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/8388608*(1603604183179*n^27-\ 507660224122323*n^26+73716826889148768*n^25-6424703884277145675*n^24+ 364789601357472283005*n^23-13182301130038689268035*n^22+ 215279674460978602199010*n^21+6918093579384696640899825*n^20-\ 670292542375863359856409935*n^19+28574172425349689530155043995*n^18-\ 848552056077586299058555352820*n^17+19383353700954259396074998831775*n^16-\ 354234206326701363781807463782965*n^15+5279828221513732089512719606081455*n^14-\ 64823303016949978821309153007848030*n^13+658556067821728689302583016866386475*n ^12-5540342695699032438888739565158498980*n^11+ 38508152870611397334284215466676790860*n^10-\ 219968485741796936921712950271093625360*n^9+ 1024051798390600835059467882202500430800*n^8-\ 3838802995062737087243168240274268557504*n^7+ 11394166929748863067667318311494698442048*n^6-\ 26159746957981932992177713822247054861568*n^5+ 44941264974355513720159586850407243596800*n^4-\ 55001943770808627026764649884834304716800*n^3+ 44340327076418689076471991160776978432000*n^2-\ 20425952855918565944404394252106792960000*n+ 3840574136549778899018604444568780800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/67108864*(18338285916919*n^28-\ 5856480734894258*n^27+833355289050211683*n^26-66696095657208645810*n^25+ 2850236142290527409055*n^24-1640717718250526361810*n^23-\ 9079896385557141910058865*n^22+741439554200428233393993750*n^21-\ 37116382196416944560167963935*n^20+1370381282989484367763789197570*n^19-\ 39734456424886087268370347749095*n^18+932552454250866091897090913035050*n^17-\ 18018174631057475954863039847221515*n^16+289479202221392695654890784805288730*n ^15-3889221732005690490305956932809327355*n^14+ 43811294780304019162884645839074914450*n^13-\ 413862958843387642049506202056186467180*n^12+ 3271765826773390295173246703217761184360*n^11-\ 21553550944083637863403129043347676650960*n^10+ 117541921307878489009776385285093345340000*n^9-\ 525690983005633985756831230006010334222144*n^8+ 1903547898968794409225402424503586215169408*n^7-\ 5484595694962312930281265098793978206225408*n^6+ 12278248044366658773895815554279286042562560*n^5-\ 20653745941335139663620723679254729030451200*n^4+ 24848780545628905619636960016951765282816000*n^3-\ 19769580815220433738732672637129723904000000*n^2+ 9024727691073509724842685628388514201600000*n-\ 1689852620081902715568185955610263552000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/67108864*( 10264786859113*n^28-2643228109887470*n^27+229610956008421197*n^26+ 4557942772298034210*n^25-3078774807639160008015*n^24+368473606646369076142290*n ^23-27086915058186682187172255*n^22+1441713281464075186846168890*n^21-\ 59266359507914470260570766545*n^20+1947213139226666297514969831390*n^19-\ 52207135882787465148597962394825*n^18+1157726684762793471150672372751590*n^17-\ 21423127998374439340129939275875205*n^16+332652362322802695129097733324858790*n ^15-4347826876274831209497811798477798965*n^14+ 47880504268878724746618850617794501790*n^13-\ 443865797204986793337216393490759806660*n^12+ 3454130047261322095542757849505830732440*n^11-\ 22456838129837979282460219392760017464560*n^10+ 121127563089837722641466259100163754715040*n^9-\ 536816568435808858166703622352048434519488*n^8+ 1929464786159870769883053598985405643346560*n^7-\ 5526592786779246077746013110599756779630592*n^6+ 12316793450573667896190388132792656798228480*n^5-\ 20652732042692339365515122728620416856883200*n^4+ 24799566119820028580005870305195718373376000*n^3-\ 19716538284569417592487464643211802869760000*n^2+ 9005707704873453676961831587329697382400000*n-\ 1689852620081902715568185955610263552000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 435/67108864*( 2680417196378*n^29+1042349101456349*n^28-647125665290355468*n^27+ 138253649124079357281*n^26-17599320087477136231860*n^25+ 1556963908004582921168325*n^24-103054336948550943034406340*n^23+ 5328218319957655216487775525*n^22-221276003425044057255282557040*n^21+ 7523387408469995521853571011835*n^20-212259780783440355884996283520860*n^19+ 5017149010796487315368076810502515*n^18-100021676732530980629714593398309900*n^ 17+1689328582008755411374914142638399255*n^16-\ 24234795952636594703671891458907634700*n^15+ 295580795120131001639633933266569717495*n^14-\ 3063104471143429801463452290713856941130*n^13+ 26913474111764469445788720578912343601740*n^12-\ 199745405766489128843938095475486060629320*n^11+ 1245335846748613894282736593752930645418960*n^10-\ 6472904487718828843928017034885433908218848*n^9+ 27765619899791922799462081895074270844901696*n^8-\ 96979109897240161827227554949104397425149312*n^7+ 270927857821230902970271090331891024474548224*n^6-\ 590925515111844262923165350477931426941337600*n^5+ 972864379766947881140620260147519156346060800*n^4-\ 1150555394372304198634721615577061330735104000*n^3+ 903699770581511232510004122697114548142080000*n^2-\ 409127638474883027251287638580033080524800000*n+ 76392992583702567589996268545001914368000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -2175/ 134217728*(4490562692047*n^29-2708784223668998*n^28+706143788158372851*n^27-\ 110335034103084655830*n^26+11832170894119171916775*n^25-\ 937160970754913843664726*n^24+57364774690977920193164943*n^23-\ 2796953845518298278611715774*n^22+110947040341758876164850357417*n^21-\ 3635297052798015631107601305786*n^20+99486323834601637323690243117993*n^19-\ 2292270048795480796355352752321154*n^18+44719503121235376577053825710950317*n^ 17-741422390373089213799666069561087186*n^16+ 10467851443047646726217763703211340453*n^15-\ 125922313781235895277719585556610853194*n^14+ 1289449190475494943355259474691185243892*n^13-\ 11213257492450986295969515526879293967896*n^12+ 82485956275392558140583553942311907288688*n^11-\ 510371716261513722079156267044057036495904*n^10+ 2635714156110989539272918752600349726805440*n^9-\ 11245102011773859220106669354816247693621888*n^8+ 39103195362763861528820074627835191759337472*n^7-\ 108856921583750411197199112049289760680070144*n^6+ 236792578707393188661653214507395919885914112*n^5-\ 389102888460449752880851758370903536556523520*n^4+ 459647804421698602807820687773882579417497600*n^3-\ 360885446170436211301472851614793134440448000*n^2+ 163442279509600234939399537679467885363200000*n-\ 30557197033481027035998507418000765747200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -87/ 268435456*(996058010313523*n^30-522071339690776455*n^29+127772543775227384555*n ^28-19539891255650713614975*n^27+2104429922052527477787687*n^26-\ 170373491222257752489386235*n^25+10799513637543830378173449975*n^24-\ 550888852714485561017722816875*n^23+23057503151168283296836324761345*n^22-\ 803118644887730095458966625318125*n^21+23522368181017139586407629296529425*n^20 -583757574622355014767339620555332125*n^19+ 12342882100104023008715894698617429165*n^18-\ 223178716747793658340584200355070300425*n^17+ 3458655662704573099450849408575543083325*n^16-\ 45978954657418046011805190453654072797625*n^15+ 524121679583860819646957080809720192489720*n^14-\ 5114456940407074803228821693326932305305800*n^13+ 42595099419518418517225102313719296778480400*n^12-\ 301426812124925125319103645198648540822096000*n^11+ 1801422496336081969302713922489701626741938112*n^10-\ 9018806866652139769912796355574767923234574720*n^9+ 37428047939673056314220926006296027272451064320*n^8-\ 126992512118376474638615921894832718420677062400*n^7+ 345947378794272777545214176335132610587337960448*n^6-\ 738399447299338309380330284386436552833257738240*n^5+ 1193664880495538036702743338225358900488235008000*n^4-\ 1390692116449871760031279282967017358629601280000*n^3+ 1079591289494908207079889209854720420535992320000*n^2-\ 484738262807270481809438890912556945611161600000*n+ 90143731248769029756195596883102258954240000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -87/134217728*(756229127708129*n^30-371803522419924885*n^29+ 86467163605343536990*n^28-12681478632747052849875*n^27+ 1318834632999662089630326*n^26-103645265497482148335576135*n^25+ 6403915885700906983506796050*n^24-319489057526178149138341668375*n^23+ 13114634081659258007754956335560*n^22-449038904305514866361000539481925*n^21+ 12954047103667953281919692178102150*n^20-317190565453423983643845841608824625*n ^19+6627010807882999438668451859475598170*n^18-\ 118561021721320400419233212096167900125*n^17+ 1820108804940398211750647236340728357350*n^16-\ 23994478463514690327164276826443055472125*n^15+ 271496681766255023828179364889331413166935*n^14-\ 2632044119073788244388293302562766292148450*n^13+ 21795345671444004977267897312479372723449700*n^12-\ 153468081595923343309355118883271833378329000*n^11+ 913231908851454476451697108972713665916988976*n^10-\ 4555327405443549745055072805764462900633795040*n^9+ 18846433303860997697526499746235152734209517760*n^8-\ 63783864639328803443039855543782853671281936000*n^7+ 173407259605003572092635741109448299111857211904*n^6-\ 369558516769378265632098819093797481011680573440*n^5+ 596770374912568749404404320285706061959050240000*n^4-\ 694829208331379206310714782167282979231825920000*n^3+ 539278702557037909218491522747336271700623360000*n^2-\ 242190825122044269197509388231674534271385600000*n+ 45071865624384514878097798441551129477120000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -2697/134217728*(63842937726100*n^31-32309886426715997*n^30+7788138031159281275 *n^29-1190971043204481286420*n^28+129828644860719782044725*n^27-\ 10746734465562801837127668*n^26+702553843285170689291084385*n^25-\ 37245054519546511151275109400*n^24+1631438750869512750433952009625*n^23-\ 59856252589307949617533299295830*n^22+1858091306932668046714476829539075*n^21-\ 49168594699667969020094826020995200*n^20+1115153666001390899619208936513230375* n^19-21759849760385516433611320704638295060*n^18+ 366178517367685554307707368207531792475*n^17-\ 5320451471831113411431933689422907581800*n^16+ 66745874931417468997526763946892893917375*n^15-\ 722160272607587645943942320227071769073205*n^14+ 6723408440769927797532261664021685696450150*n^13-\ 53674970091202477429791309990278268385293100*n^12+ 365648943678180428854857999236333311299825400*n^11-\ 2111824288704175252231958628509694787012730768*n^10+ 10254265778563607943446620712766376748593706400*n^9-\ 41409771837758101624726378213001870108395222080*n^8+ 137142243457061100813213000921925367609701526400*n^7-\ 365716106610307141989373622294110764939411881472*n^6+ 766216458998963032483215851300911477693761546240*n^5-\ 1218997183688345655457266445691818180469824512000*n^4+ 1401253252581636974538411648365221867578654720000*n^3-\ 1076022648565331458217486778825786603451514880000*n^2+ 479217008430325794132519681863076360722841600000*n-\ 88689800099595335727869861449503835422720000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), -899/268435456*(460836428705407*n^31-228782000814910325*n^30+ 54200648834593832105*n^29-8159852227983000963655*n^28+877003293887430708215553* n^27-71667930386081015638356915*n^26+4630779598425385266940042185*n^25-\ 242900022180171196271478938475*n^24+10537307451802603476891027103755*n^23-\ 383218985257417908645889123849425*n^22+11801300622648039603950294287308375*n^21 -310023544189988879600609886229982925*n^20+ 6985213532849468698529949984123999435*n^19-\ 135491176891325259454394396172909401025*n^18+ 2267830132602826280816197047896926080675*n^17-\ 32791653665657002011080817038911992864825*n^16+ 409595295484504401180498206378649720397530*n^15-\ 4414521466984372671945751915059072398701350*n^14+ 40959000424128125265850567372644409653329300*n^13-\ 326001290577748085092500557390867261836443400*n^12+ 2214959403552611314220312047683403891511131408*n^11-\ 12763468351603490649301854049825725573188300000*n^10+ 61854384298947898331989229518136196466140112320*n^9-\ 249379017046052990674526001246698789774014062720*n^8+ 824796694975100755036954999846217963447355846912*n^7-\ 2197139852518938566836090536375762985210394280960*n^6+ 4599546701852954965713743953799085683375796695040*n^5-\ 7313440096743254556298916302615264211104671744000*n^4+ 8404104069971786154068041895029633543608729600000*n^3-\ 6452818741379075769644134083258189995635507200000*n^2+ 2874175830898150379579526092937829544204697600000*n-\ 532138800597572014367219168697023012536320000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n -21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-45), -31/1073741824*(121614245103578947*n^32-63442508984546382704*n^31+ 15836873941530039997784*n^30-2518963667435070803075680*n^29+ 286799478055214312252344308*n^28-24894796240683274533463323936*n^27+ 1713289320806753254164140800536*n^26-95986829535038430368025100896480*n^25+ 4460417989721498243274604637570130*n^24-174285333467297095032544354493141760*n^ 23+5784724104323561971103352528515005560*n^22-\ 164337628106582230713392480227971098400*n^21+ 4018448814511567151178831924199687565460*n^20-\ 84915826053534928047499073144535900599520*n^19+ 1554847963565588498562856235184628085271720*n^18-\ 24706280155154995907517648531246912963588000*n^17+ 340825427706692964582095469136821149690847555*n^16-\ 4079445802071460561673619745941825147615810960*n^15+ 42296916706768773302528605193968368999076229360*n^14-\ 378863961874168847788187993165453356437779545600*n^13+ 2920433789872586787103016489436424628543379220768*n^12-\ 19273033232587465477435290073427308950006435416576*n^11+ 108159928037172418365546199063535512579423924874496*n^10-\ 511739538838552440017405428197423480844762570014720*n^9+ 2018878690934296241264044465236881816648378903384832*n^8-\ 6547804384182063523686737268996574848203574999404544*n^7+ 17138726654393432441301597508732692465527775249260544*n^6-\ 35321574049493100145815476889455649208571707566981120*n^5+ 55392955895541310661171353026127269198420074815488000*n^4-\ 62896190531522782097122930551429674748053223505920000*n^3+ 47806867882730522648607041079123559456404272578560000*n^2-\ 21122186074895865903545205567305439799420964044800000*n+ 3888870354767056280995637684837844175615426560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/ (2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37 )/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 57)/(2*n-63)/(2*n-45), -155/1073741824*(26562878528361431*n^32-\ 13745795724518019376*n^31+3405740808273596013496*n^30-537964697230279493612960* n^29+60858597394390577025701604*n^28-5251361431913351668938280032*n^27+ 359425598570524788431253563640*n^26-20034949294728185134198560315936*n^25+ 926665057104648030524956933172490*n^24-36052946308366305343243438818936000*n^23 +1191921751197975109402485960286863960*n^22-\ 33738890188361315448137899251362519520*n^21+ 822272918740489638848772771626969019780*n^20-\ 17323558624995441817416446848939765376160*n^19+ 316334826320202106357070610326902442256840*n^18-\ 5014061622865112725789887694948689689259360*n^17+ 69014994716045488759754067293426208669937815*n^16-\ 824406504761803556078653193071776631877452560*n^15+ 8532402264008388082340396972792801848584456880*n^14-\ 76305557933280856969175593492220352363045751040*n^13+ 587371642626584751376245932600551899701001608864*n^12-\ 3871557690280138765510943239299714860235634537984*n^11+ 21704237353571752416989925062739295604095314280704*n^10-\ 102597638265947435051670519516899071728597350584320*n^9+ 404456889647149088238625389567737001124507652687616*n^8-\ 1310963378387538060659360364265245459351880869285888*n^7+ 3429743321609133106709792658751713456464615217684480*n^6-\ 7065828922273097427079862482970116627536707765796864*n^5+ 11078125216909628074235100273974578561731380561510400*n^4-\ 12576820549353764924159739773223266031640945754112000*n^3+ 9559084786048997112032683741144873383719801978880000*n^2-\ 4223670935106312677008352317858164246746140508160000*n+ 777774070953411256199127536967568835123085312000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/( 2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37) /(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 57)/(2*n-63)/(2*n-45), -5/1073741824*(1743966414089719858*n^33-\ 954394575765752363913*n^32+250511568732157229312992*n^31-\ 41996710684249391271953928*n^30+5051785081753666688617631512*n^29-\ 464415279066388637194481234268*n^28+33934800965354868722802112520880*n^27-\ 2023776170434838719217028000016584*n^26+100374749390215994638471537910216460*n^ 25-4197791578175379587264057950774067190*n^24+ 149564324416140266870631435476428939920*n^23-\ 4575246994033065885212281190406187728360*n^22+ 120865069583699096370379372288387751481240*n^21-\ 2769043779800945219460162261726389649354300*n^20+ 55179984374814496778776058473288189883689680*n^19-\ 958183714865225224577466696273568033069559160*n^18+ 14510585600443523952769081266842182991091193170*n^17-\ 191615555099337308666475494124003017687934840585*n^16+ 2204055663508627722521297836639566423583860607760*n^15-\ 22038582107509011572610342564316347999297718827600*n^14+ 190984594798447274963459996418606095884198972578752*n^13-\ 1428452986965749548256773305716317529592638683904352*n^12+ 9171284182496934633256104774178734829578999317393408*n^11-\ 50196709855838590853669874477029168677860935424667392*n^10+ 232153912820070981623187641068251016527239528318427648*n^9-\ 897177117501345770233840459310878798462022107562203392*n^8+ 2856084349938222328919425006459486065377437144932495360*n^7-\ 7351590606110823854927081523185442477023663693627006976*n^6+ 14926371584684007377037272401199597674119304509342351360*n^5-\ 23101494866168682897201196860453969376777972949164032000*n^4+ 25931819419526504842645233744184316261304232797143040000*n^3-\ 19520513989679464956664515311123668966089290077962240000*n^2+ 8557955968264770175738976907968516092127618688614400000*n-\ 1567214752971123681241241986989651202773016903680000000)/(2*n-5)/(-1+2*n)/(2*n-\ 3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2* n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/( 2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), (3251776168598923056*(n-32)!*(2*n-69)!*(n+1 )+4877664252898384584*(n-33)!*((n+1)*(2*n-68)!-1/4877664252898384584*binomial(2 *n,n)*(n-35)!*(n-32)!))/(n-35)!/(n-33)!/(n-32)!/binomial(2*n,n), (( 3251776168598923056*n+3251776168598923056)*(2*n-69)!-(n-35)!*(n-33)!*binomial(2 *n,n))/(n-35)!/(n-33)!/binomial(2*n,n)] The limits, as n goes to infinity are 4294967287 17179868553 1073741469 2147479275 8589851721 274867605729 [----------, -----------, ----------, ----------, ----------, ------------, 4294967296 17179869184 1073741824 2147483648 8589934592 274877906944 274843752939 274778555313 1098476858253 4388252101005 2188365990125 ------------, ------------, -------------, -------------, -------------, 274877906944 274877906944 1099511627776 4398046511104 2199023255552 1088745175825 8633251903223 68049122578453 66457464521761 -------------, -------------, --------------, --------------, 1099511627776 8796093022208 70368744177664 70368744177664 64094093467885 243012073269265 899881069824775 201653079826435 --------------, ---------------, ----------------, ---------------, 70368744177664 281474976710656 1125899906842624 281474976710656 86372610739215 553243443196755 6326708641337055 3541351466393985 ---------------, ----------------, -----------------, -----------------, 140737488355328 1125899906842624 18014398509481984 18014398509481984 582990740212215 -9766973855202225 -86657046897276501 -----------------, -----------------, ------------------, 18014398509481984 72057594037927936 288230376151711744 -65791934110607223 -43046100761822925 -414291949406160893 ------------------, ------------------, -------------------, 144115188075855872 72057594037927936 576460752303423488 -3770041598210947357 -4117246171896021805 -4359916035224299645 --------------------, --------------------, --------------------, 4611686018427387904 4611686018427387904 4611686018427387904 -9020136026317343117 -36690252136881670541 --------------------, ---------------------] 9223372036854775808 36893488147419103232 and in Maple notation [4294967287/4294967296, 17179868553/17179869184, 1073741469/1073741824, 2147479275/2147483648, 8589851721/8589934592, 274867605729/274877906944, 274843752939/274877906944, 274778555313/274877906944, 1098476858253/ 1099511627776, 4388252101005/4398046511104, 2188365990125/2199023255552, 1088745175825/1099511627776, 8633251903223/8796093022208, 68049122578453/ 70368744177664, 66457464521761/70368744177664, 64094093467885/70368744177664, 243012073269265/281474976710656, 899881069824775/1125899906842624, 201653079826435/281474976710656, 86372610739215/140737488355328, 553243443196755/1125899906842624, 6326708641337055/18014398509481984, 3541351466393985/18014398509481984, 582990740212215/18014398509481984, -\ 9766973855202225/72057594037927936, -86657046897276501/288230376151711744, -\ 65791934110607223/144115188075855872, -43046100761822925/72057594037927936, -\ 414291949406160893/576460752303423488, -3770041598210947357/4611686018427387904 , -4117246171896021805/4611686018427387904, -4359916035224299645/ 4611686018427387904, -9020136026317343117/9223372036854775808, -\ 36690252136881670541/36893488147419103232] and in floating point [.9999999979, .9999999633, .9999996694, .9999979637, .9999903525, .9999625244, .9998757485, .9996385609, .9990588826, .9977730090, .9951536368, .9902079690, .\ 9814871081, .9670361945, .9444173731, .9108318504, .8633523168, .7992549465, .7\ 164156551, .6137143113, .4913788871, .3512028802, .1965844968, .3236248715e-1, -.1355439907, -.3006520272, -.4565232505, -.5973846523, -.7186819705, -.8174974\ 582, -.8927854488, -.9454060874, -.9779651076, -.9944912769] The cut off is at j=, 25 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 36], vs. those in the, 2, -th row from j=1 to j=, 35, are as follws 18 17 16 [(34359738331 n - 5566277597967 n + 417496587008502 n 15 14 - 19240765733251260 n + 609730278401865642 n 13 12 - 14087367442671931674 n + 245499678900052439824 n 11 10 - 3292991395022867374980 n + 34384971192200498421123 n 9 8 - 280821021317911378512351 n + 1792110218491204983899346 n 7 6 - 8878255228121957228383560 n + 33693850514674229628095104 n 5 4 - 95643742250212321522321008 n + 193296983448210866041991328 n 3 2 - 238389075173730814532467200 n + 18117657125341725689740800 n + 619181636969450730954240000 n - 1045843337171591729971200000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) 18 (2 n - 15) (2 n - 17) (2 n - 23) (2 n - 35) (2 n - 29)), (34359737701 n 17 16 15 - 5566277309427 n + 417496524965472 n - 19240757415436860 n 14 13 + 609729499766594382 n - 14087313438697865394 n 12 11 + 245496805662703288564 n - 3292871640075094514580 n 10 9 + 34381013673130341178533 n - 280716774502445523630531 n 8 7 + 1789922690195449132369956 n - 8841929936061635810593560 n 6 5 + 33223113755887972007561584 n - 90996953663349324638217648 n 4 3 + 159765190625123374864195008 n - 74326431727161766298592000 n 2 - 441748086579569555943091200 n + 973357443863129803484160000 n - 29051203810321992499200000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) 19 18 (2 n - 35) (2 n - 29)), 57 (2411209277 n - 435223153416 n 17 16 15 + 36524358819303 n - 1892236523645424 n + 67766786646170634 n 14 13 - 1780128940327156272 n + 35514877428623369726 n 12 11 - 549720445775807462208 n + 6685232679873568764801 n 10 9 - 64270513381927553051208 n + 488706844087036398045339 n 8 7 - 2921975994182099269872912 n + 13521853259241205497991688 n 6 5 - 46668750648603416226045504 n + 108790973020339328905939632 n 4 3 - 112050033400709601514131456 n - 216431391777328198309190400 n 2 + 858934473507072252425318400 n - 642340692023190464747520000 n + 37715597929189955174400000)/(262144 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 19 18 (2 n - 3) (-1 + 2 n) (2 n - 5)), 19 (904202614 n - 163208301986 n 17 16 15 + 13696556217726 n - 709578680803179 n + 25411654945090788 n 14 13 - 667490056997304072 n + 13315168084616898892 n 12 11 - 206032129123485689258 n + 2503512624796675187982 n 10 9 - 24018440691452112246198 n + 181695912353476183932438 n 8 7 - 1072683854991481482951747 n + 4812677049058197618793816 n 6 5 - 15377693885089672660085144 n + 28723056267491176087198944 n 4 3 - 155284040281315391760816 n - 142295464571051521566403200 n 2 + 278090721214280776511942400 n - 167410917709128490763520000 n + 14143349223446233190400000)/(32768 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), 589 (116670793 n - 23333999870 n 18 17 16 + 2177903893455 n - 126014819923230 n + 5063775944147448 n 15 14 - 150031185404974140 n + 3395784918621775510 n 13 12 - 60017103145844730860 n + 839261179045136952273 n 11 10 - 9345090262484592926310 n + 82850863668118145709915 n 9 8 - 580072677704685750421590 n + 3139946075722337219131438 n 7 6 - 12531093801610093325432480 n + 32736684938175726226588320 n 5 4 - 33210675639985326143764320 n - 103731893047950020367041952 n 3 2 + 440247345158927723322412800 n - 625708780512235577315827200 n + 336063216976410497333760000 n - 35586491594477618995200000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), 1767 (38889623 n - 7777727390 n + 725913909705 n 17 16 15 - 41998302049350 n + 1687360590947688 n - 49975116923670300 n 14 13 + 1130255203547273210 n - 19944289465700635100 n 12 11 + 277991884910181189423 n - 3075425323362466670310 n 10 9 + 26922379223564620623165 n - 183964244728902357066750 n 8 7 + 950716547188866332550098 n - 3462104903736047984013920 n 6 5 + 7238143558860771463194720 n + 701975679338786608231200 n 4 3 - 53399329199077668146996832 n + 150595980585440016847921920 n 2 - 185976848750220233997580800 n + 95677646556154518535680000 n - 11862163864825872998400000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 17081 (64365631 n 20 19 18 - 14191586283 n + 1465126650575 n - 94109102098305 n 17 16 15 + 4214783757889236 n - 139772313921402918 n + 3556856178938809870 n 14 13 - 71000897435696088210 n + 1126211726390445926231 n 12 11 - 14275574665626514135623 n + 144391201158977582724915 n 10 9 - 1153466171317466011552365 n + 7107888117470285869820806 n 8 7 - 32164411049246501161105368 n + 94812823160776767872809760 n 6 5 - 104106910980569018279971920 n - 461644922919523884513071904 n 4 3 + 2480708688214238440365480192 n - 5379500739056664225183045120 n 2 + 5902979533422655066848460800 n - 2919482111332364313415680000 n + 402495491137539966566400000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) 21 (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 551 (997519373 n 20 19 18 - 219908501589 n + 22697710797685 n - 1457293554401055 n 17 16 + 65214229747959348 n - 2159545344581061234 n 15 14 + 54815018292053001530 n - 1089356711596803776910 n 13 12 + 17149073646348118258693 n - 214648745894605037431689 n 11 10 + 2126621939600890100849025 n - 16428312844076012783564115 n 9 8 + 95805856840764638420397578 n - 392917246228797952425968064 n 7 6 + 914110537670729255811361120 n + 359674081657975231240129680 n 5 4 - 11437762489778341483284544992 n + 42777333939417841023503112576 n 3 2 - 82119745622823082205242959360 n + 85089414122425797616559462400 n - 41662255994176960052951040000 n + 6238680112631869481779200000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 22 21 (2 n - 35) (2 n - 29)), 1653 (5318124649 n - 1286398498979 n 20 19 18 + 146108509809647 n - 10355263990769005 n + 513262690489948014 n 17 16 - 18893601307034568114 n + 535187986482044394862 n 15 14 - 11920714835907762246890 n + 211369316034847692726389 n 13 12 - 2997965358897036130161559 n + 33932618908605311299166307 n 11 10 - 303121965249804809515031625 n + 2086134014148082669185122884 n 9 8 - 10515043419702772380540685124 n + 34019910816032156378820553712 n 7 6 - 31913772352281872257233896560 n - 317923951179848487742111964736 n 5 4 + 1941276051400575789950278003776 n - 5673478247033116654833313494528 n 3 2 + 9683161442028742243503852334080 n - 9416845788666067535021883187200 n + 4522410384953293195760517120000 n - 715368652915121033910681600000)/( 2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 22 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 513 (17121669151 n 21 20 19 - 4139199679527 n + 469691596579667 n - 33238873904351885 n 18 17 + 1643559690786450976 n - 60273951941055274422 n 16 15 + 1697482588715060093782 n - 37479026868158600942250 n 14 13 + 655915906624465929771791 n - 9126220077032118648534947 n 12 11 + 100439183200972873366945927 n - 860804426966798661315833865 n 10 9 + 5553910738601171349701473186 n - 24918298496084866656071861232 n 8 7 + 58410265727506181659842765632 n + 107828668481113669792241289520 n 6 5 - 1615458497262756618625011223904 n + 7269048858757547602565176480128 n 4 - 19015238419646011260450537835008 n 3 + 30640044111710446263908570388480 n 2 - 28965715703935892099389469491200 n + 13899897182341307844284497920000 n - 2305076770504278887045529600000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 294975 23 22 21 20 (118896369 n - 31387457386 n + 3898589147002 n - 302754054836772 n 19 18 + 16471662495545719 n - 666554385031042662 n 17 16 + 20779765697455376964 n - 509727571756849399624 n 15 14 + 9955066678046397816799 n - 155466410783677073992758 n 13 12 + 1935738774155674076560426 n - 18989546912678579659130436 n 11 10 + 142914870667062612666807369 n - 776559262843762943815987354 n 9 8 + 2520084737489844987498205048 n + 623721653117060798576770032 n 7 6 - 61200084947908293258997519856 n + 386878584399705485453768408160 n 5 4 - 1398593499413007816388679869440 n + 3266637135921966742214576140800 n 3 2 - 4912054647777795684674127974400 n + 4460526494986834048061214720000 n - 2113499228639828492585041920000 n + 360794624948495825798430720000)/( 4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 10925 ( 23 22 21 799794481 n - 210768445475 n + 26112176927045 n 20 19 18 - 2020297398625891 n + 109336973698933596 n - 4391758400441621748 n 17 16 + 135509465560150805926 n - 3277452628767331109942 n 15 14 + 62791895470718354343161 n - 955307868581928127415611 n 13 12 + 11472046114816082588063817 n - 106792293377024455727586423 n 11 10 + 738721077562262097382532266 n - 3379371614643449995036570334 n 9 8 + 5203317161148286795762521308 n + 60584215451525693980831730336 n 7 6 - 609266346058275446297886818304 n + 3081445811598086446386846603168 n 5 - 10125730949747170146684320748096 n 4 + 22417536873103133917701666193920 n 3 - 32662687508101593198522299059200 n 2 + 29212544289526655762187270144000 n - 13866375748631588427047362560000 n + 2435363718402346824139407360000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) 24 23 (2 n - 45)), 10925 (6358499707 n - 1820247620700 n 22 21 20 + 245423227627474 n - 20705223197537208 n + 1224436431463139125 n 19 18 - 53867970425405700564 n + 1825433034689138693896 n 17 16 - 48647660109384780548544 n + 1031193549554184728988133 n 15 14 - 17449644785570321707169028 n + 234712879386626220502804378 n 13 12 - 2471247460794712351096527000 n + 19626162308618511357941486971 n 11 10 - 106304849041642229014498045452 n + 236865145800893267441591124460 n 9 8 + 2029216926767761021191078770064 n - 27253383468035569647114791424176 n 7 + 174359373519264751510731628734144 n 6 - 738198518452994611354076193590208 n 5 + 2187389796970525679414557290578688 n 4 - 4521839352983371968581236640117760 n 3 + 6289934008856504063651133733785600 n 2 - 5467531652614370531492950228992000 n + 2568552411719800681335316316160000 n - 457848379059641202938208583680000 )/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 437 ( 24 23 22 157270139611 n - 44837711057220 n + 6012455235868594 n 21 20 - 503586943430969640 n + 29499851162185606741 n 19 18 - 1281975750243662928300 n + 42759211980114300167464 n 17 16 - 1116447047098412766221760 n + 23043169132875069815075461 n 15 14 - 376318927419890730610132380 n + 4816179469785623694773790394 n 13 12 - 46974679314475950165377438280 n + 323786444759393599186878405211 n 11 - 1160455332175323240370914257460 n 10 - 4675310898427739099376781237556 n 9 + 109077167546987082265537948820400 n 8 - 925966578241076124238684484571824 n 7 + 5113912277339114870115059122071360 n 6 - 20068736295081413701977245258728896 n 5 + 56764034260134898407887358775169280 n 4 - 113866482754006940251064295409075200 n 3 + 155479328946986302688024713618944000 n 2 - 133998622199101418598552811069440000 n + 63088755515391197838128762880000000 n - 11446209476491030073455214592000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 10925 (12364425859 n 24 23 22 - 3806871937094 n + 552047774137798 n - 50079487851235796 n 21 20 + 3182805316747641277 n - 150360552642238588034 n 19 18 + 5464569774037526669308 n - 155893295389428032907416 n 17 16 + 3526877348495760836627677 n - 63365639832665752353331274 n 15 14 + 895638489898405983707498038 n - 9677988208330694267035906676 n 13 12 + 73756588944792009285475474867 n - 278345444637321766862419744334 n 11 - 1765108472106572687838575797592 n 10 + 41696901856740553173319020906784 n 9 - 409116999907506061878758933357168 n 8 + 2697623161395738190265182901080096 n 7 - 12998456904967145922057137528785152 n 6 + 46627898944592322699976586723215104 n 5 - 123674228710036699923252227348992512 n 4 + 236653196468038665944138517895280640 n 3 - 312389907546151676377009802570342400 n 2 + 263397927482640505329448940679168000 n - 122837073210776148607917409075200000 n + 22434570573922418943972220600320000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 6555 (20056544124 n 24 23 22 - 6127207800295 n + 879918440011020 n - 78863052814207250 n 21 20 + 4937587517646046340 n - 228957864278879637825 n 19 18 + 8129328866168626479220 n - 225120621533510607262600 n 17 16 + 4897238208228577832507700 n - 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280716774502445523630531*n^9+1789922690195449132369956*n^8-\ 8841929936061635810593560*n^7+33223113755887972007561584*n^6-\ 90996953663349324638217648*n^5+159765190625123374864195008*n^4-\ 74326431727161766298592000*n^3-441748086579569555943091200*n^2+ 973357443863129803484160000*n-29051203810321992499200000)/(2*n-5)/(-1+2*n)/(2*n -3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-17)/(2*n-23)/(2*n-35)/(2*n-29), 57/262144*( 2411209277*n^19-435223153416*n^18+36524358819303*n^17-1892236523645424*n^16+ 67766786646170634*n^15-1780128940327156272*n^14+35514877428623369726*n^13-\ 549720445775807462208*n^12+6685232679873568764801*n^11-64270513381927553051208* n^10+488706844087036398045339*n^9-2921975994182099269872912*n^8+ 13521853259241205497991688*n^7-46668750648603416226045504*n^6+ 108790973020339328905939632*n^5-112050033400709601514131456*n^4-\ 216431391777328198309190400*n^3+858934473507072252425318400*n^2-\ 642340692023190464747520000*n+37715597929189955174400000)/(2*n-29)/(2*n-35)/(2* n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/( 2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 19/32768*( 904202614*n^19-163208301986*n^18+13696556217726*n^17-709578680803179*n^16+ 25411654945090788*n^15-667490056997304072*n^14+13315168084616898892*n^13-\ 206032129123485689258*n^12+2503512624796675187982*n^11-24018440691452112246198* n^10+181695912353476183932438*n^9-1072683854991481482951747*n^8+ 4812677049058197618793816*n^7-15377693885089672660085144*n^6+ 28723056267491176087198944*n^5-155284040281315391760816*n^4-\ 142295464571051521566403200*n^3+278090721214280776511942400*n^2-\ 167410917709128490763520000*n+14143349223446233190400000)/(2*n-29)/(2*n-35)/(2* n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/( 2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 589/65536*( 116670793*n^20-23333999870*n^19+2177903893455*n^18-126014819923230*n^17+ 5063775944147448*n^16-150031185404974140*n^15+3395784918621775510*n^14-\ 60017103145844730860*n^13+839261179045136952273*n^12-9345090262484592926310*n^ 11+82850863668118145709915*n^10-580072677704685750421590*n^9+ 3139946075722337219131438*n^8-12531093801610093325432480*n^7+ 32736684938175726226588320*n^6-33210675639985326143764320*n^5-\ 103731893047950020367041952*n^4+440247345158927723322412800*n^3-\ 625708780512235577315827200*n^2+336063216976410497333760000*n-\ 35586491594477618995200000)/(2*n-29)/(2*n-35)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/(2*n-11)/(2*n-33)/(2* n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 1767/65536*(38889623*n^20-\ 7777727390*n^19+725913909705*n^18-41998302049350*n^17+1687360590947688*n^16-\ 49975116923670300*n^15+1130255203547273210*n^14-19944289465700635100*n^13+ 277991884910181189423*n^12-3075425323362466670310*n^11+26922379223564620623165* n^10-183964244728902357066750*n^9+950716547188866332550098*n^8-\ 3462104903736047984013920*n^7+7238143558860771463194720*n^6+ 701975679338786608231200*n^5-53399329199077668146996832*n^4+ 150595980585440016847921920*n^3-185976848750220233997580800*n^2+ 95677646556154518535680000*n-11862163864825872998400000)/(2*n-29)/(2*n-35)/(2*n -39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2 *n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 17081/524288*(64365631*n^21-14191586283*n^20+1465126650575*n^19-94109102098305* n^18+4214783757889236*n^17-139772313921402918*n^16+3556856178938809870*n^15-\ 71000897435696088210*n^14+1126211726390445926231*n^13-14275574665626514135623*n ^12+144391201158977582724915*n^11-1153466171317466011552365*n^10+ 7107888117470285869820806*n^9-32164411049246501161105368*n^8+ 94812823160776767872809760*n^7-104106910980569018279971920*n^6-\ 461644922919523884513071904*n^5+2480708688214238440365480192*n^4-\ 5379500739056664225183045120*n^3+5902979533422655066848460800*n^2-\ 2919482111332364313415680000*n+402495491137539966566400000)/(2*n-5)/(-1+2*n)/(2 *n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/( 2*n-29), 551/262144*(997519373*n^21-219908501589*n^20+22697710797685*n^19-\ 1457293554401055*n^18+65214229747959348*n^17-2159545344581061234*n^16+ 54815018292053001530*n^15-1089356711596803776910*n^14+17149073646348118258693*n ^13-214648745894605037431689*n^12+2126621939600890100849025*n^11-\ 16428312844076012783564115*n^10+95805856840764638420397578*n^9-\ 392917246228797952425968064*n^8+914110537670729255811361120*n^7+ 359674081657975231240129680*n^6-11437762489778341483284544992*n^5+ 42777333939417841023503112576*n^4-82119745622823082205242959360*n^3+ 85089414122425797616559462400*n^2-41662255994176960052951040000*n+ 6238680112631869481779200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1653/2097152*( 5318124649*n^22-1286398498979*n^21+146108509809647*n^20-10355263990769005*n^19+ 513262690489948014*n^18-18893601307034568114*n^17+535187986482044394862*n^16-\ 11920714835907762246890*n^15+211369316034847692726389*n^14-\ 2997965358897036130161559*n^13+33932618908605311299166307*n^12-\ 303121965249804809515031625*n^11+2086134014148082669185122884*n^10-\ 10515043419702772380540685124*n^9+34019910816032156378820553712*n^8-\ 31913772352281872257233896560*n^7-317923951179848487742111964736*n^6+ 1941276051400575789950278003776*n^5-5673478247033116654833313494528*n^4+ 9683161442028742243503852334080*n^3-9416845788666067535021883187200*n^2+ 4522410384953293195760517120000*n-715368652915121033910681600000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29), 513/2097152*(17121669151*n^22-4139199679527*n^21+ 469691596579667*n^20-33238873904351885*n^19+1643559690786450976*n^18-\ 60273951941055274422*n^17+1697482588715060093782*n^16-37479026868158600942250*n ^15+655915906624465929771791*n^14-9126220077032118648534947*n^13+ 100439183200972873366945927*n^12-860804426966798661315833865*n^11+ 5553910738601171349701473186*n^10-24918298496084866656071861232*n^9+ 58410265727506181659842765632*n^8+107828668481113669792241289520*n^7-\ 1615458497262756618625011223904*n^6+7269048858757547602565176480128*n^5-\ 19015238419646011260450537835008*n^4+30640044111710446263908570388480*n^3-\ 28965715703935892099389469491200*n^2+13899897182341307844284497920000*n-\ 2305076770504278887045529600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 294975/ 4194304*(118896369*n^23-31387457386*n^22+3898589147002*n^21-302754054836772*n^ 20+16471662495545719*n^19-666554385031042662*n^18+20779765697455376964*n^17-\ 509727571756849399624*n^16+9955066678046397816799*n^15-155466410783677073992758 *n^14+1935738774155674076560426*n^13-18989546912678579659130436*n^12+ 142914870667062612666807369*n^11-776559262843762943815987354*n^10+ 2520084737489844987498205048*n^9+623721653117060798576770032*n^8-\ 61200084947908293258997519856*n^7+386878584399705485453768408160*n^6-\ 1398593499413007816388679869440*n^5+3266637135921966742214576140800*n^4-\ 4912054647777795684674127974400*n^3+4460526494986834048061214720000*n^2-\ 2113499228639828492585041920000*n+360794624948495825798430720000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29)/(2*n-45), 10925/1048576*(799794481*n^23-210768445475*n^ 22+26112176927045*n^21-2020297398625891*n^20+109336973698933596*n^19-\ 4391758400441621748*n^18+135509465560150805926*n^17-3277452628767331109942*n^16 +62791895470718354343161*n^15-955307868581928127415611*n^14+ 11472046114816082588063817*n^13-106792293377024455727586423*n^12+ 738721077562262097382532266*n^11-3379371614643449995036570334*n^10+ 5203317161148286795762521308*n^9+60584215451525693980831730336*n^8-\ 609266346058275446297886818304*n^7+3081445811598086446386846603168*n^6-\ 10125730949747170146684320748096*n^5+22417536873103133917701666193920*n^4-\ 32662687508101593198522299059200*n^3+29212544289526655762187270144000*n^2-\ 13866375748631588427047362560000*n+2435363718402346824139407360000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2 *n-39)/(2*n-35)/(2*n-29)/(2*n-45), 10925/4194304*(6358499707*n^24-1820247620700 *n^23+245423227627474*n^22-20705223197537208*n^21+1224436431463139125*n^20-\ 53867970425405700564*n^19+1825433034689138693896*n^18-48647660109384780548544*n ^17+1031193549554184728988133*n^16-17449644785570321707169028*n^15+ 234712879386626220502804378*n^14-2471247460794712351096527000*n^13+ 19626162308618511357941486971*n^12-106304849041642229014498045452*n^11+ 236865145800893267441591124460*n^10+2029216926767761021191078770064*n^9-\ 27253383468035569647114791424176*n^8+174359373519264751510731628734144*n^7-\ 738198518452994611354076193590208*n^6+2187389796970525679414557290578688*n^5-\ 4521839352983371968581236640117760*n^4+6289934008856504063651133733785600*n^3-\ 5467531652614370531492950228992000*n^2+2568552411719800681335316316160000*n-\ 457848379059641202938208583680000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/( 2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29) /(2*n-45), 437/4194304*(157270139611*n^24-44837711057220*n^23+6012455235868594* n^22-503586943430969640*n^21+29499851162185606741*n^20-1281975750243662928300*n ^19+42759211980114300167464*n^18-1116447047098412766221760*n^17+ 23043169132875069815075461*n^16-376318927419890730610132380*n^15+ 4816179469785623694773790394*n^14-46974679314475950165377438280*n^13+ 323786444759393599186878405211*n^12-1160455332175323240370914257460*n^11-\ 4675310898427739099376781237556*n^10+109077167546987082265537948820400*n^9-\ 925966578241076124238684484571824*n^8+5113912277339114870115059122071360*n^7-\ 20068736295081413701977245258728896*n^6+56764034260134898407887358775169280*n^5 -113866482754006940251064295409075200*n^4+155479328946986302688024713618944000* n^3-133998622199101418598552811069440000*n^2+ 63088755515391197838128762880000000*n-11446209476491030073455214592000000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 10925/4194304*(12364425859* n^25-3806871937094*n^24+552047774137798*n^23-50079487851235796*n^22+ 3182805316747641277*n^21-150360552642238588034*n^20+5464569774037526669308*n^19 -155893295389428032907416*n^18+3526877348495760836627677*n^17-\ 63365639832665752353331274*n^16+895638489898405983707498038*n^15-\ 9677988208330694267035906676*n^14+73756588944792009285475474867*n^13-\ 278345444637321766862419744334*n^12-1765108472106572687838575797592*n^11+ 41696901856740553173319020906784*n^10-409116999907506061878758933357168*n^9+ 2697623161395738190265182901080096*n^8-12998456904967145922057137528785152*n^7+ 46627898944592322699976586723215104*n^6-123674228710036699923252227348992512*n^ 5+236653196468038665944138517895280640*n^4-312389907546151676377009802570342400 *n^3+263397927482640505329448940679168000*n^2-\ 122837073210776148607917409075200000*n+22434570573922418943972220600320000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 6555/4194304*( 20056544124*n^25-6127207800295*n^24+879918440011020*n^23-78863052814207250*n^22 +4937587517646046340*n^21-228957864278879637825*n^20+8129328866168626479220*n^ 19-225120621533510607262600*n^18+4897238208228577832507700*n^17-\ 83294953991046888753384425*n^16+1081753202004046017285048020*n^15-\ 9985285514931945852974139450*n^14+48353209146806830307342498620*n^13+ 268652564929334040515758671425*n^12-8732249224953480765504165890580*n^11+ 105972927762264341913913920291700*n^10-862368373794957165406687860686320*n^9+ 5182253709866539279666972355353200*n^8-23598158086921887003164535094354880*n^7+ 81501495295037333681697390959601600*n^6-210541777126474808071337218102750464*n^ 5+395647774959394346888139216678097920*n^4-516396350518198435195908027570892800 *n^3+433259002643462810932218868924416000*n^2-\ 202443452792838519824457770434560000*n+37390950956537364906620367667200000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 6555/33554432*( 308135411579*n^26-100995753395667*n^25+15576972369568480*n^24-\ 1501081087239728410*n^23+101173953395429769465*n^22-5056999383071736345245*n^21 +193781433590056518261770*n^20-5796752653499969657427360*n^19+ 136203968671431851615178325*n^18-2495159409055851590814114925*n^17+ 34523132192037808495147240620*n^16-325293759568019163942263398610*n^15+ 1124643970953738973685410095695*n^14+25755237670177067182803054573165*n^13-\ 621307825372935911718934694363030*n^12+7983201583823839707457214410087340*n^11-\ 73058120909605112522989101508560120*n^10+509252701086308497212930680012127760*n ^9-2759473139513838869542568741267525280*n^8+ 11647584245506490991325580149114712640*n^7-\ 37953182646075596323056457248027586944*n^6+ 93693687433545288125273971184383974912*n^5-\ 169985851636745331633291710936895682560*n^4+ 216108395775931888464744463053681254400*n^3-\ 178093704256222351151597713541111808000*n^2+ 82453596760811389522903392261242880000*n-15255507990267244881901110008217600000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 2185 /33554432*(871397567207*n^26-281889810510061*n^25+42796242743956130*n^24-\ 4046125773724941790*n^23+266423744374444810985*n^22-12935277100461485364595*n^ 21+477509256638295824399600*n^20-13582938748318654368408040*n^19+ 296592769509478746791237465*n^18-4811836697312687931260092435*n^17+ 51436551757174093791701581850*n^16-142105668850329791486960033590*n^15-\ 7578451998646731970623046365625*n^14+195300388362458063631831515671235*n^13-\ 2903753507564585554577582055066700*n^12+31388585757340362288154457556461660*n^ 11-261952751957342339928865111599295120*n^10+ 1722500479423738448751078528460213520*n^9-8964191148766766699383448126612251200 *n^8+36742580059736311133794383307855498560*n^7-\ 117147258013938513182598040048605030912*n^6+ 284607112359205944367818133456534542336*n^5-\ 510576836142847003228724414938274119680*n^4+ 644573079888145177409370905361455923200*n^3-\ 529644520506230218706040907109351424000*n^2+ 245584720111998643735645709319536640000*n-\ 45766523970801734645703330024652800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 6555/67108864*(1068214776973*n^27-\ 367871921956620*n^26+59474151008038377*n^25-5988173917752961740*n^24+ 419733577575978464565*n^23-21662261175432647528460*n^22+ 847087173456284735962845*n^21-25321244209996811040916380*n^20+ 569866830897064292373302235*n^19-9005552491138746434547058260*n^18+ 70924005890590043257099461675*n^17+939726270403013010520685307420*n^16-\ 48000804514346736033197395306425*n^15+1046287800122490759489970034311740*n^14-\ 15945764944394972436962935859468865*n^13+186985466942971448601124138671825420*n ^12-1744184050105439591741891326578001380*n^11+ 13105578541808613433426156123951656240*n^10-\ 79530186360984435536425147608821240560*n^9+ 388289036514790988204314485057097101120*n^8-\ 1511117579599287682398768282470811461568*n^7+ 4617841226215027270694331938896562607360*n^6-\ 10837650830048933081122600149742938473472*n^5+ 18910688609929876323949639855061482844160*n^4-\ 23366857300098344358310918044307523174400*n^3+ 18907353952783481963157829303774445568000*n^2-\ 8688145225767376889572463889002004480000*n+ 1617083846968327957481517660871065600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/8388608*(2254215828863*n^27-\ 759446931002031*n^26+119553718550877936*n^25-11645764749144393075*n^24+ 782324306125969979385*n^23-38118788402517793610055*n^22+ 1370338298430526537316490*n^21-35622291560136441223615695*n^20+ 596794837170911845648415805*n^19-2308127855573533293652211145*n^18-\ 227761302704830369140649400820*n^17+9040076649573513622370627343255*n^16-\ 211045207955346200141646438445905*n^15+3632516977820646229014269128065075*n^14-\ 49106708534367613745942924711677590*n^13+534758530306494118830782221537216155*n ^12-4741135386653600412595190979306918660*n^11+ 34325572210382724525981296635084107660*n^10-\ 202507037473175316228742449522697658000*n^9+ 967289643739902881268467303037833296080*n^8-\ 3700631151846813770625541911734335428288*n^7+ 11160062149499558816226102482731839026496*n^6-\ 25931622712042957979978448294931782518016*n^5+ 44929035134758702257272913390813773153280*n^4-\ 55273577186634658659631795224381978931200*n^3+ 44647731886798692262044033349172097024000*n^2-\ 20538892045230639643022151041120010240000*n+ 3840574136549778899018604444568780800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/33554432*(15268627136747*n^28-\ 5404630943831854*n^27+891248011016764779*n^26-90460189763980275330*n^25+ 6269507394904408051215*n^24-308973296325505518555030*n^23+ 10730538202346281397619255*n^22-233841150977372276725637250*n^21+ 898259188019747953640586345*n^20+169969136879317636428154496910*n^19-\ 8546405162034007725714138252735*n^18+258729651824216404303439976112650*n^17-\ 5824294231020087977363919139683195*n^16+103790517866176029361868736533076990*n^ 15-1503840477445804653670133584731398115*n^14+ 17945527083736701704219581048071125850*n^13-\ 177387474529491737912961566367726204840*n^12+ 1454312417773333760325245145707886442680*n^11-\ 9868052214420258893726008660280179436480*n^10+ 55128494981240725002162191767069886596000*n^9-\ 251439371675512138766103390009236960652672*n^8+ 924967357596723636482925124085965711922304*n^7-\ 2698459268693571254613297865580024091496704*n^6+ 6098374736129229355435351041402921349678080*n^5-\ 10327241093762756836541802617195537038233600*n^4+ 12475656895105777224287678167599016347648000*n^3-\ 9940605694358955627720897803269741363200000*n^2+ 4532481138632967989987630742237236428800000*n-\ 844926310040951357784092977805131776000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1035/33554432*( 4005189636109*n^28-1353705965164798*n^27+209845407827382453*n^26-\ 19489887980523807210*n^25+1171824831933320397105*n^24-43769794491618844431510*n ^23+602893395130700098838985*n^22+41276132966095768728337350*n^21-\ 3582968075188371286690057785*n^20+160606811923364902883035518270*n^19-\ 5156603057599442357949504954945*n^18+128962625541477669463731006265650*n^17-\ 2602874329102671235661618887836165*n^16+43172764888228804038461750098795230*n^ 15-594307718294104954462206459716584605*n^14+ 6823663399555779247007737420124500050*n^13-\ 65451185213453595638968853058720108480*n^12+ 523862611883521782058098567959704160760*n^11-\ 3486102945346228579695094222571453064160*n^10+ 19168930692322145925255587590616257309600*n^9-\ 86307669820042378005320201627001518275584*n^8+ 314212195136873427528655073749529179986048*n^7-\ 909163726831472404810635762941479655017728*n^6+ 2041844765211995475721521626956301110594560*n^5-\ 3442410144977553226833813032621273576755200*n^4+ 4147247170894322383571352169805171521536000*n^3-\ 3301163766608973759967168601200008806400000*n^2+ 1506356536634053079425590707847305625600000*n-\ 281642103346983785928030992601710592000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 30015/134217728*( 765807215933*n^29-258106733425529*n^28+38451841181105107*n^27-\ 3146329475898292341*n^26+122745864257558487855*n^25+2972032559364608325495*n^24 -789871529006038953848985*n^23+62525991161971295812594455*n^22-\ 3272539706158480060002694995*n^21+129259883197432029461410976985*n^20-\ 4056950628504170768911005193455*n^19+103908451905563983695839611582065*n^18-\ 2205941556293976915302544739486155*n^17+39191972818293338718421071208027965*n^ 16-586096690110760017320682984388239795*n^15+ 7399898952066604191890528166261475485*n^14-\ 78945163865940688437555643529470624110*n^13+ 710856846220179945143419776578451267180*n^12-\ 5386328135898158124177724081380180871640*n^11+ 34174329528720451163586386266570728536720*n^10-\ 180254393873052410130137879414125198098208*n^9+ 782678464051072667661272236021507841949504*n^8-\ 2761021758395196005290574608271754555599232*n^7+ 7774527407313951815181630166925282960303616*n^6-\ 17059371525315366957155288098759165238200320*n^5+ 28204976865375645200439866508673405784678400*n^4-\ 33442039288825051372672755147636126154752000*n^3+ 26290755787413795074272191552502210560000000*n^2-\ 11892815148485115600229865959867102003200000*n+ 2214289640107320799710036769420345344000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 435/67108864* (13637308774829*n^29-3471890228865463*n^28+233907029641310991*n^27+ 29854184291520589413*n^26-8159659910406207972405*n^25+937739620400030076082545* n^24-71257440153058812923462205*n^23+4017056997539765747364158505*n^22-\ 177082623632220376586254473735*n^21+6290471171753294538405048630975*n^20-\ 183536848999208320338633785607915*n^19+4454976923190499995261169785417375*n^18-\ 90742764373909950898323101190561015*n^17+1559937301947447587095201208914996635* n^16-22710382345680668191287483495964039935*n^15+ 280433833827496845566261484895391147715*n^14-\ 2936635345956217786916721796884561171930*n^13+ 26031310169236048123136848633596186805740*n^12-\ 194648235827949767353164140206163745556520*n^11+ 1221231921247570540228372777986642445681360*n^10-\ 6381210443278684295121614731491450555835104*n^9+ 27492119494618866541252403749888663023144768*n^8-\ 96365448041136925710094572460238544961420416*n^7+ 269971021469965046891759152920537454539205632*n^6-\ 590093568958128916650281719249006648392560640*n^5+ 972948807767293668183067607714996980249804800*n^4-\ 1151686696380470864755764564052856086401024000*n^3+ 904870680703963908590742663156612260659200000*n^2-\ 409538863693760101114090931122925430374400000*n+ 76392992583702567589996268545001914368000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 435/ 536870912*(2711406021061*n^30+34334980462596045*n^29-15053033077075479895*n^28+ 3105563144058583490205*n^27-404907348660903253413171*n^26+ 37530903815552239618907745*n^25-2634303831556900969902303075*n^24+ 145579460347995587443498377825*n^23-6501025393810808429206929747885*n^22+ 238897666181949483231528160530375*n^21-7319233645084780690230852221326725*n^20+ 188736025933071795553862554263698775*n^19-4124080401083476735527395440118643345 *n^18+76721261463701187560140554604145479275*n^17-\ 1218700989429086059292297132527469217825*n^16+ 16553436746611026067482134934213971481675*n^15-\ 192264319586989676081200037961581666131860*n^14+ 1906994984706034788086701904444101533496800*n^13-\ 16108617362358281778181811774345313558892000*n^12+ 115396273893301865290232666212216095170196000*n^11-\ 696915685515186595210350792066101826671225216*n^10+ 3520300149699477122797304535620358765443182080*n^9-\ 14718455698849113186870591508195054148736410880*n^8+ 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40515470540145415003915574570865*n^22-1438593334066425028836279832471575*n^21+ 42832899056311023891670308302521725*n^20-1078142698026407656566650290978425375* n^19+23077209252804804669125563595904874305*n^18-\ 421741436984931250296018301836943483875*n^17+ 6596776035949887737923752592865337410025*n^16-\ 88409386789472871605328394024713312817875*n^15+ 1014926682567019915771143349242171290661990*n^14-\ 9964749984803140457898009036194607327948300*n^13+ 83432548900248545978457254003892406401603800*n^12-\ 593127239032939346121617873888543958080706000*n^11+ 3558636240135666918174597199915591206222620704*n^10-\ 17875462424346264231329037851872442308707159360*n^9+ 74388576502420207831417327216012773285353735040*n^8-\ 252969517398231938025536512251877567510121856000*n^7+ 690363880064471302874030886727323860285764386816*n^6-\ 1475532044817975313459512030580704556108678594560*n^5+ 2387548905261042455119470155024969517637324800000*n^4-\ 2783209159684400417341675970433300769404026880000*n^3+ 2161007831662150137319476956789525364902461440000*n^2-\ 970106100880405382239397243136678414935654400000*n+ 180287462497538059512391193766204517908480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2 *n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), -2697/1073741824*(277407094762379*n^31-152544362614652842*n^30+ 39396135805456636420*n^29-6386444298676421079920*n^28+731904894843048836216076* n^27-63269213312250828290843568*n^26+4295942658735332338058735280*n^25-\ 235470583143456978500735627400*n^24+10623259704686345928219737148810*n^23-\ 400113349376312613466026962997780*n^22+12714048882991146355571632410757800*n^21 -343524541867786791091685257883476200*n^20+ 7937670559201999192661683004695521420*n^19-\ 157485978158860510046536088525386431360*n^18+ 2689881087183028767229454943717360867600*n^17-\ 39604410975463485199819514864163463507800*n^16+ 502734212917822061325239078494147900686435*n^15-\ 5496490868387659126568196529984336125690930*n^14+ 51647127997420087949618880560481101910601300*n^13-\ 415663848890946914392591161986805688553846600*n^12+ 2851641319940763230975593240640004749207332976*n^11-\ 16570218460131596885092144915609518808506478048*n^10+ 80876925830662205345856564976303445589087468480*n^9-\ 328026791623903471032150261206440986251708062080*n^8+ 1090248305176573821006253828280385079013194811904*n^7-\ 2915609135144683649537186755726139866877736425472*n^6+ 6121681693886383107559299417889693150112096133120*n^5-\ 9753846765502133647215548208192681515459174400000*n^4+ 11222170162301826392657205016110303131431403520000*n^3-\ 8620021388871714563730349990832031295254036480000*n^2+ 3837764468619291269408236525534551719333068800000*n-\ 709518400796762685822958891596030683381760000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n -21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-45), -899/268435456*(305278824007793*n^31-158936636305681639*n^30+ 39257007539317543195*n^29-6131800290470774473385*n^28+680940444441206676321147* n^27-57292883200672606407175101*n^26+3799866520996663004552970435*n^25-\ 204040632054386937543822374325*n^24+9039957558631549528316397011745*n^23-\ 335056045814277486764510467700535*n^22+10495795060543976956623427667160525*n^21 -279999592206026748640528026021959475*n^20+ 6396630459167222096900557254034765065*n^19-\ 125627020233300937748941452282750139695*n^18+ 2126312075809400370876291370593455429025*n^17-\ 31053763996316358926102387234162898116775*n^16+ 391355203692505226789822960984065006610970*n^15-\ 4251401804306679079381555927665788830177110*n^14+ 39721900865111008304536572595615818704072500*n^13-\ 318098350516681644064995162492471958787073800*n^12+ 2172827920585553832037983943152595844086773392*n^11-\ 12578423112846612377481649492290611870964469216*n^10+ 61196805043815396989370219035060324516984364480*n^9-\ 247538245007044228822190537309323651526728602240*n^8+ 820910226509632454721601011216886447869062469888*n^7-\ 2191453426804747137036392931153566194062548296704*n^6+ 4595050806134601424084846261811852048949341859840*n^5-\ 7314526107516893309188281045686553954533222400000*n^4+ 8410934961007857540392897885353028867335127040000*n^3-\ 6459453041404901728965707262651249245782671360000*n^2+ 2876428270265759150010710089419086784469401600000*n-\ 532138800597572014367219168697023012536320000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n -21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-45), -899/1073741824*(3142239813201607*n^32-1687227817126820240*n^31+ 432355123402518936728*n^30-70427025446485251381760*n^29+ 8194438231394472393604548*n^28-725507668726683060254016960*n^27+ 50839917514673875485684847512*n^26-2895627484987307498098345883520*n^25+ 136596328210838668005766158691530*n^24-5411083506957943270618523980725600*n^23+ 181860922021312640676932097331934520*n^22-5225662002864515638467677843909520000 *n^21+129110337178300662766162680857977156260*n^20-\ 2754057576642610757841639648773533003200*n^19+ 50858941634476286543081991150566380227240*n^18-\ 814370352531497239567582601559521507049600*n^17+ 11312188192596911834338526569405184013172455*n^16-\ 136239578704254240919158066682227560787123600*n^15+ 1420386756493484961595218519047776156151435120*n^14-\ 12785112232961422151293906048459743921952969600*n^13+ 98977712427857928092335228660209596810722107808*n^12-\ 655648433032049970360332746585289326806926274560*n^11+ 3691440796540174078893805189323338619490375372032*n^10-\ 17513753037956742261696823234112428944598703523840*n^9+ 69254330258090692644126752010653227861737887745792*n^8-\ 225038712193890474419901490708503513679250642595840*n^7+ 589922513584590831056632201181632043256061694926848*n^6-\ 1217172228533214013761436751486247641548238086471680*n^5+ 1910346273075809549420053207605652413934287544320000*n^4-\ 2170125514816921853847624027426969738422143549440000*n^3+ 1649738554364939480839355807164895014733618872320000*n^2-\ 728762275311353500574725216437388038444888883200000*n+ 134098977750588147620539230511649799159152640000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/( 2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37) /(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 57)/(2*n-63)/(2*n-45), -31/1073741824*(108547406308979371*n^32-\ 57274961073495382832*n^31+14446405491719109916472*n^30-\ 2319627212266023027255040*n^29+266390384319019310563486644*n^28-\ 23305783501985876753697235008*n^27+1615477536753237451506659279928*n^26-\ 91100926440002185847896116766080*n^25+4258692811286132682366061271229090*n^24-\ 167309369047006924734926041624680480*n^23+5580692010712862799209429043442472280 *n^22-159254668132153826431545961527656016000*n^21+ 3910047072122205934269628001275337109780*n^20-\ 82930197803518933501318718527679153271360*n^19+ 1523550767524262793136609949326429938257160*n^18-\ 24281580869456000232513893597167621567094400*n^17+ 335867624187569405495974790584792940422199115*n^16-\ 4029761293377643806500699156621924133882450480*n^15+ 41870972991283411675159662245640122047922498480*n^14-\ 375756162783227504554216356487196883497422134400*n^13+ 2901271962735526989028434453609033946921478926624*n^12-\ 19174152143972032402295959596510832706972741931008*n^11+ 107738459518038350692908868438347719564862503923968*n^10-\ 510282777598836915569633730579172956746834356623360*n^9+ 2014905307415938841346781568868111429511859649989376*n^8-\ 6539623125133832280808328624611980324888440761208832*n^7+ 17127071708465827167013004092869171845933257603571712*n^6-\ 35312741727308648657967668776494892957941237761310720*n^5+ 55395674008366675999499846625090708319810442158080000*n^4-\ 62910292947069065819501200850961910102876467363840000*n^3+ 47820219160630315918457933848089284083843745710080000*n^2-\ 21126656040820885508465890208322494792726269132800000*n+ 3888870354767056280995637684837844175615426560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/ (2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37 )/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 57)/(2*n-63)/(2*n-45), -465/1073741824*(16360419778194966*n^33-\ 9063961731529470601*n^32+2406404720737364603344*n^31-407707352786936927412776*n ^30+49526172558715916303831624*n^29-4594499501792032767439066716*n^28+ 338550444395737364181807369360*n^27-20347427317706774502590904510408*n^26+ 1016435425277063202417133025947620*n^25-42789723540194870966936685491674230*n^ 24+1533838212056201899524129360178087440*n^23-\ 47182655378047775419399653267628167720*n^22+ 1252798387515923147639980866863886777480*n^21-\ 28835741020080608879722354864213872051900*n^20+ 577063452843978325246743140939315833115760*n^19-\ 10059167607564718131749283084575889775238520*n^18+ 152865751667559136187537703401059244468113590*n^17-\ 2024968979643384789143015940109598719633508745*n^16+ 23357803334107537215518486049989489426698924320*n^15-\ 234143991096970417893970720799474407907315932400*n^14+ 2033592437527168213232358373640866288622218949504*n^13-\ 15239888629932030584993189717411446478671692596704*n^12+ 98013898182603530823811236989153726558526093427456*n^11-\ 537245540172703359621567566237815430512854467960064*n^10+ 2487818081137187477677545264682142153141553358557696*n^9-\ 9624475480692241762385763173096638626205550644687104*n^8+ 30665020738530984439942576225714651833564508843192320*n^7-\ 78985874306831496090822390346925638317588521719898112*n^6+ 160452230365310764068598569521607957098817744654827520*n^5-\ 248420181176557661247873027695273725623503008210944000*n^4+ 278914074595536033938186543841872164916962867937280000*n^3-\ 209969693379269362685045108589356270109042742394880000*n^2+ 92044760729794067815941481353345189747963514060800000*n-\ 16851771537323910550981096634297324761000181760000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2* n-51)/(2*n-57)/(2*n-63)/(2*n-45), -15/1073741824*(551662488067561550*n^33-\ 303361019574002943443*n^32+79983374432121899421408*n^31-\ 13464225492387273711037656*n^30+1625806224856516943328025960*n^29-\ 149989475979351862310664602164*n^28+10995393764390862254011603471056*n^27-\ 657699710674331673828397030435608*n^26+32710334854456628550812667577406580*n^25 -1371447061271003049034482711221186610*n^24+ 48977023894962005420636414349516198000*n^23-\ 1501410622758548388079406435387043398520*n^22+ 39739840129770403150693864187418306676200*n^21-\ 912053707568582375542420622968805964591060*n^20+ 18204013458541615235392797925906845424655280*n^19-\ 316565472256635661424442975690971559918135720*n^18+ 4800288210368112969325665759116716980200011950*n^17-\ 63463374248734371462069120964397287194218689875*n^16+ 730753889959125049954938191111464323717831154480*n^15-\ 7313716027301506749716306811171484607040857330800*n^14+ 63432475724712694560964624435827884860012576219200*n^13-\ 474782067838025668460094897181490172430499967013152*n^12+ 3050230414403785865098228401403358282249364943849472*n^11-\ 16703742395391362181050389049660251846527241317373184*n^10+ 77288537657316154828088506836591975964812613890301440*n^9-\ 298802892692950455631791155540552868477871538739069696*n^8+ 951512369604997964234451929220615394224546153011810304*n^7-\ 2449812929963025016869313681840911340860864662857408512*n^6+ 4974934769603912923446225112912218747749242996484997120*n^5-\ 7700692611577580977968010859844313156957345584635904000*n^4+ 8644813687690005781035345843433355487027711527485440000*n^3-\ 6507646988293580837983080093898679648895324066938880000*n^2+ 2852919477858690438112214224824970400943507819724800000*n-\ 522404917657041227080413995663217067591005634560000000)/(2*n-5)/(-1+2*n)/(2*n-3 )/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11 )/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2* n-51)/(2*n-57)/(2*n-63)/(2*n-45), -5/4294967296*(13985098418925397067*n^34-\ 8121392020109379417943*n^33+2264944165066625348457206*n^32-\ 403977655038445368651178936*n^31+51775087879555289016550666532*n^30-\ 5078999159594717574901975939780*n^29+396660412024298828173670605864128*n^28-\ 25327573980861835792360907461950936*n^27+1347484673698429597633210785309257682* n^26-60570433448641320664557157018894156650*n^25+ 2324606899212698765639125158434238153420*n^24-\ 76778006042821914419936892522274847991480*n^23+ 2195488087053005081408761309574917205356740*n^22-\ 54597864306758614526577539230025611390397220*n^21+ 1184582763744382264280182700559417605559091920*n^20-\ 22471157861137471256644785229956161457822801000*n^19+ 373132309115575616988352217683279624312348778235*n^18-\ 5424977658200635235779419391953270671923988361015*n^17+ 69020625810175636848823733139262910470843938938350*n^16-\ 767328436529971227350882736023527778823985048064720*n^15+ 7436922480952422092399432305788698537601091457131648*n^14-\ 62630300432065011710764372559398367901086412419433632*n^13+ 456306579253251720027871392895208880245086862927185984*n^12-\ 2860027293356561520737220257239615020745540479967833344*n^11+ 15312205408443676816497905805216469876221212741098867968*n^10-\ 69402197274257302881252354882068446991281165636109633280*n^9+ 263312029545777827601484652295357178816694084713384852992*n^8-\ 824283011996634445251370970718040023060690807513484099584*n^7+ 2089691907226196391572786405822994518189705023126235824128*n^6-\ 4185118702817742666453137266204850721343560453107453460480*n^5+ 6398639549287918229581251079693485793127900171489427456000*n^4-\ 7105824937440807786824179298940263247246080047105966080000*n^3+ 5299900118303790933238249079096820717719945462777118720000*n^2-\ 2306049755577780955440709394848006345155975461417779200000*n+ 420013553796261146572652852513226522343168530186240000000)/(2*n-5)/(-1+2*n)/(2* n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n -11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/ (2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), (12465141979629205048*(n-33)!*(2* n-71)!*(n+1)+18697712969443807572*((n+1)*(2*n-70)!-1/18697712969443807572* binomial(2*n,n)*(n-36)!*(n-33)!)*(n-34)!)/(n-36)!/(n-34)!/(n-33)!/binomial(2*n, n), ((12465141979629205048*n+12465141979629205048)*(2*n-71)!-(n-36)!*(n-34)!* binomial(2*n,n))/(n-36)!/(n-34)!/binomial(2*n,n)] The limits, as n goes to infinity are 34359738331 34359737701 137438928789 8589924833 68719097077 68717963841 [-----------, -----------, ------------, ----------, -----------, -----------, 34359738368 34359738368 137438953472 8589934592 68719476736 68719476736 1099429343111 549633174523 8790860044797 8783416274463 35071456445775 -------------, ------------, -------------, -------------, --------------, 1099511627776 549755813888 8796093022208 8796093022208 35184372088832 8737754704925 69466609298975 68727051010007 135081352509575 -------------, --------------, --------------, ---------------, 8796093022208 70368744177664 70368744177664 140737488355328 32867661683205 2019827622900345 1904003684347295 7002147863058015 --------------, ----------------, ----------------, ----------------, 35184372088832 2251799813685248 2251799813685248 9007199254740992 777704460957735 5267676362177715 4145371273372815 22985703586228995 ----------------, ----------------, ----------------, -----------------, 1125899906842624 9007199254740992 9007199254740992 72057594037927936 5932229317050615 1179461619161535 -93998841333773667 -----------------, ------------------, ------------------, 36028797018963968 576460752303423488 576460752303423488 -748166934574136163 -274445662783005907 -2824873592068244693 -------------------, -------------------, --------------------, 2305843009213693952 576460752303423488 4611686018427387904 -3364969595578360501 -3803797598430329595 -4137468660506711625 --------------------, --------------------, --------------------, 4611686018427387904 4611686018427387904 4611686018427387904 -69925492094626985335 -72228833547384555833 -293589762431899175225 ---------------------, ---------------------, ----------------------] 73786976294838206464 73786976294838206464 295147905179352825856 and in Maple notation [34359738331/34359738368, 34359737701/34359738368, 137438928789/137438953472, 8589924833/8589934592, 68719097077/68719476736, 68717963841/68719476736, 1099429343111/1099511627776, 549633174523/549755813888, 8790860044797/ 8796093022208, 8783416274463/8796093022208, 35071456445775/35184372088832, 8737754704925/8796093022208, 69466609298975/70368744177664, 68727051010007/ 70368744177664, 135081352509575/140737488355328, 32867661683205/35184372088832, 2019827622900345/2251799813685248, 1904003684347295/2251799813685248, 7002147863058015/9007199254740992, 777704460957735/1125899906842624, 5267676362177715/9007199254740992, 4145371273372815/9007199254740992, 22985703586228995/72057594037927936, 5932229317050615/36028797018963968, 1179461619161535/576460752303423488, -93998841333773667/576460752303423488, -\ 748166934574136163/2305843009213693952, -274445662783005907/576460752303423488, -2824873592068244693/4611686018427387904, -3364969595578360501/ 4611686018427387904, -3803797598430329595/4611686018427387904, -\ 4137468660506711625/4611686018427387904, -69925492094626985335/ 73786976294838206464, -72228833547384555833/73786976294838206464, -\ 293589762431899175225/295147905179352825856] and in floating point [.9999999989, .9999999806, .9999998204, .9999988639, .9999944752, .9999779845, .9999251625, .9997769203, .9994050793, .9985588206, .9967907444, .9933677012, .\ 9871798923, .9766701369, .9598107376, .9341551300, .8969836531, .8455474917, .7\ 773945779, .6907403191, .5848295584, .4602286633, .3189907170, .1646524394, .\ 2046039760e-2, -.1630619968, -.3244656863, -.4760873341, -.6125468171, -.729661\ 4692, -.8248171240, -.8971705021, -.9476671305, -.9788832281, -.9947208070] The cut off is at j=, 26 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 37], vs. those in the, 2, -th row from j=1 to j=, 36, are as follws 18 17 16 [37 (928641577 n - 150439935303 n + 11283691588680 n 15 14 13 - 520020701916300 n + 16479197314826694 n - 380739702314665386 n 12 11 + 6635128675474273780 n - 88999859907913614660 n 10 9 + 929326601734430580681 n - 7589837832387925948119 n 8 7 + 48437100521612301380700 n - 239980894421875540085880 n 6 5 + 911008111793346945640048 n - 2588554260497524517502192 n 4 3 + 5250136072208855446059840 n - 6569637277627140970556160 n 2 + 844775091190634502528000 n + 16461144005434051359744000 n - 29051203810321992499200000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 17) (2 n - 23) 19 18 (2 n - 35) (2 n - 29)), 703 (97751744 n - 17644189477 n 17 16 15 + 1480718879691 n - 76712510060328 n + 2747322620966748 n 14 13 - 72168808496065134 n + 1439867826165827522 n 12 11 - 22289110732031271076 n + 271126935345816979572 n 10 9 - 2608300571445490079301 n + 19869919723011576800283 n 8 7 - 119412979660619076723564 n + 560552449366774860067736 n 6 5 - 2014076678351943969331888 n + 5291535504434223687795504 n 4 3 - 8853524267468645022484032 n + 3309313718991120231571200 n 2 + 25515246492569920489804800 n - 52758349877550094525440000 n + 1529010726859052236800000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 19 18 (2 n - 23) (2 n - 35) (2 n - 29)), 19 (3616814213 n - 652834861024 n 17 16 15 + 54786565176207 n - 2838358230625236 n + 101650486128005946 n 14 13 - 2670213463040005008 n + 53273317452901543694 n 12 11 - 824619551528433294712 n + 10029035624777989102569 n 10 9 - 96434321069814918816912 n + 733600024324861096200291 n 8 7 - 4390895195193917465141268 n + 20371532982822534591000872 n 6 5 - 70733426378652360652226656 n + 167339526376010344993685808 n 4 3 - 182475263830543198154846784 n - 303618299037890374593849600 n 2 + 1303540184189655023701017600 n - 988782187548150588257280000 n + 56573396893784932761600000)/(131072 (2 n - 29) (2 n - 35) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) 20 19 (2 n - 3) (-1 + 2 n) (2 n - 5)), 19 (14467249117 n - 2893447270850 n 18 17 16 + 270066556713735 n - 15626670209532150 n + 627982701706246722 n 15 14 - 18608783839273452900 n + 421325116683480567070 n 13 12 - 7451848786828645148300 n + 104369221427791331647857 n 11 10 - 1166149074842222250801450 n + 10415519694572559546347355 n 9 8 - 74072957706303969607741950 n + 414244769843597443230017152 n 7 6 - 1769852571678099138568370000 n + 5395024445276477179227974640 n 5 4 - 9520707988295265879001317600 n - 1006751172491994984011710848 n 3 2 + 48185733895462256347745395200 n - 90804601632050547253167052800 n + 53697508271821347511818240000 n - 4412724957715224755404800000)/(262144 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) (2 n - 19) (2 n - 3) (-1 + 2 n) 20 19 18 (2 n - 5)), 19 (3616803163 n - 723357642470 n + 67515743862405 n 17 16 15 - 3906548222293230 n + 156984609356820168 n - 4651438712813764140 n 14 13 + 105291715877726195410 n - 1861351784914228986860 n 12 11 + 26040667131802161962643 n - 290227645657172024797710 n 10 9 + 2577669065321797836836265 n - 18107920324533541562299590 n 8 7 + 98621744878086962078058058 n - 398028463253056721375149280 n 6 5 + 1063990529114877592837905120 n - 1187023902727931007343060320 n 4 3 - 2963880425009918405102584032 n + 13596202877262712558359033600 n 2 - 19699226483802542205663859200 n + 10636608260209569887208960000 n - 1103181239428806188851200000)/(65536 (2 n - 29) (2 n - 35) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 25) (2 n - 31) (2 n - 13) (2 n - 27) (2 n - 21) (2 n - 11) (2 n - 33) (2 n - 9) (2 n - 7) 21 (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), 589 (233339870 n 20 19 18 - 51450702597 n + 5312327247220 n - 341300008588035 n 17 16 + 15291920289178050 n - 507520839744436392 n 15 14 + 12934501643628820280 n - 258913526592874323870 n 13 12 + 4127772454166377473730 n - 52800640594285957253517 n 11 10 + 542647220110850455959540 n - 4455700511461449192506655 n 9 8 + 28775716642553965494862910 n - 141363050075512279041040902 n 7 6 + 490988202230299870881887200 n - 974069814709771593728305440 n 5 4 - 187629991436572468563094560 n + 7208622997524179910081573408 n 3 2 - 19663119400801044239089194240 n + 23803971567195748437883584000 n - 12056846900906860975802880000 n + 1459046155373582378803200000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 21 20 (2 n - 35) (2 n - 29)), 589 (233332408 n - 51447389469 n 19 18 17 + 5311639959710 n - 341211594409455 n + 15284026456775958 n 16 15 - 507002196400253064 n + 12908561536291630780 n 14 13 - 257905958546259124710 n + 4097047621604059423028 n 12 11 - 52062192149100448347669 n + 528688994394150744267750 n 10 9 - 4249915476967793837610315 n + 26445586742452580520889438 n 8 7 - 121592056778124375553589094 n + 369969575506796692481640320 n 6 5 - 470720784824816509522045920 n - 1470923228747598075188780832 n 4 3 + 8787944367042289911291979296 n - 19560403765755316552745698560 n 2 + 21701323650004269829738790400 n - 10752906553041461228259840000 n + 1459046155373582378803200000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 17081 ( 22 21 20 19 128723267 n - 31147248979 n + 3539714269399 n - 251115791985905 n 18 17 + 12466798786621872 n - 460138869071415534 n 16 15 + 13090729970382920174 n - 293608445411116497970 n 14 13 + 5262923250357300727507 n - 75906220578535767619199 n 12 11 + 881215025397445679053659 n - 8178326696202589975133085 n 10 9 + 59667029176302564854327402 n - 330866026761845624685065584 n 8 7 + 1297147937773975155090594064 n - 2887363881730044309041094800 n 6 5 - 1264828233910683058677015648 n + 34999381235303309858096359296 n 4 3 - 127417069820926772208889687296 n + 239843682768146762239343861760 n 2 - 244581637214251599671724134400 n + 117999055853166122025431040000 n - 17307306118914218562355200000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 551 ( 22 21 20 15957903473 n - 3860621425441 n + 438593702814421 n 19 18 - 31096948474824155 n + 1542290255446342248 n 17 16 - 56828070689603839626 n + 1612134454298666876666 n 15 14 - 35988887438207008272550 n + 640228677979959016587793 n 13 12 - 9123761339251242983400101 n + 103962540497188857340020801 n 11 10 - 937563116007337442709781695 n + 6542528792792460311854160678 n 9 8 - 33725443034962819820839166656 n + 114516084397214763732332233216 n 7 6 - 146983409341877976868727489840 n - 810006126751729154493799298592 n 5 4 + 5577221823439531950539252221824 n - 16841886724535735954935379695104 n 3 + 29186005537260267505820599818240 n 2 - 28585593758860173675906984345600 n + 13730767006821331805619240960000 n - 2146105958745363101732044800000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 342171 23 22 21 20 (51365796 n - 13577429227 n + 1689722762827 n - 131604766271055 n 19 18 17 + 7191343159298221 n - 292870331587798272 n + 9214447016162340942 n 16 15 - 228997458283380601870 n + 4554814676030730817486 n 14 13 - 72957738320813365273947 n + 940894001840571344598487 n 12 11 - 9698569040662531683161235 n + 78549540109094849395347681 n 10 9 - 482802419973648350249995882 n + 2080417143266271278847011392 n 8 7 - 4748529475510411423430875200 n - 7968058097204276507282107184 n 6 5 + 120314946580233153819185135328 n - 531305370569828752347897963648 n 4 3 + 1365077215143653737244458959360 n - 2164721959289525266028835072000 n 2 + 2017206800623813889185062912000 n - 954493194639918290105548800000 n + 155514924546765442154496000000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) 23 22 21 (2 n - 45)), 11799 (1487843321 n - 392995423308 n + 48853245308736 n 20 19 18 - 3798275263121570 n + 206993847659510681 n - 8395839039033821298 n 17 16 + 262573745914105880936 n - 6468637527312262128420 n 15 14 + 127057406484609331635531 n - 1999275780541196578552208 n 13 12 + 25144631796842244428135576 n - 250086019171677679614102570 n 11 10 + 1920787088410959194513610611 n - 10814365785821169156236857458 n 9 8 + 38521589109757648050125076096 n - 25723211077811922152645718320 n 7 6 - 648361953780808307099837152144 n + 4556513708461139183560591138272 n 5 - 17074987763430000880714279561344 n 4 + 40614310238920344143342415290880 n 3 - 61694223367945894945091935488000 n 2 + 56289343623442410178304925696000 n - 26656426006190277393903206400000 n + 4509932811856197822480384000000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) 24 23 22 (2 n - 45)), 294975 (237494791 n - 68230457700 n + 9244837036162 n 21 20 19 - 785190711708304 n + 46854766183819825 n - 2086273570987337732 n 18 17 + 71827513250503267048 n - 1954304751101449321872 n 16 15 + 42561858613108961655529 n - 746267850495605204803564 n 14 13 + 10527972951521938504900714 n - 118546480320069767822402000 n 12 11 + 1045241119876770845985071023 n - 6920684421823117854076032876 n 10 9 + 30798972118814947142050135180 n - 51108423605770281845164196368 n 8 7 - 459363542125314611137618322288 n + 4697707829666248252367731291072 n 6 - 23442745014045996703810906759104 n 5 + 75788606923170990103627664396544 n 4 - 165171181202829089603852357882880 n 3 + 237182320497328413957477257932800 n 2 - 209275830967984366992963895296000 n + 98024921093233694781323182080000 n - 16957347372579303812526243840000)/( 4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 10925 ( 24 23 22 6384352339 n - 1830459410340 n + 247304110016002 n 21 20 - 20919755835136488 n + 1241409140178893917 n 19 18 - 54856696201068725964 n + 1869376607582774620264 n 17 16 - 50171370947540700310464 n + 1072999313217797101873165 n 15 14 - 18365024925231631997296188 n + 250777923679740690104863306 n 13 12 - 2697433822334188452102948360 n + 22175256278570214532913947603 n 11 10 - 129175773389767402110725988372 n + 398745538356534771801656073388 n 9 8 + 1137903557894306570468525615664 n - 23516349519713090822448996962864 n 7 + 162826984316388783155258080851264 n 6 - 713546281862119018597754894592960 n 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768748944669851575541449655147739043624680629364326400000 n + 140004517932087048857550950837742174114389510062080000000)/(4294967296 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 65) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 57) 35 (2 n - 63) (2 n - 45)), - 5 (28032449309546944688 n 34 33 - 17245045061670605415355 n + 5100921940422707155446725 n 32 31 - 966173343581611368688879240 n + 131676261664815896153145912584 n 30 - 13755381919257991822411468638772 n 29 + 1145725202716823129349945205476300 n 28 - 78149481134978181094094986214249400 n 27 + 4449185414773949363652163966654654248 n 26 - 214412377289575973158711151476989301698 n 25 + 8839772042376835796644046904268403757630 n 24 - 314318293461929599468011327744099864785400 n 23 + 9698887206916816231921622785182780590938920 n 22 - 260933156636718526945963479748257927347503380 n 21 + 6141675470838673276872981657820790777771717100 n 20 - 126775278240218926850979936227105865770778798600 n 19 + 2298326330234176736644614824058500303188693471080 n 18 - 36617775294834718310683761478629423483742088596635 n 17 + 512628642267214175129858207934975234764351833533925 n 16 - 6299870028868231334168675679162007788964981566075200 n 15 + 67845046871167102469681137567241958414333012432673552 n 14 - 638609819175514034921953575619966022712979900839168800 n 13 + 5235467319349473858426706882078846458471814417009093600 n 12 - 37213051288997888608299083250733001503196357571151573760 n 11 + 228004265111347405334419557527121670340464188088035484416 n 10 - 1195499182726158741336904666729360100823141094782376090368 n 9 + 5315875233252853773792022643283522068410028311615120569600 n 8 - 19818390796335714506592086298233937756022170226474468454400 n 7 + 61057448492122708413761558971319690247561272234878586560512 n 6 - 152562404375376719451955616781844087361002646221076437204992 n 5 + 301572840962718679213565552332471763287848751257758665605120 n 4 - 455716693584390700793074024288392955960932404810966237184000 n 3 + 500897509997129325111986631382369882594935675003650703360000 n 2 - 370301412098549221258316243782258403960893012094044078080000 n + 159956224023205554562947787552168932295489863585418444800000 n - 28980935211942019113513046823412630041678628582850560000000)/(4294967296 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 65) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 69)), ( 47839193543441813968 (n - 34)! (2 n - 73)! (n + 1) + 71758790315162720952 (n - 35)! ((n + 1) (2 n - 72)! - 1/71758790315162720952 binomial(2 n, n) (n - 37)! (n - 34)!))/((n - 37)! (n - 35)! (n - 34)! binomial(2 n, n)), ( (47839193543441813968 n + 47839193543441813968) (2 n - 73)! - (n - 37)! (n - 35)! binomial(2 n, n))/((n - 37)! (n - 35)! binomial(2 n, n))] and in Maple notation [37/131072*(928641577*n^18-150439935303*n^17+11283691588680*n^16-\ 520020701916300*n^15+16479197314826694*n^14-380739702314665386*n^13+ 6635128675474273780*n^12-88999859907913614660*n^11+929326601734430580681*n^10-\ 7589837832387925948119*n^9+48437100521612301380700*n^8-239980894421875540085880 *n^7+911008111793346945640048*n^6-2588554260497524517502192*n^5+ 5250136072208855446059840*n^4-6569637277627140970556160*n^3+ 844775091190634502528000*n^2+16461144005434051359744000*n-\ 29051203810321992499200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/( 2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-17) /(2*n-23)/(2*n-35)/(2*n-29), 703/131072*(97751744*n^19-17644189477*n^18+ 1480718879691*n^17-76712510060328*n^16+2747322620966748*n^15-72168808496065134* n^14+1439867826165827522*n^13-22289110732031271076*n^12+271126935345816979572*n ^11-2608300571445490079301*n^10+19869919723011576800283*n^9-\ 119412979660619076723564*n^8+560552449366774860067736*n^7-\ 2014076678351943969331888*n^6+5291535504434223687795504*n^5-\ 8853524267468645022484032*n^4+3309313718991120231571200*n^3+ 25515246492569920489804800*n^2-52758349877550094525440000*n+ 1529010726859052236800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2 *n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/ (2*n-17)/(2*n-23)/(2*n-35)/(2*n-29), 19/131072*(3616814213*n^19-652834861024*n^ 18+54786565176207*n^17-2838358230625236*n^16+101650486128005946*n^15-\ 2670213463040005008*n^14+53273317452901543694*n^13-824619551528433294712*n^12+ 10029035624777989102569*n^11-96434321069814918816912*n^10+ 733600024324861096200291*n^9-4390895195193917465141268*n^8+ 20371532982822534591000872*n^7-70733426378652360652226656*n^6+ 167339526376010344993685808*n^5-182475263830543198154846784*n^4-\ 303618299037890374593849600*n^3+1303540184189655023701017600*n^2-\ 988782187548150588257280000*n+56573396893784932761600000)/(2*n-29)/(2*n-35)/(2* n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-27)/(2*n-21)/( 2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 19/262144*( 14467249117*n^20-2893447270850*n^19+270066556713735*n^18-15626670209532150*n^17 +627982701706246722*n^16-18608783839273452900*n^15+421325116683480567070*n^14-\ 7451848786828645148300*n^13+104369221427791331647857*n^12-\ 1166149074842222250801450*n^11+10415519694572559546347355*n^10-\ 74072957706303969607741950*n^9+414244769843597443230017152*n^8-\ 1769852571678099138568370000*n^7+5395024445276477179227974640*n^6-\ 9520707988295265879001317600*n^5-1006751172491994984011710848*n^4+ 48185733895462256347745395200*n^3-90804601632050547253167052800*n^2+ 53697508271821347511818240000*n-4412724957715224755404800000)/(2*n-29)/(2*n-35) /(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 19/65536*(3616803163*n^20-723357642470*n^19+67515743862405*n^18-\ 3906548222293230*n^17+156984609356820168*n^16-4651438712813764140*n^15+ 105291715877726195410*n^14-1861351784914228986860*n^13+26040667131802161962643* n^12-290227645657172024797710*n^11+2577669065321797836836265*n^10-\ 18107920324533541562299590*n^9+98621744878086962078058058*n^8-\ 398028463253056721375149280*n^7+1063990529114877592837905120*n^6-\ 1187023902727931007343060320*n^5-2963880425009918405102584032*n^4+ 13596202877262712558359033600*n^3-19699226483802542205663859200*n^2+ 10636608260209569887208960000*n-1103181239428806188851200000)/(2*n-29)/(2*n-35) /(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-25)/(2*n-31)/(2*n-13)/(2*n-\ 27)/(2*n-21)/(2*n-11)/(2*n-33)/(2*n-9)/(2*n-7)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5 ), 589/65536*(233339870*n^21-51450702597*n^20+5312327247220*n^19-\ 341300008588035*n^18+15291920289178050*n^17-507520839744436392*n^16+ 12934501643628820280*n^15-258913526592874323870*n^14+4127772454166377473730*n^ 13-52800640594285957253517*n^12+542647220110850455959540*n^11-\ 4455700511461449192506655*n^10+28775716642553965494862910*n^9-\ 141363050075512279041040902*n^8+490988202230299870881887200*n^7-\ 974069814709771593728305440*n^6-187629991436572468563094560*n^5+ 7208622997524179910081573408*n^4-19663119400801044239089194240*n^3+ 23803971567195748437883584000*n^2-12056846900906860975802880000*n+ 1459046155373582378803200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 589/65536*(233332408 *n^21-51447389469*n^20+5311639959710*n^19-341211594409455*n^18+ 15284026456775958*n^17-507002196400253064*n^16+12908561536291630780*n^15-\ 257905958546259124710*n^14+4097047621604059423028*n^13-52062192149100448347669* n^12+528688994394150744267750*n^11-4249915476967793837610315*n^10+ 26445586742452580520889438*n^9-121592056778124375553589094*n^8+ 369969575506796692481640320*n^7-470720784824816509522045920*n^6-\ 1470923228747598075188780832*n^5+8787944367042289911291979296*n^4-\ 19560403765755316552745698560*n^3+21701323650004269829738790400*n^2-\ 10752906553041461228259840000*n+1459046155373582378803200000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35) /(2*n-29), 17081/524288*(128723267*n^22-31147248979*n^21+3539714269399*n^20-\ 251115791985905*n^19+12466798786621872*n^18-460138869071415534*n^17+ 13090729970382920174*n^16-293608445411116497970*n^15+5262923250357300727507*n^ 14-75906220578535767619199*n^13+881215025397445679053659*n^12-\ 8178326696202589975133085*n^11+59667029176302564854327402*n^10-\ 330866026761845624685065584*n^9+1297147937773975155090594064*n^8-\ 2887363881730044309041094800*n^7-1264828233910683058677015648*n^6+ 34999381235303309858096359296*n^5-127417069820926772208889687296*n^4+ 239843682768146762239343861760*n^3-244581637214251599671724134400*n^2+ 117999055853166122025431040000*n-17307306118914218562355200000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-35)/(2*n-29), 551/2097152*(15957903473*n^22-3860621425441*n^21+ 438593702814421*n^20-31096948474824155*n^19+1542290255446342248*n^18-\ 56828070689603839626*n^17+1612134454298666876666*n^16-35988887438207008272550*n ^15+640228677979959016587793*n^14-9123761339251242983400101*n^13+ 103962540497188857340020801*n^12-937563116007337442709781695*n^11+ 6542528792792460311854160678*n^10-33725443034962819820839166656*n^9+ 114516084397214763732332233216*n^8-146983409341877976868727489840*n^7-\ 810006126751729154493799298592*n^6+5577221823439531950539252221824*n^5-\ 16841886724535735954935379695104*n^4+29186005537260267505820599818240*n^3-\ 28585593758860173675906984345600*n^2+13730767006821331805619240960000*n-\ 2146105958745363101732044800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 342171/ 2097152*(51365796*n^23-13577429227*n^22+1689722762827*n^21-131604766271055*n^20 +7191343159298221*n^19-292870331587798272*n^18+9214447016162340942*n^17-\ 228997458283380601870*n^16+4554814676030730817486*n^15-72957738320813365273947* n^14+940894001840571344598487*n^13-9698569040662531683161235*n^12+ 78549540109094849395347681*n^11-482802419973648350249995882*n^10+ 2080417143266271278847011392*n^9-4748529475510411423430875200*n^8-\ 7968058097204276507282107184*n^7+120314946580233153819185135328*n^6-\ 531305370569828752347897963648*n^5+1365077215143653737244458959360*n^4-\ 2164721959289525266028835072000*n^3+2017206800623813889185062912000*n^2-\ 954493194639918290105548800000*n+155514924546765442154496000000)/(2*n-5)/(-1+2* n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-35)/(2*n-29)/(2*n-45), 11799/2097152*(1487843321*n^23-392995423308*n^ 22+48853245308736*n^21-3798275263121570*n^20+206993847659510681*n^19-\ 8395839039033821298*n^18+262573745914105880936*n^17-6468637527312262128420*n^16 +127057406484609331635531*n^15-1999275780541196578552208*n^14+ 25144631796842244428135576*n^13-250086019171677679614102570*n^12+ 1920787088410959194513610611*n^11-10814365785821169156236857458*n^10+ 38521589109757648050125076096*n^9-25723211077811922152645718320*n^8-\ 648361953780808307099837152144*n^7+4556513708461139183560591138272*n^6-\ 17074987763430000880714279561344*n^5+40614310238920344143342415290880*n^4-\ 61694223367945894945091935488000*n^3+56289343623442410178304925696000*n^2-\ 26656426006190277393903206400000*n+4509932811856197822480384000000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2 *n-39)/(2*n-35)/(2*n-29)/(2*n-45), 294975/4194304*(237494791*n^24-68230457700*n ^23+9244837036162*n^22-785190711708304*n^21+46854766183819825*n^20-\ 2086273570987337732*n^19+71827513250503267048*n^18-1954304751101449321872*n^17+ 42561858613108961655529*n^16-746267850495605204803564*n^15+ 10527972951521938504900714*n^14-118546480320069767822402000*n^13+ 1045241119876770845985071023*n^12-6920684421823117854076032876*n^11+ 30798972118814947142050135180*n^10-51108423605770281845164196368*n^9-\ 459363542125314611137618322288*n^8+4697707829666248252367731291072*n^7-\ 23442745014045996703810906759104*n^6+75788606923170990103627664396544*n^5-\ 165171181202829089603852357882880*n^4+237182320497328413957477257932800*n^3-\ 209275830967984366992963895296000*n^2+98024921093233694781323182080000*n-\ 16957347372579303812526243840000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2 *n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2 *n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/ (2*n-45), 10925/4194304*(6384352339*n^24-1830459410340*n^23+247304110016002*n^ 22-20919755835136488*n^21+1241409140178893917*n^20-54856696201068725964*n^19+ 1869376607582774620264*n^18-50171370947540700310464*n^17+ 1072999313217797101873165*n^16-18365024925231631997296188*n^15+ 250777923679740690104863306*n^14-2697433822334188452102948360*n^13+ 22175256278570214532913947603*n^12-129175773389767402110725988372*n^11+ 398745538356534771801656073388*n^10+1137903557894306570468525615664*n^9-\ 23516349519713090822448996962864*n^8+162826984316388783155258080851264*n^7-\ 713546281862119018597754894592960*n^6+2155626270100569353336056063035648*n^5-\ 4509305291781851012676496414172160*n^4+6316957835610372319517522907033600*n^3-\ 5508619429948509169511777820672000*n^2+2585738912435252678442606428160000*n-\ 457848379059641202938208583680000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/( 2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29) /(2*n-45), 10925/4194304*(12671260142*n^25-3931753490275*n^24+575825590064600*n ^23-52893568770616450*n^22+3414801831679697690*n^21-164512638641512507885*n^20+ 6126935379782706259100*n^19-180236173077867061441000*n^18+ 4240064392247854001159570*n^17-80184016663826365833652285*n^16+ 1216679165870819012516840000*n^15-14650534378168882001855999050*n^14+ 136224897060494948007412530230*n^13-912840869348480094613728504835*n^12+ 3413530912402500011998941437900*n^11+8062765547314630997775878594900*n^10-\ 237820084619368366173487110090640*n^9+2028510091697317100928106649499440*n^8-\ 11063391268060066275196623605995200*n^7+42739863287055155664115742851953600*n^6 -119001607849066791572805004259956992*n^5+235188791618419399773914807729955840* n^4-316726378075328616102210404732006400*n^3+ 269453754680618400407310857822208000*n^2-125284827524211484965159535411200000*n +22434570573922418943972220600320000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29)/(2*n-45), 10925/4194304*(12511172690*n^25-3864836935339*n^24+ 562747085586008*n^23-51305705838540106*n^22+3280600964270427182*n^21-\ 156126463524920991949*n^20+5725194608974516398668*n^19-165138106524455028595576 *n^18+3788195721890512287473822*n^17-69310414949941143073985989*n^16+ 1005126734686195608934115648*n^15-11314991850908785297359897586*n^14+ 93623910556843784484792259442*n^13-473506653569430075683908620499*n^12-\ 222491783961652557651891512932*n^11+31978800304342576427649626714324*n^10-\ 361028340438437153396117292232528*n^9+2514809097996968621089410472913456*n^8-\ 12483204534594989851857206618745792*n^7+45618652625173562416039086711466944*n^6 -122497215891223799283297548183140608*n^5+236326637092078189682726639930680320* n^4-313523078133224073113837885169561600*n^3+ 264926462147137509802667188641792000*n^2-123445819066117458345457624842240000*n +22434570573922418943972220600320000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29)/(2*n-45), 6555/4194304*(40872343591*n^26-13587834786201*n^25+ 2131880466942005*n^24-209721293606451770*n^23+14491997738471466915*n^22-\ 746648819949541003495*n^21+29701611544823186779525*n^20-\ 931547167418718442183020*n^19+23297312540097953047023905*n^18-\ 466035345690074177958904535*n^17+7408640208643994941104799575*n^16-\ 91540456040037645032533547770*n^15+827323776962502068909342901385*n^14-\ 4377540783089312709504643102665*n^13-9047954914775261163015680651425*n^12+ 495270849163034630276119695474880*n^11-6176620947677820578522134959794100*n^10+ 49916138821540138331422943298513920*n^9-295583287477183508808447762763014000*n^ 8+1323965106890445492057757924012884480*n^7-\ 4498465514578532337857307382088277696*n^6+ 11441979925895891334054802253635742976*n^5-\ 21193037892639919942563887981389255680*n^4+ 27292721352286252621023945279523123200*n^3-\ 22614964714656090193328985976971264000*n^2+ 10443492368043431502718934830448640000*n-1906938498783405610237638751027200000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 6555/ 33554432*(317108429269*n^26-104567014436287*n^25+16243012553644110*n^24-\ 1578459438692523530*n^23+107453719798857216395*n^22-5435450030015823282465*n^21 +211358062682080663234600*n^20-6441234853001490707016280*n^19+ 155165956216343942878992555*n^18-2947738842985765039036999745*n^17+ 43347895397519715282565557350*n^16-466430740467867807385278746130*n^15+ 2978603694238868302433395759525*n^14+5774972567930557100402969392145*n^13-\ 445291077517960718957807270941500*n^12+6723971513129113766165386664969620*n^11-\ 65816828849584183247273825488986640*n^10+476287775391269035055943351246216240*n ^9-2643389446915254572529861363693768000*n^8+ 11342919307437068884613916563907241920*n^7-\ 37396657042932359934154591631757103104*n^6+ 93096816939989328638719845766883930112*n^5-\ 169881023092391547862120922070348226560*n^4+ 216743532028162909901726874962857574400*n^3-\ 178878561267929297574217223908343808000*n^2+ 82754240217826791308301218190458880000*n-15255507990267244881901110008217600000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 58995/33554432*(67360239592*n^27-23755528318299*n^26+3950086346467209*n^25-\ 411311881883429410*n^24+30033679173003407070*n^23-1631251137081909589285*n^22+ 68169081254826155344455*n^21-2233606762074788741023240*n^20+ 57807436192031334544507200*n^19-1175565130578197522706273445*n^18+ 18290896282464239874752488215*n^17-199981638858309233122987469290*n^16+ 1006388221402923860287288123110*n^15+13783897655212486238820735563285*n^14-\ 443100525298798188533252539876215*n^13+6747721836236507310505264654292660*n^12-\ 72434444237136209681830329532582860*n^11+595152114177356345425365135803188400*n ^10-3843529678147139330644611504151616080*n^9+ 19640444936727339361321843779657882560*n^8-\ 79099694080153909875921881946394485312*n^7+ 248059900382989897759489477511495353344*n^6-\ 593467788644622346215790950933799747584*n^5+ 1049662957754647704965120665753641246720*n^4-\ 1307936967644490986122993620798997708800*n^3+ 1061894029988573450652731043169714176000*n^2-\ 486971437855853847635457408215285760000*n+ 89837991498240442082306536715059200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 6555/33554432*(568276489043*n^27-\ 197740277603136*n^26+32358699744216651*n^25-3305354480687984400*n^24+ 235789306255223563635*n^23-12441639687198423230880*n^22+ 501053164038379852886415*n^21-15622769038756839319966320*n^20+ 376355602017579530135938005*n^19-6808157020810955413009897920*n^18+ 83334247887772940329020012705*n^17-352752498438943712388598491120*n^16-\ 12274980015890815780075646033055*n^15+384485168118066474630287302466400*n^14-\ 6615331953678176012090605679411115*n^13+82537611817928957116720190470151280*n^ 12-799753283651063539833071052827956860*n^11+ 6166165167259422234612668898393764160*n^10-\ 38119089667290892777656632595614454800*n^9+ 188694849206023720484470834773271438080*n^8-\ 742059196122876877932987463483691861568*n^7+ 2285638606700273495962655836353225317376*n^6-\ 5395602281015829997897178066308329209856*n^5+ 9453392991531999060099094293407928852480*n^4-\ 11710337833647768520241475349569465139200*n^3+ 9484757683992246359058472131687235584000*n^2-\ 4355605868549003096134636162052259840000*n+ 808541923484163978740758830435532800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 6555/67108864*(2075284847687*n^28-\ 766430236473874*n^27+133131294762894099*n^26-14432398084592225010*n^25+ 1091793780645555147615*n^24-60976584009853606318770*n^23+ 2588389361846649270759135*n^22-84303024785461343298967530*n^21+ 2077266877055748862557218145*n^20-36230494616230113021194001630*n^19+ 324589273649284406633076642345*n^18+4162317984518784977161080539370*n^17-\ 249631585011977539384258992832395*n^16+6134774399969128603763577275168250*n^15-\ 105513598177811205044914962710309835*n^14+1404203458767247442824764291169201170 *n^13-14980231661011543979654665555614262140*n^12+ 129952788573364997632774200110953500840*n^11-\ 920864623745904664185323994508537744400*n^10+ 5322491760643363722245746189869270394720*n^9-\ 24938651499673937264252257956025462005312*n^8+ 93717456087938265812649084251631799757184*n^7-\ 277989565867073455687255251527160054841344*n^6+ 636181309991123191280467699923523922657280*n^5-\ 1086965485192510615213361627100024265113600*n^4+ 1320287811127407885703671125434959187968000*n^3-\ 1054201123438543853645845632610716057600000*n^2+ 479947039948292995471895081489439129600000*n-\ 88939611583258037661483471347908608000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/33554432*( 17361134862055*n^28-6268836634384058*n^27+1059592350025518687*n^26-\ 111037879347043267710*n^25+8041020448922944139475*n^24-423254407834841892857730 *n^23+16469463088927619511301275*n^22-463909672248742494922217070*n^21+ 8390218356758118525703183125*n^20-30627104028803831649106556550*n^19-\ 4092972311866688793670944648195*n^18+176292739655428426538885316223830*n^17-\ 4547888166164611426959099867600975*n^16+87241299276863945320417573625867610*n^ 15-1324322430878535924176826506159590935*n^14+ 16321039998997535821086434484482031030*n^13-\ 165187671004121917202315868290949662400*n^12+ 1378875797625276686184504516397746449640*n^11-\ 9488373630784622166170583758200896242080*n^10+ 53598671443990131380221954201833738733280*n^9-\ 246625572235678709389168208551181048724480*n^8+ 913601746192109037097726358884063022537088*n^7-\ 2679784687589243312892138290628211280938752*n^6+ 6080904532987045076863586463584262669296640*n^5-\ 10327247949983197709886685097198100501196800*n^4+ 12497472735691260460802602369349485062144000*n^3-\ 9964479277868750694696614819273503948800000*n^2+ 4541108013870623349196237096965577113600000*n-\ 844926310040951357784092977805131776000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 30015/33554432*( 333823056302*n^29-126173435610401*n^28+22242856459724908*n^27-\ 2415737321114290929*n^26+179219232254883759120*n^25-9443140921481069647845*n^24 +348510901237532158005660*n^23-7831403378303523314329605*n^22+ 4678035248301551261345220*n^21+8471842476061757684590663965*n^20-\ 438266750070566597959059028020*n^19+14308117747671961456918200829485*n^18-\ 352462726772619923505894241138320*n^17+6935110787740313220208530961044585*n^16-\ 111790313372144151832590497347045980*n^15+1495163888946727911321372225273783465 *n^14-16695677810511231019722906450468507090*n^13+ 155980465174982857709462765384343481920*n^12-\ 1218097233750112060180336844409122359160*n^11+ 7922839535818494062545066379490224067680*n^10-\ 42654334000472587106282110982878134485152*n^9+ 188345834486468102869932224782339106098176*n^8-\ 673518846631592001731791035444535282209408*n^7+ 1917031364770096102381264474049967907739904*n^6-\ 4241070391859870974641245967879816163630080*n^5+ 7052737864888452863438982940554881078169600*n^4-\ 8391820522797732009092354968044140402688000*n^3+ 6605762401171765150141777787456870400000000*n^2-\ 2984942993780617027322806497117162700800000*n+ 553572410026830199927509192355086336000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 30015/33554432 *(257891721732*n^29-92763648399601*n^28+15292178024521678*n^27-\ 1505930348236588029*n^26+95106384085657687470*n^25-3597971574825953370045*n^24+ 31258065785654861411310*n^23+5964604326606358039273095*n^22-\ 484559023370426592359501730*n^21+22798311748204735421556642765*n^20-\ 787771688748628821067646795070*n^19+21454580063456783045816954944185*n^18-\ 475405447900065188698893421470870*n^17+8717877613532131970095526366258385*n^16-\ 133582142710333208104994443942331430*n^15+1719323386763017986922537118699619165 *n^14-18629081025834111632148515377597257690*n^13+ 169882875928018790778116553595550352720*n^12-\ 1300740479195624187650290378980453193560*n^11+ 8324156433600513128294504821372659726880*n^10-\ 44219347664425388677899033825963049932832*n^9+ 193124332827441454301292530901704182546176*n^8-\ 684485182964169773008871570597756659724928*n^7+ 1934537122526857448244508829872613918824704*n^6-\ 4256811429485033232904080445470376300846080*n^5+ 7051882959005897221274768014040677430169600*n^4-\ 8371235701021197394111252656662619199488000*n^3+ 6583917605698625595343764860083525632000000*n^2-\ 2977170829371675922374719319969320140800000*n+ 553572410026830199927509192355086336000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 1305/134217728 *(31772256209983*n^30-11239950279694875*n^29+1737085200915921785*n^28-\ 142080369459131409775*n^27+4359083517159871894317*n^26+393518613585205023400485 *n^25-65579013579866804093127675*n^24+5152460927011991687686015125*n^23-\ 282388572602086866578925681705*n^22+11905068149921986358389075080075*n^21-\ 402998730734670878972932808183325*n^20+11215040092990587809803490089867875*n^19 -260308368743273739175904598944744385*n^18+ 5085872711786643091116275839824177975*n^17-\ 84130735043335984754133865348075586025*n^16+ 1182228578020204706324576936462595354375*n^15-\ 14131553678347654932641198113572651754530*n^14+ 143632151842732607396936247709287225438900*n^13-\ 1238814133124319424946032640115929114707000*n^12+ 9033318478122792784022677605155768451390000*n^11-\ 55383367140675835980277328764708215461212448*n^10+ 283330938023892558420966470415927628749284800*n^9-\ 1197210658761067700431859324720204396723736960*n^8+ 4122510860157450021432964210070505014935718400*n^7-\ 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-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-45), -2697/536870912*(80894978067887*n^31-49795856561833876*n^30+ 13924610468270170930*n^29-2395295785495072956290*n^28+287458960589193578067498* n^27-25783217176053558087072234*n^26+1804349448481352587348388790*n^25-\ 101420986616877881663828506050*n^24+4673919550652108590938650431080*n^23-\ 179263087033968561148424686905690*n^22+5786059988451962956737614208080850*n^21-\ 158469403698377005895063985228069150*n^20+3705205391091650962468828258799416710 *n^19-74276075416959384355460841842378973630*n^18+ 1280188442799978343074392239920313808850*n^17-\ 18999231789790867178903527056598309843350*n^16+ 242861701915909652892088178530966118340105*n^15-\ 2671522721589760261716017517319038111764490*n^14+ 25236963807493537717212330338197042037336500*n^13-\ 204055972968003941467859380937940973137000200*n^12+ 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458631630845404791491895720569081699735180064186077184*n^7-\ 1182908960615624302965774926168514476050426807815659520*n^6+ 2405401835764372870159243699828544164228542062021836800*n^5-\ 3726795265399709534188543261417916051956552271585280000*n^4+ 4185996897121516830138759615312075969287322573864960000*n^3-\ 3151672591937284592679963266166345107359247224012800000*n^2+ 1381376840028816701285951888759783799780044439552000000*n-\ 252776573059858658264716449514459871415002726400000000)/(2*n-5)/(-1+2*n)/(2*n-3 )/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11 )/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2* n-51)/(2*n-57)/(2*n-63)/(2*n-45), -93/1073741824*(73385171198957986*n^33-\ 41086094028562880473*n^32+11013272343555872286336*n^31-\ 1882399388476695374126280*n^30+230510210665085939212003544*n^29-\ 21542228861214436117185205532*n^28+1598095197198659043865009140144*n^27-\ 96642373162102746844399037841480*n^26+4854991810275216419351319010516140*n^25-\ 205441606331386940147761313219046870*n^24+7399021458969165558493110874188588240 *n^23-228582516288253162217373393672357287400*n^22+ 6093137050621845494497709707225792913880*n^21-\ 140745728029522382185174116071144184127740*n^20+ 2825713887391567029768679923682622890268880*n^19-\ 49400731687691001594381377571965047843755000*n^18+ 752703099940053566543526931173210740940042690*n^17-\ 9994430002040542837865096247926423646841613145*n^16+ 115528640826955530025780109576095256311031067440*n^15-\ 1160266817023555695820596813059143776471756851600*n^14+ 10093968373322507021423126896022514973136323896384*n^13-\ 75755696063959367998556702327810007497635693372512*n^12+ 487837139691354549607689688103142975696809052726784*n^11-\ 2676932713018573386500471304864325409202075862117120*n^10+ 12407651486917196817020368879623711184032868997515776*n^9-\ 48038348496962918893419965048675098605122448324073728*n^8+ 153155452693364053924750057788999521014084040104882176*n^7-\ 394692728692029359741227166123722357233636029497221120*n^6+ 802087924854798288958190517764888311337298172320153600*n^5-\ 1242163906809003053401861651745671502755318654279680000*n^4+ 1394858437123660680429826208659435726687164654551040000*n^3-\ 1050115887038141025229907378260282784907240918220800000*n^2+ 460312364009902221492512862980269553299372376064000000*n-\ 84258857686619552754905483171486623805000908800000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2* n-51)/(2*n-57)/(2*n-63)/(2*n-45), -15/1073741824*(1022762953940104736*n^34-\ 600743868018587722724*n^33+169325351342382926235633*n^32-\ 30500352187648358742610928*n^31+3945012726663033731997870216*n^30-\ 390302181855913791498869904080*n^29+30723388041936760171527704755324*n^28-\ 1976146550794459511296755326862528*n^27+105849137371735874204271119871699336*n^ 26-4787827665341506358770583099604204600*n^25+ 184812364303543465491974833677985372710*n^24-\ 6136538723769572680115564963437641443040*n^23+ 176333891093483277840408954678125823462120*n^22-\ 4404758448612621396732510960322752025401360*n^21+ 95959381015611409773246764674968736981111260*n^20-\ 1827117715823585842733463490153513622924424000*n^19+ 30442165508157211584977217699373146570767293080*n^18-\ 443959576960235285560641607954166468582710591620*n^17+ 5664042551321377204689951454706482262695431874225*n^16-\ 63125983292934821555178762491481907877677134560560*n^15+ 613174154726312179605422276231285165213390665611984*n^14-\ 5174045920444408441077085903927017891530026115752576*n^13+ 37762073777653862130687937732603295107938970294204512*n^12-\ 237042952535671513231462050147997517522637904305536512*n^11+ 1270757911073590450699064556193780829056917099983677184*n^10-\ 5766106070788861288604300631027337913397873922877568000*n^9+ 21897066446921383468566725664581742911699322169661358336*n^8-\ 68599975540747023734527112971681067724658759568488722432*n^7+ 174017727515332624372039299026678086501653391131633721344*n^6-\ 348672876907878437823466616199708543776768805457847255040*n^5+ 533256879802431127945373684362786234842032253465305088000*n^4-\ 592303086206520554544775702663640055404073028432035840000*n^3+ 441795822604128554428100573961782235074040136201666560000*n^2-\ 192215844326913925186213319195916971736212057987481600000*n+ 35001129483021762214387737709435543528597377515520000000)/(2*n-5)/(-1+2*n)/(2*n -3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -15/4294967296*( 4433440410089246991*n^34-2586153370040273896041*n^33+724251098978277922059360*n ^32-129677692557726894018574312*n^31+16679452976393362625797189412*n^30-\ 1641635185666315806127741756220*n^29+128601689128439135098473403078648*n^28-\ 8234726860531586650557096828781992*n^27+439250242857999781599852254616410874*n^ 26-19792109561369474196040352906050531350*n^25+ 761272810672803399429445219042312637160*n^24-\ 25194650547507570592574797889671155434760*n^23+ 721787688752205449436141109498051394942340*n^22-\ 17980080186575339026286093240562222812971740*n^21+ 390709478090141227468795019500910827111324680*n^20-\ 7422078859814308093333822939112142080747179800*n^19+ 123400620707519837607706254693138766823668908975*n^18-\ 1796193785557571162321903756759123777071360653705*n^17+ 22876238691464155239743584602162950973610378104120*n^16-\ 254560161443713576471333161473670750886097350687440*n^15+ 2469231169011682291886456931969893079999811653302304*n^14-\ 20809953750009381097441533773936911151983763562514784*n^13+ 151713047159488309096589773899800405361983228691911680*n^12-\ 951436809300828161916526759474132624996297307025710848*n^11+ 5096329181114872584844678168298739070351932978642499328*n^10-\ 23108488340252530186756818048670413897742930025083296000*n^9+ 87703790141423853380062144317456930740557050716019812352*n^8-\ 274628806953012090875003757594588203693136242019179470848*n^7+ 696383782264180770602108193956554041654002490477885259776*n^6-\ 1394912763423120021513706798286460019523341869743877980160*n^5+ 2132934316073910433669349449359983435580576081633411072000*n^4-\ 2368829171979234898954138223997734887041827085979484160000*n^3+ 1766834117950612879846876922721671688454130936567562240000*n^2-\ 768748944669851575541449655147739043624680629364326400000*n+ 140004517932087048857550950837742174114389510062080000000)/(2*n-5)/(-1+2*n)/(2* n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n -11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/ (2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -5/4294967296*( 28032449309546944688*n^35-17245045061670605415355*n^34+ 5100921940422707155446725*n^33-966173343581611368688879240*n^32+ 131676261664815896153145912584*n^31-13755381919257991822411468638772*n^30+ 1145725202716823129349945205476300*n^29-78149481134978181094094986214249400*n^ 28+4449185414773949363652163966654654248*n^27-\ 214412377289575973158711151476989301698*n^26+ 8839772042376835796644046904268403757630*n^25-\ 314318293461929599468011327744099864785400*n^24+ 9698887206916816231921622785182780590938920*n^23-\ 260933156636718526945963479748257927347503380*n^22+ 6141675470838673276872981657820790777771717100*n^21-\ 126775278240218926850979936227105865770778798600*n^20+ 2298326330234176736644614824058500303188693471080*n^19-\ 36617775294834718310683761478629423483742088596635*n^18+ 512628642267214175129858207934975234764351833533925*n^17-\ 6299870028868231334168675679162007788964981566075200*n^16+ 67845046871167102469681137567241958414333012432673552*n^15-\ 638609819175514034921953575619966022712979900839168800*n^14+ 5235467319349473858426706882078846458471814417009093600*n^13-\ 37213051288997888608299083250733001503196357571151573760*n^12+ 228004265111347405334419557527121670340464188088035484416*n^11-\ 1195499182726158741336904666729360100823141094782376090368*n^10+ 5315875233252853773792022643283522068410028311615120569600*n^9-\ 19818390796335714506592086298233937756022170226474468454400*n^8+ 61057448492122708413761558971319690247561272234878586560512*n^7-\ 152562404375376719451955616781844087361002646221076437204992*n^6+ 301572840962718679213565552332471763287848751257758665605120*n^5-\ 455716693584390700793074024288392955960932404810966237184000*n^4+ 500897509997129325111986631382369882594935675003650703360000*n^3-\ 370301412098549221258316243782258403960893012094044078080000*n^2+ 159956224023205554562947787552168932295489863585418444800000*n-\ 28980935211942019113513046823412630041678628582850560000000)/(2*n-5)/(-1+2*n)/( 2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/ (2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53 )/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-69), (47839193543441813968* (n-34)!*(2*n-73)!*(n+1)+71758790315162720952*(n-35)!*((n+1)*(2*n-72)!-1/ 71758790315162720952*binomial(2*n,n)*(n-37)!*(n-34)!))/(n-37)!/(n-35)!/(n-34)!/ binomial(2*n,n), ((47839193543441813968*n+47839193543441813968)*(2*n-73)!-(n-37 )!*(n-35)!*binomial(2*n,n))/(n-37)!/(n-35)!/binomial(2*n,n)] The limits, as n goes to infinity are 34359738349 1073741813 68719470047 274877733223 68719260097 68718591715 [-----------, ----------, -----------, ------------, -----------, -----------, 34359738368 1073741824 68719476736 274877906944 68719476736 68719476736 17179098539 2198722123627 8792804813623 4393971445779 17555063344479 -----------, -------------, -------------, -------------, --------------, 17179869184 2199023255552 8796093022208 4398046511104 17592186044416 70055025975225 69749049303575 69216758525675 68342280819125 --------------, --------------, --------------, --------------, 70368744177664 70368744177664 70368744177664 70368744177664 267918212239005 2078645753858295 496739666841255 3725052385676865 ---------------, ----------------, ---------------, ----------------, 281474976710656 2251799813685248 562949953421312 4503599627370496 13603492176588285 5989591527408975 5009849517452265 1935155006946495 -----------------, ----------------, ----------------, ----------------, 18014398509481984 9007199254740992 9007199254740992 4503599627370496 41462794354027815 77181791986984485 -7770525530879757 ------------------, ------------------, ------------------, 144115188075855872 576460752303423488 288230376151711744 -218173755849091239 -1600273676773590221 -2280606963048398261 -------------------, --------------------, --------------------, 1152921504606846976 4611686018427387904 4611686018427387904 -2890560943846502021 -3412410460751546349 -479420134659424095 --------------------, --------------------, -------------------, 4611686018427387904 4611686018427387904 576460752303423488 -66501606151338704865 -8760140409233420215 -144584002993211299555 ---------------------, --------------------, ----------------------, 73786976294838206464 9223372036854775808 147573952589676412928 -587305860762240538339 ----------------------] 590295810358705651712 and in Maple notation [34359738349/34359738368, 1073741813/1073741824, 68719470047/68719476736, 274877733223/274877906944, 68719260097/68719476736, 68718591715/68719476736, 17179098539/17179869184, 2198722123627/2199023255552, 8792804813623/ 8796093022208, 4393971445779/4398046511104, 17555063344479/17592186044416, 70055025975225/70368744177664, 69749049303575/70368744177664, 69216758525675/ 70368744177664, 68342280819125/70368744177664, 267918212239005/281474976710656, 2078645753858295/2251799813685248, 496739666841255/562949953421312, 3725052385676865/4503599627370496, 13603492176588285/18014398509481984, 5989591527408975/9007199254740992, 5009849517452265/9007199254740992, 1935155006946495/4503599627370496, 41462794354027815/144115188075855872, 77181791986984485/576460752303423488, -7770525530879757/288230376151711744, -\ 218173755849091239/1152921504606846976, -1600273676773590221/ 4611686018427387904, -2280606963048398261/4611686018427387904, -\ 2890560943846502021/4611686018427387904, -3412410460751546349/ 4611686018427387904, -479420134659424095/576460752303423488, -\ 66501606151338704865/73786976294838206464, -8760140409233420215/ 9223372036854775808, -144584002993211299555/147573952589676412928, -\ 587305860762240538339/590295810358705651712] and in floating point [.9999999994, .9999999898, .9999999027, .9999993680, .9999968475, .9999871212, .9999551426, .9998630611, .9996261740, .9990734374, .9978898188, .9955417962, .\ 9911936062, .9836292993, .9712022236, .9518366974, .9231041504, .8823869046, .8\ 271277853, .7551455115, .6649782422, .5562050284, .4296907290, .2877059310, .13\ 38890665, -.2695942612e-1, -.1892355681, -.3470040394, -.4945278048, -.62679049\ 10, -.7399485670, -.8316613624, -.9012648233, -.9497763263, -.9797393134, -.994\ 9348284] The cut off is at j=, 26 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 38], vs. those in the, 2, -th row from j=1 to j=, 37, are as follws 19 18 17 [37 (3714566309 n - 670479218404 n + 56267321682111 n 16 15 14 - 2915076001975716 n + 104398322912345658 n - 2742419540769874728 n 13 12 + 54715260319817913902 n - 846999315888960785512 n 11 10 + 10303309165039504379337 n - 99129911472982092978132 n 9 8 + 755405830911175609218963 n - 4544451677097459338821428 n 7 6 + 21405441566571475889647016 n - 77803092412956769303282336 n 5 4 + 212872315410017562077823024 n - 416965137795357071224993344 n 3 2 + 499637212731046203193255680 n - 23509007044260406813670400 n - 1315704809873866220024832000 n + 2207891489584471429939200000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 35) (2 n - 29)), 37 ( 19 18 17 1857283145 n - 335239604338 n + 28133659667625 n 16 15 14 - 1457537823763572 n + 52199142679732890 n - 1371208288605649116 n 13 12 + 27357539900443125050 n - 423495319495929114304 n 11 10 + 5151487799656729444485 n - 49559790859569117624354 n 9 8 + 377573817749407594788525 n - 2269629043616019268422876 n 7 6 + 10661036756361526205044280 n - 38375668623554637309107392 n 5 4 + 101351954221051848109678800 n - 172362822055212731646541248 n 3 2 + 75050186564847336490195200 n + 474518938674126547432627200 n - 1029930840756914736245760000 n + 29051203810321992499200000)/(131072 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 35) (2 n - 29)), 37 ( 20 19 18 7429132229 n - 1485826322950 n + 138683284453695 n 17 16 15 - 8024570201158050 n + 322483827232446114 n - 9556311535596480300 n 14 13 + 216381071263275552590 n - 3827709095961280344100 n 12 11 + 53632423931422969371609 n - 599864383201017469113150 n 10 9 + 5371189137531468538363635 n - 38433329219825095704382650 n 8 7 + 218135097327453679266371024 n - 965633748689761244047846000 n 6 5 + 3210862030272112949058493680 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9280741530710406621856102224360992195219827336261446085066752 n 6 + 22870595504616030039895713248221374810875008541053788093087744 n 5 - 44646506313229310010222661825885081145321396713942907413463040 n 4 + 66715207113390337941364863666510811138138951274195228295168000 n 3 - 72608104471552676051519357457029686374025833986328640880640000 n 2 + 53222140141693621913775846028286208274924032743157855682560000 n - 22829551030710741347303839103952902604353661274508859801600000 n + 4115292800095766714118852648924593465918365258764779520000000)/( 8589934592 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 65) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69)), ( 183803217298486969456 (n - 35)! (2 n - 75)! (n + 1) + 275704825947730454184 (n - 36)! ((n + 1) (2 n - 74)! - 1/275704825947730454184 binomial(2 n, n) (n - 38)! (n - 35)!))/( (n - 38)! (n - 36)! (n - 35)! binomial(2 n, n)), ( (183803217298486969456 n + 183803217298486969456) (2 n - 75)! - (n - 38)! (n - 36)! binomial(2 n, n))/((n - 38)! (n - 36)! binomial(2 n, n))] and in Maple notation [37/262144*(3714566309*n^19-670479218404*n^18+56267321682111*n^17-\ 2915076001975716*n^16+104398322912345658*n^15-2742419540769874728*n^14+ 54715260319817913902*n^13-846999315888960785512*n^12+10303309165039504379337*n^ 11-99129911472982092978132*n^10+755405830911175609218963*n^9-\ 4544451677097459338821428*n^8+21405441566571475889647016*n^7-\ 77803092412956769303282336*n^6+212872315410017562077823024*n^5-\ 416965137795357071224993344*n^4+499637212731046203193255680*n^3-\ 23509007044260406813670400*n^2-1315704809873866220024832000*n+ 2207891489584471429939200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9) /(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-\ 37)/(2*n-17)/(2*n-23)/(2*n-35)/(2*n-29), 37/131072*(1857283145*n^19-\ 335239604338*n^18+28133659667625*n^17-1457537823763572*n^16+52199142679732890*n ^15-1371208288605649116*n^14+27357539900443125050*n^13-423495319495929114304*n^ 12+5151487799656729444485*n^11-49559790859569117624354*n^10+ 377573817749407594788525*n^9-2269629043616019268422876*n^8+ 10661036756361526205044280*n^7-38375668623554637309107392*n^6+ 101351954221051848109678800*n^5-172362822055212731646541248*n^4+ 75050186564847336490195200*n^3+474518938674126547432627200*n^2-\ 1029930840756914736245760000*n+29051203810321992499200000)/(2*n-5)/(-1+2*n)/(2* n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n -31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-35)/(2*n-29), 37/262144* (7429132229*n^20-1485826322950*n^19+138683284453695*n^18-8024570201158050*n^17+ 322483827232446114*n^16-9556311535596480300*n^15+216381071263275552590*n^14-\ 3827709095961280344100*n^13+53632423931422969371609*n^12-\ 599864383201017469113150*n^11+5371189137531468538363635*n^10-\ 38433329219825095704382650*n^9+218135097327453679266371024*n^8-\ 965633748689761244047846000*n^7+3210862030272112949058493680*n^6-\ 7260374123599158532609495200*n^5+7287267671825225198353329024*n^4+ 14010685224836952842932262400*n^3-54879519629379377098191513600*n^2+ 40736285774843614598522880000*n-2265993897205115414937600000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29) , 1/262144*(274877810573*n^20-54975533572450*n^19+5131272174181815*n^18-\ 296907746478842550*n^17+11931765226343767218*n^16-353573318072295233700*n^15+ 8005512905534018804030*n^14-141598791049805501577100*n^13+ 1983453349644422812394433*n^12-22167944375182446723189450*n^11+ 198117269358393048772422795*n^10-1410867691831044702398553150*n^9+ 7912561091367365713620172688*n^8-34004220386326642957488245200*n^7+ 104921657278955723270523784560*n^6-191017175002363019260844287200*n^5+ 3884271265935049987047305088*n^4+902379385439504035092060940800*n^3-\ 1750758300668924836905198643200*n^2+1043906453109115140827842560000*n-\ 83841774196589270352691200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-\ 37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/524288*(1099509656617*n^21-\ 242441329847661*n^20+25032899848092185*n^19-1608368785957864455*n^18+ 72070803311646533772*n^17-2392495087222154659626*n^16+61003308630686011477570*n ^15-1222312488638182980643710*n^14+19525568123179345010149457*n^13-\ 250754826032697939212796081*n^12+2597252998900287397744752405*n^11-\ 21649029468772702980280248315*n^10+143841516307217586692710737802*n^9-\ 745328862585211653519169431816*n^8+2872529921721370202082920054240*n^7-\ 7327426017790420318001698735920*n^6+7528344226645081314207486392352*n^5+ 21015587526304179296750098685184*n^4-90449930159276255961926325926400*n^3+ 127933249013342824604843901542400*n^2-67994292027823313977943838720000*n+ 6875025484120320168920678400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2* n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/65536*( 137437922168*n^21-30304791829305*n^20+3129028790076070*n^19-201034458062032035* n^18+9007722539733678558*n^17-298981120300547184360*n^16+7620989538723756905420 *n^15-152599816399817881476270*n^14+2434299733927665819046468*n^13-\ 31172683706107794204000465*n^12+321001076991244190017656270*n^11-\ 2644780319665665454521157455*n^10+17179579962574322788897627718*n^9-\ 85220164830918134800430820270*n^8+301172895159273438611056001920*n^7-\ 623558674822666022042391596640*n^6+16531965537650351072280245088*n^5+ 4089526129959449076918216554400*n^4-11591746174941342097771541479680*n^3+ 14228701396880252212119765542400*n^2-7230572919072815689754657280000*n+ 859378185515040021115084800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-\ 9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n -15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 31/65536*( 8866791937*n^22-2145705411575*n^21+243889968507878*n^20-17307625222889155*n^19+ 859742099910853467*n^18-31765132497278740560*n^17+905359599305338004938*n^16-\ 20371172474449083739430*n^15+367165875218348934962927*n^14-\ 5344830769588271217751495*n^13+63004477833559280590972218*n^12-\ 599360584050956133033082455*n^11+4549332924956133352713212197*n^10-\ 26902107846307223366285418530*n^9+118126836511010407102090999238*n^8-\ 343984039527699051107799005680*n^7+410384623824286083516962998272*n^6+ 1363776731378820026320044152160*n^5-7755477229902539206656150154272*n^4+ 16861749036309288393480348606720*n^3-18392661239646254022518206348800*n^2+ 8980741995134600426743687680000*n-1192040708940216803482214400000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29), 31/65536*(8866287719*n^22-2145471454423*n^21+ 243839178628738*n^20-17300776389457715*n^19+859099865690427669*n^18-\ 31720708241700205728*n^17+903013534463725530098*n^16-20274601651787078482150*n^ 15+364030564501402371809929*n^14-5264117193907550019944903*n^13+ 61357200205760872002649878*n^12-572850925491176348813131335*n^11+ 4216589487651471370602104459*n^10-23701140628660372105197512818*n^9+ 95132346700027308244227344998*n^8-225248401353237741466751437520*n^7-\ 5469116392247738764148282976*n^6+2255920063798775275459473787872*n^5-\ 8645579425573729410758144903712*n^4+16588560041404849625717063358720*n^3-\ 17063376420094957012813350796800*n^2+8239824461758225070193338880000*n-\ 1192040708940216803482214400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 20677/ 4194304*(1701221371*n^23-449887965842*n^22+56031120398462*n^21-4369328158190380 *n^20+239220983317722021*n^19-9772360656029895462*n^18+308934956740423863652*n^ 17-7733859905144802903320*n^16+155521801694851969880261*n^15-\ 2531671949910262426868062*n^14+33425754054956090479255422*n^13-\ 356448138073829729310534060*n^12+3033809821829233490271345931*n^11-\ 20125025229995520783219162122*n^10+99240819282164463943025404352*n^9-\ 324101700327093870255009720400*n^8+400637247935161421018645842416*n^7+ 2170481292363017405702200219488*n^6-14558746161024507093391447221888*n^5+ 43077751261478371115834590748160*n^4-73399285996608800860883373312000*n^3+ 70847616482567654990103631872000*n^2-33557597684439094236863692800000*n+ 5147042341451011085500416000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 667/2097152*(26359477163*n^23-6968971314826*n^22+867571351396486*n^21-\ 67604559890746340*n^20+3696934560520676013*n^19-150729482989602156486*n^18+ 4750237407064290935156*n^17-118337139913903125388360*n^16+ 2361781024405174219728133*n^15-38009812912389211457496086*n^14+ 493385094226884722605126566*n^13-5131228332897557449449579780*n^12+ 42081661949729614929570721043*n^11-263612163413554768683313003066*n^10+ 1176306534720418261230708115456*n^9-2995916339496791105854287290000*n^8-\ 2001172459056372406292440766352*n^7+56384666307487719295220863354464*n^6-\ 264223995017797314641345462273664*n^5+695149256369922438390141918804480*n^4-\ 1115915932372778150321513799936000*n^3+1045875356693257488192246767616000*n^2-\ 494774392042114847919935078400000*n+79779156292490671825256448000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 18009/4194304*(3902035733*n^24-\ 1122790964772*n^23+152484252620102*n^22-12994554536047568*n^21+ 779174729645897123*n^20-34931393620040489572*n^19+1214141340769964175272*n^18-\ 33470195212430797038768*n^17+742025584895599994684123*n^16-\ 13326674194357062469094252*n^15+194193734878489390327097822*n^14-\ 2285714621312963835610212688*n^13+21469153078807518959934974813*n^12-\ 157009528146044845054358431692*n^11+848801810425720600654815170132*n^10-\ 2939435236187043468362061248048*n^9+2194758204448655070343512780848*n^8+ 44054816600547496809334969988288*n^7-307095706881324117933928579595328*n^6+ 1132039535982919673430563964387072*n^5-2649736398689575288039946842352640*n^4+ 3966403950325687912919521085952000*n^3-3570395233119932990534477119488000*n^2+ 1668421777889182647831839539200000*n-277749655240523079687929856000000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/ (2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17 )/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 621/4194304*(112978067833*n^24-\ 32480045092860*n^23+4405136531939182*n^22-374647948347370720*n^21+ 22398031218019865023*n^20-999816087512521779500*n^19+34537866295073933387992*n^ 18-943863900766379016169680*n^17+20673908057243625963332983*n^16-\ 365182471214310723476093140*n^15+5201651228041030686992574982*n^14-\ 59329505029879311390252383840*n^13+532767055690081741695006556033*n^12-\ 3633421336362721660382077011780*n^11+17226472009545912698675999669332*n^10-\ 39383487184241508958180746780400*n^9-151516479215546862359968889520272*n^8+ 2015477981599067437826696347305280*n^7-10651750882774585788741018237991488*n^6+ 35343584989456504575994485842384640*n^5-78158210675511024001594973486361600*n^4 +113183391143135965773807875860992000*n^3-100256987746914986757447334379520000* n^2+46927441261778347505047228416000000*n-8054740001975169310949965824000000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2* n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 15525/4194304*(9010798549 *n^25-2807297397300*n^24+413384926297450*n^23-38247484631059400*n^22+ 2492789256359264555*n^21-121583934010646481220*n^20+4600702162735988657700*n^19 -138123393582544467207000*n^18+3335148911266655861347915*n^17-\ 65224978869827037678393020*n^16+1034386595528898425050816250*n^15-\ 13234453519597612872340216600*n^14+134731440669712412497750787685*n^13-\ 1059130915661310087529594156620*n^12+5983878485819865204267333923800*n^11-\ 18752793275149948783895345232200*n^10-37117498548582173403303274040080*n^9+ 896456371820760486612227020503680*n^8-6186413769621765519859171226694400*n^7+ 26734337691575286001528940098339200*n^6-79539219768618769191293660074458624*n^5 +163961058752807521760529326869524480*n^4-226567669920174000965802936580300800* n^3+195055492502612113419006047563776000*n^2-\ 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258062123105226384592716039126385381368388330721157120000*n^4-\ 286742265170358874927577120580590936718102991839559680000*n^3+ 213903688628851030452224413086353497211067426550579200000*n^2-\ 93050823321643505981754703661964907766282399514624000000*n+ 16936030395010530103736002117468811384805182668800000000)/(2*n-5)/(-1+2*n)/(2*n -3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -3/1073741824*(4609242104373829716 *n^34-2733772964143157342880*n^33+777437734551657048058629*n^32-\ 141189342696150380378967440*n^31+18399629004222833105328306904*n^30-\ 1832987707658805908878384261600*n^29+145204948899190188039419978849996*n^28-\ 9394209679977299611727875866034080*n^27+505882349259228003973066246535908320*n^ 26-22994803105649817830335832644666077600*n^25+ 891606353262894851200753917797800963710*n^24-\ 29727015815267872010096189816884180344000*n^23+ 857424219541541364671194871180347705172280*n^22-\ 21491723875721217704611534634266706868151200*n^21+ 469669814931979339680832617983843441893312620*n^20-\ 8968166268456928570130521061401965351812354400*n^19+ 149805835809367755003921522755913466794005274540*n^18-\ 2189800338692458503475763560908262802032789166400*n^17+ 27995901303290986034631556961246273742992832160485*n^16-\ 312599685327027785301298700761788100308865317620400*n^15+ 3041504068211553526605348267616790682061954815221904*n^14-\ 25702568449038414556557570339087325857430001379493120*n^13+ 187830734289930358228836286392340565116201918290884576*n^12-\ 1180403184971621826592099742913672243541564294448744960*n^11+ 6334193185316673206266072337881470402364616647085852416*n^10-\ 28765593848748602184836806592625650470345150596490752000*n^9+ 109315085818571499123766956404190880104534856266912969984*n^8-\ 342662544328962914134053009078395332399818339691586334720*n^7+ 869627562751410087797531866723097407343693801624599633920*n^6-\ 1743038323372814527740726273753620421849345809312936755200*n^5+ 2666421751001776386418450951632484969265502430609612800000*n^4-\ 2962080106806904085265931588391416559551155709368729600000*n^3+ 2209493682970392112152856051256953441242665366834380800000*n^2-\ 961247859206070442020292985758848417258149388484608000000*n+ 175005647415108811071938688547177717642986887577600000000)/(2*n-5)/(-1+2*n)/(2* n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n -11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/ (2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -15/8589934592*( 16490350429373349267*n^35-10252657511336672166420*n^34+ 3062770067149585365413925*n^33-585494844136156411806178060*n^32+ 80482895906139594506565420956*n^31-8474976482569648834516637151248*n^30+ 711168536111034141444999500083300*n^29-48844392630465208102734623346793200*n^28 +2798652451412592193813258484833757082*n^27-\ 135673067853517268873549249812373692632*n^26+ 5624282349752158142848408951697691185670*n^25-\ 200999820954159841396509380678040367314600*n^24+ 6231258384109760172488855813975406068514780*n^23-\ 168363690472961528208545834202453604631611920*n^22+ 3978486711757116584728274684132145639187650900*n^21-\ 82420009656479962322899999833375115663027844400*n^20+ 1499132910056891607379758208906504579159984991595*n^19-\ 23956414085760690570346189776067709379139194236340*n^18+ 336288902836896272096366358204454465833597413771325*n^17-\ 4142919151370468217297476136172529967911447280720300*n^16+ 44714692416800417277687639770132954402015004682326768*n^15-\ 421719109092825892268658849446197344596413321916129600*n^14+ 3463403592069777568613989844121484497561268402585132000*n^13-\ 24655453557083706972058845746890717470767677475416589440*n^12+ 151267735184822133784259278451570921058372324156714796544*n^11-\ 794070335554927199032803189399290416093314739422866368512*n^10+ 3534406330544828319815094242643393297813418776847072492800*n^9-\ 13187801371372860069814812103938904316739275121815647104000*n^8+ 40657404408158812878217810389102544631684184481522594603008*n^7-\ 101644586675016922911466275463810425707688484253560509923328*n^6+ 201005128347113855197753122884563967097330155637446961070080*n^5-\ 303832136539481778289997373102737885483929639659325587456000*n^4+ 334010167448071268003212293213342834904108700417906442240000*n^3-\ 246937829095683533528055530956614920332705260005983518720000*n^2+ 106660255142076445820680800775804840709391827099398963200000*n-\ 19320623474628012742342031215608420027785752388567040000000)/(2*n-5)/(-1+2*n)/( 2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/ (2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53 )/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-69), -15/4294967296*( 8904559947784058791*n^35-5500859284067650284670*n^34+1633438668101007926147395* n^33-310511834948903657972807520*n^32+42460767035954795266751249468*n^31-\ 4449439704226041333424625873864*n^30+371678033808614899515878381221580*n^29-\ 25419888701002266018747769890510960*n^28+1450781741936044872790717031534720706* n^27-70075075819224893190207851810337860916*n^26+ 2895142450958300248406394077560306322010*n^25-\ 103143357251679776488076113352268736544400*n^24+ 3188365840121425124800375614148028903062140*n^23-\ 85918384007173036883037402114756470000090760*n^22+ 2025321974406772561348765283406922565402538300*n^21-\ 41863501914554466578690224539173490343984160400*n^20+ 759894401827899156114528274736112961377431392335*n^19-\ 12120599280239625450598102870210004474586393467070*n^18+ 169855121543080533823586762279226857315248376841275*n^17-\ 2089329512973442263845389114521292863932819596202000*n^16+ 22519135685918933178437183756532781758595043741496464*n^15-\ 212123414960664553045662366061026712350012435567359040*n^14+ 1740169618753480497751829629147542832449277845534357280*n^13-\ 12376005465277184686244714709121709236151626878023265280*n^12+ 75865691078452544127760581212003497951026543587323589632*n^11-\ 397959782917210599296285336596361930390380007739160137216*n^10+ 1770208010382274203054471542265438918669847580517761637120*n^9-\ 6601635311720136981252446414359986673347713429309521981440*n^8+ 20343755339522101710156485441625040430374541772978174910464*n^7-\ 50842468380729613299636910870276481025361459260537413566464*n^6+ 100516347634326071972827581737763281190010588518984549335040*n^5-\ 151909443369008876213653992378011764391595566116673814528000*n^4+ 166980223950904018069881019280361131002013550421561835520000*n^3-\ 123446655603619039560698428930066354377065067366957711360000*n^2+ 53322905448028867343376307224320030843620933827664281600000*n-\ 9660311737314006371171015607804210013892876194283520000000)/(2*n-69)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2* n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35) /(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -5/8589934592*( 112362521261633914337*n^36-73108335079502937768006*n^35+ 22897187885387799329154375*n^34-4597650324019304816923854090*n^33+ 665093410937345829269038841556*n^32-73845192246281134847603112025656*n^31+ 6546681377618071305507046843439884*n^30-476008941541636464913157285963691720*n^ 29+28934735696069013806611464245497919742*n^28-\ 1491386097452994256479858096813193520628*n^27+ 65885345567138942003850656536956406607586*n^26-\ 2515317279393733961934608508918396041317260*n^25+ 83513602986509812174695115221899751998145780*n^24-\ 2423211740345608451042749028372239926765397560*n^23+ 61670624456703579372036939500115178408529556060*n^22-\ 1380251409298342390520872073545302877090795065000*n^21+ 27213529296695898387675184530800220819060610549545*n^20-\ 473112199951074299847728453788006067131231166940150*n^19+ 7253986486267186233473041504303853093233029771014095*n^18-\ 98036754529578036547307826019915582184860404042209850*n^17+ 1166411947866259293235607851673227784302836667126636848*n^16-\ 12192435109062899728757572384893442612481297161723054944*n^15+ 111655163877602472930136677247407346207929151539693555360*n^14-\ 892490991404644071590386607330341029415996745637410996160*n^13+ 6197407638063819131828544844356731168212332954489583814144*n^12-\ 37163990104451813726890284090452688386977826965580335586304*n^11+ 191045607129707858273733270188570283996001306643999551024896*n^10-\ 834196020462078126184506016343739217958856260933998447802880*n^9+ 3058614864862927521163691750863152684970854809196424715010048*n^8-\ 9280741530710406621856102224360992195219827336261446085066752*n^7+ 22870595504616030039895713248221374810875008541053788093087744*n^6-\ 44646506313229310010222661825885081145321396713942907413463040*n^5+ 66715207113390337941364863666510811138138951274195228295168000*n^4-\ 72608104471552676051519357457029686374025833986328640880640000*n^3+ 53222140141693621913775846028286208274924032743157855682560000*n^2-\ 22829551030710741347303839103952902604353661274508859801600000*n+ 4115292800095766714118852648924593465918365258764779520000000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31 )/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), ( 183803217298486969456*(n-35)!*(2*n-75)!*(n+1)+275704825947730454184*(n-36)!*((n +1)*(2*n-74)!-1/275704825947730454184*binomial(2*n,n)*(n-38)!*(n-35)!))/(n-38)! /(n-36)!/(n-35)!/binomial(2*n,n), ((183803217298486969456*n+ 183803217298486969456)*(2*n-75)!-(n-38)!*(n-36)!*binomial(2*n,n))/(n-38)!/(n-36 )!/binomial(2*n,n)] The limits, as n goes to infinity are 137438953433 68719476365 274877892473 274877810573 1099509656617 [------------, -----------, ------------, ------------, -------------, 137438953472 68719476736 274877906944 274877906944 1099511627776 17179740271 274870550047 274854919289 35176154288167 17581771267721 -----------, ------------, ------------, --------------, --------------, 17179869184 274877906944 274877906944 35184372088832 17592186044416 70271761515597 70159380124293 139892647473225 69566949826925 --------------, --------------, ---------------, --------------, 70368744177664 70368744177664 140737488355328 70368744177664 137858202770725 135823891474435 8495061806740995 4104209510273115 ---------------, ---------------, ----------------, ----------------, 140737488355328 140737488355328 9007199254740992 4503599627370496 15620608079440335 14558968130570835 52789659584582805 2878773683578035 -----------------, -----------------, -----------------, ----------------, 18014398509481984 18014398509481984 72057594037927936 4503599627370496 19019810092940235 14404283957242485 593402563273029165 -----------------, -----------------, -------------------, 36028797018963968 36028797018963968 2305843009213693952 120202536534958023 -252338487358941309 -987575863403295261 -------------------, -------------------, -------------------, 1152921504606846976 4611686018427387904 4611686018427387904 -1698742968208101911 -2360864065784990861 -2952358912953678323 --------------------, --------------------, --------------------, 4611686018427387904 4611686018427387904 4611686018427387904 -3456931578280372287 -247355256440600239005 -133568399216760881865 --------------------, ----------------------, ----------------------, 4611686018427387904 295147905179352825856 147573952589676412928 -561812606308169571685 -578808109277550216121 -2349695540353667171257 ----------------------, ----------------------, -----------------------] 590295810358705651712 590295810358705651712 2361183241434822606848 and in Maple notation [137438953433/137438953472, 68719476365/68719476736, 274877892473/274877906944, 274877810573/274877906944, 1099509656617/1099511627776, 17179740271/17179869184 , 274870550047/274877906944, 274854919289/274877906944, 35176154288167/ 35184372088832, 17581771267721/17592186044416, 70271761515597/70368744177664, 70159380124293/70368744177664, 139892647473225/140737488355328, 69566949826925/ 70368744177664, 137858202770725/140737488355328, 135823891474435/ 140737488355328, 8495061806740995/9007199254740992, 4104209510273115/ 4503599627370496, 15620608079440335/18014398509481984, 14558968130570835/ 18014398509481984, 52789659584582805/72057594037927936, 2878773683578035/ 4503599627370496, 19019810092940235/36028797018963968, 14404283957242485/ 36028797018963968, 593402563273029165/2305843009213693952, 120202536534958023/ 1152921504606846976, -252338487358941309/4611686018427387904, -\ 987575863403295261/4611686018427387904, -1698742968208101911/ 4611686018427387904, -2360864065784990861/4611686018427387904, -\ 2952358912953678323/4611686018427387904, -3456931578280372287/ 4611686018427387904, -247355256440600239005/295147905179352825856, -\ 133568399216760881865/147573952589676412928, -561812606308169571685/ 590295810358705651712, -578808109277550216121/590295810358705651712, -\ 2349695540353667171257/2361183241434822606848] and in floating point [.9999999997, .9999999946, .9999999474, .9999996494, .9999982072, .9999924963, .9999732358, .9999163714, .9997664361, .9994079885, .9986217935, .9970247579, .\ 9939970445, .9886058170, .9795414454, .9650867943, .9431413213, .9113175792, .8\ 671179374, .8081850817, .7326036942, .6392161652, .5279057772, .3997991926, .25\ 73473393, .1042590810, -.5471718724e-1, -.2141463793, -.3683561633, -.511930789\ 8, -.6401907895, -.7496025455, -.8380722075, -.9050946788, -.9517475755, -.9805\ 390774, -.9951347693] The cut off is at j=, 27 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 39], vs. those in the, 2, -th row from j=1 to j=, 38, are as follws 19 18 17 [3 (11453246121 n - 2067310924223 n + 173490908715414 n 16 15 14 - 8988151035629172 n + 321894832109139102 n - 8455793831003661786 n 13 12 + 168705401029349539848 n - 2611581947065720968404 n 11 10 + 31768564389348975627153 n - 305651421187848441493359 n 9 8 + 2329189494927154830061962 n - 14012492136872618028197376 n 7 6 + 66007058834749419283041504 n - 239980514537103996575542832 n 5 4 + 657203673094880305228159776 n - 1291662466009428603604722048 n 3 2 + 1569676142554171754030277120 n - 153531678752512379482060800 n - 3994743424474423907371008000 n + 6986814516382439196057600000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 35) (2 n - 29)), 111 ( 20 19 18 2476377533 n - 495275504130 n + 46227776718575 n 17 16 15 - 2674859034525870 n + 107494852841790018 n - 3185456415666779460 n 14 13 + 72128195543981598350 n - 1275959355705127589340 n 12 11 + 17879639895011811394673 n - 200021847188448408298890 n 10 9 + 1792072384485460657831875 n - 12844822516261096756390710 n 8 7 + 73252860160448010412239968 n - 328705098885421695971650320 n 6 5 + 1136292792163127656135852400 n - 2889047737049275562357674080 n 4 3 + 4698012568722444648201287808 n - 1642795227251366519394547200 n 2 - 13462681235989855464087091200 n + 27533533158748154947368960000 n - 755331299068371804979200000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 20 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 37 (7429132409 n 19 18 17 - 1485826411690 n + 138683305004475 n - 8024573170309710 n 16 15 + 322484126971471914 n - 9556333972398182580 n 14 13 + 216382360782412204550 n - 3827767217940976764220 n 12 11 + 53634503790364127732829 n - 599923806944435490286770 n 10 9 + 5372544585290306532093375 n - 38457868756753631613533430 n 8 7 + 218483311634593064803152464 n - 969423261470408547503410960 n 6 5 + 3241368047505327137728033200 n - 7430960206172358195127532640 n 4 3 + 7871321809389177467258880384 n + 13166764275121487392594752000 n 2 - 55494429356012590539351705600 n + 41754606918931743388231680000 n - 2265993897205115414937600000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 21 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 37 (14858262358 n 20 19 18 - 3276245945889 n + 338285165941940 n - 21735110594691795 n 17 16 + 973970115512860578 n - 32334054849869383674 n 15 14 + 824545138825931600680 n - 16525747203966701588790 n 13 12 + 264148854095990131013018 n - 3396970004723645685986469 n 11 10 + 35292580071849923255471220 n - 296150762636296642207522935 n 9 8 + 1996236113920293774606190798 n - 10664032995115898715941523984 n 7 6 + 43844153942360644550022139760 n - 129583875615113498333643724080 n 5 4 + 223838899975456001858669953248 n + 18461702786277271433804390016 n 3 2 - 1080698707613583971021179353600 n + 2025520643766993091654669977600 n - 1187622757307510158291937280000 n + 92905749785409732012441600000)/( 262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 21 20 (2 n - 39) (2 n - 35) (2 n - 29)), (549755255221 n - 121220868561093 n 19 18 + 12516495443591405 n - 804190724080634415 n 17 16 + 36036015107226139236 n - 1196291472066054411138 n 15 14 + 30504060553166157386410 n - 611259174603449294341230 n 13 12 + 9766263002560011755312141 n - 125470784176866151998254553 n 11 10 + 1300614753925593748825994265 n - 10857882142895788702735082595 n 9 8 + 72356210137353558601509875626 n - 376986234007884245813683978608 n 7 6 + 1467753182724809798519716673120 n - 3822732981063660479432453502960 n 5 4 + 4267014183217714992179546527776 n + 9712296962514950048237428555392 n 3 - 45069678806505053333122039795200 n 2 + 64925496543074924815584348211200 n - 34687200022523598706331535360000 n + 3437512742060160084460339200000)/(262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) 22 (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), (2199013677989 n 21 20 19 - 532158644602453 n + 60490060437125473 n - 4293042857947304615 n 18 17 + 213289773327637162464 n - 7883106666223142452098 n 16 15 + 224827209881614080210338 n - 5065042788053920566575230 n 14 13 + 91506228007590590523447349 n - 1337901935148516347854757513 n 12 11 + 15897721087421092796634221373 n - 153410954857667744503657786395 n 10 9 + 1193903992516273938300534255254 n - 7371632584565575644485092304368 n 8 + 34925501184808364843670396440848 n 7 - 118201860306899036807213312952560 n 6 + 233238755692849533202508907426144 n 5 + 14226089947781057393420847906432 n 4 - 1538143886796629244073723279948032 n 3 + 4220696872625371460513397681868800 n 2 - 5083753810853076752824051725619200 n + 2545829952876505022616108380160000 n - 295626095817173767263589171200000 )/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 22 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), (274873536569 n 21 20 19 - 66518253505033 n + 7560889856105278 n - 536576948076811565 n 18 17 + 26655809099926671819 n - 984982235698683048288 n 16 15 + 28080043500988875119438 n - 632078343119073473215450 n 14 13 + 11400712818323038934833879 n - 166167826574729910229612313 n 12 11 + 1962895764943451349189432618 n - 18737196852433720384405266585 n 10 9 + 143000185648444670289690754709 n - 852924918293893735670522492878 n 8 7 + 3798130063798684051861398005338 n - 11366785543776633726061820156720 n 6 + 15185057030604645844075512073824 n 5 + 36992845088409839724864747028512 n 4 - 235147650352618588812813455112672 n 3 + 524332339556945000591565650200320 n 2 - 578045954667914633909393156620800 n + 282791513393017298033237153280000 n - 36953261977146720907948646400000)/ (65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 23 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 713 (771006836 n 22 21 20 - 203921708587 n + 25403686787677 n - 1981850866937480 n 19 18 + 108587961886619061 n - 4441591975339872507 n 17 16 + 140716238705945616692 n - 3535325313656918652970 n 15 14 + 71508888555604285951726 n - 1174975949413554169492457 n 13 12 + 15741702842855988195197637 n - 171685572067020898752307260 n 11 10 + 1512127554920490066898992521 n - 10571558275061547738607641217 n 9 8 + 56761246417123715024268636442 n - 218593377626439983755290891650 n 7 6 + 496829870583261413886597611856 n + 36937686095136159185891652768 n 5 4 - 4852354078710156104814131588448 n + 18103528971071295179630194539360 n 3 - 34088567535046738654253838432000 n 2 + 34538662336192394868035214912000 n - 16448070055400940083756428800000 n + 2332253560969989398117376000000)/(65536 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) 23 22 21 (2 n - 45)), 713 (3083736449 n - 815545750273 n + 101582684994703 n 20 19 18 - 7922869591370195 n + 433905071135602374 n - 17733810626905951578 n 17 16 + 561048102046218607238 n - 14061977480454993328030 n 15 14 + 283291156971872021756509 n - 4624167975882154191745253 n 13 12 + 61297085771481029757785043 n - 657419494698628019611795815 n 11 10 + 5641349252698856013702443564 n - 37870619603925798826986193168 n 9 8 + 190314162761032636951142746888 n - 645924524599775828772734831000 n 7 6 + 963012217710885107290952839104 n + 3277972161373518581291869822272 n 5 - 25275320796032377144451477983872 n 4 + 77286767089785095163949067975040 n 3 - 133664273193122439916760108928000 n 2 + 129918882724265639020169357568000 n - 61554925766508101941376563200000 n + 9329014243879957592469504000000)/( 262144 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 20677 ( 24 23 22 3401957917 n - 979490438796 n + 133149911288998 n 21 20 - 11363767987629024 n + 682947470620103227 n 19 18 - 30723560390487600636 n + 1073420430477393509368 n 17 16 - 29816603983796274560784 n + 668325877043839248768787 n 15 14 - 12192097979536191648902916 n + 181605428101732057621596958 n 13 12 - 2204148757365532464655799904 n + 21618265352142655958228615557 n 11 10 - 168448441767160887433323089076 n + 1009320113411667446115524037988 n 9 8 - 4336204484825132028511747341744 n + 10725794383859628234396875210032 n 7 + 6468014340608091501203172623424 n 6 - 187922273293232990281201984149312 n 5 + 864771715647371552887850441331456 n 4 - 2236483663581403761264915195755520 n 3 + 3536303022200852564482918199808000 n 2 - 3269685723124731007469810909184000 n + 1526356894460027538518043033600000 n - 241910990048197521018519552000000 )/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 667 ( 24 23 22 105406007167 n - 30338171990916 n + 4121843403678178 n 21 20 - 351471830554773984 n + 21093573991691085577 n 19 18 - 946869104719431079956 n + 32971420263860963620648 n 17 16 - 911246280077561137418544 n + 20272977278130341207947537 n 15 14 - 365825379776935173060520236 n + 5364503608841951435175894538 n 13 12 - 63676305259300357876011168864 n + 605006164398282483445787521207 n 11 - 4498703536317527926932741567996 n 10 + 25001663892284195497768472232268 n 9 - 92299661154409411283404097890704 n 8 + 119945885330117185704383521026832 n 7 + 981357707137472405290100487271104 n 6 - 7799129490529647336646123348381632 n 5 + 29855866884219083118225497817252096 n 4 - 71171744142858359773673393829688320 n 3 + 107612536340316620333684520396288000 n 2 - 97327061400627774581573895634944000 n + 45458277573958891545378572697600000 n - 7499240691494123151574106112000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 24 (2 n - 35) (2 n - 29) (2 n - 45)), 450225 (311977250 n - 97379205333 n 23 22 21 + 14378004812936 n - 1335333430218022 n + 87489922217798254 n 20 19 - 4298208740619455123 n + 164244269902292907756 n 18 17 - 4996090432421941797912 n + 122751680521817115942734 n 16 15 - 2456208757068295850318203 n + 40144775058280201921236016 n 14 13 - 534733704347369071033914782 n + 5755796376881116623900709074 n 12 11 - 49181906618408829437404204173 n + 321835974352555443657558305756 n 10 9 - 1480436748034625082527082220612 n + 3411426472371202537984748554384 n 8 + 10792861493751641079645399180112 n 7 - 149789458207776456879575153795264 n 6 + 782796114749151887189771300267328 n 5 - 2558492647514153412636162582141696 n 4 + 5575036249402078708435479038622720 n 3 - 7963826108556965234400808166707200 n 2 + 6964888381749692116641662091264000 n - 3219113606258050590154434969600000 n + 544389324271425236188342517760000 )/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 5175 ( 25 24 23 27085178104 n - 8445274820351 n + 1245002636437972 n 22 21 - 115367362397552354 n + 7534487031614233768 n 20 19 - 368480531229968980361 n + 13992125005082006139412 n 18 17 - 421968890543955319078184 n + 10247433372359526351861688 n 16 15 - 201868704081029903550021521 n + 3231235621161632523604137532 n 14 13 - 41846999115557706852890884874 n + 433207838534904476215696564888 n 12 - 3493784661695328540322626182711 n 11 + 20713325125477886198745621235612 n 10 - 75360305191649361987510665982284 n 9 - 8194294581994505304862427240432 n 8 + 2263878372734287903504226353683184 n 7 - 17264507897450477518685727320411328 n 6 + 77469954960283894114704172208373696 n 5 - 235142675399156384269209278016758016 n 4 + 490563938877786881013988000857461760 n 3 - 682709128563376184358508785088819200 n 2 + 589663684212762819560314261929984000 n - 273107643916171422976190375362560000 n + 47361871211613995548385799045120000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 5175 (53963286569 n 25 24 23 - 18172293227647 n + 2898209960347325 n - 291055591434479950 n 22 21 + 20639389924670133305 n - 1098225722618792552785 n 20 19 + 45475432368557254076585 n - 1499398457476761594910300 n 18 17 + 39933155689244668346654015 n - 865983260415671558099747665 n 16 15 + 15331976002518608921428540535 n - 220984204617955851830457795550 n 14 + 2567223334592140939162584422435 n 13 - 23514693498115537613144016212575 n 12 + 161608613860523475482305661775635 n 11 - 720457670990699962189064117579800 n 10 + 574633159152742753646738292729020 n 9 + 21993405887664070674397747371727520 n 8 - 218497002295811736828223793148945840 n 7 + 1239166683087346715500249439101027200 n 6 - 4842321577236654607398395092849297344 n 5 + 13496619701389231764534676707010513152 n 4 - 26569053754103605827822148845824394240 n 3 + 35531707095782122263244587770664038400 n 2 - 29941299065830424553505349568995328000 n + 13744507186171934358309287385661440000 n - 2415455431792313772967675751301120000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 575 ( 26 25 24 482464929946 n - 162054067884098 n + 25756021986031675 n 23 22 - 2574719933338014800 n + 181480298438299786120 n 21 20 - 9581110771373347396190 n + 392744933467083309234265 n 19 18 - 12783044326305043958581700 n + 334878131607427055903050510 n 17 16 - 7110348181327471907086656110 n + 122480700100048552698875133565 n 15 - 1701612022502746050829115820200 n 14 + 18758521221812773999345005913540 n 13 - 158006529692478757870731543546050 n 12 + 917336631410916838280301548153215 n 11 - 2146785580296032938509505513669700 n 10 - 22768146950038241773584350715604820 n 9 + 339347312222468218231101810296705680 n 8 - 2516185501873853302831848080168872560 n 7 + 12736367155127008196447177886784660800 n 6 - 46753736036328399813063005497940623296 n 5 + 125271510367484174286292808068579776768 n 4 - 240300949352529925338030388959919580160 n 3 + 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53) (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69)), - 185 ( 37 36 6085427693261833654 n - 4181381303688555739289 n 35 34 + 1384456916775830264468196 n - 294214660322970381240408015 n 33 32 + 45097925375747587197898961022 n - 5312348878035304559566197545460 n 31 + 500327373668281670130150029920784 n 30 - 38701854293786906563840673233761196 n 29 + 2506542565863028374477645586381041804 n 28 - 137874673598020240301290639742365259406 n 27 + 6511357122151316903332882680583528193880 n 26 - 266236901252037116708385712453393042783314 n 25 + 9486182935583093661918879259819052018177100 n 24 - 296018884266774048753108001851145344077114580 n 23 + 8121062254925413569962908842049480496044938480 n 22 - 196426339822362773793439437637939588363006217340 n 21 + 4196952611528875395916738980282246413055902571390 n 20 - 79311500307623068657770160714910318843616658630305 n 19 + 1326229323966582170809854202672560313953334326864740 n 18 - 19620077423461568622111610324299983486826090800748055 n 17 + 256573952473827002372460695351886173372377085770566966 n 16 - 2961349823780854014762806330931575638999487022984318576 n 15 + 30099394801818888927624916646926136392811414524147792224 n 14 - 268599962973321779458860981207135567228059357109962282400 n 13 + 2096271333283106480842346115951087811586019484627451748928 n 12 - 14238466288353318796862491978350725160314091701199097740800 n 11 + 83660937556017653818400559487175616525477010950130034535936 n 10 - 422060456885193318079860107513800167267226346728973992621824 n 9 + 1811305982107836872083335652207206259220759806519534494782976 n 8 - 6536549701611463976015402354080186149971302302026619040579584 n 7 + 19547513255384440727350081746776918992026161574909903053045760 n 6 - 47537188043419137551874392608121385592915466372616690687737856 n 5 + 91693092422635812758392930206234302316613224221218085293916160 n 4 - 135551763991444861019590967960167276041473506604915310723072000 n 3 + 146129708787133574459504809315697194205787149527197013770240000 n 2 - 106239009366233811483172158441972168050798611682060339773440000 n + 45264259193636582567315703425156301502561713154609800806400000 n - 8119361470459215408937195766797170892217315240265646080000000)/( 8589934592 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 65) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 73) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69)), ( 706935451148026805600 (n - 36)! (2 n - 77)! (n + 1) + 1060403176722040208400 (n - 37)! ((n + 1) (2 n - 76)! - 1/1060403176722040208400 binomial(2 n, n) (n - 39)! (n - 36)!))/( (n - 37)! (n - 39)! (n - 36)! binomial(2 n, n)), ( (706935451148026805600 n + 706935451148026805600) (2 n - 77)! - (n - 39)! (n - 37)! binomial(2 n, n))/((n - 39)! (n - 37)! binomial(2 n, n))] and in Maple notation [3/65536*(11453246121*n^19-2067310924223*n^18+173490908715414*n^17-\ 8988151035629172*n^16+321894832109139102*n^15-8455793831003661786*n^14+ 168705401029349539848*n^13-2611581947065720968404*n^12+31768564389348975627153* n^11-305651421187848441493359*n^10+2329189494927154830061962*n^9-\ 14012492136872618028197376*n^8+66007058834749419283041504*n^7-\ 239980514537103996575542832*n^6+657203673094880305228159776*n^5-\ 1291662466009428603604722048*n^4+1569676142554171754030277120*n^3-\ 153531678752512379482060800*n^2-3994743424474423907371008000*n+ 6986814516382439196057600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9) /(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-\ 37)/(2*n-17)/(2*n-23)/(2*n-35)/(2*n-29), 111/262144*(2476377533*n^20-\ 495275504130*n^19+46227776718575*n^18-2674859034525870*n^17+107494852841790018* n^16-3185456415666779460*n^15+72128195543981598350*n^14-1275959355705127589340* n^13+17879639895011811394673*n^12-200021847188448408298890*n^11+ 1792072384485460657831875*n^10-12844822516261096756390710*n^9+ 73252860160448010412239968*n^8-328705098885421695971650320*n^7+ 1136292792163127656135852400*n^6-2889047737049275562357674080*n^5+ 4698012568722444648201287808*n^4-1642795227251366519394547200*n^3-\ 13462681235989855464087091200*n^2+27533533158748154947368960000*n-\ 755331299068371804979200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/ (2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37 )/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 37/262144*(7429132409*n^20-\ 1485826411690*n^19+138683305004475*n^18-8024573170309710*n^17+ 322484126971471914*n^16-9556333972398182580*n^15+216382360782412204550*n^14-\ 3827767217940976764220*n^13+53634503790364127732829*n^12-\ 599923806944435490286770*n^11+5372544585290306532093375*n^10-\ 38457868756753631613533430*n^9+218483311634593064803152464*n^8-\ 969423261470408547503410960*n^7+3241368047505327137728033200*n^6-\ 7430960206172358195127532640*n^5+7871321809389177467258880384*n^4+ 13166764275121487392594752000*n^3-55494429356012590539351705600*n^2+ 41754606918931743388231680000*n-2265993897205115414937600000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29) , 37/262144*(14858262358*n^21-3276245945889*n^20+338285165941940*n^19-\ 21735110594691795*n^18+973970115512860578*n^17-32334054849869383674*n^16+ 824545138825931600680*n^15-16525747203966701588790*n^14+ 264148854095990131013018*n^13-3396970004723645685986469*n^12+ 35292580071849923255471220*n^11-296150762636296642207522935*n^10+ 1996236113920293774606190798*n^9-10664032995115898715941523984*n^8+ 43844153942360644550022139760*n^7-129583875615113498333643724080*n^6+ 223838899975456001858669953248*n^5+18461702786277271433804390016*n^4-\ 1080698707613583971021179353600*n^3+2025520643766993091654669977600*n^2-\ 1187622757307510158291937280000*n+92905749785409732012441600000)/(2*n-5)/(-1+2* n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29), 1/262144*(549755255221*n^21-121220868561093*n^20+ 12516495443591405*n^19-804190724080634415*n^18+36036015107226139236*n^17-\ 1196291472066054411138*n^16+30504060553166157386410*n^15-\ 611259174603449294341230*n^14+9766263002560011755312141*n^13-\ 125470784176866151998254553*n^12+1300614753925593748825994265*n^11-\ 10857882142895788702735082595*n^10+72356210137353558601509875626*n^9-\ 376986234007884245813683978608*n^8+1467753182724809798519716673120*n^7-\ 3822732981063660479432453502960*n^6+4267014183217714992179546527776*n^5+ 9712296962514950048237428555392*n^4-45069678806505053333122039795200*n^3+ 64925496543074924815584348211200*n^2-34687200022523598706331535360000*n+ 3437512742060160084460339200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2* n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/524288*( 2199013677989*n^22-532158644602453*n^21+60490060437125473*n^20-\ 4293042857947304615*n^19+213289773327637162464*n^18-7883106666223142452098*n^17 +224827209881614080210338*n^16-5065042788053920566575230*n^15+ 91506228007590590523447349*n^14-1337901935148516347854757513*n^13+ 15897721087421092796634221373*n^12-153410954857667744503657786395*n^11+ 1193903992516273938300534255254*n^10-7371632584565575644485092304368*n^9+ 34925501184808364843670396440848*n^8-118201860306899036807213312952560*n^7+ 233238755692849533202508907426144*n^6+14226089947781057393420847906432*n^5-\ 1538143886796629244073723279948032*n^4+4220696872625371460513397681868800*n^3-\ 5083753810853076752824051725619200*n^2+2545829952876505022616108380160000*n-\ 295626095817173767263589171200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/65536 *(274873536569*n^22-66518253505033*n^21+7560889856105278*n^20-\ 536576948076811565*n^19+26655809099926671819*n^18-984982235698683048288*n^17+ 28080043500988875119438*n^16-632078343119073473215450*n^15+ 11400712818323038934833879*n^14-166167826574729910229612313*n^13+ 1962895764943451349189432618*n^12-18737196852433720384405266585*n^11+ 143000185648444670289690754709*n^10-852924918293893735670522492878*n^9+ 3798130063798684051861398005338*n^8-11366785543776633726061820156720*n^7+ 15185057030604645844075512073824*n^6+36992845088409839724864747028512*n^5-\ 235147650352618588812813455112672*n^4+524332339556945000591565650200320*n^3-\ 578045954667914633909393156620800*n^2+282791513393017298033237153280000*n-\ 36953261977146720907948646400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2* n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 713/ 65536*(771006836*n^23-203921708587*n^22+25403686787677*n^21-1981850866937480*n^ 20+108587961886619061*n^19-4441591975339872507*n^18+140716238705945616692*n^17-\ 3535325313656918652970*n^16+71508888555604285951726*n^15-\ 1174975949413554169492457*n^14+15741702842855988195197637*n^13-\ 171685572067020898752307260*n^12+1512127554920490066898992521*n^11-\ 10571558275061547738607641217*n^10+56761246417123715024268636442*n^9-\ 218593377626439983755290891650*n^8+496829870583261413886597611856*n^7+ 36937686095136159185891652768*n^6-4852354078710156104814131588448*n^5+ 18103528971071295179630194539360*n^4-34088567535046738654253838432000*n^3+ 34538662336192394868035214912000*n^2-16448070055400940083756428800000*n+ 2332253560969989398117376000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 713/262144*(3083736449*n^23-815545750273*n^22+101582684994703*n^21-\ 7922869591370195*n^20+433905071135602374*n^19-17733810626905951578*n^18+ 561048102046218607238*n^17-14061977480454993328030*n^16+ 283291156971872021756509*n^15-4624167975882154191745253*n^14+ 61297085771481029757785043*n^13-657419494698628019611795815*n^12+ 5641349252698856013702443564*n^11-37870619603925798826986193168*n^10+ 190314162761032636951142746888*n^9-645924524599775828772734831000*n^8+ 963012217710885107290952839104*n^7+3277972161373518581291869822272*n^6-\ 25275320796032377144451477983872*n^5+77286767089785095163949067975040*n^4-\ 133664273193122439916760108928000*n^3+129918882724265639020169357568000*n^2-\ 61554925766508101941376563200000*n+9329014243879957592469504000000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2 *n-39)/(2*n-35)/(2*n-29)/(2*n-45), 20677/4194304*(3401957917*n^24-979490438796* n^23+133149911288998*n^22-11363767987629024*n^21+682947470620103227*n^20-\ 30723560390487600636*n^19+1073420430477393509368*n^18-29816603983796274560784*n ^17+668325877043839248768787*n^16-12192097979536191648902916*n^15+ 181605428101732057621596958*n^14-2204148757365532464655799904*n^13+ 21618265352142655958228615557*n^12-168448441767160887433323089076*n^11+ 1009320113411667446115524037988*n^10-4336204484825132028511747341744*n^9+ 10725794383859628234396875210032*n^8+6468014340608091501203172623424*n^7-\ 187922273293232990281201984149312*n^6+864771715647371552887850441331456*n^5-\ 2236483663581403761264915195755520*n^4+3536303022200852564482918199808000*n^3-\ 3269685723124731007469810909184000*n^2+1526356894460027538518043033600000*n-\ 241910990048197521018519552000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/( 2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29) /(2*n-45), 667/4194304*(105406007167*n^24-30338171990916*n^23+4121843403678178* n^22-351471830554773984*n^21+21093573991691085577*n^20-946869104719431079956*n^ 19+32971420263860963620648*n^18-911246280077561137418544*n^17+ 20272977278130341207947537*n^16-365825379776935173060520236*n^15+ 5364503608841951435175894538*n^14-63676305259300357876011168864*n^13+ 605006164398282483445787521207*n^12-4498703536317527926932741567996*n^11+ 25001663892284195497768472232268*n^10-92299661154409411283404097890704*n^9+ 119945885330117185704383521026832*n^8+981357707137472405290100487271104*n^7-\ 7799129490529647336646123348381632*n^6+29855866884219083118225497817252096*n^5-\ 71171744142858359773673393829688320*n^4+107612536340316620333684520396288000*n^ 3-97327061400627774581573895634944000*n^2+45458277573958891545378572697600000*n -7499240691494123151574106112000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47) /(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13) /(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), 450225/4194304*(311977250*n^25-97379205333*n^24+14378004812936*n^ 23-1335333430218022*n^22+87489922217798254*n^21-4298208740619455123*n^20+ 164244269902292907756*n^19-4996090432421941797912*n^18+122751680521817115942734 *n^17-2456208757068295850318203*n^16+40144775058280201921236016*n^15-\ 534733704347369071033914782*n^14+5755796376881116623900709074*n^13-\ 49181906618408829437404204173*n^12+321835974352555443657558305756*n^11-\ 1480436748034625082527082220612*n^10+3411426472371202537984748554384*n^9+ 10792861493751641079645399180112*n^8-149789458207776456879575153795264*n^7+ 782796114749151887189771300267328*n^6-2558492647514153412636162582141696*n^5+ 5575036249402078708435479038622720*n^4-7963826108556965234400808166707200*n^3+ 6964888381749692116641662091264000*n^2-3219113606258050590154434969600000*n+ 544389324271425236188342517760000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/( 2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/( 2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35) /(2*n-29)/(2*n-45), 5175/4194304*(27085178104*n^25-8445274820351*n^24+ 1245002636437972*n^23-115367362397552354*n^22+7534487031614233768*n^21-\ 368480531229968980361*n^20+13992125005082006139412*n^19-\ 421968890543955319078184*n^18+10247433372359526351861688*n^17-\ 201868704081029903550021521*n^16+3231235621161632523604137532*n^15-\ 41846999115557706852890884874*n^14+433207838534904476215696564888*n^13-\ 3493784661695328540322626182711*n^12+20713325125477886198745621235612*n^11-\ 75360305191649361987510665982284*n^10-8194294581994505304862427240432*n^9+ 2263878372734287903504226353683184*n^8-17264507897450477518685727320411328*n^7+ 77469954960283894114704172208373696*n^6-235142675399156384269209278016758016*n^ 5+490563938877786881013988000857461760*n^4-682709128563376184358508785088819200 *n^3+589663684212762819560314261929984000*n^2-\ 273107643916171422976190375362560000*n+47361871211613995548385799045120000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 5175/4194304*( 53963286569*n^26-18172293227647*n^25+2898209960347325*n^24-291055591434479950*n ^23+20639389924670133305*n^22-1098225722618792552785*n^21+ 45475432368557254076585*n^20-1499398457476761594910300*n^19+ 39933155689244668346654015*n^18-865983260415671558099747665*n^17+ 15331976002518608921428540535*n^16-220984204617955851830457795550*n^15+ 2567223334592140939162584422435*n^14-23514693498115537613144016212575*n^13+ 161608613860523475482305661775635*n^12-720457670990699962189064117579800*n^11+ 574633159152742753646738292729020*n^10+21993405887664070674397747371727520*n^9-\ 218497002295811736828223793148945840*n^8+1239166683087346715500249439101027200* n^7-4842321577236654607398395092849297344*n^6+ 13496619701389231764534676707010513152*n^5-\ 26569053754103605827822148845824394240*n^4+ 35531707095782122263244587770664038400*n^3-\ 29941299065830424553505349568995328000*n^2+ 13744507186171934358309287385661440000*n-2415455431792313772967675751301120000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 575/ 4194304*(482464929946*n^26-162054067884098*n^25+25756021986031675*n^24-\ 2574719933338014800*n^23+181480298438299786120*n^22-9581110771373347396190*n^21 +392744933467083309234265*n^20-12783044326305043958581700*n^19+ 334878131607427055903050510*n^18-7110348181327471907086656110*n^17+ 122480700100048552698875133565*n^16-1701612022502746050829115820200*n^15+ 18758521221812773999345005913540*n^14-158006529692478757870731543546050*n^13+ 917336631410916838280301548153215*n^12-2146785580296032938509505513669700*n^11-\ 22768146950038241773584350715604820*n^10+339347312222468218231101810296705680*n ^9-2516185501873853302831848080168872560*n^8+ 12736367155127008196447177886784660800*n^7-\ 46753736036328399813063005497940623296*n^6+ 125271510367484174286292808068579776768*n^5-\ 240300949352529925338030388959919580160*n^4+ 316244525864640591192022685321801625600*n^3-\ 264541190356170135108122803366551552000*n^2+ 121709804704290007763546491071528960000*n-\ 21739098886130823956709081761710080000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 5175/4194304*(106053085928*n^27-\ 38295752835126*n^26+6552011330958756*n^25-706061608929129375*n^24+ 53731284626181370260*n^23-3067940355516936078660*n^22+136279862146982645648700* n^21-4817756599470689540499405*n^20+137457159799701211795619580*n^19-\ 3188923619696623041630304050*n^18+60250272653799034478064707340*n^17-\ 922259722801382707161803966505*n^16+11260779616502418988448637580620*n^15-\ 105669026437027072035659901905880*n^14+687042608192235470207873590073220*n^13-\ 1776806539134459428079584298714555*n^12-22976275234895954651220388528969860*n^ 11+392362123868774106474762688096638180*n^10-\ 3422925478144196321271011360565997040*n^9+ 20811579808371332103561514337327043120*n^8-\ 94004359789232146393251577831844507328*n^7+ 319491075068245059415510910106280441536*n^6-\ 809448975779908401039154384306040990976*n^5+ 1489655569786678668337140080140619166720*n^4-\ 1902647744559080823116416595539587379200*n^3+ 1560950722850263742016754859113242624000*n^2-\ 712187736035368693047214371708764160000*n+ 128019137884992629967286814818959360000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1035/16777216*(2085002019997*n^27-\ 749075177473599*n^26+127344395994056124*n^25-13614461795872682550*n^24+ 1025937476623599679815*n^23-57874567127030420701545*n^22+ 2532842252740639911224010*n^21-87909452241125542073998380*n^20+ 2451231554508860538027693795*n^19-55224380970778813925091007905*n^18+ 1003603825819561654340979981720*n^17-14539861399932630301800617017830*n^16+ 162725473133760381328769203990305*n^15-1288427490485146780642093393250775*n^14+ 4777097018267909557899840378856290*n^13+44258672977047201454650057534644520*n^ 12-1037210405874074991862571693986172040*n^11+ 11211882223830577703659446054813685440*n^10-\ 83978422009828519292611546527261474400*n^9+ 471615062071813583447515326227620590720*n^8-\ 2027400884123086252507440765940514451072*n^7+ 6662713760177914142542236258777806832384*n^6-\ 16486960555656121766110013739647889183744*n^5+ 29852171612818131356085900405456425963520*n^4-\ 37742550470854484420303412913129133260800*n^3+ 30828233251786168271342182231223894016000*n^2-\ 14092757788843023579444410678234972160000*n+ 2560382757699852599345736296379187200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 3105/67108864*(5417416382537*n^28-\ 2081706054742734*n^27+378904167481236789*n^26-43420711054293573910*n^25+ 3511643398493535988865*n^24-212902164224515858944670*n^23+ 10029516571010043840403385*n^22-375337639257246846365815630*n^21+ 11304349291312443052851058895*n^20-275526745229943492684690220930*n^19+ 5422297720727232583115369823695*n^18-84982039938421586641487852229730*n^17+ 1021449793124194556542423473025355*n^16-8400417025018980456358817440539850*n^15 +22847655826503160335269811006553715*n^14+614904864381825408617485453636876070* n^13-13028767096276302321539777076403655140*n^12+ 153529713895483298798148904176459302840*n^11-\ 1307086061835755481873433261190661606000*n^10+ 8555601585645439270013735142581474489120*n^9-\ 43859021514346649030735961422857020670912*n^8+ 176260108969064462298895112981299919497344*n^7-\ 549928784796516659708987781410234675011584*n^6+ 1306557469427147527002822419840740775854080*n^5-\ 2292081718193937989062948631752853535129600*n^4+ 2830039168013658752254216011676341301248000*n^3-\ 2274542024063708742424539056032754073600000*n^2+ 1031340401302557422335215468356606361600000*n-\ 187761402231322523952020661734473728000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/67108864*( 46934087998843*n^28-17878704554009306*n^27+3220491614843415591*n^26-\ 364486134549049621290*n^25+29041322976708799355235*n^24-\ 1729345690965077067248730*n^23+79702258524650206439080515*n^22-\ 2902624058824424998397303970*n^21+84420676164956211516104073405*n^20-\ 1963030052503961077112199794070*n^19+36071134959688944581179375794405*n^18-\ 504519934337743663128757390383870*n^17+4760849918271941845766482425741345*n^16-\ 12715527528545314516900535766724350*n^15-537273904637185759683906049685305215*n ^14+12819035674148857914480815165872404330*n^13-\ 176251785352127621117950772977703990460*n^12+ 1773203488269273479049292517007556764360*n^11-\ 13867993504753544476847271498620380742800*n^10+ 86016305295757618905471695401175286445280*n^9-\ 424784651779860905973042978060509167675968*n^8+ 1661319373717593695587504722985433406900096*n^7-\ 5079678806272154071625699149020672977642496*n^6+ 11890401620071285284342539602414787962859520*n^5-\ 20641617084455174388817191569579098918502400*n^4+ 25319994179865014280517365900912400576512000*n^3-\ 20295459435431931440468479573390685798400000*n^2+ 9216644480628845604888859618134550118400000*n-\ 1689852620081902715568185955610263552000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 10005/67108864*( 3065423676034*n^29-1241787206541647*n^28+238039169399884356*n^27-\ 28690331815766150523*n^26+2436127738052240641500*n^25-154683174341684326833495* n^24+7603740081334996021407660*n^23-295239107652052146995838855*n^22+ 9138509851460725933938140400*n^21-225030366992327059141667714025*n^20+ 4322808910341803868445206083700*n^19-60860249884693337058918705283425*n^18+ 486972895395824163029375108318820*n^17+2765747205754305655897448502177315*n^16-\ 182468556298340471733721086021058620*n^15+3812905051688957573435018580425335635 *n^14-54545943436710681801028617541113740370*n^13+ 599842625741951748558951281666788714860*n^12-\ 5265371175339893219077659119355564282280*n^11+ 37401940559040094008016014725486999737040*n^10-\ 215602488798880935167579306086716031101024*n^9+ 1004594414841749838213543474025632469161792*n^8-\ 3747801735987431547129071963067657635538816*n^7+ 11024928050200014777019485961542623767600128*n^6-\ 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(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 10005/ 134217728*(10224982676093*n^30-4311251676211185*n^29+857841648310443955*n^28-\ 106919655317104179165*n^27+9337050186644980702287*n^26-604675081827475200880065 *n^25+29910699681344905141392375*n^24-1141499981919104203491918225*n^23+ 33166243657244908137407553645*n^22-687417358199192082290046192975*n^21+ 7429425909153902523138988289025*n^20+109732838055437926961726198074425*n^19-\ 8142552915371542570178509055667435*n^18+245953025859288792667217693779135125*n^ 17-5245740522688214600713733111461947075*n^16+ 87569672587116382230970849171543858725*n^15-\ 1187643560559466510314497908693911958830*n^14+ 13302079085532977196541752358721407185900*n^13-\ 123908284536452327087369579458207107401000*n^12+ 961496035147885725400472528256182319282000*n^11-\ 6202048105597345820064469344120374635712608*n^10+ 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*n^25+4810824369362704167140250*n^24-156113644749194118969224775*n^23+ 3169833634348446705570567780*n^22-3821358721251010885885464525*n^21-\ 2749294329332916344576463061650*n^20+133649525411022880924685774146575*n^19-\ 4104230226359139139917354987846090*n^18+95600010736755815060198900764672875*n^ 17-1788121418705174274460962193563544050*n^16+ 27533079431031164997146018614341160275*n^15-\ 353296053993161873722161508478308942245*n^14+ 3799695221141201304983829991206811540350*n^13-\ 34311008226529538106471183289530558860000*n^12+ 259800784612630363451316596567067536121000*n^11-\ 1643181363884875369593425509322532527377312*n^10+ 8624968823269478741992203317566368752636640*n^9-\ 37221369358630391179260704100715783230616320*n^8+ 130377605226777668934572785251765536327852160*n^7-\ 364258752473118252623586941016496072873398528*n^6+ 792579176961889844223284791836232417481948160*n^5-\ 1298769352194566234347145486881445364587827200*n^4+ 1525587302495844034841546939869736635570176000*n^3-\ 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21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 13485/134217728*(7920795156383*n^31-3196047836585419*n^30+ 588714402066777835*n^29-64067685410876027255*n^28+4324859682431780478567*n^27-\ 150270222726590691999771*n^26-3029246784854301559933905*n^25+ 806658456283609893105420525*n^24-62186593970349489624791002455*n^23+ 3216233706554854111545812479515*n^22-127336871920334349995538537623175*n^21+ 4057880366837016143850486708668475*n^20-106763361211610554167682396173835635*n^ 19+2353572531930067753486580484441965655*n^18-\ 43867462669758141324691816979561334075*n^17+ 695104161084357584013485426516276110575*n^16-\ 9391672625156409143069697729532202562780*n^15+ 108307697288213075809256035072463758469940*n^14-\ 1065361475793687424611348400329133134882200*n^13+ 8917770534264949114958869575444978464596000*n^12-\ 63274152506926553851338857886280909057490848*n^11+ 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4749010992868771192*n^30-336215698055963570720*n^29-8593896108251076810684*n^28 +4734923315339769354655776*n^27-594380944348262343143237832*n^26+ 47974817345967768185565149280*n^25-2892492740347825008419997005190*n^24+ 138132778083327923088885827328960*n^23-5389531256319078110471617589084520*n^22+ 175093940961990267832861452916164000*n^21-4796063985657417668504584416962365980 *n^20+111708325354876412394304391386452407520*n^19-\ 2225237084843851043367895141250371576440*n^18+ 38051587079469478653200823827482654893600*n^17-\ 559727567767738799550970955553330185214465*n^16+ 7087207075884958029208617469001480975867760*n^15-\ 77195247179565977635755675587244487294583120*n^14+ 721928546531516434921077631567260008633843200*n^13-\ 5778127594786199697186824969278340223639298144*n^12+ 39395687426688228527727314516382816284248987136*n^11-\ 227377115802099122789397881078422011935200964352*n^10+ 1101786742093000098798634232806685646240724254720*n^9-\ 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6004711632226956102802579944*n^26+401283583881970366442522847840*n^25-\ 21542417791996336994324774353470*n^24+950045777735988382052783499128640*n^23-\ 34962654060749780885541987788626440*n^22+1085858576221072000786561768552452000* n^21-28695557221050057018744509156553745740*n^20+ 649076512481181107029560101900829120480*n^19-\ 12618271562002823115396984286186066798680*n^18+ 211377794142411104817841334582562182407200*n^17-\ 3055205410603053782075948079837467623989045*n^16+ 38105661467413273582225768548909342952574640*n^15-\ 409687007067696206113148566383331409257905040*n^14+ 3788494073301011543618787742063299324942963200*n^13-\ 30028374410962865992112911310251727353891738592*n^12+ 203024145917062335931841212858768285728809004544*n^11-\ 1163373547070899008192605439392487944555252792064*n^10+ 5602889852291713906508585452483876297597381754880*n^9-\ 22435614510347333856152485262793904381179341284608*n^8+ 73659656251339479410812228472144315868753399115776*n^7-\ 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n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -5/1073741824*(4816926137401582664 *n^35-3062848150244373928444*n^34+934037520600949739471798*n^33-\ 181978334834958645326312431*n^32+25456371168494578673805545024*n^31-\ 2724161981864769139664883816904*n^30+232018457793295254940025208062472*n^29-\ 16155389712269078411970687431140644*n^28+937432152123233827579494992467695552*n ^27-45977175424175787867411809787620578272*n^26+ 1926521392921312800422174125349678047140*n^25-\ 69532602436193371450128859161035098845690*n^24+ 2175243072839373493438221080075717737274640*n^23-\ 59264645603160159940218125804606030516936040*n^22+ 1411163407667526845957969236275847769224652040*n^21-\ 29438814976669788465827849950942703356678338180*n^20+ 538877320565657686356909476825012598300777622200*n^19-\ 8661316376696468304701140025376149634962001634500*n^18+ 122222841729522525079579165393889741208938438538070*n^17-\ 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6163159137492339302701230835756328726213668228173145977344*n^12-\ 37003170927472192026331393809723022782771767032094416917504*n^11+ 190417352939579737034820827977797588706013801149605259658496*n^10-\ 832193441311491191011571133959872144438867113668496001774080*n^9+ 3053553731026272259713833859217849891985698531423739518107648*n^8-\ 9271067371852169851701187234086975580813823421231535445655552*n^7+ 22857885190431487169334655779074218226742162770660629658271744*n^6-\ 44638131927279394236605129294204045332218921610584283669463040*n^5+ 66719772765432549193265577477018197441304157035502671495168000*n^4-\ 72623867853343131715315521849871882571002878528621395312640000*n^3+ 53236009397442318868857846911241359951452810807230395842560000*n^2-\ 22834005871473797233677346706032320424035748313910843801600000*n+ 4115292800095766714118852648924593465918365258764779520000000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31 )/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -5/ 8589934592*(107278396441733721557*n^36-70073112562022522678346*n^35+ 22026506089355791814615475*n^34-4437827631886481620262138790*n^33+ 644001128093177400890014586916*n^32-71713370603531639611400636147976*n^31+ 6375042264901627158608141422153084*n^30-464703769194830464076058336626486520*n^ 29+28313920467749314990175797033326429462*n^28-\ 1462563681030237949849988289237619129068*n^27+ 64742113801390444307664593686302969411786*n^26-\ 2476266651697810299723366246705214080010260*n^25+ 82357950917169183778327188481399422472376580*n^24-\ 2393448451158397171998804007354577531805521160*n^23+ 61001374041993194527192453382757286275421986060*n^22-\ 1367084712172807340698304170057000769492853951000*n^21+ 26986621385554642760027336685031342471653745069245*n^20-\ 469685961218168446678931385044509516292755251553050*n^19+ 7208685372760633933227864945465638463455025665349595*n^18-\ 97513077849020360865416879789090847062265290731375350*n^17+ 1161132252248324895522795132295987737015137590093063728*n^16-\ 12146173768001952799184210703181139152184260140000038304*n^15+ 111304544019191849059291344329709992394270536528661545760*n^14-\ 890206592784519263695863315991941877827037948487630352960*n^13+ 6184715771236279924994645337837801210486244921525553206784*n^12-\ 37104489431907842748206565299316988389052197791514090537984*n^11+ 190813516529108558563404818079514819829808963590612529789696*n^10-\ 833457295113726941333512102133989520705949879423725587822080*n^9+ 3056750479081160526936789058288975298417565288552541307977728*n^8-\ 9277182788666874254824508991203774597971785646829839883354112*n^7+ 22865927195580652738763768782605326367992119374039527540588544*n^6-\ 44643439536886960650119690982647943853310852515923146466263040*n^5+ 66716895427992911194080670129061035438831953343095766253568000*n^4-\ 72613897105533409744358212861055989229649293372084626391040000*n^3+ 53227229660320702515719221822038355943187994039835751874560000*n^2-\ 22831184472323861838974125224715355804903759855622920601600000*n+ 4115292800095766714118852648924593465918365258764779520000000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31 )/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -185/ 8589934592*(6085427693261833654*n^37-4181381303688555739289*n^36+ 1384456916775830264468196*n^35-294214660322970381240408015*n^34+ 45097925375747587197898961022*n^33-5312348878035304559566197545460*n^32+ 500327373668281670130150029920784*n^31-38701854293786906563840673233761196*n^30 +2506542565863028374477645586381041804*n^29-\ 137874673598020240301290639742365259406*n^28+ 6511357122151316903332882680583528193880*n^27-\ 266236901252037116708385712453393042783314*n^26+ 9486182935583093661918879259819052018177100*n^25-\ 296018884266774048753108001851145344077114580*n^24+ 8121062254925413569962908842049480496044938480*n^23-\ 196426339822362773793439437637939588363006217340*n^22+ 4196952611528875395916738980282246413055902571390*n^21-\ 79311500307623068657770160714910318843616658630305*n^20+ 1326229323966582170809854202672560313953334326864740*n^19-\ 19620077423461568622111610324299983486826090800748055*n^18+ 256573952473827002372460695351886173372377085770566966*n^17-\ 2961349823780854014762806330931575638999487022984318576*n^16+ 30099394801818888927624916646926136392811414524147792224*n^15-\ 268599962973321779458860981207135567228059357109962282400*n^14+ 2096271333283106480842346115951087811586019484627451748928*n^13-\ 14238466288353318796862491978350725160314091701199097740800*n^12+ 83660937556017653818400559487175616525477010950130034535936*n^11-\ 422060456885193318079860107513800167267226346728973992621824*n^10+ 1811305982107836872083335652207206259220759806519534494782976*n^9-\ 6536549701611463976015402354080186149971302302026619040579584*n^8+ 19547513255384440727350081746776918992026161574909903053045760*n^7-\ 47537188043419137551874392608121385592915466372616690687737856*n^6+ 91693092422635812758392930206234302316613224221218085293916160*n^5-\ 135551763991444861019590967960167276041473506604915310723072000*n^4+ 146129708787133574459504809315697194205787149527197013770240000*n^3-\ 106239009366233811483172158441972168050798611682060339773440000*n^2+ 45264259193636582567315703425156301502561713154609800806400000*n-\ 8119361470459215408937195766797170892217315240265646080000000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31 )/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), ( 706935451148026805600*(n-36)!*(2*n-77)!*(n+1)+1060403176722040208400*(n-37)!*(( n+1)*(2*n-76)!-1/1060403176722040208400*binomial(2*n,n)*(n-39)!*(n-36)!))/(n-37 )!/(n-39)!/(n-36)!/binomial(2*n,n), ((706935451148026805600*n+ 706935451148026805600)*(2*n-77)!-(n-39)!*(n-37)!*binomial(2*n,n))/(n-39)!/(n-37 )!/binomial(2*n,n)] The limits, as n goes to infinity are 34359738363 274877906163 274877899133 274877853623 549755255221 [-----------, ------------, ------------, ------------, ------------, 34359738368 274877906944 274877906944 274877906944 549755813888 2199013677989 274873536569 137431968517 2198704088137 70342283849809 -------------, ------------, ------------, -------------, --------------, 2199023255552 274877906944 137438953472 2199023255552 70368744177664 70305806780389 70229978690625 17520724586025 279260007994575 --------------, --------------, --------------, ---------------, 70368744177664 70368744177664 17592186044416 281474976710656 138708667359475 68603089959675 2157977090696895 16821077867777385 ---------------, --------------, ----------------, -----------------, 140737488355328 70368744177664 2251799813685248 18014398509481984 16192260359600835 15334781939360085 28419703833231345 102300951674310465 -----------------, -----------------, -----------------, ------------------, 18014398509481984 18014398509481984 36028797018963968 144115188075855872 22104717305004885 9006931044503055 106811922683824755 -----------------, -----------------, ------------------, 36028797018963968 18014398509481984 288230376151711744 1051036861083322515 349219365768257379 -374878050033000301 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 -1096986187219419361 -896059510215241843 -2436696738498925151 --------------------, -------------------, --------------------, 4611686018427387904 2305843009213693952 4611686018427387904 -3010578835875989165 -55980434366092531275 -498261046059417161935 --------------------, ---------------------, ----------------------, 4611686018427387904 73786976294838206464 590295810358705651712 -536391982208668607785 -562902061626719612995 -1158499887869035465749 ----------------------, ----------------------, -----------------------, 590295810358705651712 590295810358705651712 1180591620717411303424 -4700274750021269376021 -----------------------] 4722366482869645213696 and in Maple notation [34359738363/34359738368, 274877906163/274877906944, 274877899133/274877906944, 274877853623/274877906944, 549755255221/549755813888, 2199013677989/ 2199023255552, 274873536569/274877906944, 137431968517/137438953472, 2198704088137/2199023255552, 70342283849809/70368744177664, 70305806780389/ 70368744177664, 70229978690625/70368744177664, 17520724586025/17592186044416, 279260007994575/281474976710656, 138708667359475/140737488355328, 68603089959675/70368744177664, 2157977090696895/2251799813685248, 16821077867777385/18014398509481984, 16192260359600835/18014398509481984, 15334781939360085/18014398509481984, 28419703833231345/36028797018963968, 102300951674310465/144115188075855872, 22104717305004885/36028797018963968, 9006931044503055/18014398509481984, 106811922683824755/288230376151711744, 1051036861083322515/4611686018427387904, 349219365768257379/4611686018427387904 , -374878050033000301/4611686018427387904, -1096986187219419361/ 4611686018427387904, -896059510215241843/2305843009213693952, -\ 2436696738498925151/4611686018427387904, -3010578835875989165/ 4611686018427387904, -55980434366092531275/73786976294838206464, -\ 498261046059417161935/590295810358705651712, -536391982208668607785/ 590295810358705651712, -562902061626719612995/590295810358705651712, -\ 1158499887869035465749/1180591620717411303424, -4700274750021269376021/ 4722366482869645213696] and in floating point [.9999999999, .9999999972, .9999999716, .9999998060, .9999989838, .9999956446, .9999841007, .9999491778, .9998548595, .9996239761, .9991056058, .9980280238, .\ 9959378864, .9921308503, .9855843598, .9749085444, .9583343411, .9337573974, .8\ 988510136, .8512513993, .7888052387, .7098554499, .6135291526, .4999851113, .37\ 05782996, .2279072896, .7572487901e-1, -.8128871925e-1, -.2378709615, -.3886038\ 671, -.5283743795, -.6528152229, -.7586763570, -.8440870447, -.9086833631, -.95\ 35931846, -.9812875744, -.9953218936] The cut off is at j=, 28 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 40], vs. those in the, 2, -th row from j=1 to j=, 39, are as follws 20 19 18 [3 (183251937949 n - 36650387584470 n + 3420855549662875 n 17 16 - 197939580392811750 n + 7954620471772265874 n 15 14 - 235723891918182838620 n + 5337494292308211347350 n 13 12 - 94421406904525148514300 n + 1323111051542594197149409 n 11 10 - 14802230568917646860761470 n + 132630720563062114772902575 n 9 8 - 950917263196810854258321150 n + 5428202562696750027084679264 n 7 6 - 24436777780391121636770674320 n + 85423482270489876164197994800 n 5 4 - 226027951064914226606608600800 n + 430230891296607092610796197504 n 3 2 - 501948847982213908036895301120 n + 9781367938204721570543462400 n + 1340271810354738649924343808000 n - 2235780645242380542738432000000)/( 524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 20 19 (2 n - 35) (2 n - 29)), 3 (183251937689 n - 36650387436530 n 18 17 16 + 3420855509871175 n - 197939573668162650 n + 7954619671312528914 n 15 14 - 235723820616324482580 n + 5337489363859030645550 n 13 12 - 94421136426106169416100 n + 1323099092893946741592349 n 11 10 - 14801800946952364869110730 n + 132618130790539986753381675 n 9 8 - 950616444632959716225242850 n + 5422369501373929026869395304 n 7 6 - 24345876963262819004437125680 n + 84303430047750719133936458800 n 5 4 - 215396570448114500653997018400 n + 355742460751272507387443775744 n 3 2 - 145090541760539385471208404480 n - 976600110120793372951227187200 n + 2090581244071870003995340800000 n - 55894516131059513568460800000)/( 524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 21 20 (2 n - 35) (2 n - 29)), 111 (19811020017 n - 4368329807415 n 19 18 17 + 451047358361985 n - 28980221204243365 n + 1298634933733316172 n 16 15 - 43112738332582225230 n + 1099435642780517346450 n 14 13 - 22036436173147187965530 n + 352282804571666877420057 n 12 11 - 4532016098096361046276195 n + 47127833220878837295142365 n 10 9 - 396363124872813944656482945 n + 2686833208834446888636815802 n 8 7 - 14553438382123407261936410760 n + 61890529651428753204619008480 n 6 5 - 198929061369828855290393420560 n + 437409844594856675371025777952 n 4 3 - 428421970846277851181668070400 n - 822392826662619450399696929280 n 2 + 3180218552688508184839116902400 n - 2344450685122476332316794880000 n + 123874333047212976016588800000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 37 ( 21 20 19 29716527299 n - 6552493214274 n + 676570650148225 n 18 17 - 43470269006086230 n + 1947945254835719454 n 16 15 - 64668501456190840404 n + 1649113760985539269010 n 14 13 - 33052599961895224117260 n + 528339105900503822552719 n 12 11 - 6795180615149627019877194 n + 70614935622173294577379245 n 10 9 - 592871217127720067609007390 n + 4001069519713143612406205504 n 8 7 - 21428540200582677470032490304 n + 88571053674107197956222008080 n 6 5 - 264721296468487219289643815520 n + 470620800613050044155004145024 n 4 3 - 14672102767003691447743597824 n - 2132346977371153962583282114560 n 2 + 4108291386255722295170131046400 n - 2428130326031330471729418240000 n + 185811499570819464024883200000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 37 ( 22 21 20 237732107323 n - 57531127210601 n + 6539578130848301 n 19 18 - 464129714026664455 n + 23060152095292333098 n 17 16 - 852369325024464182166 n + 24314225579430941277706 n 15 14 - 547981446213741481352510 n + 9908125925189522310099743 n 13 12 - 145113001133689229774954821 n + 1730329709445947055645691401 n 11 10 - 16814308436865143123426421915 n + 132657348712105474394391018028 n 9 8 - 840874229479957761477658554956 n + 4187824024191058203703107271376 n 7 - 15632824469648977078215961322320 n 6 + 38997185004449872994547579110208 n 5 - 40370135558574139236719581467456 n 4 - 102310010456292356835240589238784 n 3 + 445885841237385050668543001011200 n 2 - 627768698596982082281292847718400 n + 330437079328803266670582251520000 n - 31959577926180947812279910400000)/ (2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 22 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 5 (1759214173265 n 21 20 19 - 425728521426241 n + 48392422733663701 n - 3434488666600201595 n 18 17 + 170637329638429121160 n - 6306898814150072988234 n 16 15 + 179885550325580746982522 n - 4053105363201487898095846 n 14 13 + 73243231101109635206274865 n - 1071410916443444704867897157 n 12 11 + 12742938411302590128613401153 n - 123175921345541211421088221407 n 10 9 + 961474422218492169401901177350 n - 5967081207061569248567099053120 n 8 + 28518277760954550911326398730624 n 7 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n 9 + 26157556996887519203381023408832 n 8 - 111680800411327330857219761379472 n 7 + 320981698078931127061662562789296 n 6 - 404890900772356138956956328656160 n 5 - 1051075713663993459648210118899840 n 4 + 6340325536202299330168310521996800 n 3 - 13820636632267165456419050578176000 n 2 + 14992987144151572567149211637760000 n - 7234147644894966063806086348800000 n + 925438212992891793172974796800000 )/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 23 ( 23 22 21 95606767571 n - 25287223019683 n + 3150268771901581 n 20 19 - 245779431084353705 n + 13467855998252531106 n 18 17 - 550972282953537544638 n + 17460804827869177942706 n 16 15 - 438903875802997609441690 n + 8885215202748529263546631 n 14 13 - 146196874150976329426146863 n + 1962991341473765828156343321 n 12 11 - 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(2 n - 79)! (n + 1) + 4082552230379854802340 (n - 38)! ((n + 1) (2 n - 78)! - 1/4082552230379854802340 binomial(2 n, n) (n - 40)! (n - 37)!))/( (n - 38)! (n - 40)! (n - 37)! binomial(2 n, n)), ( (2721701486919903201560 n + 2721701486919903201560) (2 n - 79)! - (n - 40)! (n - 38)! binomial(2 n, n))/((n - 40)! (n - 38)! binomial(2 n, n))] and in Maple notation [3/524288*(183251937949*n^20-36650387584470*n^19+3420855549662875*n^18-\ 197939580392811750*n^17+7954620471772265874*n^16-235723891918182838620*n^15+ 5337494292308211347350*n^14-94421406904525148514300*n^13+ 1323111051542594197149409*n^12-14802230568917646860761470*n^11+ 132630720563062114772902575*n^10-950917263196810854258321150*n^9+ 5428202562696750027084679264*n^8-24436777780391121636770674320*n^7+ 85423482270489876164197994800*n^6-226027951064914226606608600800*n^5+ 430230891296607092610796197504*n^4-501948847982213908036895301120*n^3+ 9781367938204721570543462400*n^2+1340271810354738649924343808000*n-\ 2235780645242380542738432000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 3/524288*(183251937689*n^20 -36650387436530*n^19+3420855509871175*n^18-197939573668162650*n^17+ 7954619671312528914*n^16-235723820616324482580*n^15+5337489363859030645550*n^14 -94421136426106169416100*n^13+1323099092893946741592349*n^12-\ 14801800946952364869110730*n^11+132618130790539986753381675*n^10-\ 950616444632959716225242850*n^9+5422369501373929026869395304*n^8-\ 24345876963262819004437125680*n^7+84303430047750719133936458800*n^6-\ 215396570448114500653997018400*n^5+355742460751272507387443775744*n^4-\ 145090541760539385471208404480*n^3-976600110120793372951227187200*n^2+ 2090581244071870003995340800000*n-55894516131059513568460800000)/(2*n-5)/(-1+2* n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13 )/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29), 111/1048576*(19811020017*n^21-4368329807415*n^20+451047358361985*n^19-\ 28980221204243365*n^18+1298634933733316172*n^17-43112738332582225230*n^16+ 1099435642780517346450*n^15-22036436173147187965530*n^14+ 352282804571666877420057*n^13-4532016098096361046276195*n^12+ 47127833220878837295142365*n^11-396363124872813944656482945*n^10+ 2686833208834446888636815802*n^9-14553438382123407261936410760*n^8+ 61890529651428753204619008480*n^7-198929061369828855290393420560*n^6+ 437409844594856675371025777952*n^5-428421970846277851181668070400*n^4-\ 822392826662619450399696929280*n^3+3180218552688508184839116902400*n^2-\ 2344450685122476332316794880000*n+123874333047212976016588800000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29), 37/524288*(29716527299*n^21-6552493214274*n^20+676570650148225* n^19-43470269006086230*n^18+1947945254835719454*n^17-64668501456190840404*n^16+ 1649113760985539269010*n^15-33052599961895224117260*n^14+ 528339105900503822552719*n^13-6795180615149627019877194*n^12+ 70614935622173294577379245*n^11-592871217127720067609007390*n^10+ 4001069519713143612406205504*n^9-21428540200582677470032490304*n^8+ 88571053674107197956222008080*n^7-264721296468487219289643815520*n^6+ 470620800613050044155004145024*n^5-14672102767003691447743597824*n^4-\ 2132346977371153962583282114560*n^3+4108291386255722295170131046400*n^2-\ 2428130326031330471729418240000*n+185811499570819464024883200000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29), 37/2097152*(237732107323*n^22-57531127210601*n^21+ 6539578130848301*n^20-464129714026664455*n^19+23060152095292333098*n^18-\ 852369325024464182166*n^17+24314225579430941277706*n^16-\ 547981446213741481352510*n^15+9908125925189522310099743*n^14-\ 145113001133689229774954821*n^13+1730329709445947055645691401*n^12-\ 16814308436865143123426421915*n^11+132657348712105474394391018028*n^10-\ 840874229479957761477658554956*n^9+4187824024191058203703107271376*n^8-\ 15632824469648977078215961322320*n^7+38997185004449872994547579110208*n^6-\ 40370135558574139236719581467456*n^5-102310010456292356835240589238784*n^4+ 445885841237385050668543001011200*n^3-627768698596982082281292847718400*n^2+ 330437079328803266670582251520000*n-31959577926180947812279910400000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29), 5/2097152*(1759214173265*n^22-425728521426241*n^21+ 48392422733663701*n^20-3434488666600201595*n^19+170637329638429121160*n^18-\ 6306898814150072988234*n^17+179885550325580746982522*n^16-\ 4053105363201487898095846*n^15+73243231101109635206274865*n^14-\ 1071410916443444704867897157*n^13+12742938411302590128613401153*n^12-\ 123175921345541211421088221407*n^11+961474422218492169401901177350*n^10-\ 5967081207061569248567099053120*n^9+28518277760954550911326398730624*n^8-\ 98031907003642897972880309669168*n^7+200814788120390291382904101865440*n^6-\ 24473977047529401773235683461248*n^5-1186855814670571826804406068928000*n^4+ 3379871445249182262782829188438016*n^3-4125470379856430816274556749742080*n^2+ 2072745506303505226610186305536000*n-236500876653739013810871336960000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/ (2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23 )/(2*n-39)/(2*n-35)/(2*n-29), 115/4194304*(305948183911*n^23-80922495003194*n^ 22+10081682727963638*n^21-786619043225070748*n^20+43110383541789755961*n^19-\ 1764153425449507983198*n^18+55937585729831534092276*n^17-\ 1407484403254939117475576*n^16+28545741913190409330961241*n^15-\ 471252976235105711242500982*n^14+6364784023248919305707855094*n^13-\ 70364142556928214068395871004*n^12+633690220571194703063420545591*n^11-\ 4592996836372705446534433336466*n^10+26157556996887519203381023408832*n^9-\ 111680800411327330857219761379472*n^8+320981698078931127061662562789296*n^7-\ 404890900772356138956956328656160*n^6-1051075713663993459648210118899840*n^5+ 6340325536202299330168310521996800*n^4-13820636632267165456419050578176000*n^3+ 14992987144151572567149211637760000*n^2-7234147644894966063806086348800000*n+ 925438212992891793172974796800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45) , 23/262144*(95606767571*n^23-25287223019683*n^22+3150268771901581*n^21-\ 245779431084353705*n^20+13467855998252531106*n^19-550972282953537544638*n^18+ 17460804827869177942706*n^17-438903875802997609441690*n^16+ 8885215202748529263546631*n^15-146196874150976329426146863*n^14+ 1962991341473765828156343321*n^13-21482541373868998242909159525*n^12+ 190191079192229256371172347636*n^11-1340016275183346507628480364128*n^10+ 7280817986912591335772460791416*n^9-28613342132658053086895700219560*n^8+ 68367331898830674940423992085056*n^7-16079188276356744294968236250688*n^6-\ 562976936184193789010665636209024*n^5+2212678094456541166934317410084480*n^4-\ 4244736354196421390149160581248000*n^3+4337150189383182970619378562816000*n^2-\ 2067527226273347915783881574400000*n+289199441560278685366554624000000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/ (2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23 )/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 713/2097152*(49342517597*n^24-\ 14209562240388*n^23+1932278829616478*n^22-165005731147765992*n^21+ 9925978131852831587*n^20-447224713045180479948*n^19+15663903113224278618728*n^ 18-436810918446644995595232*n^17+9850978882696716415194467*n^16-\ 181396660699281603649798428*n^15+2740119368987451640230604598*n^14-\ 33951918369284042246365543752*n^13+343208070956377188441024578957*n^12-\ 2795375191170482588288069009748*n^11+17920196678640649249722691148708*n^10-\ 86457226206571949635730972606832*n^9+282936248292951936679951266202352*n^8-\ 406468538609188622194690195159488*n^7-1375814594051620859708855804296512*n^6+ 10308329456550618803832087067991808*n^5-30903118318829666887986229132584960*n^4 +52590886849551869740549837905408000*n^3-50416123292095068373285730322432000*n^ 2+23572734239254419973739214028800000*n-3507709355698864054768533504000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 713/2097152*(49335225829*n^ 24-14205865314012*n^23+1931399194475566*n^22-164875257953336568*n^21+ 9912463023683011099*n^20-446186784085280776692*n^19+15602672877039485282056*n^ 18-433974313192561402877088*n^17+9746365830914253545133019*n^16-\ 178301627901860624916026052*n^15+2666436790264082629399857286*n^14-\ 32541695259179114303548927128*n^13+321609855295835992801727370709*n^12-\ 2532902175431890315720719065772*n^11+15421049097403864885151695168996*n^10-\ 68140299764360624613477845168208*n^9+182093236137156020593994641855984*n^8-\ 3819949883510369326654263953472*n^7-2478180585446950279336321040195904*n^6+ 12158196914493066894536064632308992*n^5-32198919300450999141533642493296640*n^4 +51528000990401088330663056871936000*n^3-47914592487338460439561038501888000*n^ 2+22364523238958144577096719155200000*n-3507709355698864054768533504000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 103385/8388608*(2721019451* n^25-849983612845*n^24+125647147679030*n^23-11689858996072750*n^22+ 767915414318044685*n^21-37870885239896522275*n^20+1455146976207104003780*n^19-\ 44612378780532963459400*n^18+1108211353339331172124325*n^17-\ 22513457791789283917532275*n^16+375653284038721062557658230*n^15-\ 5146282649963309063788884550*n^14+57564697263571899782367447155*n^13-\ 519316342587191643775879809325*n^12+3691805855483327201352289692080*n^11-\ 19749653816473118038080809577700*n^10+70804593845131833466503117806720*n^9-\ 93091332946305692620347585521200*n^8-679462428558423800716705435749120*n^7+ 5372607485092087574194002885230400*n^6-20250612475855730716922456760942336*n^5+ 47545961430676170836211714111697920*n^4-70900655752998299712351030557184000*n^3 +63312071779312675697240701784064000*n^2-29199318707847136318914886041600000*n+ 4741455404944671411962983219200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/ (2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/ (2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35 )/(2*n-29)/(2*n-45), 3335/4194304*(42145350073*n^25-13159550741765*n^24+ 1943972540890090*n^23-180673336095140750*n^22+11849707600069871455*n^21-\ 582989994585588249275*n^20+22321539675018280964740*n^19-\ 680817696915338951388200*n^18+16787458286964592019847175*n^17-\ 337498911096175857506788475*n^16+5550224537322199781138560090*n^15-\ 74526466668226525753108346150*n^14+810805734005827079258428287265*n^13-\ 7032078000538311784415337353525*n^12+47091058859192203593501346959640*n^11-\ 226577806928355896237328843388100*n^10+613036772609256905682708830577760*n^9+ 794586590267523980964590576808400*n^8-18105164341362858793661301292248960*n^7+ 100994129002581664806022090993531200*n^6-339119599881698192020950381903913728*n ^5+749862001079377608948739007965424640*n^4-\ 1080188396095730202161289865212825600*n^3+948400734035938633040379552325632000* n^2-438057956758976318854471454392320000*n+73492558776642406885426239897600000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 30015/16777216 *(37408848677*n^26-12625347515677*n^25+2019717795615260*n^24-203683346889708790 *n^23+14525773388134733855*n^22-778805423124812998915*n^21+ 32573945351661890998250*n^20-1088196104104626845511040*n^19+ 29478637053852697204986635*n^18-653439412110516609362636995*n^17+ 11901244725678485207344940600*n^16-178008964491517006629165537790*n^15+ 2173999231753890257860009379945*n^14-21399610720941463504635443801005*n^13+ 165455197893331342288978626478850*n^12-949948326937835003712925778669740*n^11+ 3424352283703595479131463721983400*n^10-704167774020362556095187799762960*n^9-\ 86902238953336485218509738354528800*n^8+664911660725283131618018129163840960*n^ 7-2947502807283061541196420017030704512*n^6+ 8820882028079800357300475549182935552*n^5-\ 18151360706841710631616065850163804160*n^4+ 24938288252027011802593498372995686400*n^3-\ 21280193533504644396576881791868928000*n^2+ 9738455738441091885850412700794880000*n-1665831332270561222736328104345600000)/ (2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 1035/ 16777216*(1081957636687*n^26-364702984331309*n^25+58239608038905670*n^24-\ 5858727355930334990*n^23+416378135635737053545*n^22-22218967867420310941955*n^ 21+923397171965087942593900*n^20-30585664253963312630519240*n^19+ 819260140245344332675222225*n^18-17893806184740831739276254515*n^17+ 319665470545372077607515464950*n^16-4660970580368556593750959556390*n^15+ 54995517275036674123871021714935*n^14-515339467866382060006834590239885*n^13+ 3683370627610856840136890738087800*n^12-18052513525055023289582622449016740*n^ 11+34962133304281713380314961948719840*n^10+ 321086523997148484472276396915577680*n^9-3908181561698534692607095351843094400* n^8+23452885375580588594171245116725194560*n^7-\ 94181244398126345136926844140656511232*n^6+ 266738770261600521030089731267602729984*n^5-\ 530390324006970239330832895464587857920*n^4+ 713608445833457192973721783289187532800*n^3-\ 602967602355103587203784279293607936000*n^2+ 276562382099294904393624907848744960000*n-\ 48309108635846275459353515026022400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 5175/33554432*(861489795599*n^27-\ 312637274605008*n^26+53832100136784723*n^25-5848436060899965000*n^24+ 449649266644303492455*n^23-26005040544540687065280*n^22+ 1173701370293498939724975*n^21-42319253709677138126601240*n^20+ 1237336246287247863677900265*n^19-29597544908296697335995248400*n^18+ 581448931138219533537968502345*n^17-9371032338780587945097404258040*n^16+ 123030565791705999153333365735085*n^15-1294344019255955188017540819443040*n^14+ 10528385694029244526461129462570885*n^13-60365894529526329741085092378502440*n^ 12+159977071832928430358760126210757620*n^11+ 1081581113422688217034739024336509440*n^10-\ 17647715385454514670759328210739442320*n^9+ 130900573009753472467152958431836568960*n^8-\ 655154494802980816769215420790840687424*n^7+ 2372267794353670480230745676564604300288*n^6-\ 6269280763942449056434722757948575940608*n^5+ 11869168607460003359528864721442163957760*n^4-\ 15428947644605851829113455975482077593600*n^3+ 12757038523593639308767548658594824192000*n^2-\ 5803012166759591821823281029839585280000*n+ 1024153103079941039738294518551674880000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 5175/8388608*(213673984837*n^27-\ 77323674332379*n^26+13264710949966164*n^25-1434156712758689400*n^24+ 109579448204537859015*n^23-6287369405960320021305*n^22+280940376041748465286530 *n^21-10002716666258259460824450*n^20+287866999663974758281878195*n^19-\ 6749764257791230731413437365*n^18+129257137797007592226577856640*n^17-\ 2014316534998201233516454120500*n^16+25238520937191272691688733753505*n^15-\ 247205576276462607552672179675535*n^14+1763813700974766585905209415912010*n^13-\ 7022952372170501030778374836948050*n^12-20835815636046000657303215170688640*n^ 11+638435563109065796681517015633236280*n^10-\ 6171072154247323387446747968264970880*n^9+ 39215182838291058060328026958085402400*n^8-\ 181603776607227626497702755259183610112*n^7+ 627117269318533033864412339488552274304*n^6-\ 1605942254165992726194529374980470270464*n^5+ 2976743304047941771920232386253236480000*n^4-\ 3818791333480210865451307930107984076800*n^3+ 3138891932883966900076679933489135616000*n^2-\ 1430940556064839572246597298023628800000*n+ 256038275769985259934573629637918720000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 5175/134217728*(6751337800829*n^28-\ 2618758450031458*n^27+482107226188472493*n^26-56009441846860240830*n^25+ 4604910060841802899005*n^24-284753151837353294924010*n^23+ 13737098013835727270267385*n^22-529141775535579623101434750*n^21+ 16514085814608273767834473515*n^20-421076481203220357133456467630*n^19+ 8796646364979325144731833212455*n^18-150082679574776723161196597383050*n^17+ 2066583570305074070146333811411535*n^16-22319880222962755455669536099785470*n^ 15+175612041203847210725896950228751395*n^14-\ 750109272156479449402539441894217050*n^13-3309701214118349853989496316514688180 *n^12+100686729652162093010795903929425335160*n^11-\ 1110199667112245841403148953550699242160*n^10+ 8265268599755890329136724514042949626400*n^9-\ 45838063802324126971774452595696653722304*n^8+ 194270247150843657264279999799199815905408*n^7-\ 629416081844799108465893001552038159981568*n^6+ 1536416171259258234584812713254264628449280*n^5-\ 2746501165995659186324662718309783055974400*n^4+ 3431097852363187759293602059310436569088000*n^3-\ 2771209324103812570728400072765474897920000*n^2+ 1253338808994349162598510428779121213440000*n-\ 225313682677587028742424794081368473600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1035/134217728*( 33103372112533*n^28-12771707022592574*n^27+2335737226704812973*n^26-\ 269165781948770422170*n^25+21912142605587781140685*n^24-\ 1338781515379638157308390*n^23+63647927770804896283343865*n^22-\ 2408233520876171395831834410*n^21+73516449340841556132388860555*n^20-\ 1822936829308335216281716760850*n^19+36716350954616130060660160516935*n^18-\ 595425436451746151032163762417310*n^17+7584992459551647492849577454212095*n^16-\ 71065776307576550174975231608340370*n^15+381182571569610771470628723483874275*n ^14+1295821849456868559572923294503657090*n^13-\ 58789227232375803855608177834541428460*n^12+ 792563576889639118975794786106659675720*n^11-\ 7151130145166959488102220147118229569680*n^10+ 48371245319960931249610899145307898806240*n^9-\ 253279421407979258641306561667709158630208*n^8+ 1032924896115696612327066103008715445230464*n^7-\ 3256753364616520028430632264524880896378368*n^6+ 7796175658131170541298446376922919190210560*n^5-\ 13748257776644715831423905514410284162867200*n^4+ 17029638460109552988735628908572582608896000*n^3-\ 13704889743900156548134442256350729011200000*n^2+ 6209537265174857927024804677749964800000000*n-\ 1126568413387935143712123970406842368000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 10005/268435456*( 13305581476603*n^29-5475521684358017*n^28+1069112707699967637*n^27-\ 131670667645325270973*n^26+11468747708240636755305*n^25-\ 750661991862416453444625*n^24+38283399731214917514639105*n^23-\ 1556088404253532483181358225*n^22+51101608383857058865648725915*n^21-\ 1364674648732225950628804149855*n^20+29612675788927400732077888653735*n^19-\ 516471935630319930697655138177895*n^18+7020887228319734722141754605319955*n^17-\ 68108306743730674811748687027698715*n^16+307909107232600981623535942043874795*n ^15+3846192063619712539019108951310561765*n^14-\ 113468585084730105141724871525200638450*n^13+ 1600808247937383331494365431002702192780*n^12-\ 16045177270119315396623577898547885031400*n^11+ 123956674707724330851406481130711748347920*n^10-\ 757415526548099124306933495016245524172768*n^9+ 3682824091747067234777770570324604907431232*n^8-\ 14186552373820169312765464773912916072927872*n^7+ 42755624366589593172812759787167582812497408*n^6-\ 98735859755736440655506109642711794704066560*n^5+ 169257153866958514119709053086019940216627200*n^4-\ 205203170963481961062627062413304545714176000*n^3+ 162705823312529453336927555637451279564800000*n^2-\ 73146650908519214087724703225383380582400000*n+ 13285737840643924798260220616522072064000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 10005/ 134217728*(6379391821703*n^29-2601881580629554*n^28+502672845595183347*n^27-\ 61136696487814898706*n^26+5246331685870468386375*n^25-337319461482119142316290* n^24+16835689107410313124128495*n^23-666299175656890386662204010*n^22+ 21149627210697794524112021625*n^21-539709740632276492300180908750*n^20+ 10970415056781774155024622657225*n^19-172114917719594590161308990834550*n^18+ 1890496359850922702307574360026765*n^17-8480257468630475275983263821458870*n^16 -184881095066565639657172420566077115*n^15+ 5684213629179009102898209829680418770*n^14-\ 91585590520769938485009995470381856340*n^13+ 1068454595484973192211212399893521851320*n^12-\ 9719662798032602564012910811607378410160*n^11+ 70719325932762644219727408601036337675680*n^10-\ 414725172219762680995797347961416846966208*n^9+ 1957233356294624926169203978359453853167744*n^8-\ 7372641060421674899087682679969178665569792*n^7+ 21847225536265703952368062473940882260642816*n^6-\ 49820802256368296085662082412770373130833920*n^5+ 84647107926421053283921988867008306990694400*n^4-\ 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27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 10005/536870912*(44311583975321*n^30-18899658541255005*n^29+3811407709359139675 *n^28-482688749765053518225*n^27+42982119254083281899319*n^26-\ 2853229777859629609176645*n^25+145872058262013320457590175*n^24-\ 5837323330156022638338719925*n^23+183038668817790036724467673965*n^22-\ 4402323220710477742078060136475*n^21+75025469217033065285520325767225*n^20-\ 605737354719761253541760287572675*n^19-12225000956502782941848149432024595*n^18 +648312967303784538635014504064454825*n^17-\ 16294612246097695556970295110276307275*n^16+ 294801127599821645966928525101039628825*n^15-\ 4196009571868985738010968597471663408010*n^14+ 48544661330394825763940472615166565850900*n^13-\ 462842886272160396634776351676566526543400*n^12+ 3654916086588591522347749920971478036337200*n^11-\ 23896762232531612308650469930922923319044576*n^10+ 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916869186878531802965509200*n^24+28710016472245533975610949850*n^23-\ 641317348172489807084286074340*n^22+7371264943961104229704395352800*n^21+ 129546457895359377653270036601600*n^20-10121188325132107947222080904186300*n^19 +333719722628399828273102732920044120*n^18-\ 7833257261980149919537798814103894600*n^17+ 144798635943569410940565368124499664400*n^16-\ 2187850736283640813527169336512989749425*n^15+ 27480358505891728738429874500361709618810*n^14-\ 289170273133665741207073386372334560845300*n^13+ 2556054609057853590530643371477104707593800*n^12-\ 18963352107001836143944142058586774021395760*n^11+ 117650488422683076337876085740922293746335456*n^10-\ 606497334616246719671140052039768347346857920*n^9+ 2573718248510932957910729026532498429898848640*n^8-\ 8875493579152301124448249605136968838599352320*n^7+ 24441412952903043338721615923730293782960226304*n^6-\ 52477394090932467720574833807033154894754611200*n^5+ 84946231190224109484687875763925673656071782400*n^4-\ 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19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 13485/1073741824*(161583821705363*n^32-73251417729865936*n^31+ 15516829649735157496*n^30-2025780258901178817920*n^29+180117700133305064110932* n^28-11240464385650238281658304*n^27+471303378014243760203356344*n^26-\ 9524968366940989373872106880*n^25-336576097738749942765924397230*n^24+ 43279643082080869603300406038560*n^23-2421829226336080330487769307761960*n^22+ 96253818142419701318286179861212800*n^21-3008615126138915956525691656370221260* n^20+77007015423430225472006375448564755520*n^19-\ 1647253440212479670084027417675560934520*n^18+ 29787496822506643085188741692299812464000*n^17-\ 458364753061361603689726720839138554677005*n^16+ 6022839790979726129517086179588056835226160*n^15-\ 67659731521952968548711573577890276664380560*n^14+ 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2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 2697/1073741824*(584872073626111*n^32-248711301256658672*n^ 31+48108036492477952472*n^30-5446778234802879975040*n^29+ 371470800938303010720804*n^28-10907048441125779636955968*n^27-\ 697299538877078425401396072*n^26+118816963865740196791802920320*n^25-\ 9152498134046216077931393030310*n^24+496127474927128040488163364781920*n^23-\ 20936407947229597802470163063019720*n^22+717678051251610842947616386602096000*n ^21-20443891958714861835356731781005393020*n^20+ 490640462494282778177993348205836725440*n^19-\ 10007120820881579292921344271301201194840*n^18+ 174418041338549609602126207553551593513600*n^17-\ 2606212160015424159514034801213538402634785*n^16+ 33433815813433427568906076442080951324411920*n^15-\ 368193157520430149728000736600890706884173520*n^14+ 3475504227175010147845338469666050581838905600*n^13-\ 28037335646157097856300034998496681441425054816*n^12+ 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42120543163063431*n^32-92888776691800148224*n^31+29187988231914675518520*n^30-\ 5130614674672715584088920*n^29+618338496603211621650438564*n^28-\ 55699280315266992811793404656*n^27+3934797906009865179265481512440*n^26-\ 224754322550499989430784838565900*n^25+10599294071604390131901470944914090*n^24 -418901732398611495466085193659673360*n^23+ 14026973512451688731615810833486791000*n^22-\ 401186135255212084288126960709167143000*n^21+ 9859163914261651737557454028884240109380*n^20-\ 209070344449951667588712274210537647845520*n^19+ 3836539259947457222454304927119329430976200*n^18-\ 61022605733458346907959547409817925092545650*n^17+ 841736940158995166248440838792622401669750215*n^16-\ 10064056632887083973606994621906210770940053360*n^15+ 104137219984080439771743164409653829341694673200*n^14-\ 930103360883730618200065689180648912261330073280*n^13+ 7143252611206123794572701615298676473075773781664*n^12-\ 46932389096753405658599082561407822449124723910656*n^11+ 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1860541533018019675*n^33+680793304375990241183*n^32-149821516947556928780680*n^ 31+22768694628302710340063228*n^30-2575455939123741070561070500*n^29+ 227144110370708058270262340172*n^28-16115398935723815180762206657320*n^27+ 940322518125374295802023030530490*n^26-45861937380083109834210486944540850*n^25 +1892564244079228390346580844072006770*n^24-\ 66693238661399076749469957650140252200*n^23+ 2021137399336453891222151150556461575260*n^22-\ 52952791208639847334442407833214356502500*n^21+ 1204045009846567324846212185699380107792540*n^20-\ 23824275822582095476695080526220419468749400*n^19+ 410890088134647530842827238757750774137219755*n^18-\ 6180870247303689133662217937252771853786412875*n^17+ 81074555440069299932407076869778286206441832295*n^16-\ 926257590656114643344266059014715457938419205200*n^15+ 9197914828366382027645905412196025805306590349008*n^14-\ 79145154338091399313964556798548729967659227767200*n^13+ 587658360061535582786730345215168831136598102382752*n^12-\ 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n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -31/1073741824*(155000788671106969*n ^34-105372213529769088467*n^33+33560414988242363525821*n^32-\ 6709124149277212217999048*n^31+949672945388576907928694836*n^30-\ 101671826317646808649693748228*n^29+8580682808298110989963442663364*n^28-\ 587157412641588145053913677440232*n^27+33237946134115209656955784771785870*n^26 -1579855311398976107267600882896544130*n^25+ 63763982609683813939372375107011391990*n^24-\ 2204051304647718082290822967082359782120*n^23+ 65672862263586182828967465148114985336820*n^22-\ 1695109260480395192965168360578626610947460*n^21+ 38037350183845359467593880227588750211956980*n^20-\ 743849383088329399813046698228406004131193240*n^19+ 12695430014842686794124270452388765677388668385*n^18-\ 189200192062440156513690212390108907579342677955*n^17+ 2461202362237473019799576481083217035565270266165*n^16-\ 27911420374617854531711856391721114731916875389520*n^15+ 275351043923930368237779325133346532810957781456336*n^14-\ 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11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -31/2147483648*(970708880859300329 *n^35-649795341398863862860*n^34+207268465845630192936275*n^33-\ 42014053053695878019152750*n^32+6087612202234781676468850532*n^31-\ 672239526711278083771898349280*n^30+58891447265541675960094578169500*n^29-\ 4206008165426388962156053189113000*n^28+249718811599771662630069747428335374*n^ 27-12504680127357423508133196628335032760*n^26+ 533932717953859098237254275413495825450*n^25-\ 19603512285053411729402601668976555352500*n^24+ 622887746038714617879214420501909781554260*n^23-\ 17212385781401174024844173455445709493934400*n^22+ 415151989981275904088528376818968629556387500*n^21-\ 8762393983035850314287984836772513304562905000*n^20+ 162104273092545672942287059815895550761749082065*n^19-\ 2630609711304814063153504202177338241513828699100*n^18+ 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29916351501723504866402713639081718970178599309253017600000*n^2+ 12917037196121491781970073365868337614349816259674112000000*n-\ 2337172194511453154315568292210695971103115208294400000000)/(2*n-69)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2* n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35) /(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 1/1073741824*(-\ 20067456005368465222*n^35+13004365309279186985870*n^34-\ 4033344900724446990383050*n^33+797781327281523102580121675*n^32-\ 113123903644577796198027529576*n^31+12254643460012819112032621072160*n^30-\ 1055326881932561601080579006831400*n^29+74220596612975173607206732178645700*n^ 28-4345951371814993778618139743676454932*n^27+ 214913134363840020922975196029101127020*n^26-\ 9072853009668742045153465747982784677100*n^25+ 329695259780549996931750421819209451223250*n^24-\ 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41152928000957667141188526489245934659183652587647795200000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 1/ 17179869184*(-905770464083615882747*n^36+600188998419324984424014*n^35-\ 191192185822965377134991025*n^34+39000633673644373065253534350*n^33-\ 5725085330203712563789519062396*n^32+644366424770646056147869736454072*n^31-\ 57852245121371007675983329119999220*n^30+4256045534364083993721216051495905400* n^29-261535918986952394289194805343173261162*n^28+ 13616765209091199728366600777237905315524*n^27-\ 607179714928508528605265601222038206431630*n^26+ 23380724145801226197000842973031146813296100*n^25-\ 782473538389867621831974818770536159067333980*n^24+ 22870657019426373902601166362383120463267829560*n^23-\ 585981657645500691461993427604385748761206083300*n^22+ 13195954560027504994220040556046601472347448935000*n^21-\ 261647755488869535159709088961630655434888016033395*n^20+ 4572267650428435240740142846674474013110059859754590*n^19-\ 70433273076486944218237861708744787578650594245305225*n^18+ 955946781030594764871343311425655302338209769422138750*n^17-\ 11417250837261049281490826364832379516601604824886340688*n^16+ 119755774198632501923778181807160900988992622884851748256*n^15-\ 1100076340682521304495326402082739629259277838957277218400*n^14+ 8817324947730218060867686318365289607536142249102913710400*n^13-\ 61375067433370499621720675782793764699293019630837813743104*n^12+ 368826019071937676672999397371062037112980544283850053091328*n^11-\ 1899459182680404394158058797567540836686975578448066082023680*n^10+ 8306894552868306481284613818326724666810386224197727943769600*n^9-\ 30497496008409552217185171723232106583317770904404653881622528*n^8+ 92637900258426716119844343431179847503958483727695531647942656*n^7-\ 228483152043039801417140113048721995386547144126397411783147520*n^6+ 446318158718463902045228008849558060921195238332310369412710400*n^5-\ 667231967969857184122956172210872202242460794294668887326720000*n^4+ 726357330432367008607547195465627956472136448649001503293440000*n^3-\ 532464572846170423290928107673929347212879427604300221644800000*n^2+ 228373633364622203533741802687995083208028145792734724096000000*n-\ 41152928000957667141188526489245934659183652587647795200000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -185/ 34359738368*(21690702831242234095*n^37-15042135171707984126321*n^36+ 5023688535861698057729523*n^35-1076270585378389509235659735*n^34+ 166226810105842157225821781010*n^33-19719874279780958434220547171540*n^32+ 1869571434206787234028844663692988*n^31-145511670889488668222464214790290764*n^ 30+9478455550899414498468323831596124250*n^29-\ 524168852661937711913926581829599290334*n^28+ 24878219188202650996096949048674510793514*n^27-\ 1021933171151085895757536967747322757157826*n^26+ 36568381440258388134040912129019845100945760*n^25-\ 1145657687416972434000056183744315169618781620*n^24+ 31545544416278924410269128404839910488925827180*n^23-\ 765579148285374908054857311664638020367095580060*n^22+ 16408605479404459519194839799574141610527968640575*n^21-\ 310963444280435343256671243270797584325071015481145*n^20+ 5213407527787334721752695685764176255815290957478075*n^19-\ 77309640504218514448031020946718412183349887054390495*n^18+ 1013166546381192271099836971701499417188930650598655430*n^17-\ 11716663912683422165378383278601573671871084218530759664*n^16+ 119298120566082396244726705101548569662238089567555562752*n^15-\ 1066261419132571535020978540675716572161644641064586404000*n^14+ 8333203704199307500524511077028628864759962726367908863040*n^13-\ 56671359527161106321727477178857059194251087282303128153600*n^12+ 333343768227779726889869288444620461256352784002255735899392*n^11-\ 1683257925565829744529628416570689344094050204913428001308416*n^10+ 7229614469299953364367478219001963541183378959531706534131200*n^9-\ 26107401704104430869927792450454709207757122253859048849227776*n^8+ 78117104900192467701341244310192724888843799473387350613016576*n^7-\ 190054653859839920660158064581233903779705406377945324738248704*n^6+ 366712484043345470480161811915125405880119607925750358343024640*n^5-\ 542243529158551813811298754495013692155771885521734666027008000*n^4+ 584636265873693813179996746717880711832311066042677985280000000*n^3-\ 425057734364287241544709029044203791598940059823997148200960000*n^2+ 181089402253366303439929768693446957284003805932799236505600000*n-\ 32477445881836861635748783067188683568869260961062584320000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -37/17179869184*(58203268990813236019*n^37-40138376017664733863618*n^36+ 13335185944764277994873373*n^35-2842924996104775526716154670*n^34+ 437065567922891298448985730852*n^33-51627167616369376498991557836264*n^32+ 4874895837383624178049046032745924*n^31-377993684870424689785129376548283800*n^ 30+24535575320860622078786911257165880314*n^29-\ 1352399056520744869382246781256085281788*n^28+ 63992011442285610684526627707620371102998*n^27-\ 2621167174281039289927556832197206247394180*n^26+ 93547456317472102143753098514140471811085260*n^25-\ 2923606885166378060646248309012322682756911720*n^24+ 80319243485277607750364576704155580484273111220*n^23-\ 1945202256070310269372475634242654465325021497400*n^22+ 41611292234111867869443652218164961241262293244715*n^21-\ 787194130861543110331018550059722406873098252693330*n^20+ 13176248807228332401009174428085941561689374214863605*n^19-\ 195102416930489235852458765794699581484599710949529150*n^18+ 2553446729029489107600798337483554092644205267313800376*n^17-\ 29493201337113505045275942527491870831592291656493011872*n^16+ 299968471699752236212628044376348371619688612230577252192*n^15-\ 2678422226954090953291845254827007510035488175140474281280*n^14+ 20914488892861309697867053329579964047824824513862182766848*n^13-\ 142122486152435931816763823606030669699584422575929562817536*n^12+ 835404741321910385984568518959749081642625640072537347435776*n^11-\ 4215992745818947079341040587258964276597874821010513203904000*n^10+ 18098635147019380405943621725296199702115241522896391276374016*n^9-\ 65329691243488382162101004801529462187213679068501379806543872*n^8+ 195407896540062192032853651304487810071121538693732232458534912*n^7-\ 475285278504556672801340757256264935060512263569640103154155520*n^6+ 916875967305198942964893440733085579793048149218821715433881600*n^5-\ 1355551407185386272017234528504941438073392541295367201751040000*n^4+ 1461405218126062343467792203438838795352554588566434957230080000*n^3-\ 1062483615792229473779445220740089018311981187773418491084800000*n^2+ 452672333187173368586742884785507327006907304861996417024000000*n-\ 81193614704592154089371957667971708922173152402656460800000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -111/68719476736*(162572628258293908609*n^38-117797180526607342931523*n^37+ 41170792528759094615608003*n^36-9245450675260538223592806345*n^35+ 1499198815495012061512891230072*n^34-187042767412419876798710974297704*n^33+ 18681145729922589952537168964218364*n^32-1534449399337586269065591474573746100* n^31+105677068793781884311576260680682771854*n^30-\ 6190543467898255022354271030806994015018*n^29+ 311855045270798343653623252370235173382378*n^28-\ 13624932135725795432591664482085436278448830*n^27+ 519691169634953917166462294494046025701770860*n^26-\ 17395037619961321074506482505383248347546340420*n^25+ 512984436162182711726077447631217494119117809420*n^24-\ 13368644047071927211649822414712556547282688852900*n^23+ 308543722745652306154759993028234282629183039595865*n^22-\ 6315569110099152119343494752228598637581271609175755*n^21+ 114736516707680710715543005772883396126269362243866155*n^20-\ 1850270465229661572777314912136812637126692178894881025*n^19+ 26472568333154540318018685414201151070628830351267421236*n^18-\ 335659087394522398745206393841669519150071183922675502492*n^17+ 3765116767814975973470585239375128654939860179851833447712*n^16-\ 37271448050736567026646581558473398472823777421721975176480*n^15+ 324569035155138426526636183146193134007733018369955270752128*n^14-\ 2476394768720175969807967200198473275528418515477981746324096*n^13+ 16471743277521629605751953485700158348368909943439606338670336*n^12-\ 94926434116389803135687422485463573060045849513284502291027200*n^11+ 470401984341572628660327588393514963912632331015410574465483776*n^10-\ 1985738158187148176023166729260173680561041877309598373193812992*n^9+ 7058122573861799232644140486395205248457176707739519521218117632*n^8-\ 20815741525760428933818503397556939542232146031436241100176261120*n^7+ 49983064963760954990812012249532125488971959474597269564103065600*n^6-\ 95308260351498342517204952101926656297199478786155351493017600000*n^5+ 139448262817371195090125306455471754811661831288222728655994880000*n^4-\ 148961837560095988735233452217765298538818976164492214324428800000*n^3+ 107445273622398269300079148469469503352280155088985819971584000000*n^2-\ 45480527655758765453940813224488023459843423450051064627200000000*n+ 8119361470459215408937195766797170892217315240265646080000000000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69), (2721701486919903201560*(n-37)!*(2*n-79)!*(n+1)+ 4082552230379854802340*(n-38)!*((n+1)*(2*n-78)!-1/4082552230379854802340* binomial(2*n,n)*(n-40)!*(n-37)!))/(n-38)!/(n-40)!/(n-37)!/binomial(2*n,n), (( 2721701486919903201560*n+2721701486919903201560)*(2*n-79)!-(n-40)!*(n-38)!* binomial(2*n,n))/(n-40)!/(n-38)!/binomial(2*n,n)] The limits, as n goes to infinity are 549755813847 549755813067 2199023221887 1099511510063 8796087970951 [------------, ------------, -------------, -------------, -------------, 549755813888 549755813888 2199023255552 1099511627776 8796093022208 8796070866325 35184041149765 2198955654133 35181215046661 -------------, --------------, -------------, --------------, 8796093022208 35184372088832 2199023255552 35184372088832 35176016016077 281312595941635 140554742493455 1122826593040155 --------------, ---------------, ---------------, ----------------, 35184372088832 281474976710656 140737488355328 1125899906842624 1119826153971045 4458209692224825 1105762871531475 34938173119290075 ----------------, ----------------, ----------------, -----------------, 1125899906842624 4503599627370496 1125899906842624 36028797018963968 34261990136471655 133122342673413015 63825815176138515 -----------------, ------------------, -----------------, 36028797018963968 144115188075855872 72057594037927936 481263609868469655 443337397673086605 1584063313462661925 ------------------, ------------------, -------------------, 576460752303423488 576460752303423488 2305843009213693952 169474199051241285 2178957835696820055 1577399982569621367 ------------------, -------------------, -------------------, 288230376151711744 4611686018427387904 4611686018427387904 459723040355873901 222502318003037285 -1968882489303173593 -------------------, -------------------, --------------------, 2305843009213693952 4611686018427387904 18446744073709551616 -4805024448804316039 -30091975306638310199 -10033728002684232611 --------------------, ---------------------, ---------------------, 18446744073709551616 73786976294838206464 18446744073709551616 -784768293369991294583 -905770464083615882747 -4012780023779813307575 ----------------------, ----------------------, -----------------------, 1180591620717411303424 1180591620717411303424 4722366482869645213696 -2153520952660089732703 -18045561736670623855599 -18549253245613592954589 -----------------------, ------------------------, ------------------------, 2361183241434822606848 18889465931478580854784 18889465931478580854784 -75217651040049335518941 ------------------------] 75557863725914323419136 and in Maple notation [549755813847/549755813888, 549755813067/549755813888, 2199023221887/ 2199023255552, 1099511510063/1099511627776, 8796087970951/8796093022208, 8796070866325/8796093022208, 35184041149765/35184372088832, 2198955654133/ 2199023255552, 35181215046661/35184372088832, 35176016016077/35184372088832, 281312595941635/281474976710656, 140554742493455/140737488355328, 1122826593040155/1125899906842624, 1119826153971045/1125899906842624, 4458209692224825/4503599627370496, 1105762871531475/1125899906842624, 34938173119290075/36028797018963968, 34261990136471655/36028797018963968, 133122342673413015/144115188075855872, 63825815176138515/72057594037927936, 481263609868469655/576460752303423488, 443337397673086605/576460752303423488, 1584063313462661925/2305843009213693952, 169474199051241285/288230376151711744, 2178957835696820055/4611686018427387904, 1577399982569621367/ 4611686018427387904, 459723040355873901/2305843009213693952, 222502318003037285 /4611686018427387904, -1968882489303173593/18446744073709551616, -\ 4805024448804316039/18446744073709551616, -30091975306638310199/ 73786976294838206464, -10033728002684232611/18446744073709551616, -\ 784768293369991294583/1180591620717411303424, -905770464083615882747/ 1180591620717411303424, -4012780023779813307575/4722366482869645213696, -\ 2153520952660089732703/2361183241434822606848, -18045561736670623855599/ 18889465931478580854784, -18549253245613592954589/18889465931478580854784, -\ 75217651040049335518941/75557863725914323419136] and in floating point [.9999999999, .9999999985, .9999999847, .9999998929, .9999994257, .9999974812, .9999905941, .9999692584, .9999102715, .9997625061, .9994231076, .9987015126, .\ 9972703490, .9946054238, .9899214098, .9821147198, .9697291059, .9509612580, .9\ 237218120, .8857611197, .8348592822, .7690677915, .6869779543, .5879817433, .47\ 24861638, .3420440976, .1993730876, .4824749931e-1, -.1067333336, -.2604808973, -.4078223125, -.5439294849, -.6647246004, -.7672174257, -.8497392226, -.9120515\ 998, -.9553240839, -.9819892904, -.9954973226] The cut off is at j=, 29 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 41], vs. those in the, 2, -th row from j=1 to j=, 40, are as follws 20 19 18 [41 (13408678387 n - 2681735677190 n + 250306503708525 n 17 16 - 14483383943797950 n + 582045401875381062 n 15 14 - 17248089786324557340 n + 390548372098441694650 n 13 12 - 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(2 n - 81)! (n + 1) + 15732762253658952652920 (n - 39)! ((n + 1) (2 n - 80)! - 1/15732762253658952652920 binomial(2 n, n) (n - 41)! (n - 38)!))/( (n - 41)! (n - 39)! (n - 38)! binomial(2 n, n)), ( (10488508169105968435280 n + 10488508169105968435280) (2 n - 81)! - (n - 41)! (n - 39)! binomial(2 n, n))/((n - 41)! (n - 39)! binomial(2 n, n))] and in Maple notation [41/524288*(13408678387*n^20-2681735677190*n^19+250306503708525*n^18-\ 14483383943797950*n^17+582045401875381062*n^16-17248089786324557340*n^15+ 390548372098441694650*n^14-6908883939502626212300*n^13+96813026207898351115167* n^12-1083090847673083507619790*n^11+9704710491054305073476025*n^10-\ 69579876328779501790070550*n^9+397196497197928840631365432*n^8-\ 1788227456383455997122682640*n^7+6252600114018469657474640400*n^6-\ 16558576870822616442045679200*n^5+31620062272257056654867549952*n^4-\ 37397492265530988622897943040*n^3+2566331759191336277247590400*n^2+ 96660959043372750455869440000*n-167683548393178540705382400000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-21)/(2*n-27)/(2*n-13) /(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29), 123/524288*(8939118918*n^21-1971075718689*n^20+203521388888580*n^19-\ 13076444633309575*n^18+585969821060474718*n^17-19453344367276122114*n^16+ 496089275644793799480*n^15-9943405054850038399950*n^14+158962847266021802068338 *n^13-2045148642061508003296549*n^12+21271285126308513686824980*n^11-\ 178997001713217317740785675*n^10+1215264543756634815102171498*n^9-\ 6612190348780122701045195304*n^8+28485557751512589621464795280*n^7-\ 95069881791970931993951230000*n^6+234566652205379856751515066528*n^5-\ 371523909281583563326665817344*n^4+121509596829256735437533191680*n^3+ 1060279452911686266949268275200*n^2-2146475760202929517563801600000*n+ 55894516131059513568460800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9 )/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-\ 15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 3/524288*( 366503872908*n^21-80814102861009*n^20+8344376498194170*n^19-536134152015352835* n^18+24024753071256737898*n^17-797586235632395981394*n^16+ 20339597157884064403860*n^15-407676023291398507982070*n^14+ 6517312834266638774942208*n^13-83845002637899793195913189*n^12+ 871937981760811014185695650*n^11-7334309647636778296282811655*n^10+ 49734171228022695295913085258*n^9-269620636350539007601726352904*n^8+ 1149031503217734406045879795440*n^7-3712217379923882273657446726640*n^6+ 8268527605124120781855680461728*n^5-8523135077402338059730751541504*n^4-\ 14359351396728904547214741189120*n^3+59463785206891171158583497523200*n^2-\ 44407783586093689636716441600000*n+2291675161373440056306892800000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29), 111/1048576*(39622038305*n^22-9588532451993*n^21+ 1089932611961347*n^20-77355428429891535*n^19+3843413135332454530*n^18-\ 142066199643156026238*n^17+4052677330483163361782*n^16-91346219622330787905230* n^15+1652022380933763264477205*n^14-24208069019560011038506053*n^13+ 289006108947508920022083847*n^12-2816067260913098651779395075*n^11+ 22353143536104292316128590680*n^10-143590323771841783355536934708*n^9+ 735923019893957804163005977072*n^8-2921505534262390940723685269520*n^7+ 8392952996660783680726337466880*n^6-14209144418021455293308680771008*n^5-\ 916005184679332125711012354048*n^4+65811089697614391085975335951360*n^3-\ 122695398373224153767096527257600*n^2+71376345334961753561124249600000*n-\ 5326596321030157968713318400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 37/ 2097152*(237732166921*n^22-57531158916737*n^21+6539586065249237*n^20-\ 464130954744384235*n^19+23060287911785959296*n^18-852380373648696968922*n^17+ 24314917547413662853882*n^16-548015546192490593964230*n^15+ 9909465659428073353452761*n^14-145155253606613543284828837*n^13+ 1731401163080048415979571457*n^12-16836084762427178797507733775*n^11+ 133009090145051264309314229806*n^10-845323499359576545212285284112*n^9+ 4230883606387212903983187324992*n^8-15940333077170920151489790662320*n^7+ 40526181187595147711912526558816*n^6-45148271901534404542965038251392*n^5-\ 94822872175908935472459653549568*n^4+444486603953543014295100300994560*n^3-\ 636828245456933939295352261017600*n^2+336926432651619276195880550400000*n-\ 31959577926180947812279910400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2* n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 4255/ 2097152*(4134467804*n^23-1093564856821*n^22+136243403982037*n^21-\ 10630689213749297*n^20+582650432871716979*n^19-23846307580625752992*n^18+ 756319536456090364274*n^17-19040403249878906394994*n^16+ 386567802582796378303474*n^15-6394555524393853828513733*n^14+ 86695084917374271071451081*n^13-965247927174971526323196141*n^12+ 8805537778265048892199089599*n^11-65299006177994839663997513974*n^10+ 387061315543714920830720603968*n^9-1773925229421332532364606326848*n^8+ 5860028578188475855704369943344*n^7-11482894297616601479255683296480*n^6+ 474027506331165486740126528640*n^5+68382307585353027328808749217280*n^4-\ 188676027558842346984200481331200*n^3+226185068359802201825161211904000*n^2-\ 112082099382273616481146060800000*n+12505921797201240448283443200000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 115/2097152*(152974682969*n^23-\ 40461553646590*n^22+5040915847049170*n^21-393320826719707964*n^20+ 21556390908037710879*n^19-882171065791847864802*n^18+27974497786851942340124*n^ 17-704013095998463630422648*n^16+14283106747347903609850519*n^15-\ 235936403063316143494241474*n^14+3189925963575883459440716658*n^13-\ 35328746256045081839931936732*n^12+319115730371789134700793389249*n^11-\ 2324107976637043474346376890686*n^10+13337625565146631219422493032832*n^9-\ 57661884266554752804771230492336*n^8+169744554812065296338227777362384*n^7-\ 234445519521641323919423915732448*n^6-457592414067499489258491467558784*n^5+ 3103141382408479695023080326919680*n^4-6931819673567234305055691399936000*n^3+ 7597364267508424801677327200256000*n^2-3673032738781910940128441548800000*n+ 462719106496445896586487398400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45) , 115/4194304*(611890795409*n^24-176221406236860*n^23+23965884786189758*n^22-\ 2046937577014673376*n^21+123175546187461216295*n^20-5553073873670954027628*n^19 +194694781304088119955032*n^18-5438953093478871123981648*n^17+ 123026696989576994226867551*n^16-2276697849627982214604061716*n^15+ 34670086930897565580278097686*n^14-435151581710318351903537516640*n^13+ 4488088469011609533549110242457*n^12-37702906273146918942518146327044*n^11+ 253510428629516160591328159670900*n^10-1320886558400493846411739741871472*n^9+ 4996534138650856032106843078765808*n^8-11493493058462340112434003096322752*n^7+ 2106490383163632589802781791730624*n^6+92822663173353242443597397731643136*n^5-\ 355133894727191148438978336003947520*n^4+669076656482325445361481388018176000*n ^3-673910950249182853483979688566784000*n^2+ 317058021668992706968776899788800000*n-43495596010665914279129815449600000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 23/2097152*(1529670866363*n ^24-440523209626020*n^23+59907015709253522*n^22-5116122800672100360*n^21+ 307803224212472820293*n^20-13871484760524746387340*n^19+ 486024542367193701107192*n^18-13561686563564600400947040*n^17+ 306138152104558976490741893*n^16-5645716597307828022507907260*n^15+ 85477449850338820267069034042*n^14-1062724988074662958393832713320*n^13+ 10795905566749661990571061236683*n^12-88557959845171835967478991335380*n^11+ 573629678297063174455490884292252*n^10-2812860040533895402893249878842800*n^9+ 9501520638795446307262126262781968*n^8-15517271790824516583201412429656000*n^7-\ 34664820721834079879526793956067008*n^6+306157953762608034554426861629643520*n^ 5-948610011992462705739680864863027200*n^4+ 1638016933803614841926224827919872000*n^3-1581020667152818157530790163671040000 *n^2+739506924028789306815252731904000000*n-\ 108738990026664785697824538624000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13 )/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), 3565/2097152*(19735833242*n^25-6166728937585*n^24+911994481365320 *n^23-84910683596962390*n^22+5584242668111207870*n^21-275890785126877307695*n^ 20+10630319877414158198420*n^19-327291021472179980481640*n^18+ 8181923815296419990534150*n^17-167774865198203219601489295*n^16+ 2837460909612326225665512320*n^15-39625731834867958520970694990*n^14+ 455411677738912401880395687410*n^13-4268974907243281392556945873345*n^12+ 32090117805267878875979237014820*n^11-187539451048171269119888646667540*n^10+ 799741709028429891873754400605040*n^9-2075116043427436727582193378076720*n^8+ 79598300820289881744809622333120*n^7+26599346190659539061464219495018560*n^6-\ 128182861913369114091458195467667712*n^5+333936840868886695434874754467184640*n ^4-526770618731208391774508363303424000*n^3+ 483581022571398256989539603724288000*n^2-222885746665123703723717983027200000*n +34375551685848867736731628339200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29)/(2*n-45), 3565/2097152*(19731475966*n^25-6164419581305*n^24+ 911419146642280*n^23-84821163840842750*n^22+5574494004299214610*n^21-\ 275101674000967912175*n^20+10581103663088213076580*n^19-\ 324871647573646202209400*n^18+8086824928633876029960850*n^17-\ 164759843764156923955335575*n^16+2760027955842407503220541280*n^15-\ 38013598705749264030662254550*n^14+428269882104492749532541291630*n^13-\ 3901513952113172900731455405425*n^12+28124754048702817920804717047380*n^11-\ 153855235285244151713673195727700*n^10+578318412107610080967866926655920*n^9-\ 975064904839836812746053649617200*n^8-3910145732831740985585431026016320*n^7+ 36574503744720105870636209832590400*n^6-143431743235743340075795429526558976*n^ 5+342977786373098987698363807412551680*n^4-516547918219398378733833534864691200 *n^3+463430815082669170060304412063744000*n^2-\ 213644405940906338738667095408640000*n+34375551685848867736731628339200000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 103385/8388608*( 5440553467*n^26-1837922228441*n^25+294432083698090*n^24-29753529889685030*n^23+ 2128136821952719045*n^22-114578101778420214935*n^21+4820358684309173022160*n^20 -162336310155008457061880*n^19+4446083264821255078632325*n^18-\ 100016895245050060454037575*n^17+1857695584542479171130526210*n^16-\ 28517546979297413008292405630*n^15+360578409270590592604072653835*n^14-\ 3722182193805839716457747032505*n^13+30843597734692917921756892220260*n^12-\ 198695047522705913454721563611780*n^11+927326038644239730092337355597840*n^10-\ 2482892157728975166079547231910320*n^9-2547174754728704249899647561618240*n^8+ 65103508320211403200363678857933120*n^7-359815888090056905455076605156672512*n^ 6+1191282428920558004446626032631883776*n^5-\ 2597821333544774071436386226787348480*n^4+3694250438161230175619283578817331200 *n^3-3204666162115447847943538398633984000*n^2+ 1462496670088367978087405972029440000*n-241814225652178242010112144179200000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2* n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/( 2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 3335/ 16777216*(336986033819*n^26-113779487129737*n^25+18212571578758310*n^24-\ 1838232773636044630*n^23+131246601176645230445*n^22-7047972891388280066215*n^21 +295411151758700024290100*n^20-9896499050323115240175880*n^19+ 269072484266080701848961605*n^18-5992557781872974273850097495*n^17+ 109803219965992154787009681350*n^16-1655062485858699204218855153230*n^15+ 20416854758921108772796184045075*n^14-203720400224505406998293893720105*n^13+ 1606924353778557504350119529264000*n^12-9553175945935218233702502143972980*n^11 +37619764683418483065747948439756160*n^10-42431786899261368547844324022363760*n ^9-635419205729027585856717171903744000*n^8+ 5543425972632296874771648107005792320*n^7-\ 25607625925466836100450718141829519104*n^6+ 78232397693248918348314952914704397312*n^5-\ 162939390868800694845670033175637749760*n^4+ 225459430651814162653580230916798054400*n^3-\ 193018116854508501226592658116395008000*n^2+ 88264693890697126841340558097121280000*n-14992481990435051004626952939110400000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 30015/16777216*(74746990648*n^27-27194687443053*n^26+4698625131610839*n^25-\ 512809377686785530*n^24+39665836187635133970*n^23-2312246057477637913515*n^22+ 105432727358142090554505*n^21-3851637490958906318901660*n^20+ 114506361750491110431757440*n^19-2797446234776872611942787635*n^18+ 56448552070728888764221364265*n^17-941631884947687674394303290810*n^16+ 12938112652051979840863455057930*n^15-145047433245090184349186420200725*n^14+ 1301844886131571657335686182478535*n^13-8996290828690107919741787909212560*n^12 +43351296514657355377648349371845900*n^11-91393060624106055241099328461840320*n ^10-611798421799933841476639538549012080*n^9+ 7831625613130852143343279346428611840*n^8-\ 46738674007589753024889576253120049088*n^7+ 185377152592257399327440507455137109248*n^6-\ 518040200858369793771813233950891736064*n^5+ 1016836348905161938164710696033787678720*n^4-\ 1351532117490079523762904353507679436800*n^3+ 1128882598740636993472369765931237376000*n^2-\ 511903957808715863232374670688911360000*n+ 88289060610339744805025389530316800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1035/16777216*(2160435493343*n^27-\ 784895000650506*n^26+135344076723553581*n^25-14731535835165544200*n^24+ 1135316451731713811235*n^23-65857730587896253826430*n^22+ 2983584269102612122054965*n^21-108078370257316895596351620*n^20+ 3178161689597866674738426105*n^19-76560661271323130257520577270*n^18+ 1517257093406101545249363001455*n^17-24725325506729645763884469301920*n^16+ 329383671131438334216416986147545*n^15-3537655887727186572267693757876050*n^14+ 29751739870445604399202380189509535*n^13-182795234961723024048513938696104020*n ^12+640295692808493113360629729409509740*n^11+ 1249744829165262340818583510915526160*n^10-\ 37168527100778761198834408860410121200*n^9+ 301607861477843784317798358257141321280*n^8-\ 1567494657230619056228535581002499352768*n^7+ 5796129194218842126511264213453486500096*n^6-\ 15520597660216570571381746235117321358336*n^5+ 29635577941732046667354878258370856980480*n^4-\ 38724276028166893935612495605279114035200*n^3+ 32091610140163116666861885584239509504000*n^2-\ 14585813896685795605287327438712995840000*n+ 2560382757699852599345736296379187200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 5175/33554432*(1718505588301*n^28-\ 670209208383302*n^27+124228966270724217*n^26-14556000206720970270*n^25+ 1209450651517228510845*n^24-75766950815720022503190*n^23+ 3713804669984192826893565*n^22-145863127873222309417617750*n^21+ 4662075411305691803695306035*n^20-122427277986413711992651940970*n^19+ 2654223691837738092643191106395*n^18-47523318046424200215859573035450*n^17+ 699345768256362328023133100372415*n^16-8354572520084518929106124648442930*n^15+ 78892661238686746923646436965944255*n^14-552744229841731562261847760485486450*n ^13+2310891383915759649322003106489648580*n^12+ 3146839063921592939186592316061090040*n^11-\ 152737191245415365661974925701556633040*n^10+ 1504382456221974450588468421438720301600*n^9-\ 9497032496700087494197926922805597952576*n^8+ 43448750922465384740943848417666215988352*n^7-\ 148048170411492058621102497358580689915392*n^6+ 374169164103150255199797894351196718008320*n^5-\ 684930161908735642344228964511167499673600*n^4+ 868403601335806756985088458902491881472000*n^3-\ 705893780334747460444131810655395348480000*n^2+ 318343581119087510419689133994099343360000*n-\ 56328420669396757185606198520342118400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 5175/134217728*( 6810226385969*n^28-2647790522505478*n^27+488849439189736233*n^26-\ 56990270393276058930*n^25+4705180107907343551305*n^24-292413449529406218541710* n^23+14191173088450539471518085*n^22-550547295501801013528162050*n^21+ 17330192153141660915317610415*n^20-446535520461078425423777861130*n^19+ 9451667440243208635033970509755*n^18-164052962417705338755867471277350*n^17+ 2314275678807477509085864867871635*n^16-25973494336022314632537678364578570*n^ 15+220399526811713752595995319636057895*n^14-\ 1204978092803709135043873209484729350*n^13+497786907569502362109293907681790020 *n^12+74625177230687612010420190621832222760*n^11-\ 965967213436225614581992636216656468560*n^10+ 7630256195407612740235538396457670933600*n^9-\ 43666723713574940310579420370262718778944*n^8+ 188720129302778272629622927857438531616128*n^7-\ 619530396576119806961546005979399073873408*n^6+ 1526131910224300921422752756832858834094080*n^5-\ 2745081319265388039470950827065651184230400*n^4+ 3442441774044039078124198472861089824768000*n^3-\ 2784834281169143467831414324234737745920000*n^2+ 1258490798689720817219145142058420797440000*n-\ 225313682677587028742424794081368473600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 150075/134217728*( 462862165862*n^29-192362866182661*n^28+38004240735184668*n^27-\ 4746622367459751429*n^26+420381366960183163080*n^25-28065160512790219063245*n^ 24+1465518782171172426934860*n^23-61286914729357842058181505*n^22+ 2083943642451671025659453820*n^21-58140519680561669984870872635*n^20+ 1336038136910945909082436392780*n^19-25246880929308448759152735492015*n^18+ 388807299382751458393412013140280*n^17-4772085047382007484679247877365215*n^16+ 44176010275415218371882109093220820*n^15-256578918200910607122983462167348635*n ^14-141134294579582205701101217106863490*n^13+ 24790024174200566915206973740972451820*n^12-\ 350499536255300074124842699337270746760*n^11+ 3165307590046124530258748848270749166480*n^10-\ 21196717092463085716492410108926136479712*n^9+ 109526474228065823411883903571211459324736*n^8-\ 440469759699561900137044795277918177962368*n^7+ 1369855992944153546763205356892479005167104*n^6-\ 3236680795711356826735939204732237481379840*n^5+ 5638169301612308374354347294052994895667200*n^4-\ 6904198135138661007180496494923204296704000*n^3+ 5496824624922433871005766212995147694080000*n^2-\ 2465297591553610665978797037274996408320000*n+ 442857928021464159942007353884069068800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 30015/ 134217728*(2263851394592*n^29-935575424859945*n^28+183579602699258178*n^27-\ 22740041400170727885*n^26+1994026448286015706290*n^25-131541241986458464225065* n^24+6770704717622986694447010*n^23-278253439643237120640076785*n^22+ 9261539585771697637154111370*n^21-251575954460282545421182669815*n^20+ 5584419360123982876257371132430*n^19-100648883925905845575797969679735*n^18+ 1444215602758286726194962734619030*n^17-15678055967049822012086802783971715*n^ 16+108643456886584290206681832819918870*n^15+4714399279119806371228809070929765 *n^14-13015179505386517661258255361710547090*n^13+ 221451759459709058127581628326474902860*n^12-\ 2387079909240103899316709264996187773960*n^11+ 19180618061438565698235357605566800276240*n^10-\ 120142383457096415052278899421759265607392*n^9+ 594209004585015695948444042430347367823680*n^8-\ 2317161574605117287295187497918293246022528*n^7+ 7046520578617318367003731243035073827878400*n^6-\ 16379358263817268523418170787355517500876800*n^5+ 28206390513283033441199096522626523827200000*n^4-\ 34292842320173757906018392098984320860160000*n^3+ 27221308994482659695166692990637101875200000*n^2-\ 12229063449451518890117168647500437913600000*n+ 2214289640107320799710036769420345344000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 50025/ 268435456*(5260884541063*n^30-2313105019601643*n^29+483302981903869991*n^28-\ 63807362087982290547*n^27+5969495686065874083891*n^26-420605400047141749924575* n^25+23150489448306888599405955*n^24-1018575652113352226895575535*n^23+ 36335510216535205013279277765*n^22-1058597310720858832130081659785*n^21+ 25199237449699027385337605201445*n^20-485909749118228593623343465233465*n^19+ 7399064689533591595176655122658545*n^18-82904738041697922758857087192115925*n^ 17+512236817001787927027582306680021825*n^16+ 3234960872319192010150321264674930075*n^15-\ 148386319244441262416469572661799396080*n^14+ 2500797697824231063944937912637272539880*n^13-\ 29237141849279305697822592850654403204560*n^12+ 263344760465317138850174257821072895581120*n^11-\ 1889459088339356036275364020079083697970368*n^10+ 10913457377566570242401118149893778712789888*n^9-\ 50727993137823338964723940331179182615161856*n^8+ 188307417908675384708711112607687979208452352*n^7-\ 550326906400608098023753115649536392457634816*n^6+ 1238785753303129640667110369408664767059292160*n^5-\ 2079438578997679630812378536965414995248332800*n^4+ 2479118445345054046944703819971985655463936000*n^3-\ 1941024445911127999497903472133698617999360000*n^2+ 865474699987654978528770065998045910138880000*n-\ 156771706519598312619470603274960450355200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 10005/536870912*(50248279155121*n^30-21891752911874205*n^29+4525176570826493675 *n^28-589929152126008920225*n^27+54374270974406717160519*n^26-\ 3763865950781186818103445*n^25+202790711057824382014892175*n^24-\ 8691251971256133665921849925*n^23+299894590932784447490557345965*n^22-\ 8359624962062457326497493714475*n^21+186877496345554148539435240617225*n^20-\ 3261350226476694218295748748922675*n^19+40962119922945520045612469851629405*n^ 18-252477593891761940004795424030031175*n^17-\ 3383543270743080493979674493653057275*n^16+ 138310893696504326039070151867455918825*n^15-\ 2595830890314008614883239620078549911010*n^14+ 34797491657442783408908786245212781332900*n^13-\ 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310155/536870912*(3040742850692*n^31-1401858405339001*n^30+306794348424857605*n ^29-42364947110058126275*n^28+4137689162032265953113*n^27-\ 303545937973547226859839*n^26+17327933561774004597548445*n^25-\ 785984800853560434528488775*n^24+28625542934362717875466513605*n^23-\ 837087239154045875293639162965*n^22+19356155282425416581709378288675*n^21-\ 336646744364178270575513963463825*n^20+3660786649153467250024659605258235*n^19+ 5678860100288771782361126625910395*n^18-1379218687368515502150935378996678625*n ^17+38374115144248825186476058246489926675*n^16-\ 705851733409252726916255664714361156845*n^15+ 9986538722772541160317518614650946705010*n^14-\ 113936482698012104606369746947779881129700*n^13+ 1068172654426657134391733110372832660729800*n^12-\ 8287685575669856006221963114732978286951152*n^11+ 53251018870266376276238031893387346258133856*n^10-\ 282269874599152257267486009125466995843897280*n^9+ 1224847993346213244886085798727321871008067200*n^8-\ 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8939931850039209841369282034109986610610176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 13485/1073741824*(192411460675379*n^32-89497583467064368*n^ 31+19590763794900711928*n^30-2672902958209725484160*n^29+ 253247788713628472787156*n^28-17500786737529360233794112*n^27+ 893375567381081588244893112*n^26-32529120374319586465252421760*n^25+ 695810080887200692922938267410*n^24+4617875506598733245428514266080*n^23-\ 1201801170528467854467180845066280*n^22+63579408962161898732471925519484800*n^ 21-2262160836900340388262434343478978380*n^20+ 62411149151794143651637716860180852160*n^19-\ 1402501313896007507299307725726798813560*n^18+ 26265793791268604865716363769007999209600*n^17-\ 414912698508138234599386717999273711609965*n^16+ 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n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 2697/1073741824*(1523355300143582*n^33-730279032542153919*n ^32+163585202505307646624*n^31-22564986968942791252440*n^30+ 2113976393369796930987848*n^29-137901562071026098104364836*n^28+ 5879205278761822778781419376*n^27-97106748575369948062571266200*n^26-\ 7853114435448356221036558631820*n^25+859484270144409210464873384407590*n^24-\ 49483305109678939943797109323872240*n^23+2085561378406680961172844336885606600* n^22-69979514815541560702723476553907061240*n^21+ 1937216255445880421863354694584913536380*n^20-\ 45080264716658072422568393698120302407280*n^19+ 891524110019548510030633801367439481709400*n^18-\ 15081672252679362599651741961950322711379170*n^17+ 219046519794937645507495931360188665984282465*n^16-\ 2735883250389459234087360014739409181980337840*n^15+ 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(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*(1074541144550702*n^33-\ 479840733721326879*n^32+96969512647278839744*n^31-11318947444996210730040*n^30+ 760502330979518878012328*n^29-14238750920604323000579076*n^28-\ 3040044422023817234769084144*n^27+424240581428315305144499665800*n^26-\ 33012517204171473935975384539020*n^25+1875551688265960530977588637533190*n^24-\ 84168039032229334529433987106711440*n^23+3093800256141396960951857410276530600* n^22-95071675618632389792777562334950551640*n^21+ 2473866270414366658728081961952647535580*n^20-\ 54966826700882023743868135836177960577680*n^19+ 1048589506141632619860034471816426833545400*n^18-\ 17233294458858063159043344296829901897784370*n^17+ 244430773507896926793620017863708660947668065*n^16-\ 2993164685584144755137199792942025947539653040*n^15+ 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(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*(1174573775337234*n^34-\ 463537260603039682*n^33+67851560022915712461*n^32-1100308254414112863328*n^31-\ 1269881177186003455945344*n^30+262547276293690546092448712*n^29-\ 31205141469437332233346935876*n^28+2668322011879633832208323899248*n^27-\ 177377659580819479808110098668940*n^26+9534838402184956688480903907133620*n^25-\ 424463461026200822290270496368189410*n^24+ 15898815326050887022472780491073713680*n^23-\ 506640117741743671990830934662215602080*n^22+ 13844823892631084989916895990060880968840*n^21-\ 326280003686686494626816403274274615402820*n^20+ 6657658299604783408115846965996134016809360*n^19-\ 117919782978414987543636703557549750433393590*n^18+ 1815404245013095043493416020034529747187763070*n^17-\ 24299743825321249727910736015302528493355543235*n^16+ 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)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11 )/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 899/1073741824*(447058732556251*n^34 +378469209427765279*n^33-282508783095718833401*n^32+81670958075328998218216*n^ 31-14427341157834022967847716*n^30+1797658038725164288200770836*n^29-\ 169681700397052836758874640884*n^28+12661871295872889321441750123144*n^27-\ 768278814333286455396119439569910*n^26+38659789584306265073719449473225610*n^25 -1636610687742154159634430293792624190*n^24+ 58912050224430378001730435393005436040*n^23-\ 1817637599166216064646040145754446507620*n^22+ 48355855287158882670445953433213616944020*n^21-\ 1114110234507689738124042675832445624161380*n^20+ 22298133926717722521361240795435098151705080*n^19-\ 388417744528898508235577467428744317471319885*n^18+ 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3216921268965368698988078265777037676901942951936000000*n-\ 584001048103811382887448348878234875338109747200000000)/(2*n-5)/(-1+2*n)/(2*n-3 )/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11 )/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -31/1073741824*(156033209753245544*n ^35-130598144600353205335*n^34+48714977889064479930455*n^33-\ 11103869484260891576580265*n^32+1763858306212795174605088712*n^31-\ 209823466608880783804483478140*n^30+19550280851800400476304649937620*n^29-\ 1470811276332132602662671214444260*n^28+91299327542014882304911677302071704*n^ 27-4751382523745434299517538564598427650*n^26+ 209820382150843991129403226826004109050*n^25-\ 7935090716830177487888422825186754530350*n^24+ 258823559548505736708445670436192907226760*n^23-\ 7320580469638856503408413357634649860722300*n^22+ 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12130255346290667773460362314238516472800505066590996070400*n^5-\ 18389350678013841702276785369065707645472182264803205120000*n^4+ 20250620038819419017294696424409852530138552017983242240000*n^3-\ 14979420798138066968884830130438683398880148234371072000000*n^2+ 6465455747096179030954567403229593742300799420071936000000*n-\ 1168586097255726577157784146105347985551557604147200000000)/(2*n-69)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2* n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35) /(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), -31/1073741824*( 335662971940375337*n^35-237477853101695432170*n^34+79169407770270465035675*n^33 -16637924753293448694899425*n^32+2484388595401061617799834996*n^31-\ 281419929328896231725379626560*n^30+25197474466149369522892157655900*n^29-\ 1833892509440761983431232596666700*n^28+110690279208547448967141526169696622*n^ 27-5623623217370311965405305275714970820*n^26+ 243210655195836903462160283317020017850*n^25-\ 9031433614113580481541108272290984245750*n^24+ 289880792961066290459879858071266454125780*n^23-\ 8082900005596942612068290406413630752852800*n^22+ 196533403910762237050081615201180135380835500*n^21-\ 4178198065228093629128571425593950915318751500*n^20+ 77798481978172517929230900157908568039897968945*n^19-\ 1269843810966077403674192097070082602759266948450*n^18+ 18169265782032162590423349864268714657260796487875*n^17-\ 227716190995351013104274530328610445282607099872625*n^16+ 2495898092874307602252275605803075445753914926795248*n^15-\ 23865458661013709823997736591259959859939078165405680*n^14+ 198403495450634144745778016107648633426452758851007200*n^13-\ 1427688130360296202321189332315299553042780203290527200*n^12+ 8842111598178763928144246045453637741017284428001999104*n^11-\ 46796200801042455786044499461410845701744649617790493440*n^10+ 209747620901122506942263638166727006195523672030246905600*n^9-\ 787228104200356806188703622657911816019494247177458796800*n^8+ 2438727340564885913397659907159656881500639838088480083968*n^7-\ 6120368234594768582454713154376512213795190408896945070080*n^6+ 12138586657338217928665181479064464708004625460581566054400*n^5-\ 18385715441158521652246947913204341357871952418020106240000*n^4+ 20236038529177978724905037094161746240626399398833029120000*n^3-\ 14966192576791129743983747392781328435778897671762739200000*n^2+ 6461131290554610320324210590577900276248295758626816000000*n-\ 1168586097255726577157784146105347985551557604147200000000)/(2*n-69)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2* n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/( 2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35) /(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 1/2147483648*(-\ 62878397046083567293*n^36+44188703826989429737674*n^35-\ 14832933808483681936042515*n^34+3170979061124309356475083950*n^33-\ 485558114479237656051203702694*n^32+56777217698004834990371295906792*n^31-\ 5277243801640470715521812361439820*n^30+400665852441739366087730014204162200*n^ 29-25338959160575584803024480394898149758*n^28+ 1354346319244539335789552677900518030444*n^27-\ 61857732271094755783836466736759603616090*n^26+ 2434829268833428060489006069691274275069700*n^25-\ 83138978426164078097240238778258337651119920*n^24+ 2475120735724445184315949583631545271759343560*n^23-\ 64491419430303310495718223154594725122276291100*n^22+ 1474784368970309115651548923688920023071661459000*n^21-\ 29654492498423948690727787698611098394547307168605*n^20+ 524865776519977638460301500082665742506192821316890*n^19-\ 8179602040480634206041282456478278280692525118215275*n^18+ 112189937976669237285169640109225220574814901523634750*n^17-\ 1352715250987813374097556552649156476931014552494413522*n^16+ 14310428637777217092351632192848607808662976586918250496*n^15-\ 132465460729496978863642331426974816112616042043141306560*n^14+ 1068996468099100769068774061416726208287413624498100948800*n^13-\ 7485982628864558343859072012779027793597283609960667211456*n^12+ 45224515047938367369100358021645133182453330862926304657408*n^11-\ 233977936040803471956885065825595659031061639270899401959680*n^10+ 1027284844741223543939633018688741239718101203580173421452800*n^9-\ 3784031228020406489178580397829190866260951391334671357706752*n^8+ 11525610312391394940323738495108595816724666948356480710516736*n^7-\ 28488821682893873429273978342374375449909184964023060210728960*n^6+ 55742047293023455416887090747096562656121929330822852906188800*n^5-\ 83428987749815137935460787423491285969982947918769693736960000*n^4+ 90883245264851298926955482239614019923131141743669106442240000*n^3-\ 66636453429450332938526516985363793900028158444547368550400000*n^2+ 28571964523715954766988232458424585421916549340863660032000000*n-\ 5144116000119708392648565811155741832397956573455974400000000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31 )/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 1/ 17179869184*(-659550019147181180699*n^36+450240747453036250876782*n^35-\ 147348942295359131914813905*n^34+30804030158946167446659738990*n^33-\ 4624201077478764228292982942652*n^32+531211875387309940658703655575096*n^31-\ 48593951638357900228506798364554100*n^30+3636767651694926328772018442856032760* n^29-227023605186400679944746422302717550634*n^28+ 11991724399030278798693432560101596626052*n^27-\ 541848896054382747743482821721963107393870*n^26+ 21120257010748913242162013852787761008049700*n^25-\ 714751745648666320204581300990165707704634460*n^24+ 21105975439066453011910339278101612950312714680*n^23-\ 545856299916459359513469990666053405405123168100*n^22+ 12398106327932863667048207957444860320013393354200*n^21-\ 247758405852181285453471148231650564972185626343315*n^20+ 4360518600587867011855310484620526505260193754041470*n^19-\ 67607912224865109868033796617578405626764085956045225*n^18+ 923002063479847670697530799615721620503986423119616350*n^17-\ 11082368826353959405596399918717276168213225947551892496*n^16+ 116798640803624878063419832327172497268091935410511668128*n^15-\ 1077498887495612925758144295505249828944008460085879463520*n^14+ 8669203156015034501216641782478257849378086343546004618560*n^13-\ 60546721554459323112937973011367845797946074494589860853248*n^12+ 364918627864994519120522469996143849534555531303869814882304*n^11-\ 1884128785900014331967163552821132915216182760260954204473600*n^10+ 8257828508034479546216929203067922983586019058944962441879040*n^9-\ 30373003276964549568156066189884530571180826198757871727722496*n^8+ 92399003824001164216259272227073762788237942269514547301695488*n^7-\ 228167898125340574437821963350443578882270229447686829506887680*n^6+ 446108742602238727946759399174669509564547931778897917060710400*n^5-\ 667343047028665588686561733444862750311815667933138228346880000*n^4+ 726747748915321593980516583981812236640344347517206968401920000*n^3-\ 532809422094340037804241502418506523031626484298078696243200000*n^2+ 228484651980441490662393198195523351989629047738000736256000000*n-\ 41152928000957667141188526489245934659183652587647795200000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -37/ 17179869184*(43137784833725713546*n^37-30497306720622462561551*n^36+ 10368561605860268352052908*n^35-2258342108600846321052077925*n^34+ 354174008082096886769255425878*n^33-42617002401606634459606784657868*n^32+ 4093889326233224248662771197987024*n^31-322546070208874907633224982703767620*n^ 30+21249327207899020417569419303871146916*n^29-\ 1187497216459761362830698525118930505346*n^28+ 56911288673624561975577957217298949890088*n^27-\ 2358875562425331872501950852136649117390630*n^26+ 85113251107924563632042017058046137513946540*n^25-\ 2687069615182284242540946008065369968282695340*n^24+ 74513527174219938578907319837346385960585079920*n^23-\ 1820187646180556540792745718085500904918927613300*n^22+ 39246142227810669168748962908515726673808693434610*n^21-\ 747855114678556171786220500819225385693044463138535*n^20+ 12601105959284990420143305084660677396936178717287180*n^19-\ 187718534187633010499087858302792687342166088071285725*n^18+ 2470355211859213354603117324127263177910660475768933534*n^17-\ 28675855986251119297122410593537101612952423996783628304*n^16+ 292966734899093431079497079659276923006863167370668561632*n^15-\ 2626444527251358679204282082045937971723884465306098156000*n^14+ 20582233601053364826253198402065381148151899489091294999872*n^13-\ 140308629969306314071007436157651074429878788417471219461632*n^12+ 827037748827973228906991521555994437814559102980108553432576*n^11-\ 4183841230873665227745357539671163706426120797196458851073280*n^10+ 17997716594076863407252743665280798316294368349479056195157504*n^9-\ 65078329142436981405560927849215635760108765349808446893031424*n^8+ 194934272313765610883125187077638077623801247005449440883228672*n^7-\ 474672866696798862832016649871310774067662486314959448633835520*n^6+ 916484407170533289102259210422225544413695538318543425657241600*n^5-\ 1355786490607662947247924294373738466695487277140423057080320000*n^4+ 1462168857510000250758786313059280118660620120256824847892480000*n^3-\ 1063146324452358821349069742959191369488245213062899748044800000*n^2+ 452883503050331790049939668200349309983493167635469697024000000*n-\ 81193614704592154089371957667971708922173152402656460800000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -37/17179869184*(49474340401942783084*n^37-34578048506554255344023*n^36+ 11631849006717864420496278*n^35-2508720900024478994364191745*n^34+ 389873645014458823214868290622*n^33-46518060969478443884122848283404*n^32+ 4433748211334866900782604725033464*n^31-346791107109369281721539240901207700*n^ 30+22692909956834092202312541719707312704*n^29-\ 1260254587302914306851302161710040524818*n^28+ 60048555040819550014146555567765042548628*n^27-\ 2475555413883497593634037389402061322882830*n^26+ 88879523524439129880861058581380049487063860*n^25-\ 2793079080382104211666987175630904919735053420*n^24+ 77124501533759883232748527624033083323876308920*n^23-\ 1876595068701564661294916806015938677601718344900*n^22+ 40316656424256630883319512102782896623096111588740*n^21-\ 765713720753227800647100506902513622838028392714255*n^20+ 12862939077999073774266023959042159049650271436689030*n^19-\ 191089041069638858391134150952960079231910701236852025*n^18+ 2508380078373738895732297563320566300681544836251737886*n^17-\ 29050793044761176585161342776542741595682235561006422992*n^16+ 296185902782455599527920353009935560113142150082968689312*n^15-\ 2650393419123857040693164735744216115722710333113382954080*n^14+ 20735631163283902742012911687366420883599395420817866795328*n^13-\ 141147660296143616179689146549935012278703852007259884304896*n^12+ 830915012840898283828700791975899970297334635047639281780736*n^11-\ 4198765578223982576376961096407703728484522679902589104953600*n^10+ 18044637399720484563205978895998244179575937348114897503346176*n^9-\ 65195378353477224872946440636033315233460542262480936111112192*n^8+ 195155162733664824955901245265206351378168280134616451157573632*n^7-\ 474958987115626955640728473785869427621185744393300805159813120*n^6+ 916667963702389820798942172804439583247989084672477233437081600*n^5-\ 1355677441672932458017017752795998787059815602900422121226240000*n^4+ 1461812295936684222818962196171220955853532073654810580090880000*n^3-\ 1062836416052326466643090627284914419816449640475654933708800000*n^2+ 452784662947217639347165770769163569891765814552243994624000000*n-\ 81193614704592154089371957667971708922173152402656460800000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -37/34359738368*(218264861648468855851*n^38-159522142614060797658347*n^37+ 56207520059333205652297317*n^36-12718448622843952816294317705*n^35+ 2077109584148677260579892491108*n^34-260879118944281806141932920105056*n^33+ 26218896405251488210091622085573796*n^32-2166206453344720460030125844256190900* n^31+150002099000835502198522994595140832906*n^30-\ 8831940704887518492128568507233163081402*n^29+ 447033956989133673484723212678192859527942*n^28-\ 19617321616721352497245151659014837162759870*n^27+ 751333680334461985502693083906483908091122540*n^26-\ 25244483455374002720951973861344907965959459380*n^25+ 747096861192428747793544007143319555845395895380*n^24-\ 19533314857538968503917081547675820448646395288100*n^23+ 452180040379673032449044053550574012429204916187235*n^22-\ 9281306979157297009720390687829968845260536937010195*n^21+ 169044708736624830893695540327320658662336438067682045*n^20-\ 2732403811048464852276891676076309380755107469076636225*n^19+ 39176613739150236732349445421168935479009966442862766104*n^18-\ 497698690678464326881817477278349406951932438803714247988*n^17+ 5592480002603908361094230085131368768671784248406006481568*n^16-\ 55447899411081999978597017633192240807202981943405514178720*n^15+ 483534894288216615144896202723990231745189398123244171432192*n^14-\ 3693903297435733261720254139042190410964486101162751630390144*n^13+ 24597268875494779926758836371925094342082781939413471920565504*n^12-\ 141891552288313386254610136512494668796363268108331891045612800*n^11+ 703727250051706768619069999126294750034153394289077690462193664*n^10-\ 2972830309863237896302889910731170079532658480127478824051647488*n^9+ 10573052985905544900932886012670453143165604831777117789239656448*n^8-\ 31197460193683488499838420576688289412577384341361268844351815680*n^7+ 74941459126860264182682739174835260495778764271418167978518118400*n^6-\ 142942023911174779920065354415177758498674817326386963674726400000*n^5+ 209186152367624177182029347453877124269151368848004558233272320000*n^4-\ 223484372925113670899958986664529206807027284612237263582003200000*n^3+ 161203410932588684588137699164334915211552783888071237042176000000*n^2-\ 68231989789963549247459550754238506316866770554690784460800000000*n+ 12179042205688823113405793650195756338325972860398469120000000000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69), -37/68719476736*(467242620183704120177*n^38-339714064832915388220369 *n^37+119112325126691581103061609*n^36-26828409382661835544174265535*n^35+ 4362554988727203199463419069266*n^34-545704569997951333296781323504012*n^33+ 54636100362273823518843823423275892*n^32-4497973859618494505091341623373000300* n^31+310431533809931386579750793021650708662*n^30-\ 18220940246525170875902147144089495657854*n^29+ 919583987136312343469126265354208900072734*n^28-\ 40245057681696868325267443437152049378965490*n^27+ 1537478976746460784011087835765258948940989080*n^26-\ 51537560626945687948512601392638796615672328260*n^25+ 1521910415609622930849417767035824166765381672260*n^24-\ 39711206894660132594322278866722546947189019968700*n^23+ 917572058649097415802022130550051178649163310887345*n^22-\ 18801520921156420573833715140822448532505429136550265*n^21+ 341901273402724431062155228721491825295094285401134465*n^20-\ 5518442544527140769001353036103203853855363399482445575*n^19+ 79017877504859661839981415314562646398100904620815887358*n^18-\ 1002636069351578160730661530399517167401394552204195363176*n^17+ 11254041180192727195703049064192663006035416267635316713536*n^16-\ 111471208528455491394646176091172304345979258648382121265440*n^15+ 971231697791842612186577766317259694107854591705330826123584*n^14-\ 7413776276794916025915488014332767986046348194075613349253888*n^13+ 49333170157781149401312795134626584376020776021820493060670208*n^12-\ 284409401739879846961427323952334002917263065012627750814969600*n^11+ 1409814826193602602848185589523898610835537112768923769669897728*n^10-\ 5952935411311576910530505979214431879551414751243893577990322176*n^9+ 21163913457481081709495456482704587766111094001945363349352759296*n^8-\ 62427900440439259896415619609406253491411856502755145406808719360*n^7+ 149924742777454615496222289002960152196003267188238980994010316800*n^6-\ 285909795161237608064155794064824733773561244293820037868748800000*n^5+ 418354993659112885470515330083307466042242465974063409544560640000*n^4-\ 446916236046177076149648653660443019186172458267630327719526400000*n^3+ 322361997741752912859663973765597594520523345086761823895552000000*n^2-\ 136449834350884486621384367717942522788973580629868911001600000000*n+ 24358084411377646226811587300391512676651945720796938240000000000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69), -3/68719476736*(12050849755704926842716*n^39-\ 9195414848997250038768993*n^38+3387696032617574835634877049*n^37-\ 802705684566222367343608728377*n^36+137485954361726221928793630742683*n^35-\ 18138204138351021683304985417707474*n^34+1917884592582176570702695825748657532* n^33-166985579009642140855822707925005766836*n^32+ 12206462926080415337241609716078339688996*n^31-\ 760033050563123952758390794061884876241878*n^30+ 40757115980859347429266553611944549850971054*n^29-\ 1898580289193752227990327616551476792773718142*n^28+ 77344662621689781909143242300438720965537420810*n^27-\ 2770143257376256619292336655209608146238393973720*n^26+ 87586214870084496809503041509915140451442648870660*n^25-\ 2452479346107460049143627417708535728239526485954180*n^24+ 60957955326858835760233134897201026634495057867336360*n^23-\ 1347158918097106894555520455255387296303150671786921505*n^22+ 26497046261265034437427423986811284943175045120123514465*n^21-\ 464014456685803608146938400367357358726087310110643796945*n^20+ 7233282841126807747588483107180362211643667961662860795539*n^19-\ 100294247558268436760469212891927257332037114435942652409022*n^18+ 1235286961776617892520506769559271540516645683543073363471096*n^17-\ 13488428765373546173877408363243840583811493904159937599334208*n^16+ 130233360948731369835048123977017411486144419365759485227909792*n^15-\ 1108156408462513106212970942484795530527154770515880741422739776*n^14+ 8275426176556317783254634022771960162150345506351390745802611968*n^13-\ 53959241217839605676301435718018601838728910219624088483027153664*n^12+ 305285031112617335245884396027674840410749112870361620707783469824*n^11-\ 1487231272924235128497752014032504837340220031645852620247041183232*n^10+ 6180009025275302419244607926691549897767641558761780823716131506176*n^9-\ 21649780388311412112920411690931456998518013409605762566758403227648*n^8+ 63004306824860437022933290408762105781691134438467227167660538593280*n^7-\ 149456461146423870744252506993114542588052182230505330747992254054400*n^6+ 281853548898478068268520389597930614549265897393953426379567595520000*n^5-\ 408310460221832802752819590633545814902968659194121391900794552320000*n^4+ 432340833580722924264242516957246561185832215896793631198753587200000*n^3-\ 309475119726014235607381709556582690830561550127419728137814016000000*n^2+ 130174220683532874068552460903884528647590859845437030714572800000000*n-\ 23132060829338304700062070739605139871927131119516825681920000000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2 *n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/ (2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55 )/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-\ 71)/(2*n-69)/(2*n-77), (10488508169105968435280*(n-38)!*(2*n-81)!*(n+1)+ 15732762253658952652920*(n-39)!*((n+1)*(2*n-80)!-1/15732762253658952652920* binomial(2*n,n)*(n-41)!*(n-38)!))/(n-41)!/(n-39)!/(n-38)!/binomial(2*n,n), (( 10488508169105968435280*n+10488508169105968435280)*(2*n-81)!-(n-41)!*(n-39)!* binomial(2*n,n))/(n-41)!/(n-39)!/binomial(2*n,n)] The limits, as n goes to infinity are 549755813867 549755813457 274877904681 4398046251855 8796090176077 [------------, ------------, ------------, -------------, -------------, 549755813888 549755813888 274877906944 4398046511104 8796093022208 4398040126505 17592088541435 70367441472035 35182429926349 -------------, --------------, --------------, --------------, 4398046511104 17592186044416 70368744177664 35184372088832 35179122753865 35171355909395 562471620185795 1123848422786365 --------------, --------------, ---------------, ----------------, 35184372088832 35184372088832 562949953421312 1125899906842624 280441365537465 2236050735610005 8893266419457675 35242921547389575 ---------------, ----------------, ----------------, -----------------, 281474976710656 2251799813685248 9007199254740992 36028797018963968 34732019770869825 2123421862771215 263175749166676575 502734032946985605 -----------------, ----------------, ------------------, ------------------, 36028797018963968 2251799813685248 288230376151711744 576460752303423488 235775399714094315 863591103001089735 3062343097656984795 ------------------, -------------------, -------------------, 288230376151711744 1152921504606846976 4611686018427387904 2594668547207485815 2054244622243620327 1449018733426621647 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 1583912736042260049 401905800568069649 -604628687793826483 -------------------, --------------------, -------------------, 9223372036854775808 18446744073709551616 4611686018427387904 -10405552130151635447 -62878397046083567293 -659550019147181180699 ---------------------, ---------------------, ----------------------, 36893488147419103232 147573952589676412928 1180591620717411303424 -798049019423925700601 -457637648717970743527 -8075799880993347666487 ----------------------, ----------------------, -----------------------, 1180591620717411303424 590295810358705651712 9444732965739290427392 -17287976946797052446549 -9038137316778695132037 -37123400102388038682363 ------------------------, -----------------------, ------------------------, 18889465931478580854784 9444732965739290427392 37778931862957161709568 -150460195691259523811067 -------------------------] 151115727451828646838272 and in Maple notation [549755813867/549755813888, 549755813457/549755813888, 274877904681/ 274877906944, 4398046251855/4398046511104, 8796090176077/8796093022208, 4398040126505/4398046511104, 17592088541435/17592186044416, 70367441472035/ 70368744177664, 35182429926349/35184372088832, 35179122753865/35184372088832, 35171355909395/35184372088832, 562471620185795/562949953421312, 1123848422786365/1125899906842624, 280441365537465/281474976710656, 2236050735610005/2251799813685248, 8893266419457675/9007199254740992, 35242921547389575/36028797018963968, 34732019770869825/36028797018963968, 2123421862771215/2251799813685248, 263175749166676575/288230376151711744, 502734032946985605/576460752303423488, 235775399714094315/288230376151711744, 863591103001089735/1152921504606846976, 3062343097656984795/4611686018427387904 , 2594668547207485815/4611686018427387904, 2054244622243620327/ 4611686018427387904, 1449018733426621647/4611686018427387904, 1583912736042260049/9223372036854775808, 401905800568069649/ 18446744073709551616, -604628687793826483/4611686018427387904, -\ 10405552130151635447/36893488147419103232, -62878397046083567293/ 147573952589676412928, -659550019147181180699/1180591620717411303424, -\ 798049019423925700601/1180591620717411303424, -457637648717970743527/ 590295810358705651712, -8075799880993347666487/9444732965739290427392, -\ 17287976946797052446549/18889465931478580854784, -9038137316778695132037/ 9444732965739290427392, -37123400102388038682363/37778931862957161709568, -\ 150460195691259523811067/151115727451828646838272] and in floating point [1.000000000, .9999999992, .9999999918, .9999999411, .9999996764, .9999985483, .9999944576, .9999814874, .9999448004, .9998508049, .9996300579, .9991503095, .\ 9981779161, .9963278754, .9930060044, .9873509143, .9781875739, .9640072010, .9\ 429887372, .9130743008, .8721045291, .8180102419, .7490458800, .6640398079, .56\ 26290552, .4454432964, .3142058518, .1717281629, .2178735711e-1, -.1311079474, -.2820430556, -.4260805917, -.5586605966, -.6759738130, -.7752683327, -.8550585\ 718, -.9152178791, -.9569500111, -.9826482188, -.9956620547] The cut off is at j=, 30 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 42], vs. those in the, 2, -th row from j=1 to j=, 41, are as follws 21 20 19 [41 (53634713549 n - 11826454337103 n + 1221128340539485 n 18 17 - 78458669111276685 n + 3515819095006208844 n 16 15 - 116720082496465045038 n + 2976536880797470922570 n 14 13 - 59660504068607286145770 n + 953780675477675702405749 n 12 11 - 12271035031924272591976443 n + 127632405991299042563671305 n 10 9 - 1074108859464127701103831305 n + 7294404562381930149778702274 n 8 7 - 39724269608341070238112484088 n + 171663344513778313710587782240 n 6 5 - 579160722484829462324038125840 n + 1486205643478087865726169031584 n 4 3 - 2755267365879645130696499267328 n + 3135554745606175562175657254400 n 2 + 21567829781043522724014489600 n - 8489884791977072265073324032000 n + 14085418065026997419252121600000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 41 ( 21 20 19 26817356764 n - 5913227161947 n + 610564168300310 n 18 17 - 39229334185461105 n + 1757909498317771134 n 16 15 - 58360036337820765702 n + 1488268058229445432780 n 14 13 - 29830228292599393831410 n + 476889142177700040307064 n 12 11 - 6135468240054934394282487 n + 63814531966536669437669550 n 10 9 - 537007735625385036486420765 n + 3646129437372466101435730414 n 8 7 - 19841989243957386502919932632 n + 85525817447190815635215546320 n 6 5 - 285889673850443660223958282320 n + 708640268467042329759880134624 n 4 3 - 1139148083900218045425647807232 n + 435375841281647241844297751040 n 2 + 3122258404018897503361857945600 n - 6599125898126101448601292800000 n + 167683548393178540705382400000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 123 ( 22 21 20 17878237771 n - 4326533509837 n + 491799021412622 n 19 18 - 34904306676005305 n + 1734226453160705581 n 17 16 - 64103360185593080352 n + 1828669951204669005532 n 15 14 - 41218498983135501474370 n + 745484540133196756150181 n 13 12 - 10925386870296405765455197 n + 130473327315009834625837822 n 11 10 - 1272361409463125301372737685 n + 10120541250752566755912408211 n 9 8 - 65351679988918939964769661702 n + 339331477014369008637810537272 n 7 6 - 1391272759968006027692694452800 n + 4334958235803265296790022391856 n 5 4 - 9288850386706641874830637842912 n + 8888949945103280115906384106752 n 3 + 17078166644709012774852904220160 n 2 - 65220024324912917337854019993600 n + 47774483124899133875566694400000 n - 2403464193635559083443814400000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), ( 22 21 20 2199023184343 n - 532163584324031 n + 60491268884193686 n 19 18 - 4293227778277161955 n + 213309606275430796593 n 17 16 - 7884689702676095845536 n + 224924656389051932658796 n 15 14 - 5069772548327888995531190 n + 91689720151843045538080313 n 13 12 - 1343634336774130955372850031 n + 16042286287862821229367661446 n 11 10 - 156347743297796144493127303575 n + 1241629967963107392447775717423 n 9 8 - 7984322952375016405801740748706 n + 41015192509021439629113476797976 n 7 - 163618778699365878024955973344960 n 6 + 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69547403473467106752537226144881762974114722376471168989105068441600000 n + 2 49417474855629656974446271052538270047032738261639952362245193728000000 n - 20659964537180530260550787857721837873464248759428828157601382400000000 n + 3654865611035452142609807176857612099764486716883658457743360000000000)/ (137438953472 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 65) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 75) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 79) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 73) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69) (2 n - 77)), ( 40455674366551592536080 (n - 39)! (2 n - 83)! (n + 1) + 60683511549827388804120 (n - 40)! ((n + 1) (2 n - 82)! - 1/60683511549827388804120 binomial(2 n, n) (n - 42)! (n - 39)!))/( (n - 42)! (n - 40)! (n - 39)! binomial(2 n, n)), ( (40455674366551592536080 n + 40455674366551592536080) (2 n - 83)! - (n - 42)! (n - 40)! binomial(2 n, n))/((n - 42)! (n - 40)! binomial(2 n, n))] and in Maple notation [41/1048576*(53634713549*n^21-11826454337103*n^20+1221128340539485*n^19-\ 78458669111276685*n^18+3515819095006208844*n^17-116720082496465045038*n^16+ 2976536880797470922570*n^15-59660504068607286145770*n^14+ 953780675477675702405749*n^13-12271035031924272591976443*n^12+ 127632405991299042563671305*n^11-1074108859464127701103831305*n^10+ 7294404562381930149778702274*n^9-39724269608341070238112484088*n^8+ 171663344513778313710587782240*n^7-579160722484829462324038125840*n^6+ 1486205643478087865726169031584*n^5-2755267365879645130696499267328*n^4+ 3135554745606175562175657254400*n^3+21567829781043522724014489600*n^2-\ 8489884791977072265073324032000*n+14085418065026997419252121600000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29), 41/524288*(26817356764*n^21-5913227161947*n^20+610564168300310 *n^19-39229334185461105*n^18+1757909498317771134*n^17-58360036337820765702*n^16 +1488268058229445432780*n^15-29830228292599393831410*n^14+ 476889142177700040307064*n^13-6135468240054934394282487*n^12+ 63814531966536669437669550*n^11-537007735625385036486420765*n^10+ 3646129437372466101435730414*n^9-19841989243957386502919932632*n^8+ 85525817447190815635215546320*n^7-285889673850443660223958282320*n^6+ 708640268467042329759880134624*n^5-1139148083900218045425647807232*n^4+ 435375841281647241844297751040*n^3+3122258404018897503361857945600*n^2-\ 6599125898126101448601292800000*n+167683548393178540705382400000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29), 123/524288*(17878237771*n^22-4326533509837*n^21+491799021412622 *n^20-34904306676005305*n^19+1734226453160705581*n^18-64103360185593080352*n^17 +1828669951204669005532*n^16-41218498983135501474370*n^15+ 745484540133196756150181*n^14-10925386870296405765455197*n^13+ 130473327315009834625837822*n^12-1272361409463125301372737685*n^11+ 10120541250752566755912408211*n^10-65351679988918939964769661702*n^9+ 339331477014369008637810537272*n^8-1391272759968006027692694452800*n^7+ 4334958235803265296790022391856*n^6-9288850386706641874830637842912*n^5+ 8888949945103280115906384106752*n^4+17078166644709012774852904220160*n^3-\ 65220024324912917337854019993600*n^2+47774483124899133875566694400000*n-\ 2403464193635559083443814400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 1/524288* (2199023184343*n^22-532163584324031*n^21+60491268884193686*n^20-\ 4293227778277161955*n^19+213309606275430796593*n^18-7884689702676095845536*n^17 +224924656389051932658796*n^16-5069772548327888995531190*n^15+ 91689720151843045538080313*n^14-1343634336774130955372850031*n^13+ 16042286287862821229367661446*n^12-156347743297796144493127303575*n^11+ 1241629967963107392447775717423*n^10-7984322952375016405801740748706*n^9+ 41015192509021439629113476797976*n^8-163618778699365878024955973344960*n^7+ 474938175479016651911868263494128*n^6-825849353049171215480026897281696*n^5+ 32108064670427656689932852388096*n^4+3606439702861731799476614880391680*n^3-\ 6901842744154750732859139386572800*n^2+4046195116922946336015341875200000*n-\ 295626095817173767263589171200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2 *n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 851/ 4194304*(41344730543*n^23-10935678599926*n^22+1362442146913606*n^21-\ 106308264743898140*n^20+5826667697109775593*n^19-238477595032618955706*n^18+ 7564194287875789897916*n^17-190458456589935375839800*n^16+ 3868059886194037003734913*n^15-64029987343254136952466026*n^14+ 869383645468736385166108566*n^13-9709419162803651768830795260*n^12+ 89132601469574465695621857623*n^11-669304354787226112646345197126*n^10+ 4065330392653661680631266066696*n^9-19527490392513821755209162433520*n^8+ 70783276208285970771038234642928*n^7-172917166684369262364728359749216*n^6+ 179708510871773065430881519113216*n^5+418757922651872678937297855966720*n^4-\ 1845434848296328347861647724441600*n^3+2586808669923568226185992327168000*n^2-\ 1349493444245776500464829235200000*n+125059217972012404482834432000000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/ (2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23 )/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 851/2097152*(20672351791*n^23-\ 5467831346468*n^22+681218858691536*n^21-53153745541991710*n^20+ 2913286350240540351*n^19-119234442284583968958*n^18+3781787989668684497656*n^17 -95211849397293632101340*n^16+1933245175122462756565901*n^15-\ 31986290914607155839727168*n^14+433838530267829815973642616*n^13-\ 4834105494026467113943048470*n^12+44164202512223564344884691381*n^11-\ 328371668684846965528590245918*n^10+1955380926901652580711035046656*n^9-\ 9032347559801245333103015381200*n^8+30260887841573161121844990748176*n^7-\ 61300625967474291698585281099488*n^6+12066697364344656255494023441536*n^5+ 330217741972674979830632386782720*n^4-944384083633189949887527435417600*n^3+ 1146669898162363185979961929728000*n^2-570084067082025487909280563200000*n+ 62529608986006202241417216000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2* n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 4255/4194304*(16537841417*n^24-4762881521460*n^23+647763647473694*n^22-\ 55328736405381648*n^21+3329773908288846095*n^20-150143845070661150324*n^19+ 5266068493997718621896*n^18-147211835037714036027504*n^17+ 3334043035447079382038183*n^16-61839522068673137714222748*n^15+ 945539354614328028272809238*n^14-11952155432234200946315563920*n^13+ 124774302041451599418900129521*n^12-1069566760589008417102777876092*n^11+ 7433707083308145862561213860260*n^10-40898294322096288358761154396656*n^9+ 170102232734714770880298261173744*n^8-482344873502427730830079690637376*n^7+ 632481259818433755259824004446912*n^6+1319645563624034480249631465769728*n^5-\ 8462718105932825319127045021608960*n^4+18493450172114623436493548241408000*n^3-\ 19960500145104678762975693791232000*n^2+9524872833740429440798185062400000*n-\ 1175556648936916602138643660800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/ (2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/ (2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29 )/(2*n-45), 115/4194304*(611895330533*n^24-176223855203820*n^23+ 23966506705884374*n^22-2047036255145144736*n^21+123186507097051537139*n^20-\ 5553978918006520789788*n^19+194752349861100370361528*n^18-\ 5441837632479483439861008*n^17+123142138635587782839104075*n^16-\ 2280417286134012720786061476*n^15+34766876221960603453702437902*n^14-\ 437184225443743234438083866400*n^13+4522376598771557965331066959181*n^12-\ 38163456239829380468887146285204*n^11+258370743237736542572970948752516*n^10-\ 1360430506317046510099505356080432*n^9+5238135327162508054167851679503792*n^8-\ 12561489895256791320343788576011712*n^7+5333241630011564820431771564287680*n^6+ 86834679672001733074269448858872576*n^5-350222761379758030488090657227934720*n^ 4+671878653040078881440469561119232000*n^3-682295330731618326306537996263424000 *n^2+321352011983634521967297264844800000*n-43495596010665914279129815449600000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37 )/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 115/8388608*( 2447525766863*n^25-764828885596225*n^24+113127116123597150*n^23-\ 10535341511492352550*n^22+693171749158919661785*n^21-34271601457584702318415*n^ 20+1322139560068102045070900*n^19-40788831405788866418689000*n^18+ 1023005722648650812312489105*n^17-21086055066915861320151154015*n^16+ 359494643128299624772486229150*n^15-5082378464356246493239606721950*n^14+ 59492635583901053293951612797095*n^13-572983781375135948917485136678465*n^12+ 4482651609737259475348869735724400*n^11-27836548488609910232093954811766900*n^ 10+131454168527364277598995296518407040*n^9-\ 429156063462243699212788177048576240*n^8+676429862201311519677523484818694400*n ^7+1518669650567803532502441669093518400*n^6-\ 12978315750435053238874105003436421888*n^5+ 39441637493292406412460494197579023360*n^4-\ 67063277888812705873644019765449216000*n^3+ 63886693980975555607048128180621312000*n^2-\ 29509822922909995226904797867212800000*n+4262568409045259599354721914060800000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 345/2097152*( 203948355414*n^25-63728914002445*n^24+9425449803144840*n^23-877644853672734230* n^22+57729373118561511290*n^21-2852938165926542501115*n^20+ 109975538342228538607540*n^19-3388343677327733979499480*n^18+ 84795614604083955037021050*n^17-1741570091756316424479125315*n^16+ 29523229992534300985899503840*n^15-413688149060375009334171668430*n^14+ 4777006325201765337484659806470*n^13-45075138447807049011202820488165*n^12+ 341983360493057527077242525258340*n^11-2026128225924483659826556482311780*n^10+ 8843915818562589339829929333261680*n^9-24323952097431988819110649909901040*n^8+ 11271845387088790573418424938357440*n^7+248734497995228038182536960088217920*n^ 6-1284618693450976920414660978249755904*n^5+ 3427002022181749373089329705113518080*n^4-5470070132991512939633591923855872000 *n^3+5049778252851120073706572835690496000*n^2-\ 2327356226960180894391647673139200000*n+355214034087104966612893492838400000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2* n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/( 2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 10695/2097152*( 13156080970*n^26-4445998758006*n^25+712649183543765*n^24-72079898102686760*n^23 +5162561116910236260*n^22-278522429185724653930*n^21+11753593034499196667215*n^ 20-397618418680333848400260*n^19+10961340467835019824328910*n^18-\ 248882717106646133526042170*n^17+4683261183445075126421685315*n^16-\ 73197481651292231453031753760*n^15+948603168318358710755588619760*n^14-\ 10129205177406064656118340335350*n^13+88019209170690404485525193281865*n^12-\ 608770316702020561082970992219060*n^11+3213415109334742674743190160639860*n^10-\ 11733328508695801385446705874277040*n^9+19591565102674810676638897775582640*n^8 +72283848631577039753056213641714240*n^7-674877485620936679767130436822941760*n ^6+2607782566020777402990219347105546496*n^5-\ 6146190533978383841108493915514060800*n^4+9135664360394773944063474153345945600 *n^3-8097898330320438343158544721393664000*n^2+ 3688536819519432166636894702878720000*n-584384378659430751524437681766400000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2* n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/( 2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 3565/ 2097152*(39455736962*n^26-13331067978826*n^25+2136140378615555*n^24-\ 215944737397517440*n^23+15453916308600904160*n^22-832686574281895603270*n^21+ 35070853556543958092225*n^20-1182950826455504495757940*n^19+ 32469270746841139071021590*n^18-732568552073760419516780710*n^17+ 13660218212063029484377848725*n^16-210788837987682687249236869240*n^15+ 2683443703492335652402966757900*n^14-27952067183358647066536038210490*n^13+ 234522465925538062318409175325175*n^12-1539321844556494601135668293326740*n^11+ 7432915555618037336025766099773980*n^10-21983379817818466087972703301209680*n^9 -1413646581217108584524622568753200*n^8+418178704722275891904454528226660160*n^ 7-2490842300657080137326592222310310592*n^6+ 8480623070382105458215748061883742976*n^5-\ 18767259375734444074463297032964628480*n^4+ 26911319123603994067340354195349811200*n^3-\ 23436372003788756374230179197200384000*n^2+ 10689175845811442405422215160381440000*n-1753153135978292254573313045299200000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 2791395/33554432*(1611436863*n^27-586954477036*n^26+101579722375651*n^25-\ 11112312249045540*n^24+862342252482745255*n^23-50495113009982170140*n^22+ 2316595771098217246495*n^21-85328883943332643245060*n^20+ 2564648849462078851582825*n^19-63561416985418238627263380*n^18+ 1306779919263652476067563385*n^17-22334331879079964550877525260*n^16+ 316768849506129723068952651565*n^15-3705113651146828594139967456660*n^14+ 35297339265489246004154602313365*n^13-267628706688284894333735201528460*n^12+ 1541292901809082109223939444329860*n^11-5962483324336677008685331489486800*n^10 +7475339033232041164618027217208880*n^9+86581343072710470581656432510920640*n^8 -764480383269649203832822288374063168*n^7+3494281895452516282017927152563190016 *n^6-10535557040719770814926605699723467776*n^5+ 21661194652657051067697640897732423680*n^4-\ 29611214401200865923316480811045683200*n^3+ 25061534884013451301993266744459264000*n^2-\ 11329673163032717336329213020733440000*n+1898689475491177307634954613555200000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 45), 90045/16777216*(24945405259*n^27-9080331761668*n^26+1569978138903863*n^25-\ 171512328026258700*n^24+13283644151885394755*n^23-775676278743390603540*n^22+ 35448624445650279172595*n^21-1298772649169394236554860*n^20+ 38755300835210516348785565*n^19-951269236848283362371136060*n^18+ 19308986149687556456028753365*n^17-324502805915703697199542768260*n^16+ 4501270455120491556701905638785*n^15-51101077239640879449147376395900*n^14+ 466877356395173983159354320506705*n^13-3320151260058390317378941281141060*n^12+ 16989081990962205671525467907380820*n^11-46514327151509993486993866838345520*n^ 10-123988885996765273101217861646013200*n^9+ 2303995703566049182921655843004139840*n^8-\ 14706829150634789203742048587821835584*n^7+ 60063199962672723454762105377002450688*n^6-\ 170633022362979863142048187240661603328*n^5+ 338339178756525788192441556812122583040*n^4-\ 452445501628567299087683058767759769600*n^3+ 378952574984802274136209006865006592000*n^2-\ 171702145935279626955859614555832320000*n+ 29429686870113248268341796510105600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 30015/33554432*(298643808523*n^28-\ 116803374307886*n^27+21732470739033831*n^26-2559023466515997390*n^25+ 213991934191990607835*n^24-13515940069931891403630*n^23+ 669419780344344389103315*n^22-26638093672465333552163670*n^21+ 865440131241091667823767205*n^20-23194816464930234999223519170*n^19+ 515854185004020309951631568805*n^18-9539282573655200395747828134570*n^17+ 146387598457239258555689508517545*n^16-1851529389269210508552059593973850*n^15+ 19030152247757942325574122254882385*n^14-154512191285631473170736133552284370*n ^13+929081088882040054495949234677796940*n^12-\ 3332824457981319096537861165375168840*n^11-\ 3533684169410891510842174130035423600*n^10+ 153050101539515486418731832366467394080*n^9-\ 1253626972491334423455562700560158675648*n^8+ 6458201459519224790232666480933539381376*n^7-\ 23587837071091774605050998508227674164736*n^6+ 62361580574150558916155070083169973585920*n^5-\ 117621586607265458106421842234416990822400*n^4+ 151921942791689722726524099702799982592000*n^3-\ 124520483208794858813962667814314803200000*n^2+ 55983208344867578050741572923444428800000*n-\ 9711796667137371928552792848334848000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1035/33554432*( 8625337296083*n^28-3368041287587914*n^27+625293278955817983*n^26-\ 73414490970888874170*n^25+6115459113783521664435*n^24-384314135998672389272490* n^23+18910532511256740507296715*n^22-746238321875814062361276210*n^21+ 23988261938170170829629245805*n^20-634339951266993577273057940550*n^19+ 13870378227415530098635906688685*n^18-251008749541373753020334287336110*n^17+ 3745184072593911424519094519151345*n^16-45601614697499271774386702217228270*n^ 15+443408452666993684097507262203976825*n^14-\ 3281091974298312327082476544834000710*n^13+ 16052285928312616881844785056133456540*n^12-\ 16170969753249816005956432260739736280*n^11-\ 581105039879015264458055405610990626480*n^10+ 6692510147699733142031771684546452505440*n^9-\ 44564742852587791013613096664807250179008*n^8+ 209567975725684032105140510220847479429504*n^7-\ 726181455807668055831556821141149916553728*n^6+ 1855697404269734184249729436621546580981760*n^5-\ 3421893892705163398529926838487904317235200*n^4+ 4358049645632901873402322195354739367936000*n^3-\ 3549392762332515437392431411304496332800000*n^2+ 1599372290285704010391960101128313241600000*n-\ 281642103346983785928030992601710592000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 150075/268435456*( 945312621599*n^29-395166290868697*n^28+78637445442782961*n^27-\ 9909208044843539733*n^26+887166105585655089285*n^25-60012514511554115375865*n^ 24+3184000671198816383512845*n^23-135735628059301533049457385*n^22+ 4724158566529088068939572015*n^21-135608115047129599241508035895*n^20+ 3228698652428087865370490217435*n^19-63856006102597905027555662684655*n^18+ 1045884294531809170144683869284935*n^17-14055106204882331460159236404953555*n^ 16+151862331343455248892777823578276015*n^15-\ 1260374164382842500108363673875535395*n^14+ 7036403994943842472233838840795419270*n^13-\ 9748741366073355372005855095674013860*n^12-\ 306265240925078616914119709894471848520*n^11+ 4200329807606383399221956075253108922960*n^10-\ 33222130105953385821507105553145400227424*n^9+ 188316416478494669047739990997057810833472*n^8-\ 803805107903420035376996320394444221892736*n^7+ 2604758873775904115701080605115272864414208*n^6-\ 6336053897542414112271141490453499660359680*n^5+ 11261220448408967299285730556451966495334400*n^4-\ 13963862913514161814014538947066514833408000*n^3+ 11176647414653403162811302920894820188160000*n^2-\ 4999144065608664915845590959918824816640000*n+ 885715856042928319884014707768138137600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 150075/ 134217728*(467563355432*n^29-194807484759061*n^28+38604389894501298*n^27-\ 4839144457875406329*n^26+430432334647906775730*n^25-28883597833674625686645*n^ 24+1517402868497216722608210*n^23-63912486220372011901426605*n^22+ 2191848483428688804337387770*n^21-61785859178312521336098049035*n^20+ 1438128632975922306829779016830*n^19-27630625095827367847969416341115*n^18+ 435374463434503374459135366967830*n^17-5534417588052213889459630169556615*n^16+ 54632835849063522873689132010567270*n^15-376542488724234163829767262158746735*n ^14+1005714279875878039586422007381605110*n^13+ 15706020578979490186801996136474379420*n^12-\ 291376968432604093190759242662441488360*n^11+ 2852815654219297538307666857816116764880*n^10-\ 19877574422141875796944256486833411336032*n^9+ 105187036329492539664459258424538471324736*n^8-\ 429772917205092230577110305694498703781248*n^7+ 1351479538233294792118388171311173526715904*n^6-\ 3218424185283361386698193108735871323555840*n^5+ 5636721102368070029376642941348502720307200*n^4-\ 6925454193106320677666591871534853423104000*n^3+ 5521310413443816758570561685759616942080000*n^2-\ 2474374409098629402603966280027283128320000*n+ 442857928021464159942007353884069068800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 50025/ 134217728*(2759151768487*n^30-1225669510202925*n^29+259220208713362790*n^28-\ 34716083066760739875*n^27+3302986238549527183038*n^26-237387928106596685127255* n^25+13376653865467710237730050*n^24-605319171247483235979330375*n^23+ 22343993785843424633775502380*n^22-679322863188302006993584990725*n^21+ 17091299620268030636516069691750*n^20-355732882644070099998186387746625*n^19+ 6083910786258026301891194085884610*n^18-83997187266098703475186849612622925*n^ 17+897216407391876452548096145919184950*n^16-\ 6533781590374425459026341166063962125*n^15+ 13039649018646075544441690375719194205*n^14+ 467921848223480171501750674629467474550*n^13-\ 8771966819471468902395602900313241317700*n^12+ 95599703300700771458720673620354725323000*n^11-\ 763171719048674565525957305856657562557872*n^10+ 4723958383283205676793395227966079370186400*n^9-\ 23051445264187813552248009157847586138613440*n^8+ 88671995711711559763113143844391433608368000*n^7-\ 266110720705796522933150496905320201450894848*n^6+ 610856104806151890922672098398469369611642880*n^5-\ 1039627208036406425256063218843240647656038400*n^4+ 1250122743702726604953045523223446270476288000*n^3-\ 982157080522750457602592343736343226286080000*n^2+ 436961947705019468571173342651836598845440000*n-\ 78385853259799156309735301637480225177600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 50025/134217728*(2691998986945*n^30-1188869785917909*n^29+249679410123001598*n^ 28-33159149050042983891*n^27+3123495887685767591382*n^26-\ 221833680911184600207495*n^25+12324027285129208008910290*n^24-\ 548260166926494637109669655*n^23+19822621047438866521119701160*n^22-\ 587355353022970246995286211205*n^21+14297625751474265259812383850310*n^20-\ 284619485770257845011860820800945*n^19+4560820463094436507159238980404090*n^18-\ 56487635562889775001227676590951325*n^17+477927890516202438091882359564982950*n ^16-1145682323915444059056403542352163325*n^15-\ 45196269541720962516027859403923749225*n^14+ 995116720089427466555727047323940933790*n^13-\ 12744726472224650255549429845185062355580*n^12+ 120303928627941855963328416482419621561560*n^11-\ 888412391926019336074376865775754975017040*n^10+ 5232811832855673517638831066121365203226144*n^9-\ 24667302905827924318610679439600427462299968*n^8+ 92524779542319137150971199101479985857872256*n^7-\ 272511366887836485985929694022404220369997312*n^6+ 616940443233764216020331793931112697593088000*n^5-\ 1039757782403356757193924138001254071137689600*n^4+ 1242663853005296341429987212125141343043584000*n^3-\ 973880730242856990131517980821968404152320000*n^2+ 433949625858778231152419939247221652848640000*n-\ 78385853259799156309735301637480225177600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 310155/1073741824*(6704838695263*n^31-3142820839348178*n^30+701101223660388020* n^29-98987965686179456560*n^28+9921963944614505297052*n^27-\ 750555711329216388754032*n^26+44456997593378710936106160*n^25-\ 2110690450765038985493874600*n^24+81507457983420315197221588770*n^23-\ 2580486034033026132168895474020*n^22+67077440727023861159641443664200*n^21-\ 1421911354202623052116639357739400*n^20+24059795052028124765699435772970140*n^ 19-306611196418637709239792390189447040*n^18+ 2380545224696498787599346104748925200*n^17+ 6090266032159948515966380713635913800*n^16-\ 586425722861300112207094619221393814505*n^15+ 11808874167354213956220248455405669348230*n^14-\ 159780424076190034143374810795291223432700*n^13+ 1660689288039315390804435538417624360789400*n^12-\ 13817750474412226947868122343380214130101328*n^11+ 93398949512060980798315135047313782115319968*n^10-\ 514402382092909918694226552426455325681625920*n^9+ 2299154598213445914625309727446117673799749760*n^8-\ 8259123576881402236894779788371504997317627392*n^7+ 23478964621837309260299821398158474917429635072*n^6-\ 51637989674327122049734298541883790158198824960*n^5+ 85027444867600728879177179016139521336670617600*n^4-\ 99808707772219024485449205577127643424555008000*n^3+ 77225412282409993097069704952193376879902720000*n^2-\ 34162714563861909857830092736797721441075200000*n+ 6169725224319675528895294709530701594624000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 310155/536870912*(3189160230187*n^31-1480964868609836*n^30+ 326793145059670370*n^29-45556489653016230610*n^28+4498691116979430783258*n^27-\ 334353817320928308280314*n^26+19389652915689272131060470*n^25-\ 897010419561837653193683250*n^24+33524412824847372138518461680*n^23-\ 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-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n -21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-63)/(2*n-45), 13485/1073741824*(249241824532919*n^32-120413301405566128*n^ 31+27587932596933740728*n^30-3982340464709521830560*n^29+ 405663624120779751564516*n^28-30928623726827236194886752*n^27+ 1824253091249829967665845112*n^26-84650192017041581116604489760*n^25+ 3096530910422368851077189528010*n^24-87565662887320218374821321360320*n^23+ 1777983036475688136143900198613720*n^22-18084669433304319528377965151263200*n^ 21-354961108412242122730283048534925180*n^20+ 24326284491821529297923773127529579360*n^19-\ 750963921711657480421006906079219749560*n^18+ 16711040274492791090365448975755013949600*n^17-\ 294880108665182271011756460766100681369865*n^16+ 4274839626734939138605440470131996295015280*n^15-\ 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n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 13485/1073741824*(438496042771768*n^33-221934571723741041*n ^32+53223310256744812096*n^31-8029612819315300140840*n^30+ 852626451208833108192352*n^29-67455336433792351273226844*n^28+ 4095146837622915588961037184*n^27-192573837615018060912343861800*n^26+ 6905545936905294428159557839120*n^25-175203126302519074514831705990790*n^24+ 2112907697185977848107566186599040*n^23+63421174138588057851789448774855800*n^ 22-5081813809845372980022272775373836960*n^21+ 194465097532200364438322099392151052420*n^20-\ 5419098558815120870330996717727825697920*n^19+ 120517399089707159111977664600952134442600*n^18-\ 2216187401649313865073008077810408552358280*n^17+ 34272443136722913496972302863661403289231535*n^16-\ 449571507303287212142053838439378541182093760*n^15+ 5022269869747696535403246269719075184197398000*n^14-\ 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/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*(1838237612480174*n^33-\ 905983362825972255*n^32+210321924596178650816*n^31-30455065492092614412600*n^30 +3063556608570595184027816*n^29-224661820052750581589878020*n^28+ 12136837608519367042256534544*n^27-462877415022399919360230555000*n^26+ 9798403062608180334523697004660*n^25+146624272033733522105158873484550*n^24-\ 25148936230683266282128061736546960*n^23+1378193784821372275613600244125625000* n^22-52375173426706407595098229069428929880*n^21+ 1560709419547355157269371437161836667100*n^20-\ 38143978372170792289212066674276675563920*n^19+ 781329022581514896467784442100054704707000*n^18-\ 13572121435011131413538903498431633821869490*n^17+ 201237249729655102128977127813274637771875425*n^16-\ 2555377862522711614557154964000430435349834160*n^15+ 27818165505243534807017116963325016202194018000*n^14-\ 259395981902928166296523763073973285359962999744*n^13+ 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)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*(2865047727785284*n^34-\ 1449083574880252832*n^33+342343958525620279261*n^32-49731660061565321497328*n^ 31+4886990292744239873136856*n^30-330680619959264379177689888*n^29+ 14034025054051768366898841324*n^28-135217155131175039447444550752*n^27-\ 33522087572120509363709743189440*n^26+3338691632741048847882802911705120*n^25-\ 198145895983885765132069535298727410*n^24+8835217838668400218324943237573023680 *n^23-317190470922244689096389793890340283080*n^22+ 9460750989466014943196610109944933831840*n^21-\ 238509159519164252068339170359116221608820*n^20+ 5135114190200580977660498020815616787499360*n^19-\ 95025818535908754195292724885384553552290340*n^18+ 1517189604470769911834781062980279444290328320*n^17-\ 20940417836024747270575581020062738967712471235*n^16+ 249952609054343947218281621890643814061010527280*n^15-\ 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2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31) /(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 1/ 2147483648*(-44658806881388974003*n^36+32928997105208171084454*n^35-\ 11493884177770597907564085*n^34+2538194147161678472417414730*n^33-\ 399451259684597520826686384894*n^32+47814937572756328449703929012312*n^31-\ 4535063764840730191207999724366900*n^30+350445226475907351264746293342256520*n^ 29-22508998833382399440843989700942002898*n^28+ 1219674443098363619004927042917745795444*n^27-\ 56388341617741615929860784555735804006390*n^26+ 2243740893585847358470491590878224170493900*n^25-\ 77360786885075288153396288530585437424110120*n^24+ 2323214861798732941869281131784542535900955960*n^23-\ 61008092560310065722967257443248146829196489700*n^22+ 1404962727299681621892882444881845933232072903400*n^21-\ 28429655201993402074810990734903378225511552047555*n^20+ 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33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69), -37/137438953472*(1756356525483339649633*n^39-\ 1351080942521906390987634*n^38+501555933078994468181166062*n^37-\ 119695239521890205866617977676*n^36+20639319600684302645889698984229*n^35-\ 2740112100232875735141502017433962*n^34+291449331584915269178000006565533816*n^ 33-25516681931047024986158544535497755568*n^32+ 1874933556031457088687495626397245682898*n^31-\ 117309493362686210387411390401381543117764*n^30+ 6319312594214360185505500008312959516007252*n^29-\ 295616749046898534098340125376653684700599496*n^28+ 12090342343490241582998231867936679114817042530*n^27-\ 434609670570728985671794488101381440055133649860*n^26+ 13788228596918618483158625368219206956186548123080*n^25-\ 387298606293542847052310708825901705219766314837840*n^24+ 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/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-\ 71)/(2*n-69)/(2*n-77), 1/68719476736*(-34688567848845581724173*n^39+ 26554137647111964959504604*n^38-9812304974202660050682985372*n^37+ 2331561578466705191848148881506*n^36-400401272665588298122159854848499*n^35+ 52954725530314869246441840100001922*n^34-5612248353606354414640424832303677896* n^33+489701787583242074893955689057159468008*n^32-\ 35868951663173606593930445360173437917138*n^31+ 2237579890305245234983439981153659836510384*n^30-\ 120201794404206680741841972302718503479519512*n^29+ 5608478516217868007477006576549954052141863676*n^28-\ 228825791216965226768525482979912444065020893430*n^27+ 8207066869825518827330491606501640924545205103660*n^26-\ 259829059094564577053330626573782485054195304618480*n^25+ 7284185960508982480020516422334854110788365431040040*n^24-\ 181254797439375445944889345549019075340092264215398705*n^23+ 4009805486258779854503400216266750525911246987497355140*n^22-\ 78942567581123077176674369445484909496970004025736575020*n^21+ 1383631705720180569318456177206864917768650075899965740210*n^20-\ 21585763258330849591561257853666313194031732605335402196967*n^19+ 299516313202950099790807313600818739013837023462301834120066*n^18-\ 3691441751726222485559195554678181533232921356392310014947688*n^17+ 40331655319829779183566848624592477906655357675412931275910624*n^16-\ 389616912963350215925177214302584389364532484938957284662630176*n^15+ 3316830567598882614773595106631438786009296875304243562309147328*n^14-\ 24779723711035750622469307151931634492473632330392992087041903104*n^13+ 161634590710391280166269730628575588670810994450267895148469460992*n^12-\ 914778908524787908041328385914346789655832571696895386374100451072*n^11+ 4457714723527105863049691605067575271559438263231921690444803293696*n^10-\ 18527981462289419854799902093467456982042764731140737762190233212928*n^9+ 64920357593723817146492517277609024155145268347344785013506270674944*n^8-\ 188960153033868991358273902866755566395854335574696937567574962339840*n^7+ 448303756219886377653264324348107746455720879226028369887225606963200*n^6-\ 845521814940041419804285793170192608527131042349893715460478402560000*n^5+ 1224960523371528381527963233963859849530865072874411922330704936960000*n^4-\ 1297105485736420091828421300531663462001257294383718129471691161600000*n^3+ 928495057264460752212087984736734250245858340514786954390274048000000*n^2-\ 390544445703324244490900872749076700303702918674760585340518400000000*n+ 69396182488014914100186212218815419615781393358550477045760000000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2 *n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/ (2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55 )/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-\ 71)/(2*n-69)/(2*n-77), -3/137438953472*(48280450676412823097389*n^40-\ 38746102153807381325820700*n^39+15026413338124526295170747690*n^38-\ 3751566466260588240890229030400*n^37+677724182606416198522235888599873*n^36-\ 94402563528235031632124418386389860*n^35+10550887070926825222769920410626032820 *n^34-972148396295644998235025076571905647600*n^33+ 75295866773175472387554945986504919911722*n^32-\ 4974111438966594599438657970350488739147480*n^31+ 283399400395975046083734304552431943253174540*n^30-\ 14047143222850508239397248761483702514178688000*n^29+ 609886016305993654627567278379870182836173746426*n^28-\ 23319809220442721304513976463724491408132998644360*n^27+ 788611454924609993338219960106305513854571622834400*n^26-\ 23664575614616497104268067448838199580161870399292000*n^25+ 631713346554964529598321932627545040936871722434610505*n^24-\ 15028391439441988084901662178030024431542097875945021900*n^23+ 318999852071626329807350224368406987483830267072082566650*n^22-\ 6045314756515106732639990976784358447488188855204619376000*n^21+ 102288394823374241277879176231795287259783761990437292919941*n^20-\ 1544581950994834188473264100977841529021180456262193528914900*n^19+ 20794079332990789463539307474894314541260904023579310102979260*n^18-\ 249196041471198639266710178413398689832761433820749698535447600*n^17+ 2652717821198146921196356275109778158651606954956445176936252832*n^16-\ 25014402404725046979667525420069612482603138842682509588154701440*n^15+ 208224628231930609405860297364811698821059132169645755625115575680*n^14-\ 1523560667273317187844643580307930199836227634414270702409948710400*n^13+ 9747767799359660696252030551119828708075906262360680606807192471808*n^12-\ 54188644096726145969180385986837143083469660403149315224106745963520*n^11+ 259719755861969246804674592671900510875440946216664987934021636213760*n^10-\ 1063092157390985163535700674498735636632410020320042119082312058368000*n^9+ 3672805816524557253296303191653754057910584974701949523491635803029504*n^8-\ 10552729178441837084795788054506037473664499655409378078285169733795840*n^7+ 24741809971575558919970253342750727232292839138816997475799023661875200*n^6-\ 46166262491371602841744407824912542314696449911412353818940721725440000*n^5+ 66242290509437178784248436637789514801550349164954927644093563535360000*n^4-\ 69547403473467106752537226144881762974114722376471168989105068441600000*n^3+ 49417474855629656974446271052538270047032738261639952362245193728000000*n^2-\ 20659964537180530260550787857721837873464248759428828157601382400000000*n+ 3654865611035452142609807176857612099764486716883658457743360000000000)/(2*n-5) /(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/ (2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59 )/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2* n-45)/(2*n-71)/(2*n-69)/(2*n-77), (40455674366551592536080*(n-39)!*(2*n-83)!*(n +1)+60683511549827388804120*(n-40)!*((n+1)*(2*n-82)!-1/60683511549827388804120* binomial(2*n,n)*(n-42)!*(n-39)!))/(n-42)!/(n-40)!/(n-39)!/binomial(2*n,n), (( 40455674366551592536080*n+40455674366551592536080)*(2*n-83)!-(n-42)!*(n-40)!* binomial(2*n,n))/(n-42)!/(n-40)!/binomial(2*n,n)] The limits, as n goes to infinity are 2199023255509 274877906831 2199023245833 2199023184343 35184365692093 [-------------, ------------, -------------, -------------, --------------, 2199023255552 274877906944 2199023255552 2199023255552 35184372088832 17592171374141 70368515229335 70367963011295 281465463189245 --------------, --------------, --------------, ---------------, 17592186044416 70368744177664 70368744177664 281474976710656 35181091308915 70352142987075 70329851134765 4498156802193885 --------------, --------------, --------------, ----------------, 35184372088832 70368744177664 70368744177664 4503599627370496 2246209016546655 8963793912817845 8927224101445905 141867791686469925 ----------------, ----------------, ----------------, ------------------, 2251799813685248 9007199254740992 9007199254740992 144115188075855872 8771196320807175 138026567218562175 134667249321923625 ----------------, ------------------, ------------------, 9007199254740992 144115188075855872 144115188075855872 2079539245529295765 989133991193648985 3692895899570221515 -------------------, -------------------, -------------------, 2305843009213693952 1152921504606846976 4611686018427387904 3361026003826412715 739139892097161435 2478863420429514639 -------------------, -------------------, -------------------, 4611686018427387904 1152921504606846976 4611686018427387904 1931758430459227737 1323863997158908957 10695644980112044357 -------------------, -------------------, --------------------, 4611686018427387904 4611686018427387904 73786976294838206464 -136328582778319229 -22795298683755462055 -44658806881388974003 --------------------, ---------------------, ---------------------, 36893488147419103232 147573952589676412928 147573952589676412928 -2094096861215799823181 -338018997685301573677 -3242435911129423853741 -----------------------, ----------------------, -----------------------, 4722366482869645213696 590295810358705651712 4722366482869645213696 -3696986785053566699171 -64985191442883567036421 -34688567848845581724173 -----------------------, ------------------------, ------------------------, 4722366482869645213696 75557863725914323419136 37778931862957161709568 -144841352029238469292167 -148587247803919172304767 -------------------------, -------------------------, 151115727451828646838272 151115727451828646838272 -601934430159405112819583 -------------------------] 604462909807314587353088 and in Maple notation [2199023255509/2199023255552, 274877906831/274877906944, 2199023245833/ 2199023255552, 2199023184343/2199023255552, 35184365692093/35184372088832, 17592171374141/17592186044416, 70368515229335/70368744177664, 70367963011295/ 70368744177664, 281465463189245/281474976710656, 35181091308915/35184372088832, 70352142987075/70368744177664, 70329851134765/70368744177664, 4498156802193885/ 4503599627370496, 2246209016546655/2251799813685248, 8963793912817845/ 9007199254740992, 8927224101445905/9007199254740992, 141867791686469925/ 144115188075855872, 8771196320807175/9007199254740992, 138026567218562175/ 144115188075855872, 134667249321923625/144115188075855872, 2079539245529295765/ 2305843009213693952, 989133991193648985/1152921504606846976, 3692895899570221515/4611686018427387904, 3361026003826412715/ 4611686018427387904, 739139892097161435/1152921504606846976, 2478863420429514639/4611686018427387904, 1931758430459227737/ 4611686018427387904, 1323863997158908957/4611686018427387904, 10695644980112044357/73786976294838206464, -136328582778319229/ 36893488147419103232, -22795298683755462055/147573952589676412928, -\ 44658806881388974003/147573952589676412928, -2094096861215799823181/ 4722366482869645213696, -338018997685301573677/590295810358705651712, -\ 3242435911129423853741/4722366482869645213696, -3696986785053566699171/ 4722366482869645213696, -64985191442883567036421/75557863725914323419136, -\ 34688567848845581724173/37778931862957161709568, -144841352029238469292167/ 151115727451828646838272, -148587247803919172304767/151115727451828646838272, -\ 601934430159405112819583/604462909807314587353088] and in floating point [1.000000000, .9999999996, .9999999956, .9999999676, .9999998182, .9999991661, .9999967464, .9999888990, .9999662012, .9999067546, .9997640829, .9994472966, .\ 9987914500, .9975171873, .9951810390, .9911209743, .9844055549, .9737984109, .9\ 577517058, .9344417554, .9018563871, .8579369777, .8007691514, .7288063390, .64\ 11016614, .5375178211, .4188833374, .2870672444, .1449530190, -.3695193640e-2, -.1544669522, -.3026198465, -.4434422590, -.5726264557, -.6866125115, -.7828674\ 031, -.8600718474, -.9181987457, -.9584796664, -.9832679252, -.9958169813] The cut off is at j=, 30 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 43], vs. those in the, 2, -th row from j=1 to j=, 42, are as follws 21 20 19 [43 (25570037855 n - 5638193346912 n + 582165836814625 n 18 17 - 37404714352356960 n + 1676146313879363130 n 16 15 - 55645620839254539072 n + 1419046661360870789330 n 14 13 - 28442799003445432912320 n + 454709419601243946749155 n 12 11 - 5850145754194297442726112 n + 60848046368621935451149485 n 10 9 - 512076239840030155504512480 n + 3477589915642978303346907380 n 8 7 - 18938783082121048569707958912 n + 81846614356794273774119253040 n 6 5 - 276197337170487791827461450240 n + 709341354524948903499759756480 n 4 3 - 1319103874408658830213972948992 n + 1521192414336000920089407083520 n 2 - 62077534386441477352501248000 n - 3992754761241684095766011904000 n + 6875025484120320168920678400000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 1763 ( 22 21 20 2494637839 n - 603702356807 n + 68623121234129 n 19 18 17 - 4870368743163385 n + 241985135618404794 n - 8944659814591509162 n 16 15 + 255163630751370875314 n - 5751442178175311186690 n 14 13 + 104022292860386596570259 n - 1524521807891077787138347 n 12 11 + 18207249734165040449937309 n - 177585443717140728970145445 n 10 9 + 1413240202346743616057729044 n - 9138962318843905301568027092 n 8 7 + 47654094288104539168189141424 n - 197817683038235509939546318000 n 6 5 + 639302282129077379174566083264 n - 1534340153996412129788313638592 n 4 3 + 2371466084734609812848138701824 n - 725028784009368006259578516480 n 2 - 6748892077531499080265016115200 n + 13533618893038559978613350400000 n - 335367096786357081410764800000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 41 ( 22 21 20 53634713423 n - 12979600596391 n + 1475397083467846 n 19 18 - 104712923503635755 n + 5202679802171447673 n 17 16 - 192310122775297761696 n + 5486012979927284932556 n 15 14 - 123655680896808611195590 n + 2236462347205565501846593 n 13 12 - 32776497369873702007954391 n + 391430595490438069941217206 n 11 10 - 3817357412765347760246512575 n + 30367342188535801016448227303 n 9 8 - 196151522410874911504085242066 n + 1019287059243010332592031288536 n 7 6 - 4187248996856967749234157498560 n + 13109099550924616962987785405808 n 5 - 28433061436534497601458177495456 n 4 + 28565298455304966998106544793856 n 3 + 48540716024355082639103063892480 n 2 - 197664044795507156485291764940800 n + 146597271033802315992852787200000 n - 7210392580906677250331443200000)/( 524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 23 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 943 (4663888052 n 22 21 20 - 1233598357579 n + 153690254921269 n - 11992115283223460 n 19 18 + 657279666609827877 n - 26901735846793668399 n 17 16 + 853300347494657930384 n - 21485965960954276335400 n 15 14 + 436399284600633162603982 n - 7225358743107225386333129 n 13 12 + 98149226577558211781907909 n - 1097317709711018878512606540 n 11 10 + 10098168923933846550108512297 n - 76251460512560261865815877229 n 9 8 + 468922576703584537762167888454 n - 2314199962583307489793513675880 n 7 6 + 8896507809884007690170442456192 n - 24896360743657430571030880595664 n 5 4 + 41358267744067612209446690901984 n + 2090103947978077190839287041280 n 3 - 184638460718773871586579763238400 n 2 + 342455086343250353962688274432000 n - 197773492888992240152297164800000 n + 14107289832208716359344128000000)/ (524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 23 ( 23 22 21 382438788089 n - 101155045083403 n + 12602594880802333 n 20 19 - 983352325995495995 n + 53896783897931677164 n 18 17 - 2205927629286451784868 n + 69969498152182405834838 n 16 15 - 1761780129096977120525350 n + 35781333363051102404708449 n 14 13 - 592342464757878475695411203 n + 8043733511643660207424123713 n 12 11 - 89858845826145348166825647855 n + 825388297935423288592300046954 n 10 - 6205305970133113492036176314878 n 9 + 37779543719696484041454287947228 n 8 - 182287687748297674272635977369760 n 7 + 666378982847588435811364771398144 n 6 - 1656527997505630981517189481879648 n 5 + 1838682490958676040232044906941888 n 4 + 3599515874923845130512019418088960 n 3 - 17021144456048246214097483349068800 n 2 + 24260692001163786101372076417024000 n - 12720130844540505171776544153600000 n + 1156797766241114741466218496000000)/(1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) 24 23 (2 n - 29) (2 n - 45)), 851 (82689436633 n - 23814543958812 n 22 21 20 + 3238856492460862 n - 276650419879569648 n + 16649704697946885103 n 19 18 - 750796622134165098492 n + 26335903441993876974232 n 17 16 - 736376162999114228781168 n + 16684762047443212561141303 n 15 14 - 309739817838970465454357652 n + 4744181786311696437308247382 n 13 12 - 60169746662693549157981567408 n + 632121897877477973926759033393 n 11 - 5482167216709518070807731425172 n 10 + 38912325508179034536987325055092 n 9 - 222209854650069616635399961295088 n 8 + 987695757708281652904132168175728 n 7 - 3189586660790542354913274628596672 n 6 + 6200558710204985859413082264426432 n 5 - 757746659141260041786150933266688 n 4 - 33714002974969646235814765346012160 n 3 + 93490094456326886530763814426316800 n 2 - 111579177342344273253587856973824000 n + 54753210469453592733682190745600000 n - 5877783244684583010693218304000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), 851 (82689318757 n 23 22 21 - 23814471818700 n + 3238835638546198 n - 276646635325837920 n 20 19 + 16649221343993899507 n - 750750466041862537020 n 18 17 + 26332486293635458659448 n - 736175460395917526490000 n 16 15 + 16675272239466888870570187 n - 309375401748177162399955140 n 14 13 + 4732768398624419992349895118 n - 59878114964634311721707866080 n 12 11 + 626063895713354536935622457677 n - 5380636451827924200953787871860 n 10 + 37556359110602702669591975442628 n 9 - 208045108438400422239728036090160 n 8 + 875061100511293353666681631076272 n 7 - 2534637633194302252298518452665280 n 6 + 3582605484749621895466136687356608 n 5 + 5714325636308254789503201872444160 n 4 - 41450748867797998835626155539942400 n 3 + 92755134658902470338509409202688000 n 2 - 101110481531040947022403182305280000 n + 48346882374859388304973148160000000 n - 5877783244684583010693218304000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 4255 (33075571162 n 24 23 22 - 10336047589025 n + 1528887941962600 n - 142394085580335950 n 21 20 + 9370159386244169590 n - 463398132633985191335 n 19 18 + 17885510779071685575100 n - 552239693715272526227000 n 17 16 + 13870710950350634597773270 n - 286621565420098585260890735 n 15 14 + 4907431886205097735943455600 n - 69869891785020955589562136550 n 13 12 + 827262282386580462849768803530 n - 8112844782527532638076794913785 n 11 + 65281031946154507149817611429100 n 10 - 423491237889336049054590122914100 n 9 + 2145542745846398130224511821338960 n 8 - 7973838980275460172652328244139760 n 7 + 18456912117172356892768656210561600 n 6 - 6878628324828064596303341238974400 n 5 - 125762117400093430905997918797856512 n 4 + 493651445168048594594615043383024640 n 3 - 930685771800170048060568806275584000 n 2 + 932300675101092769997712188940288000 n - 433666490049450061855259435827200000 n + 57602275797908913504793539379200000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 115 (1223779033219 n 24 23 22 - 382423535128475 n + 56565968675979550 n - 5268070695576040850 n 21 20 + 346632411082947209605 n - 17139834623061854533445 n 19 18 + 661339838697838291126300 n - 20408701394411226190098200 n 17 16 + 512108703266146285591936765 n - 10563807096262588951589502245 n 15 14 + 180326251519623640461166665550 n - 2554299877993486217052258333050 n 13 + 29987014162516474078935899408635 n 12 - 290052802527052533530857140787595 n 11 + 2283375894510447666632474762603800 n 10 - 14309305984938349580093916187619900 n 9 + 68538240462397135183967833433117920 n 8 - 229813979956038638724008612523891920 n 7 + 398281115396010763478965495563292800 n 6 + 596602104179918724816506041252536000 n 5 - 6218654281187346566962748265741606144 n 4 + 19534084485384785091236374437375943680 n 3 - 33689520351125198404892090486650368000 n 2 + 32308444526862008295986824857747456000 n - 14930453280625950649278538159718400000 n + 2131284204522629799677360957030400000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 345 (1631644367317 n 25 24 23 - 551469622730871 n + 88413142347906350 n - 8945443827832933850 n 22 21 + 641051690071779725115 n - 34615991859553445446105 n 20 19 + 1462880562605041003361080 n - 49600338899001561163607400 n 18 17 + 1372152052486188929327052395 n - 31322144547368137007173859945 n 16 15 + 594120350613408189987900189030 n - 9395439729643077900005533312450 n 14 + 123839361532513770766781442261205 n 13 - 1354652727149335947692640291871575 n 12 + 12182442572533076368942112928308180 n 11 - 88564676483267307402331931482017500 n 10 + 505237097839900380220593053152929360 n 9 - 2133124005505977929328882459609388240 n 8 + 5700362991119634180841528166711340480 n 7 - 2600819152190877813308234074611028800 n 6 - 55418171465201910394478099628836731392 n 5 + 281105035669547866825392842447566196736 n 4 - 738110294888234549008211453737993605120 n 3 + 1162091844081713294033445846523238400000 n 2 - 1059802417865896933323155528311087104000 n + 482624488505521280123959354830028800000 n - 72463662953769413189030272539033600000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 345 ( 26 25 24 407873701226 n - 137845457398610 n + 22097209987854495 n 23 22 - 2235306836210631040 n + 160134413510210495200 n 21 20 - 8642301497927433606270 n + 364895825724856246940405 n 19 18 - 12354041583272794486049540 n + 340968643560009669638795790 n 17 16 - 7754983567719693760433419870 n + 146277315429377228995382758105 n 15 - 2293894871722837660167074260440 n 14 + 29863588031287529290714402823420 n 13 - 320858880271960217714554525239170 n 12 + 2811706055267460941091237618411555 n 11 - 19679453812448628485100068356069540 n 10 + 105827191702370722295472856148198860 n 9 - 400924097506791738443556187519243920 n 8 + 776705712801482535960629068795518480 n 7 + 1674274046692076197691728846998173760 n 6 - 19609309237990742145779896789585154496 n 5 + 78991717219883807326216430167513347840 n 4 - 189606994445662122360696435807514583040 n 3 + 284600398634266356441476030382265036800 n 2 - 253447315250349393225931166619125760000 n + 115403910896785405199071753062236160000 n - 18115915738442353297257568134758400000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 32085 ( 27 26 25 8769654248 n - 3195775965150 n + 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(2 n - 85)! (n + 1) + 234266579471426663755440 ((n + 1) (2 n - 84)! - 1/234266579471426663755440 binomial(2 n, n) (n - 43)! (n - 40)!) (n - 41)!)/((n - 41)! (n - 43)! (n - 40)! binomial(2 n, n)), ( (156177719647617775836960 n + 156177719647617775836960) (2 n - 85)! - (n - 43)! (n - 41)! binomial(2 n, n))/((n - 43)! (n - 41)! binomial(2 n, n))] and in Maple notation [43/524288*(25570037855*n^21-5638193346912*n^20+582165836814625*n^19-\ 37404714352356960*n^18+1676146313879363130*n^17-55645620839254539072*n^16+ 1419046661360870789330*n^15-28442799003445432912320*n^14+ 454709419601243946749155*n^13-5850145754194297442726112*n^12+ 60848046368621935451149485*n^11-512076239840030155504512480*n^10+ 3477589915642978303346907380*n^9-18938783082121048569707958912*n^8+ 81846614356794273774119253040*n^7-276197337170487791827461450240*n^6+ 709341354524948903499759756480*n^5-1319103874408658830213972948992*n^4+ 1521192414336000920089407083520*n^3-62077534386441477352501248000*n^2-\ 3992754761241684095766011904000*n+6875025484120320168920678400000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29), 1763/1048576*(2494637839*n^22-603702356807*n^21+68623121234129* n^20-4870368743163385*n^19+241985135618404794*n^18-8944659814591509162*n^17+ 255163630751370875314*n^16-5751442178175311186690*n^15+104022292860386596570259 *n^14-1524521807891077787138347*n^13+18207249734165040449937309*n^12-\ 177585443717140728970145445*n^11+1413240202346743616057729044*n^10-\ 9138962318843905301568027092*n^9+47654094288104539168189141424*n^8-\ 197817683038235509939546318000*n^7+639302282129077379174566083264*n^6-\ 1534340153996412129788313638592*n^5+2371466084734609812848138701824*n^4-\ 725028784009368006259578516480*n^3-6748892077531499080265016115200*n^2+ 13533618893038559978613350400000*n-335367096786357081410764800000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-35)/(2*n-29), 41/524288*(53634713423*n^22-12979600596391*n^21+ 1475397083467846*n^20-104712923503635755*n^19+5202679802171447673*n^18-\ 192310122775297761696*n^17+5486012979927284932556*n^16-123655680896808611195590 *n^15+2236462347205565501846593*n^14-32776497369873702007954391*n^13+ 391430595490438069941217206*n^12-3817357412765347760246512575*n^11+ 30367342188535801016448227303*n^10-196151522410874911504085242066*n^9+ 1019287059243010332592031288536*n^8-4187248996856967749234157498560*n^7+ 13109099550924616962987785405808*n^6-28433061436534497601458177495456*n^5+ 28565298455304966998106544793856*n^4+48540716024355082639103063892480*n^3-\ 197664044795507156485291764940800*n^2+146597271033802315992852787200000*n-\ 7210392580906677250331443200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 943/ 524288*(4663888052*n^23-1233598357579*n^22+153690254921269*n^21-\ 11992115283223460*n^20+657279666609827877*n^19-26901735846793668399*n^18+ 853300347494657930384*n^17-21485965960954276335400*n^16+ 436399284600633162603982*n^15-7225358743107225386333129*n^14+ 98149226577558211781907909*n^13-1097317709711018878512606540*n^12+ 10098168923933846550108512297*n^11-76251460512560261865815877229*n^10+ 468922576703584537762167888454*n^9-2314199962583307489793513675880*n^8+ 8896507809884007690170442456192*n^7-24896360743657430571030880595664*n^6+ 41358267744067612209446690901984*n^5+2090103947978077190839287041280*n^4-\ 184638460718773871586579763238400*n^3+342455086343250353962688274432000*n^2-\ 197773492888992240152297164800000*n+14107289832208716359344128000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 23/1048576*(382438788089*n^23-\ 101155045083403*n^22+12602594880802333*n^21-983352325995495995*n^20+ 53896783897931677164*n^19-2205927629286451784868*n^18+69969498152182405834838*n ^17-1761780129096977120525350*n^16+35781333363051102404708449*n^15-\ 592342464757878475695411203*n^14+8043733511643660207424123713*n^13-\ 89858845826145348166825647855*n^12+825388297935423288592300046954*n^11-\ 6205305970133113492036176314878*n^10+37779543719696484041454287947228*n^9-\ 182287687748297674272635977369760*n^8+666378982847588435811364771398144*n^7-\ 1656527997505630981517189481879648*n^6+1838682490958676040232044906941888*n^5+ 3599515874923845130512019418088960*n^4-17021144456048246214097483349068800*n^3+ 24260692001163786101372076417024000*n^2-12720130844540505171776544153600000*n+ 1156797766241114741466218496000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/( 2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/ (2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45 ), 851/4194304*(82689436633*n^24-23814543958812*n^23+3238856492460862*n^22-\ 276650419879569648*n^21+16649704697946885103*n^20-750796622134165098492*n^19+ 26335903441993876974232*n^18-736376162999114228781168*n^17+ 16684762047443212561141303*n^16-309739817838970465454357652*n^15+ 4744181786311696437308247382*n^14-60169746662693549157981567408*n^13+ 632121897877477973926759033393*n^12-5482167216709518070807731425172*n^11+ 38912325508179034536987325055092*n^10-222209854650069616635399961295088*n^9+ 987695757708281652904132168175728*n^8-3189586660790542354913274628596672*n^7+ 6200558710204985859413082264426432*n^6-757746659141260041786150933266688*n^5-\ 33714002974969646235814765346012160*n^4+93490094456326886530763814426316800*n^3 -111579177342344273253587856973824000*n^2+54753210469453592733682190745600000*n -5877783244684583010693218304000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47) /(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13) /(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), 851/4194304*(82689318757*n^24-23814471818700*n^23+ 3238835638546198*n^22-276646635325837920*n^21+16649221343993899507*n^20-\ 750750466041862537020*n^19+26332486293635458659448*n^18-\ 736175460395917526490000*n^17+16675272239466888870570187*n^16-\ 309375401748177162399955140*n^15+4732768398624419992349895118*n^14-\ 59878114964634311721707866080*n^13+626063895713354536935622457677*n^12-\ 5380636451827924200953787871860*n^11+37556359110602702669591975442628*n^10-\ 208045108438400422239728036090160*n^9+875061100511293353666681631076272*n^8-\ 2534637633194302252298518452665280*n^7+3582605484749621895466136687356608*n^6+ 5714325636308254789503201872444160*n^5-41450748867797998835626155539942400*n^4+ 92755134658902470338509409202688000*n^3-101110481531040947022403182305280000*n^ 2+48346882374859388304973148160000000*n-5877783244684583010693218304000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 4255/4194304*(33075571162*n ^25-10336047589025*n^24+1528887941962600*n^23-142394085580335950*n^22+ 9370159386244169590*n^21-463398132633985191335*n^20+17885510779071685575100*n^ 19-552239693715272526227000*n^18+13870710950350634597773270*n^17-\ 286621565420098585260890735*n^16+4907431886205097735943455600*n^15-\ 69869891785020955589562136550*n^14+827262282386580462849768803530*n^13-\ 8112844782527532638076794913785*n^12+65281031946154507149817611429100*n^11-\ 423491237889336049054590122914100*n^10+2145542745846398130224511821338960*n^9-\ 7973838980275460172652328244139760*n^8+18456912117172356892768656210561600*n^7-\ 6878628324828064596303341238974400*n^6-125762117400093430905997918797856512*n^5 +493651445168048594594615043383024640*n^4-930685771800170048060568806275584000* n^3+932300675101092769997712188940288000*n^2-\ 433666490049450061855259435827200000*n+57602275797908913504793539379200000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 115/4194304*( 1223779033219*n^25-382423535128475*n^24+56565968675979550*n^23-\ 5268070695576040850*n^22+346632411082947209605*n^21-17139834623061854533445*n^ 20+661339838697838291126300*n^19-20408701394411226190098200*n^18+ 512108703266146285591936765*n^17-10563807096262588951589502245*n^16+ 180326251519623640461166665550*n^15-2554299877993486217052258333050*n^14+ 29987014162516474078935899408635*n^13-290052802527052533530857140787595*n^12+ 2283375894510447666632474762603800*n^11-14309305984938349580093916187619900*n^ 10+68538240462397135183967833433117920*n^9-229813979956038638724008612523891920 *n^8+398281115396010763478965495563292800*n^7+ 596602104179918724816506041252536000*n^6-6218654281187346566962748265741606144* n^5+19534084485384785091236374437375943680*n^4-\ 33689520351125198404892090486650368000*n^3+ 32308444526862008295986824857747456000*n^2-\ 14930453280625950649278538159718400000*n+2131284204522629799677360957030400000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 345/8388608*( 1631644367317*n^26-551469622730871*n^25+88413142347906350*n^24-\ 8945443827832933850*n^23+641051690071779725115*n^22-34615991859553445446105*n^ 21+1462880562605041003361080*n^20-49600338899001561163607400*n^19+ 1372152052486188929327052395*n^18-31322144547368137007173859945*n^17+ 594120350613408189987900189030*n^16-9395439729643077900005533312450*n^15+ 123839361532513770766781442261205*n^14-1354652727149335947692640291871575*n^13+ 12182442572533076368942112928308180*n^12-88564676483267307402331931482017500*n^ 11+505237097839900380220593053152929360*n^10-\ 2133124005505977929328882459609388240*n^9+5700362991119634180841528166711340480 *n^8-2600819152190877813308234074611028800*n^7-\ 55418171465201910394478099628836731392*n^6+ 281105035669547866825392842447566196736*n^5-\ 738110294888234549008211453737993605120*n^4+ 1162091844081713294033445846523238400000*n^3-\ 1059802417865896933323155528311087104000*n^2+ 482624488505521280123959354830028800000*n-\ 72463662953769413189030272539033600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 345/2097152*(407873701226*n^26-\ 137845457398610*n^25+22097209987854495*n^24-2235306836210631040*n^23+ 160134413510210495200*n^22-8642301497927433606270*n^21+364895825724856246940405 *n^20-12354041583272794486049540*n^19+340968643560009669638795790*n^18-\ 7754983567719693760433419870*n^17+146277315429377228995382758105*n^16-\ 2293894871722837660167074260440*n^15+29863588031287529290714402823420*n^14-\ 320858880271960217714554525239170*n^13+2811706055267460941091237618411555*n^12-\ 19679453812448628485100068356069540*n^11+105827191702370722295472856148198860*n ^10-400924097506791738443556187519243920*n^9+ 776705712801482535960629068795518480*n^8+1674274046692076197691728846998173760* n^7-19609309237990742145779896789585154496*n^6+ 78991717219883807326216430167513347840*n^5-\ 189606994445662122360696435807514583040*n^4+ 284600398634266356441476030382265036800*n^3-\ 253447315250349393225931166619125760000*n^2+ 115403910896785405199071753062236160000*n-\ 18115915738442353297257568134758400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 32085/2097152*(8769654248*n^27-3195775965150* n^26+553460022216972*n^25-60610082316481815*n^24+4710906561922706340*n^23-\ 276488617549000082460*n^22+12727077366086978028420*n^21-\ 471022846610991823410405*n^20+14252024160733679388757260*n^19-\ 356497157928872513774145210*n^18+7422252317680675049423187300*n^17-\ 129024646435981657621302849105*n^16+1871925631637927078937137639100*n^15-\ 22569984277242080205872558499360*n^14+224103093977913299463322223464860*n^13-\ 1802900353258783992704269112156355*n^12+11411984303372457537160788732535020*n^ 11-53489793533562868100103004410522060*n^10+ 155486646367793156860518843967609840*n^9-8747006342243856092866525038539280*n^8 -2676561214519071330136955122351013568*n^7+ 15824383398724639711138273265119266240*n^6-\ 53166790398896997938933673014953627392*n^5+ 116116340469165048720741953269124136960*n^4-\ 164482091966038901425713198748568678400*n^3+ 141615227744318860925128860157544448000*n^2-\ 63852489062155666785347343191162880000*n+10324124022983276610265065711206400000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 45), 96255/2097152*(2921933557*n^27-1064514914689*n^26+184283590564424*n^25-\ 20168596921863975*n^24+1566119140476031115*n^23-91786850609218641045*n^22+ 4216161027216789532310*n^21-155558967482695655819655*n^20+ 4686094156826971918667495*n^19-116487409691692916153103255*n^18+ 2404292480949346101242343020*n^17-41300253846909368287441576605*n^16+ 589586584087182067717328965805*n^15-6954664994420936503856271100575*n^14+ 67008428083986529570802153319590*n^13-516371269293825475870364257576005*n^12+ 3054385401482511820871857438302860*n^11-12541751423279443653588599447465460*n^ 10+22613401803413902648621234463940400*n^9+120574604327405519918270438553650320 *n^8-1277638189648861434632069058915460032*n^7+ 6110351640213896883002565496987209024*n^6-\ 18818619964753118318220416477160639744*n^5+ 39163012397114046175432597389743185920*n^4-\ 53917035929707382263361095654452940800*n^3+ 45783089547990313256900372020156416000*n^2-\ 20682710646346967011759958583951360000*n+3441374674327758870088355237068800000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 45), 2791395/33554432*(3221391581*n^28-1261644248102*n^27+235187615281297*n^26-\ 27766093995287830*n^25+2330153357153030045*n^24-147882105032135019910*n^23+ 7371190268606132688005*n^22-295789204957405533564190*n^21+ 9715077748885308839474435*n^20-264050412567737178829135690*n^19+ 5978628351687516429083541635*n^18-113112955543266216024769863490*n^17+ 1787460356372053515761048332415*n^16-23492804558829233024156077516450*n^15+ 254455857639759537397344225423095*n^14-2232851965271267217055449568114090*n^13+ 15362810665165462616798055329934980*n^12-76907385380031954381564708573675880*n^ 11+214301880361919484352399997882180400*n^10+ 420349278461446677216755402135174560*n^9-8884727748868711751710807237764654656* n^8+56619073574489049520041819979820092032*n^7-\ 228695342531586219703737748840188114432*n^6+ 641654515586326117874072412123018455040*n^5-\ 1256911765797550526274576437301858508800*n^4+ 1661597587268030669277117708939976704000*n^3-\ 1376556571804988063295543552366182400000*n^2+ 616932735706385898966940564920729600000*n-\ 104427921152014751919922503745536000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 90045/33554432*( 99700103061*n^28-39017389055862*n^27+7265467733322057*n^26-856441024403157830*n ^25+71719563670302665845*n^24-4538251022575714281110*n^23+ 225304617215251948722205*n^22-8992484258718765838022990*n^21+ 293258260232049882113779435*n^20-7896591389326371346444119290*n^19+ 176643010581448022739955608235*n^18-3290141758378723381081975134290*n^17+ 50949424647960145072378420271815*n^16-652028907496747314106599649360850*n^15+ 6810507090846043838688972203770495*n^14-56670995600744515151954556697368890*n^ 13+356597510459674612043601264055056580*n^12-\ 1457543292747470008034763116861232680*n^11+883380896872077552560069242787345200 *n^10+41309305151183094801555965888224281760*n^9-\ 382520099140784632394740631674391183936*n^8+ 2056781479617405191970780376186395825792*n^7-\ 7681155936935817235770324806779821768192*n^6+ 20583870666752305273166106891592328202240*n^5-\ 39160391568940057467671650194892607692800*n^4+ 50846016655481845928939888083648487424000*n^3-\ 41772552330246587168015488562181734400000*n^2+ 18765034787287710138703459336460697600000*n-\ 3237265555712457309517597616111616000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 30015/33554432*( 596465915858*n^29-250134781586407*n^28+49982129236533732*n^27-\ 6331699002617008083*n^26+570691304054591117580*n^25-38931847743154870679175*n^ 24+2087420602758473167786380*n^23-90155976722389746657092175*n^22+ 3188630988754910084297751840*n^21-93358061511025764196303064505*n^20+ 2277695524043666260461013419060*n^19-46442117554316595314921143352745*n^18+ 790909846564923325585098086841780*n^17-11196083385873817448532156150287565*n^16 +130347433050778717991158553793149220*n^15-\ 1222061166821134517841406968891558885*n^14+ 8821694864547130656327034629372763950*n^13-\ 43314627225578421633989180375804961420*n^12+ 65088701299276652809317919018803254200*n^11+ 1194400771528973892271179051279763435120*n^10-\ 14409966040279348846710212996379652743648*n^9+ 95773723850322530298648559176164630407872*n^8-\ 445807152323726288858561410941474374366592*n^7+ 1526180103738712357202325675487754983776768*n^6-\ 3852688783043402070416797573480173647247360*n^5+ 7021571590268636403580108273843945189171200*n^4-\ 8843944196606456463011323259878487678976000*n^3+ 7127282516732903098237093700417305804800000*n^2-\ 3178465937647983020062535840515797811200000*n+ 553572410026830199927509192355086336000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 30015/ 134217728*(2374154421503*n^29-993841439962240*n^28+198118631130411327*n^27-\ 25019512856884574280*n^26+2245990518050270899635*n^25-152428033603280713037760* n^24+8119258596136138095242715*n^23-347783235099225685517111640*n^22+ 12173819379693220407840887205*n^21-351868441294432350348686786160*n^20+ 8447957320355325832113011293245*n^19-168814417496315494483335958970040*n^18+ 2801657559020746905228973926992145*n^17-38325436215017590952000763138478560*n^ 16+425125285980046599542964088798545705*n^15-\ 3692437124452170773884891928054283240*n^14+ 22954879114692554214107845067450836440*n^13-\ 68308595690836509609630730628017250160*n^12-\ 470527494320790792551494535381391928640*n^11+ 8894002684406690684279168474332729994560*n^10-\ 76081707588386632252551331310725517075328*n^9+ 447245775645434118388975690594791546842880*n^8-\ 1950012784952492247903546732226808433284352*n^7+ 6407074177381563475164598388332956456944640*n^6-\ 15732542504046366267988348308223578148761600*n^5+ 28140005106710389651480354542882304899072000*n^4-\ 35030229674556097303721793969141949562880000*n^3+ 28084684126829992249732007100291455385600000*n^2-\ 12551654160677093463527198808024258969600000*n+ 2214289640107320799710036769420345344000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 150075/ 268435456*(1884229438783*n^30-842242960545075*n^29+179486975824411235*n^28-\ 24260102559835093875*n^27+2333905882271254935567*n^26-169983601420028185721295* n^25+9731877927827309581408575*n^24-448832988000992889229323375*n^23+ 16949878779452355084146200545*n^22-529770933451553120351073074025*n^21+ 13791069177222116820615324647625*n^20-299751801576310643601989564969625*n^19+ 5430898505092984641138901491655365*n^18-81454130939144946833107019274716325*n^ 17+995598264605966610931669897817963925*n^16-\ 9584037811465918399617598347878809125*n^15+ 66486371792274360141009234543762998220*n^14-\ 222642365028952795924933070709391162800*n^13-\ 1735827330813852737074690569024111236800*n^12+ 37658439833399291695241799349416831132000*n^11-\ 374132016459767882381806098185248038645248*n^10+ 2592614220069483607448013214720278944697600*n^9-\ 13570437995275324685824197401222727492168960*n^8+ 54760679178921595851518945984828667096672000*n^7-\ 170054145851535689312016385600807208640863232*n^6+ 400091378594810073308106921699629648519761920*n^5-\ 692721580785297979249780646868995174384025600*n^4+ 841952479288014495884747992374526667784192000*n^3-\ 664423624526210465029655431021198460190720000*n^2+ 294853963463779181227081388353450105896960000*n-\ 52257235506532770873156867758320150118400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 50025/134217728*(2791404115537*n^30-1243343796386325*n^29+263802493172838590*n^ 28-35463852185002331475*n^27+3389192423241423852438*n^26-\ 244858370389529688501255*n^25+13882212637622198163604050*n^24-\ 632723647539161278914708375*n^23+23554966536846624940662767880*n^22-\ 723493305380953762301160238725*n^21+18433054006190213198633613447750*n^20-\ 389887446360006218419323540728625*n^19+6815425349961469327672466760407610*n^18-\ 97209560429925034137604045957712925*n^17+1098593602049050524203001655037734950* n^16-9121594685994259801782377941860832125*n^15+ 41009377346425997767619684455931932455*n^14+ 214719053343475151126989524953153123550*n^13-\ 6863917441253307989606858917690273680700*n^12+ 83734681025231089821017258191152677079000*n^11-\ 703020738647523133766099579909669422094672*n^10+ 4479564663822212527752361837547716609120800*n^9-\ 22275376123865212324722965291484998040786240*n^8+ 86821568694989012483808213361967710333833600*n^7-\ 263036598424230312164626615580128180230901248*n^6+ 607933899797945769161915419553165445656954880*n^5-\ 1039564495372805406377919703575236222829158400*n^4+ 1253705122955789673884746936541641255845888000*n^3-\ 986132071809858038036121486693196654510080000*n^2+ 438408715022476079060463399898744687165440000*n-\ 78385853259799156309735301637480225177600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 1550775/134217728*(176750362004*n^31-83737758339997*n^30+18914650810440955*n^29 -2709710939931229940*n^28+276260704615366984821*n^27-21317716246009144202388*n^ 26+1292591805230577794681505*n^25-63100285951511201333155800*n^24+ 2520094787483166902014890585*n^23-83185131848657937204864130230*n^22+ 2281787950310571440862383825475*n^21-52052002430305532172013818504000*n^20+ 982579512739634434010082685083495*n^19-15132336505502320960173638629078260*n^18 +183834174017548993136052731857506075*n^17-\ 1606291360438270905825166591331260200*n^16+ 6441687608249480787233446834393140735*n^15+ 81041205680809492884205158960195109995*n^14-\ 2164320574102876199730887855184470735450*n^13+ 28436989913701093737890790698689638291700*n^12-\ 267963803482295880948757653467733981228424*n^11+ 1959319731736152696688892518902926715867632*n^10-\ 11395172605934746198177615064336497017552480*n^9+ 53000926525460005581825792912792332000571840*n^8-\ 196179803622464723798724481044773751187545216*n^7+ 570483975478570588188997240687766291255413248*n^6-\ 1276043048206264003401776376283522327793326080*n^5+ 2126380627366910758506959626377756523395686400*n^4-\ 2514595460305048713868132457223293848485888000*n^3+ 1951297830075628323900872758108739124264960000*n^2-\ 861439295870285751268736664309492052131840000*n+ 154243130607991888222382367738267539865600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21 )/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-\ 17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2* n-45), 310155/268435456*(1719871995923*n^31-809894136284461*n^30+ 181629047028144025*n^29-25800041925866846075*n^28+2604126912578508916137*n^27-\ 198583728784683601609719*n^26+11873475622333072114850145*n^25-\ 570017967736325819742362175*n^24+22310286261411405143283579195*n^23-\ 718327230991995572258330012565*n^22+19089558559556433928914961510575*n^21-\ 417447540205732222672707593415225*n^20+7416197873896789307772709927024515*n^19-\ 103555081693658145411118998305096205*n^18+1034270244073470873939684301116640875 *n^17-4571682153729628863727917173474704725*n^16-\ 74805455827036963030468497591486597930*n^15+ 2236146917458988292955447487266194883110*n^14-\ 33927741225702201310218614158403939882900*n^13+ 372844378007580543865139281509085310763400*n^12-\ 3207451630411883701116262243325894227836688*n^11+ 22169126765105462902135457119277030513218016*n^10-\ 124058027272156500231614294294804838075864000*n^9+ 561078398152331434886936333075926512866985600*n^8-\ 2033637698243852615781015636645688031668945152*n^7+ 5820513590360684000778323063484387819757261824*n^6-\ 12865807195661574351358073128603030407168798720*n^5+ 21260014905337782736140225998518709838838579200*n^4-\ 25010198302561246311727444575878085253595136000*n^3+ 19367360859414320513462342346184081014128640000*n^2-\ 8562286344201259273826439429513816676761600000*n+ 1542431306079918882223823677382675398656000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 310155/1073741824*(13235028102097*n^32-6600287905537232*n^31+ 1568689586282774216*n^30-236331963685187581600*n^29+25320257294669646542268*n^ 28-2051259693900558180385248*n^27+130404010264544841726276744*n^26-\ 6661465321287597082519314720*n^25+277583697831456130615683381030*n^24-\ 9515896746166381726215824599680*n^23+268996866021024256207446151510440*n^22-\ 6237132676381725421687339300581600*n^21+116478644158183201519181713145340060*n^ 20-1668265574814833360994307183132016160*n^19+ 15541684859742111587651154700766295480*n^18-\ 2755070093845826001906354482164514400*n^17-\ 3379492768067772471170608964931985781295*n^16+ 81520701044337188151254816045993226461520*n^15-\ 1268039429212727205848228514342291138099760*n^14+ 15073232996810613568842884881505433141344000*n^13-\ 143913511936632588226483096957168236626044832*n^12+ 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1793968154677710799427041924880180625445676790235279363276800*n^5-\ 2693307428991374703167111679054964712152096770041513164800000*n^4+ 2939285660321440390397707887892940128752485859101099950080000*n^3-\ 2156317814063571663981114668523451478577708155890434048000000*n^2+ 923864073562549661409274613335948611581941584035315712000000*n-\ 165939225810313173956405348746959413948321179788902400000000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31) /(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -31/ 2147483648*(134432413133057917*n^36-240642103241806540650*n^35+ 122943484039034967272475*n^34-34132753835815363399485990*n^33+ 6284847191585493659048965506*n^32-844830661658954864621396434920*n^31+ 87669631515509947042771992171020*n^30-7281515711758472152315022862018360*n^29+ 496295789870115413551198924138360782*n^28-\ 28265501444904366264896810912254812780*n^27+ 1363396366565812272038152724935713242730*n^26-\ 56270689546884654647621422053785785929300*n^25+ 2002839173523946559292464916911677127648280*n^24-\ 61848351902759836570009129113192742784545800*n^23+ 1664603444329925734060835308563378094261566300*n^22-\ 39179205016564259159746253273981805198927082200*n^21+ 808313554626639183837520690591245471579435556845*n^20-\ 14638888617921061559603489580852971145343205417850*n^19+ 232865574494389527550223218035235963187773619875475*n^18-\ 3253079312333679258265905969602468516899750920151350*n^17+ 39871506232385810367109542888429871512523590487643118*n^16-\ 428007565933821868578218442424808725604208415132766400*n^15+ 4013642157433251653811549677787570379976489129915011200*n^14-\ 32764409159328772285352447545608336879502530246886378560*n^13+ 231775972775958374953227376068839719637314287803923687744*n^12-\ 1412650482156629535672296145994517492709005564885911572480*n^11+ 7364843528375832415417453626395755628061720527286563738880*n^10-\ 32548304668798092302612528423090727874917547022936802301440*n^9+ 120557079243124991639605844444922589267916017377750534839808*n^8-\ 368876717794618562318425345478278323395148859197598635909120*n^7+ 915111139850978319582016621150488958701816525833606410321920*n^6-\ 1795509561355469240648918832781844116348010865928343784652800*n^5+ 2692573174102405867301569786440527955908469426618863370240000*n^4-\ 2936513120511670947282429796352512575198715574849667072000000*n^3+ 2153831863723861759267541421739556114227000287362993356800000*n^2-\ 923056777676490399242472534387219752631431173591007232000000*n+ 165939225810313173956405348746959413948321179788902400000000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31) /(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -37/ 2147483648*(1410824194451417594*n^37-1206970431794486073775*n^36+ 476993746630486574236956*n^35-117498120661930029804906585*n^34+ 20430250775497947572070428742*n^33-2684855447363166047715065190870*n^32+ 278380060525257765824109181344688*n^31-23449977360142207236550112474800580*n^30 +1638934343618953001487983917312493124*n^29-\ 96534511365741411528656413174908447930*n^28+ 4849245332753741429549064908768242357896*n^27-\ 209667829066123569224345915192076492520110*n^26+ 7858986643333664859195805954491512554477260*n^25-\ 256802062132039894477455084185819558796505800*n^24+ 7346664261954677576768333503974459059032824240*n^23-\ 184603459772452048653746423244388826076324750900*n^22+ 4083633685087109741934359056179460291599039544290*n^21-\ 79644660967324867987520539906838339524903762542975*n^20+ 1370538459231312484657116771683819707652661858236860*n^19-\ 20809686179806301571026869829578134438298017409028225*n^18+ 278608588322122790362556332792485988301044527094256526*n^17-\ 3284651334854063888882232238758315267865731983282974650*n^16+ 34028540339597688534421801860957661405376502313106658624*n^15-\ 308892359963965547160559621186486244307756177130865811840*n^14+ 2447652353228577427765070617573094736910632622890570099008*n^13-\ 16850107919049721387796554305841688019682698561458017017280*n^12+ 100180295152667000053418408884350849072568202611651401318912*n^11-\ 510599233003815545158513656523944628853607551720132393137920*n^10+ 2210592303468163991226281545395918910358934828053057819168256*n^9-\ 8036717735560223618260921813033124741948793348184646565726720*n^8+ 24180794492715095948670654635031891683188263843546360886861824*n^7-\ 59091855262535806806640850658599929481649786358289836513443840*n^6+ 114403479007613713350654739971483351900153878422230363960115200*n^5-\ 169563542344486159713498596890108536142545177011187431915520000*n^4+ 183072439088683594883043536603155452133095642273870253916160000*n^3-\ 133156475127213134344893783978655519767988176739944077721600000*n^2+ 56694616860348399937661649929642372666790306484246806528000000*n-\ 10149201838074019261171494708496463615271644050332057600000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -37/17179869184*(20565873197630215009*n^37-15817205136656421096848*n^36+ 5780512662453918557216403*n^35-1340598562937949735226773120*n^34+ 222151275881350701326442091272*n^33-28065991829250549807268649979504*n^32+ 2815681670479359577405814133859964*n^31-230631166550969654645552334619763200*n^ 30+15734456933898260305997916971390890254*n^29-\ 907487219307058423929636194306182020768*n^28+ 44751514224634770143547108274193400546378*n^27-\ 1903547310000315779374594263451880690734080*n^26+ 70319629402985906442796488223133459366470860*n^25-\ 2268060580988231125267175724847231157186173920*n^24+ 64131510181943920354420654316357169905934921420*n^23-\ 1594606542532441900733891976722925801354547142400*n^22+ 34941513385975818068312038050235300110016663679865*n^21-\ 675668000048868105789031952814179701689758894876880*n^20+ 11537452598462952052428465273130034383534991648652155*n^19-\ 173961295311685895891722545256755376361829567748566400*n^18+ 2314448812788755680508842445469432674038624619102416836*n^17-\ 27131963574319873184770849418480697545836483918808751792*n^16+ 279656994205573710763004797527390276995313049386346377312*n^15-\ 2527044536075844356591857314975099670166192387140990222080*n^14+ 19943227504552928653232086545193370059788722712447827820928*n^13-\ 136801402196140091479787065500096683469044015793461957751296*n^12+ 810777395267729286785240495818017905739331351599233221076736*n^11-\ 4121058099512399691030488265797534907975726521744728726425600*n^10+ 17799750450387685051801285667314354100486820609128677631333376*n^9-\ 64583077117063760153666696046632951147722530733616027281108992*n^8+ 193996993148387045461835785617788985849126121327240396326469632*n^7-\ 473454894675833192396118299746792209027414047729986515741573120*n^6+ 915698204870612054635648523767245697192095017150700558621081600*n^5-\ 1356244582859653560000248296664596872839189373048870180618240000*n^4+ 1463685041672287802426838011210858001609486364256249975930880000*n^3-\ 1064467847563554771480558230352700944185150739157115260108800000*n^2+ 453305679323405021980198931376088431222316477190057754624000000*n-\ 81193614704592154089371957667971708922173152402656460800000000)/(2*n-5)/(-1+2*n )/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33) /(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-\ 31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2* n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), -37/34359738368*(117412279313319626161*n^38-90782108159615359095617*n^37+ 33659659380773767965464187*n^36-7977634185264660594377673555*n^35+ 1359120233436871459340889617088*n^34-177431869344445721734080182373816*n^33+ 18475817812509563640580723218468956*n^32-1576995191186495501133810988851739900* n^31+112520964698206020456588685170774482766*n^30-\ 6810282478352642653804975896897504577422*n^29+ 353572687303763676701007046182707758483962*n^28-\ 15883313252151316614415364510999942173501770*n^27+ 621583649179899066776635227965523100918502940*n^26-\ 21303989415755833892085397002419234494565187180*n^25+ 642118319876751906242408873252901280189962291180*n^24-\ 17073466875558957941903265166425663589747030145100*n^23+ 401394870909438235373705827173781164401910735612585*n^22-\ 8356612922386982656156038697520275704263366804309145*n^21+ 154193105121753221781577849999507468819876866785544995*n^20-\ 2522111929107934500347163951957034538913817127162604475*n^19+ 36554936852006669759722262814154613264629981688372366844*n^18-\ 468982055715791087844095888681572779165840191136598510068*n^17+ 5316930131337322891876484176368667343048128196390890326048*n^16-\ 53140797246793518345454217856110486722482465317956637505120*n^15+ 466765514770543021850344490073205526791856831505179399811712*n^14-\ 3588776127968241734138324125583294350462650912885033744704384*n^13+ 24033591646107102149630789942367898657775446627922872306585344*n^12-\ 139334365517451670595760561453489501193822395796069776505132800*n^11+ 694051668707882316379296498924994913996670621805034519864085504*n^10-\ 2942896042444951502889789005385673709657124522917286671146042368*n^9+ 10499510257640640179546701778068145592675026947514903581433315328*n^8-\ 31060754179681849255487032326282570490245377741553818427894497280*n^7+ 74767360364234893835775289560167946151939398033201386336609894400*n^6-\ 142833943157502153742298984802504612999775599609944343986995200000*n^5+ 209257068111468669279526925659082964747499638883316816499179520000*n^4-\ 223702625536526328999033499862423983968646903554389779100467200000*n^3+ 161390392985773648010165713786614456781323237038693145378816000000*n^2-\ 68291122863613790462046712726886033892247583827507858636800000000*n+ 12179042205688823113405793650195756338325972860398469120000000000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69), -37/17179869184*(74776841516014329143*n^38-56463767660902941114871*n ^37+20499686954883503207216406*n^36-4768118965347211253934484215*n^35+ 798710501700572209858139081919*n^34-102690429964699225934182476373158*n^33+ 10545923233701469061471391639284728*n^32-888861750341137732420897742232034700*n ^31+62695784030581236498867622630093159758*n^30-\ 3754891458146902431325608850805117162586*n^29+ 193072813507674705910170049899333303208356*n^28-\ 8596853547453745194045508222001460184079010*n^27+ 333709295362387456480672131197521542780076470*n^26-\ 11352382974683837331306692703874682279635434840*n^25+ 339830218913391337468662296042184178632335616840*n^24-\ 8979089750444369728032166240855033678349659386300*n^23+ 209879845135801674383773114700006838963052368455355*n^22-\ 4346363978494299420531166274696307171906520725666135*n^21+ 79809002561843409612342240347366929441277281390080310*n^20-\ 1299642140037695530995836744112538918249179359861998175*n^19+ 18760644400613462523615849151213015028079127890959189547*n^18-\ 239805699225283765117663164079382113655711580656574107034*n^17+ 2709664257894673305244386991109214004370381947756653699024*n^16-\ 27000684005592613303051696187752670988173148292579512058560*n^15+ 236521415394434388480820789768299005090758074712398321901856*n^14-\ 1814130159321307881779706421907761140203950206417074941792192*n^13+ 12122984467690323920337853817174984501798374523540712587388672*n^12-\ 70150355203936796911834059944598888058412908438435674690950400*n^11+ 348859153580591958891213901339958284624640845246322568793258752*n^10-\ 1477135111703969099882196670716221173640419650911098109310749184*n^9+ 5263763244888916257487454310993149675953234034064575311573745664*n^8-\ 15556483622022911110956559520625848457769545707859037397591408640*n^7+ 37417025440688329837530893574028701583669945858126442587722547200*n^6-\ 71437793157568660021636468938663678775547790783491304685977600000*n^5+ 104615104163950927561812044494388822629131809713062761048309760000*n^4-\ 111809553850546616062586519859987276194822797905542182836633600000*n^3+ 80659330311816365407425651683415740405980480864765668753408000000*n^2-\ 34134201797863024297886862960585602672707867924641506918400000000*n+ 6089521102844411556702896825097878169162986430199234560000000000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69), -37/17179869184*(177838118304492606556*n^39-139106714385257956224693 *n^38+52449136009960595507185259*n^37-12699430432120606383255020052*n^36+ 2219532694068720177367362572703*n^35-298396934634863401314913543379499*n^34+ 32112753477577975347453076543808462*n^33-2842369974171678545605913220973161936* n^32+210989203904589206776433214043317456436*n^31-\ 13326680993697038198643613537271129543778*n^30+ 724249948956667696482319930501562085991314*n^29-\ 34159378485848325411874115864301650470465992*n^28+ 1407764899631843323932933277921539592991706210*n^27-\ 50964111744313123262893719666482298164271274470*n^26+ 1627509541209553081875598546145196897701396605560*n^25-\ 45993876572778973696513420989274328344556722951680*n^24+ 1153002696734101972783839978740794968452613692540760*n^23-\ 25682571617172024904287526251590591139514381309472505*n^22+ 508822724074280374365196892316521448467035582881266315*n^21-\ 8970028242244847046137010942277583872771266279897842820*n^20+ 140685208371538244137075987761994350899745035541595549399*n^19-\ 1961597699509763627407303506121496802467610102348945744447*n^18+ 24283154047471885283968984607896279800187804475224244067186*n^17-\ 266376467226226543167793360229859342667980018780074450931408*n^16+ 2582610512313561873683815652764565658225954427722239554238272*n^15-\ 22057448648979692750905239206035823968383798318029727637769376*n^14+ 165267504114865896760221046067655697789233706548578054889860288*n^13-\ 1080788334404103594998185925010254187530194287992975978034680064*n^12+ 6130584955321075339766841462691330310133392728883767108726781184*n^11-\ 29932913509044825832384448621373311533022878825913566756140248832*n^10+ 124621890590429316138319384208581466266935126553822815695470895616*n^9-\ 437282275978396735679737184044280238386277630214592653480954626048*n^8+ 1274258244821355344102302546316293887979676790760214088674211348480*n^7-\ 3025959883844654543850888711735691226426073333320727888445130342400*n^6+ 5711125962148875109712721298288864731595248088665018870621880320000*n^5-\ 8278128624515353246048284883956891917075859787865973013126840320000*n^4+ 8768165824696061027285228888879711235906079621318627819074355200000*n^3-\ 6276933736853272736773298198564656545334420261237342743822336000000*n^2+ 2639852813101681670254126704777328541136834937770145008844800000000*n-\ 468893124919019689866123055532536619025549955125341061120000000000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n -33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2 *n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71 )/(2*n-69)/(2*n-77), -37/34359738368*(403340747901470755567*n^39-\ 312388883367228046389621*n^38+116703770512332717684082463*n^37-\ 28015908645165214994073066369*n^36+4857464736518748000513348082146*n^35-\ 648194160565087126752193249841478*n^34+69273844442745204401634948209083184*n^33 -6091956057152210226655547050029887892*n^32+ 449480444505209497352147316127751538202*n^31-\ 28230908592604453992866092225701876426366*n^30+ 1526196567440364183190487664770097397528098*n^29-\ 71632139326953201230157124833528476532672174*n^28+ 2938675806717280622769540864747920562224401920*n^27-\ 105937264736009622820701982405850964700581951840*n^26+ 3369783171645954746826232154073196867992662262420*n^25-\ 94884809227922738119195402407894710143176880463460*n^24+ 2370624035027524058825840435889794326291877279605695*n^23-\ 52640223180914536257911979989124614189635834087504485*n^22+ 1039915085164512858982219490190248259895563109404954455*n^21-\ 18284250984037844712078468271740639124479962847131968665*n^20+ 286074032599180911272306079419803616846703553042734012118*n^19-\ 3979938474585733022521147803566960175902636385015439680834*n^18+ 49169194882590455202496546775684943997643355180499342511652*n^17-\ 538376694113739885197289273771089563915538868161705716589376*n^16+ 5211087540255203127581437245453378950452747707176010194706304*n^15-\ 44440251908925394838830111778083164231178629751636281896686272*n^14+ 332528945222287238265610212175912890797720268841780526094386816*n^13-\ 2172044803274256760022099559904786049979293457841066765414948608*n^12+ 12307736020156832577046392172815253908960527564666367069117669888*n^11-\ 60038814231923693875772168348051329558340413006428681761550242304*n^10+ 249769811331646481600311728928421339961228176241321608346378150912*n^9-\ 875835780208933955538829548612092660509027160345176202366601363456*n^8+ 2550841072943669157350823672590175476626762823209261920427220828160*n^7-\ 6054825550786793468049774026228328778640167723643485153878019276800*n^6+ 11423989467944839608503191556120147388665504940520821427975127040000*n^5-\ 16554989846841300601498135542398168833622567958796913258767319040000*n^4+ 17532664173886178433009222303193578126458036764134146779014758400000*n^3-\ 12550774078162567232493745031828063468244498438521974582935552000000*n^2+ 5278736338089835651226876410990248737841575715426335824281600000000*n-\ 937786249838039379732246111065073238051099910250682122240000000000)/(2*n-5)/(-1 +2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n -33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2 *n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71 )/(2*n-69)/(2*n-77), 1/137438953472*(-130685350555154417209667*n^40+ 105677929361755441123917100*n^39-41278155434561373724631838070*n^38+ 10375376377872639915065617611200*n^37-1886240622797955612389822263177119*n^36+ 264311440858466334879385725640334580*n^35-\ 29706565481233246205960339913591858460*n^34+ 2751554959388882209551434880917814702800*n^33-\ 214169202113294456805227494592561708290166*n^32+ 14213682240394265155196895726099318123272440*n^31-\ 813329826995204979747827314996496849125093620*n^30+ 40477168999213391855232005925330717346711584000*n^29-\ 1764041122202406750796113779034801171741025414278*n^28+ 67688066801454715600069983683404301284992311383080*n^27-\ 2296528404394435335635311740326992506054898022803200*n^26+ 69123978786006797781883510117404018138696561560676000*n^25-\ 1850443602113794581417833656120654025262511975820319015*n^24+ 44137047858164279656102965435769009835912972856177740700*n^23-\ 939139046769531399413176344842116912754668101009923774950*n^22+ 17837126202458920833427884909099587038118739528526378328000*n^21-\ 302427226486376687752064455946465043184519556767182383367323*n^20+ 4575303980062577144978200889791728897874751779729883602689700*n^19-\ 61701090310597106340772376710634439608283230819284440871317780*n^18+ 740579504968168828619874155931653102938141751577104006906422800*n^17-\ 7894701670087615673159306301065647477710298975571048901589318496*n^16+ 74539921875639614511723050739569886786934773722291008907413064320*n^15-\ 621195372670151426837404992934106653896168185138484207148988967040*n^14+ 4549874344529863553025323093471953401800299142220375416803456371200*n^13-\ 29136507045199893274058043582691681576282948142542198003056459335424*n^12+ 162100802408426054221930677574989719021223674631748439013173369410560*n^11-\ 777465278238309287779203089547899644738507642987696392492704314721280*n^10+ 3184220726227997618600712666245247351492829688749636471742623977984000*n^9-\ 11006416664388014352191243915351927076905277754509139761008356727488512*n^8+ 31636633276965824953998921807289503247445521225479518840367688913387520*n^7-\ 74199036490948854872752960938553606527655042231941840977691230345625600*n^6+ 138483674404109806966990862502663777179719152637827278376022728376320000*n^5-\ 198739227591882764948062905855340557588908403766555883662181746606080000*n^4+ 208675783804147592955582001663260347235920722350978868145857285324800000*n^3-\ 148280267040631121863186933905953201301475458252630378129269981184000000*n^2+ 61988529749996705907956361643081482704242607780053670331364147200000000*n-\ 10964596833106356427829421530572836299293460150650975373230080000000000)/(2*n-5 )/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-\ 59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77), 1/137438953472*(-139178951439604848459167*n ^40+112031142823324363698543100*n^39-43570799146897540830084375070*n^38+ 10907051773335935685464744025200*n^37-1975291702119109516631636199086619*n^36+ 275789832031900017895963198061342580*n^35-\ 30891272766325453315134979551597924460*n^34+ 2852142418119015778727838921897126654800*n^33-\ 221331976471179220438505475763268197525166*n^32+ 14647666146779231196744644443089564306788440*n^31-\ 835949527583377502994543495774160522138195620*n^30+ 41500020406945918637424267995948538855669828000*n^29-\ 1804436590789771155506961586922807770377970045278*n^28+ 69088611127116678569501257806386737397699431663080*n^27-\ 2339328501264404500985162691238632640924951769683200*n^26+ 70280333947474392324801460546201246799790025979436000*n^25-\ 1878124909243386081106596368199540168512719364817466515*n^24+ 44725021675365325413617399584008125839344870184670750700*n^23-\ 950229186544772575002694507548105501595061511439762319950*n^22+ 18022896289723872696766044285842358519470431378784687318000*n^21-\ 305189413076603498852682542661694580356865761613932211962823*n^20+ 4611718181397060062474073702340429453083936915237971933193700*n^19-\ 62125946067875467465972895248288771176773011794061765311915780*n^18+ 744955368091347240776114994004627921325222970053726016330478800*n^17-\ 7934355431289336888509991051280149698489355627190796737396726496*n^16+ 74854727827064074032893391654818080708059244951271830445677256320*n^15-\ 623373032359652552786140938940477519245435861506587481851922151040*n^14+ 4562911054370009436161482500528497859143254470645123548046178419200*n^13-\ 29203469427031729102247408858616883393582868170041583642511670215424*n^12+ 162392653156757089436113166150123347074946277720796781628054030914560*n^11-\ 778528914933787019670180065338382753097762362581169398493595296609280*n^10+ 3187397237535344072562948598365106326938224059026934160005458987520000*n^9-\ 11013961481018489012881417829047349988976278624879602630883554122432512*n^8+ 31650192953441216980606205295364996708761497728947235280609562464747520*n^7-\ 74215652796863961710545048246313964349959106270803782178932863932825600*n^6+ 138493204288709645946629323835897868410610283990826116707424231096320000*n^5-\ 198731467933872507873041391457054714504837702713114223604326811566080000*n^4+ 208654653292315151735656965919830303367317453309090315359422827724800000*n^3-\ 148262732759277408107427345833052451847250682264099075261867229184000000*n^2+ 61983088982769983378384842859034422181417195033940343240471347200000000*n-\ 10964596833106356427829421530572836299293460150650975373230080000000000)/(2*n-5 )/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-\ 59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77), -41/137438953472*(7076055447185429310374*n^ 41-5965065159042442692579847*n^40+2432098308067077911612769640*n^39-\ 638950197822087522468251138270*n^38+121575469068628354537814982500718*n^37-\ 17854478152177331296119322049018739*n^36+2106095918775270292480568153877373900* n^35-205035276553562714385828383065098616460*n^34+ 16798984128491891273681136035586465421052*n^33-\ 1175392137106364674605496498807814328525886*n^32+ 71022266532768927448671544818414511938271680*n^31-\ 3738708122057783823242136890457594434234352420*n^30+ 172651426493753329249763719588634899334481695916*n^29-\ 7032812006280980668936305493926581612137187561358*n^28+ 253801149745363399386937336068753298353752639968680*n^27-\ 8142440622678364551365362343529108880697285260603200*n^26+ 232842092594537782714387480921832761966794711344170830*n^25-\ 5946595445123284740313835170377428384114957577747103515*n^24+ 135821050192371505151454257091268940727419273190254332600*n^23-\ 2776578858584514011400488617305093248626385463499800423950*n^22+ 50818905759499942702924447369570131493364601319579819810406*n^21-\ 832578229979547457080934315313011333295515781693693931057343*n^20+ 12201337907121280531344006588846515577290088623237465312962860*n^19-\ 159754559569254212814342080622813759022850721767963995394204580*n^18+ 1865589749259623279570126791145627321455913968120300055702183312*n^17-\ 19386820083934942343829026154109931895943572338208974488560695776*n^16+ 178759072948047179431994558916783915973214696709169977629615152000*n^15-\ 1457284188777982662668506497345824706675335922656683073084047207040*n^14+ 10457642245742880105089239105904051510718214520143090933800705404928*n^13-\ 65709779640731813068358097403649859198274140708652842215458191170304*n^12+ 359202126869138360814887370750742079333407299206563087626080545029120*n^11-\ 1694988433103033927223353067567409034216919953558941923375651416724480*n^10+ 6838543438764051228846105063549778146422596331470131202494413490622464*n^9-\ 23313087013798112106113473247603864905371096603527198602021512765407232*n^8+ 66166210113942115380587643161391125128447856239458444986787168719339520*n^7-\ 153398018206878913192156323966357525311432182553759650279177280952729600*n^6+ 283314389327563321800353812693899882922624071622919600664732128378880000*n^5-\ 402783693130431762817104478584529249488777502811631306015035495546880000*n^4+ 419425501745974286469730112035279644441178185597837407922893409484800000*n^3-\ 295910773253057178120245456063554184889206861645754010464675692544000000*n^2+ 122982455282447580787745165066009001862311841361622196903162675200000000*n-\ 21661764475161338308638613267717066835189518834212902566625280000000000)/(2*n-5 )/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-\ 59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), (156177719647617775836960*(n-40)!* (2*n-85)!*(n+1)+234266579471426663755440*((n+1)*(2*n-84)!-1/ 234266579471426663755440*binomial(2*n,n)*(n-43)!*(n-40)!)*(n-41)!)/(n-41)!/(n-\ 43)!/(n-40)!/binomial(2*n,n), ((156177719647617775836960*n+ 156177719647617775836960)*(2*n-85)!-(n-43)!*(n-41)!*binomial(2*n,n))/(n-43)!/(n -41)!/binomial(2*n,n)] The limits, as n goes to infinity are 1099511627765 4398046510157 2199023250343 1099511608259 8796092126047 [-------------, -------------, -------------, -------------, -------------, 1099511627776 4398046511104 2199023255552 1099511627776 8796093022208 70368710574683 70368610262207 70368277647155 140734588820185 --------------, --------------, --------------, ---------------, 70368744177664 70368744177664 70368744177664 140737488355328 562917306724365 70358213461485 35171794568385 281250714529035 ---------------, --------------, --------------, ---------------, 562949953421312 70368744177664 35184372088832 281474976710656 8992176352245495 8977495780127745 8951462232238935 71260244961412545 ----------------, ----------------, ----------------, -----------------, 9007199254740992 9007199254740992 9007199254740992 72057594037927936 282775733025358725 139639990879738425 68525010659188275 ------------------, ------------------, -----------------, 288230376151711744 144115188075855872 72057594037927936 533426898895498065 4104910141005895035 3889093157396732715 ------------------, -------------------, -------------------, 576460752303423488 4611686018427387904 4611686018427387904 3611857901772314715 3266964042096699465 5702044094655814947 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 9223372036854775808 2364353073691547613 1811593251783583327 2403863367005480729 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 9223372036854775808 17565138048944315639 -4167404807124795427 -26100247597351225489 ---------------------, ---------------------, ---------------------, 147573952589676412928 147573952589676412928 147573952589676412928 -760937308312317955333 -4344254334592826167957 -2766743136092530178291 ----------------------, -----------------------, -----------------------, 2361183241434822606848 9444732965739290427392 4722366482869645213696 -1645002594316556610643 -14923607672354417955979 -----------------------, ------------------------, 2361183241434822606848 18889465931478580854784 -130685350555154417209667 -139178951439604848459167 -------------------------, -------------------------, 151115727451828646838272 151115727451828646838272 -145059136667301300862667 -297350901164669238181639 -------------------------, -------------------------, 151115727451828646838272 302231454903657293676544 -1204045265875641119211271 --------------------------] 1208925819614629174706176 and in Maple notation [1099511627765/1099511627776, 4398046510157/4398046511104, 2199023250343/ 2199023255552, 1099511608259/1099511627776, 8796092126047/8796093022208, 70368710574683/70368744177664, 70368610262207/70368744177664, 70368277647155/ 70368744177664, 140734588820185/140737488355328, 562917306724365/ 562949953421312, 70358213461485/70368744177664, 35171794568385/35184372088832, 281250714529035/281474976710656, 8992176352245495/9007199254740992, 8977495780127745/9007199254740992, 8951462232238935/9007199254740992, 71260244961412545/72057594037927936, 282775733025358725/288230376151711744, 139639990879738425/144115188075855872, 68525010659188275/72057594037927936, 533426898895498065/576460752303423488, 4104910141005895035/4611686018427387904, 3889093157396732715/4611686018427387904, 3611857901772314715/ 4611686018427387904, 3266964042096699465/4611686018427387904, 5702044094655814947/9223372036854775808, 2364353073691547613/ 4611686018427387904, 1811593251783583327/4611686018427387904, 2403863367005480729/9223372036854775808, 17565138048944315639/ 147573952589676412928, -4167404807124795427/147573952589676412928, -\ 26100247597351225489/147573952589676412928, -760937308312317955333/ 2361183241434822606848, -4344254334592826167957/9444732965739290427392, -\ 2766743136092530178291/4722366482869645213696, -1645002594316556610643/ 2361183241434822606848, -14923607672354417955979/18889465931478580854784, -\ 130685350555154417209667/151115727451828646838272, -139178951439604848459167/ 151115727451828646838272, -145059136667301300862667/151115727451828646838272, -\ 297350901164669238181639/302231454903657293676544, -1204045265875641119211271/ 1208925819614629174706176] and in floating point [1.000000000, .9999999998, .9999999976, .9999999822, .9999998981, .9999995225, .9999980969, .9999933702, .9999793976, .9999420078, .9998503495, .9996425254, .\ 9992032607, .9983321228, .9967022519, .9938119474, .9889345587, .9810754050, .9\ 689470815, .9509755575, .9253481642, .8901104985, .8433126500, .7831968368, .70\ 84099024, .6182168595, .5126873478, .3928266678, .2606273885, .1190260052, -.\ 2823943341e-1, -.1768621572, -.3222694855, -.4599658191, -.5858806482, -.696685\ 6979, -.7900492119, -.8648031066, -.9210090424, -.9599208442, -.9838516023, -.9\ 959629006] The cut off is at j=, 31 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 44], vs. those in the, 2, -th row from j=1 to j=, 43, are as follws 22 21 20 [43 (204560302841 n - 49503593287027 n + 5627095950399847 n 19 18 - 399370238794103885 n + 19842781385805328566 n 17 16 - 733462133365686366642 n + 20923420131277469303102 n 15 14 - 471618421571425089581290 n + 8529836994232348537677781 n 13 12 - 125011195680156432664504487 n + 1493009796919653246778640187 n 11 10 - 14562485025297444483683379945 n + 115898121529657490236808968076 n 9 8 - 749661940147616857257254443492 n + 3912353316249522218763466132432 n 7 - 16288771280008641366833093721200 n 6 + 53198717727022042525416852937536 n 5 - 132720107100842736505599717500352 n 4 + 240117195874042548098089347474432 n 3 - 266967652570695964206600709063680 n 2 - 8594596266367505663397084364800 n + 732992053129802947432091492352000 n - 1210004485205176349730039398400000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) 22 21 (2 n - 29)), 43 (204560302819 n - 49503593271803 n 20 19 18 + 5627095945392383 n - 399370237752920665 n + 19842781232277239544 n 17 16 - 733462116294398239518 n + 20923418645471900664238 n 15 14 - 471618317871095622639170 n + 8529831096241291159563779 n 13 12 - 125010919428847373166348103 n + 1492999073436826098664392963 n 11 10 - 14562139044287935759663129125 n + 115888846076418283194696135634 n 9 8 - 749456031498703698815783192528 n + 3908594998524797168068606915328 n 7 - 16233019316128965581309359592080 n 6 + 52538405228193535399453579806624 n 5 - 126644120241230951586094548498048 n 4 + 198536804833054212649570725785088 n 3 - 71112148448869357381227984168960 n 2 - 543852265788750210716350608998400 n + 1136006846532166777073082777600000 n - 27500101936481280675682713600000) /(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) 23 (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29)), 40549 (867700117 n 22 21 20 - 229506680474 n + 28593538664834 n - 2231091772400260 n 19 18 + 122284665675654867 n - 5004982017808189494 n 17 16 + 158754190059682512724 n - 3997430005489223553800 n 15 14 + 81192703762289295890747 n - 1344344231312506705331974 n 13 12 + 18263540212102903720421874 n - 204245611165924432565637540 n 11 10 + 1880964449007224094787893037 n - 14230090732187949074235849674 n 9 8 + 87936358429950366409845086744 n - 439354005159170623089325006480 n 7 6 + 1742059317199626131886105943632 n - 5274699573446369761957101300384 n 5 + 11035049039294903974490472013824 n 4 - 10328607502064094333641226001920 n 3 - 19901448142258765410356142182400 n 2 + 75081705799758503451824074752000 n - 54673041690696217634618572800000 n + 2624612061806272811040768000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) 23 22 21 (2 n - 45)), 943 (9327776179 n - 2467196762333 n + 307380523881563 n 20 19 - 23984233194019945 n + 1314559679989059504 n 18 17 - 53803505982982742448 n + 1706603329820037409618 n 16 15 - 42972092946853246234850 n + 872806513197072425275139 n 14 13 - 14451036722058073164237133 n + 196308949816459344679648743 n 12 11 - 2194917951753408852296094405 n + 20202541936042872886969001194 n 10 9 - 152613209584320588955316429558 n + 939411837719517439289005890908 n 8 7 - 4645828151701658196114841207360 n + 17940182653714169680533452534784 n 6 - 50690095807767724371216406242528 n 5 4 + 86335872234183868150694276319168 n - 3810764950639749583037504893440 n 3 - 364892793985079704142861291596800 n 2 + 693816058651495446674953150464000 n - 403699189909352448884342169600000 n + 28214579664417432718688256000000)/ (1048576 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 943 ( 24 23 22 74622205907 n - 21491193651516 n + 2922876427196258 n 21 20 - 249661252173091704 n + 15025491671901165917 n 19 18 - 677562955279256082036 n + 23767692627449027242568 n 17 16 - 664605276415704155150784 n + 15060506780368877298250637 n 15 14 - 279664859357339820626031396 n + 4286127983555514366504885098 n 13 12 - 54430933407373142565312955224 n + 573402320432049413807623699667 n 11 - 5001261536166183399853779222636 n 10 + 35910050762784705323905312149068 n 9 - 209790655062479469810046285542384 n 8 + 974797857326833505602095286841552 n 7 - 3438673936693943282450572844202816 n 6 + 8236722661034280285478058893459008 n 5 - 8571374812077368740683560460059904 n 4 - 18558528654396684458647052062663680 n 3 + 82592118679490102111260180441190400 n 2 - 115285063416327302029823469330432000 n + 59645559790778935352335449292800000 n - 5304340976910477351113392128000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), 23 (3059509782307 n 23 22 21 - 881138510130276 n + 119837801125961698 n - 10236085662488922984 n 20 19 + 616041644388239263117 n - 27779720485454990448396 n 18 17 + 974446600370043218175208 n - 27246985403902408927088064 n 16 15 + 617386664279636586207538237 n - 11462311291070217060782373756 n 14 13 + 175595424564415102001320026538 n - 2227831576913885445210955405704 n 12 + 23420727778430432404742946933667 n 11 - 203380146086031554200472944355796 n 10 + 1446967319644280224617860840886508 n 9 - 8297095317814634712977690170144464 n 8 + 37143754621208586203397513533439952 n 7 - 121497844459648253132060546375530176 n 6 + 243343423522051639697456436170922048 n 5 - 62456283595844495876126415361238784 n 4 - 1206272688734744308262513015407457280 n 3 + 3463042144715851924115250470644838400 n 2 - 4184103989390487618066778296041472000 n + 2059938804963761200153466420428800000 n - 217477980053329571395649077248000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 4255 (33075753757 n 24 23 22 - 10336160250140 n + 1528920804315130 n - 142400109535096760 n 21 20 + 9370937434515137635 n - 463473374088982079900 n 19 18 + 17891161577901602803780 n - 552577057240105645910360 n 17 16 + 13886965556433215730099715 n - 287259562243333062954550100 n 15 14 + 4927934196304271938692645130 n - 70410040988683428629131244360 n 13 12 + 838905138902696985939953111725 n - 8317040840582991150012759302900 n 11 + 68167859921596696235441130873880 n 10 - 455935291440075708843800687365160 n 9 + 2429727689375893203459528760628080 n 8 - 9860762899040568032040607521113600 n 7 + 27584955634696700191502915494606080 n 6 - 37180758417448403559313131304671360 n 5 - 63649971414150458239306160761830912 n 4 + 434867166796757967663024975510896640 n 3 - 952594654292946247001037021010944000 n 2 + 1023306656274165176794481302130688000 n - 483284880495021202615894663987200000 n + 57602275797908913504793539379200000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 4255 (33075659992 n 24 23 22 - 10336100709365 n + 1528902929893180 n - 142396737377200310 n 21 20 + 9370489184727738760 n - 463428762682793490275 n 19 18 + 17887713688826563887580 n - 552365237990876979631160 n 17 16 + 13876464726438048595309240 n - 286835543728052136965116475 n 15 14 + 4913918786308811895781072180 n - 70030339151526353718193516910 n 13 12 + 830491842312746712442005407800 n - 8165428961366317957962427490525 n 11 + 65966862011075385580522945768180 n 10 - 430555496678775447677581906603460 n 9 + 2201889972516110315337961712021680 n 8 - 8312634109180153095582860880551600 n 7 + 19935481638935299446293200544210880 n 6 - 11299985998586802855418820029076160 n 5 - 117629878764332965329681943368137472 n 4 + 487059336003544765549235210488458240 n 3 - 934560333906565338311732210668032000 n 2 + 943660054071889778797340896124928000 n - 439459772393365395046771440844800000 n + 57602275797908913504793539379200000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 4255 (264603014771 n 25 24 23 - 89434788342523 n + 14339347545173300 n - 1450986041629889050 n 22 21 + 104001530346177988745 n - 5617867204864961411365 n 20 19 + 237551417050634686360790 n - 8062398626873497238675200 n 18 17 + 223409845773213952512401885 n - 5113717691685113992408069285 n 16 15 + 97425758043677350587589591640 n - 1551477851809598227942840080850 n 14 + 20671839609922941445401178738415 n 13 - 229861658503300554666897455826475 n 12 + 2118335185455121444830936628170590 n 11 - 15969532558373134375692646602656500 n 10 + 96265627059901112538426939615486680 n 9 - 445262617436318648581980688934731120 n 8 + 1446068829735282801961044307652574240 n 7 - 2423242041037581174340055900257598400 n 6 - 3691232660891304313448196127817978496 n 5 + 36978068403758481196736650928506080768 n 4 - 113966033581141023525392160581874370560 n 3 + 193661467036096687761180893195750400000 n 2 - 183419041468563676889855280780337152000 n + 83758277684526649053086808745574400000 n - 11750864262773418354977882033356800000)/(16777216 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 115 ( 26 25 24 9790082513177 n - 3308944709929651 n + 530514002181532850 n 23 22 - 53678761186198608850 n + 3847053580971290790815 n 21 20 - 207763616668664882831005 n + 8782052139667293976611980 n 19 18 - 297868398675284714613021400 n + 8244883874364045095673695495 n 17 16 - 188369948262602527364960776045 n + 3577754670265427428779587930930 n 15 - 56690618067788651452793665555450 n 14 + 749378124949534189730330557859105 n 13 - 8230916454307268896453559795941075 n 12 + 74447822472438670658289069549732080 n 11 - 545629387129833091156454730711149500 n 10 + 3149913974893311875801019351959092160 n 9 - 13564445129802742953841601362351967440 n 8 + 37953165752378120841039475454959130880 n 7 - 29052313130648216927065529703498004800 n 6 - 299216991930193403003130909197291626752 n 5 + 1636210207308627792407664875437998345216 n 4 - 4399276690763106140528190204567537438720 n 3 + 7006813822096175441447119722726051840000 n 2 - 6425362461338252575752159622933069824000 n + 2926131624096899915308571500481740800000 n - 434781977722616479134181635234201600000)/(16777216 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 1035 ( 27 26 25 4350893019377 n - 1585787008086324 n + 274708189293302349 n 24 23 - 30096540816801985500 n + 2340833663508878379465 n 22 21 - 137531781293217397478340 n + 6341084565187139351094225 n 20 19 - 235263782986251875170277820 n + 7145030881299310508039197095 n 18 17 - 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(n - 40)! (2 n - 87)! (n + 1) + /85 894472394345447261611680 (n - 42)! |-- (n - 40)! (n + 1) (2 n - 86)! + \84 (n - 41)! ((2 n - 85)! (n + 1) \\ - 1/894472394345447261611680 binomial(2 n, n) (n - 44)! (n - 40)!)||/( // (n - 42)! (n - 41)! (n - 44)! (n - 40)! binomial(2 n, n)), ( 603413916820341406642800 (n - 41)! (2 n - 87)! (n + 1) + 905120875230512109964200 (n - 42)! ((2 n - 86)! (n + 1) - 1/905120875230512109964200 binomial(2 n, n) (n - 44)! (n - 41)!))/( (n - 42)! (n - 44)! (n - 41)! binomial(2 n, n)), ( (603413916820341406642800 n + 603413916820341406642800) (2 n - 87)! - (n - 44)! (n - 42)! binomial(2 n, n))/((n - 44)! (n - 42)! binomial(2 n, n))] and in Maple notation [43/2097152*(204560302841*n^22-49503593287027*n^21+5627095950399847*n^20-\ 399370238794103885*n^19+19842781385805328566*n^18-733462133365686366642*n^17+ 20923420131277469303102*n^16-471618421571425089581290*n^15+ 8529836994232348537677781*n^14-125011195680156432664504487*n^13+ 1493009796919653246778640187*n^12-14562485025297444483683379945*n^11+ 115898121529657490236808968076*n^10-749661940147616857257254443492*n^9+ 3912353316249522218763466132432*n^8-16288771280008641366833093721200*n^7+ 53198717727022042525416852937536*n^6-132720107100842736505599717500352*n^5+ 240117195874042548098089347474432*n^4-266967652570695964206600709063680*n^3-\ 8594596266367505663397084364800*n^2+732992053129802947432091492352000*n-\ 1210004485205176349730039398400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/( 2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/ (2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 43/ 2097152*(204560302819*n^22-49503593271803*n^21+5627095945392383*n^20-\ 399370237752920665*n^19+19842781232277239544*n^18-733462116294398239518*n^17+ 20923418645471900664238*n^16-471618317871095622639170*n^15+ 8529831096241291159563779*n^14-125010919428847373166348103*n^13+ 1492999073436826098664392963*n^12-14562139044287935759663129125*n^11+ 115888846076418283194696135634*n^10-749456031498703698815783192528*n^9+ 3908594998524797168068606915328*n^8-16233019316128965581309359592080*n^7+ 52538405228193535399453579806624*n^6-126644120241230951586094548498048*n^5+ 198536804833054212649570725785088*n^4-71112148448869357381227984168960*n^3-\ 543852265788750210716350608998400*n^2+1136006846532166777073082777600000*n-\ 27500101936481280675682713600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2* n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 40549/ 4194304*(867700117*n^23-229506680474*n^22+28593538664834*n^21-2231091772400260* n^20+122284665675654867*n^19-5004982017808189494*n^18+158754190059682512724*n^ 17-3997430005489223553800*n^16+81192703762289295890747*n^15-\ 1344344231312506705331974*n^14+18263540212102903720421874*n^13-\ 204245611165924432565637540*n^12+1880964449007224094787893037*n^11-\ 14230090732187949074235849674*n^10+87936358429950366409845086744*n^9-\ 439354005159170623089325006480*n^8+1742059317199626131886105943632*n^7-\ 5274699573446369761957101300384*n^6+11035049039294903974490472013824*n^5-\ 10328607502064094333641226001920*n^4-19901448142258765410356142182400*n^3+ 75081705799758503451824074752000*n^2-54673041690696217634618572800000*n+ 2624612061806272811040768000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 943/1048576*(9327776179*n^23-2467196762333*n^22+307380523881563*n^21-\ 23984233194019945*n^20+1314559679989059504*n^19-53803505982982742448*n^18+ 1706603329820037409618*n^17-42972092946853246234850*n^16+ 872806513197072425275139*n^15-14451036722058073164237133*n^14+ 196308949816459344679648743*n^13-2194917951753408852296094405*n^12+ 20202541936042872886969001194*n^11-152613209584320588955316429558*n^10+ 939411837719517439289005890908*n^9-4645828151701658196114841207360*n^8+ 17940182653714169680533452534784*n^7-50690095807767724371216406242528*n^6+ 86335872234183868150694276319168*n^5-3810764950639749583037504893440*n^4-\ 364892793985079704142861291596800*n^3+693816058651495446674953150464000*n^2-\ 403699189909352448884342169600000*n+28214579664417432718688256000000)/(2*n-5)/( -1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2 *n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/ (2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 943/4194304*(74622205907*n^24-\ 21491193651516*n^23+2922876427196258*n^22-249661252173091704*n^21+ 15025491671901165917*n^20-677562955279256082036*n^19+23767692627449027242568*n^ 18-664605276415704155150784*n^17+15060506780368877298250637*n^16-\ 279664859357339820626031396*n^15+4286127983555514366504885098*n^14-\ 54430933407373142565312955224*n^13+573402320432049413807623699667*n^12-\ 5001261536166183399853779222636*n^11+35910050762784705323905312149068*n^10-\ 209790655062479469810046285542384*n^9+974797857326833505602095286841552*n^8-\ 3438673936693943282450572844202816*n^7+8236722661034280285478058893459008*n^6-\ 8571374812077368740683560460059904*n^5-18558528654396684458647052062663680*n^4+ 82592118679490102111260180441190400*n^3-115285063416327302029823469330432000*n^ 2+59645559790778935352335449292800000*n-5304340976910477351113392128000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 23/4194304*(3059509782307*n ^24-881138510130276*n^23+119837801125961698*n^22-10236085662488922984*n^21+ 616041644388239263117*n^20-27779720485454990448396*n^19+ 974446600370043218175208*n^18-27246985403902408927088064*n^17+ 617386664279636586207538237*n^16-11462311291070217060782373756*n^15+ 175595424564415102001320026538*n^14-2227831576913885445210955405704*n^13+ 23420727778430432404742946933667*n^12-203380146086031554200472944355796*n^11+ 1446967319644280224617860840886508*n^10-8297095317814634712977690170144464*n^9+ 37143754621208586203397513533439952*n^8-121497844459648253132060546375530176*n^ 7+243343423522051639697456436170922048*n^6-62456283595844495876126415361238784* n^5-1206272688734744308262513015407457280*n^4+ 3463042144715851924115250470644838400*n^3-4184103989390487618066778296041472000 *n^2+2059938804963761200153466420428800000*n-\ 217477980053329571395649077248000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13 )/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-\ 29)/(2*n-45), 4255/4194304*(33075753757*n^25-10336160250140*n^24+ 1528920804315130*n^23-142400109535096760*n^22+9370937434515137635*n^21-\ 463473374088982079900*n^20+17891161577901602803780*n^19-\ 552577057240105645910360*n^18+13886965556433215730099715*n^17-\ 287259562243333062954550100*n^16+4927934196304271938692645130*n^15-\ 70410040988683428629131244360*n^14+838905138902696985939953111725*n^13-\ 8317040840582991150012759302900*n^12+68167859921596696235441130873880*n^11-\ 455935291440075708843800687365160*n^10+2429727689375893203459528760628080*n^9-\ 9860762899040568032040607521113600*n^8+27584955634696700191502915494606080*n^7-\ 37180758417448403559313131304671360*n^6-63649971414150458239306160761830912*n^5 +434867166796757967663024975510896640*n^4-952594654292946247001037021010944000* n^3+1023306656274165176794481302130688000*n^2-\ 483284880495021202615894663987200000*n+57602275797908913504793539379200000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 4255/4194304*( 33075659992*n^25-10336100709365*n^24+1528902929893180*n^23-142396737377200310*n ^22+9370489184727738760*n^21-463428762682793490275*n^20+17887713688826563887580 *n^19-552365237990876979631160*n^18+13876464726438048595309240*n^17-\ 286835543728052136965116475*n^16+4913918786308811895781072180*n^15-\ 70030339151526353718193516910*n^14+830491842312746712442005407800*n^13-\ 8165428961366317957962427490525*n^12+65966862011075385580522945768180*n^11-\ 430555496678775447677581906603460*n^10+2201889972516110315337961712021680*n^9-\ 8312634109180153095582860880551600*n^8+19935481638935299446293200544210880*n^7-\ 11299985998586802855418820029076160*n^6-117629878764332965329681943368137472*n^ 5+487059336003544765549235210488458240*n^4-934560333906565338311732210668032000 *n^3+943660054071889778797340896124928000*n^2-\ 439459772393365395046771440844800000*n+57602275797908913504793539379200000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2* n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 4255/16777216*( 264603014771*n^26-89434788342523*n^25+14339347545173300*n^24-\ 1450986041629889050*n^23+104001530346177988745*n^22-5617867204864961411365*n^21 +237551417050634686360790*n^20-8062398626873497238675200*n^19+ 223409845773213952512401885*n^18-5113717691685113992408069285*n^17+ 97425758043677350587589591640*n^16-1551477851809598227942840080850*n^15+ 20671839609922941445401178738415*n^14-229861658503300554666897455826475*n^13+ 2118335185455121444830936628170590*n^12-15969532558373134375692646602656500*n^ 11+96265627059901112538426939615486680*n^10-\ 445262617436318648581980688934731120*n^9+1446068829735282801961044307652574240* n^8-2423242041037581174340055900257598400*n^7-\ 3691232660891304313448196127817978496*n^6+ 36978068403758481196736650928506080768*n^5-\ 113966033581141023525392160581874370560*n^4+ 193661467036096687761180893195750400000*n^3-\ 183419041468563676889855280780337152000*n^2+ 83758277684526649053086808745574400000*n-11750864262773418354977882033356800000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 115/ 16777216*(9790082513177*n^26-3308944709929651*n^25+530514002181532850*n^24-\ 53678761186198608850*n^23+3847053580971290790815*n^22-207763616668664882831005* n^21+8782052139667293976611980*n^20-297868398675284714613021400*n^19+ 8244883874364045095673695495*n^18-188369948262602527364960776045*n^17+ 3577754670265427428779587930930*n^16-56690618067788651452793665555450*n^15+ 749378124949534189730330557859105*n^14-8230916454307268896453559795941075*n^13+ 74447822472438670658289069549732080*n^12-545629387129833091156454730711149500*n ^11+3149913974893311875801019351959092160*n^10-\ 13564445129802742953841601362351967440*n^9+ 37953165752378120841039475454959130880*n^8-\ 29052313130648216927065529703498004800*n^7-\ 299216991930193403003130909197291626752*n^6+ 1636210207308627792407664875437998345216*n^5-\ 4399276690763106140528190204567537438720*n^4+ 7006813822096175441447119722726051840000*n^3-\ 6425362461338252575752159622933069824000*n^2+ 2926131624096899915308571500481740800000*n-\ 434781977722616479134181635234201600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-51)/(2*n-45), 1035/33554432*(4350893019377*n^27-\ 1585787008086324*n^26+274708189293302349*n^25-30096540816801985500*n^24+ 2340833663508878379465*n^23-137531781293217397478340*n^22+ 6341084565187139351094225*n^21-235263782986251875170277820*n^20+ 7145030881299310508039197095*n^19-179705309840680371389386730700*n^18+ 3771207386793884096471218266135*n^17-66299293195177428693747033858420*n^16+ 977173126812929925699447370837155*n^15-12041698073337932475775854118516620*n^14 +123219713531814413387820886311373755*n^13-\ 1034028346198145845185453725267072820*n^12+ 6965462757416180311012499096873073660*n^11-\ 36224872849817906498004851987441653680*n^10+ 133430757346918010247277360821016415440*n^9-\ 254111351038796720530841545253293840320*n^8-\ 511060274904077630946700764303250210752*n^7+ 5996756049577648119614212381056055889664*n^6-\ 23814256625375191244368818356109206651904*n^5+ 56375679787685724917541323977091815034880*n^4-\ 83566815750762880562650406683489591296000*n^3+ 73570923572605088469878616198575947776000*n^2-\ 33117748551854941380511384362398515200000*n+ 5120765515399705198691472592758374400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1035/2097152*(271893963881*n^27-\ 99089135909397*n^26+17162743943439432*n^25-1879851175768945275*n^24+ 146151045684055413495*n^23-8581299735511099990185*n^22+395246746908625803764430 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-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 32085/8388608*(70145673253*n^28-\ 27488202152306*n^27+5128554987018981*n^26-606228462430508190*n^25+ 50966615077626979185*n^24-3242853578669352432330*n^23+162222201092144227785465* n^22-6542133904006314629485470*n^21+216342913682650044130095255*n^20-\ 5934350413339302280370714670*n^19+136018505400423471911661121455*n^18-\ 2615174172344456050526723531370*n^17+42205888623173402475596886007995*n^16-\ 570243175199239942425689640625950*n^15+6407499630132123244185675409450035*n^14-\ 59156685260296912854384412062730170*n^13+439312687859881624228215769276122840*n ^12-2519904640896657849258225229329141240*n^11+ 10131059446938973262172207299087628800*n^10-\ 18703963035683075039595946156078882720*n^9-\ 84084893572677429836809605348931456128*n^8+ 913756852430283603803121171690939674496*n^7-\ 4330166680512997195757527063959442004736*n^6+ 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12878155599*n^29-5409139465167*n^28+1083183244017441*n^27-137614275720789203*n^ 26+12451369048614257125*n^25-853735898921138983295*n^24+46078336847337694350685 *n^23-2007131095981699393189055*n^22+71762231615405438141008975*n^21-\ 2130093561401738530465492625*n^20+52872358705996948884428264875*n^19-\ 1101654338003994757482828698825*n^18+19281338420287031557581146669095*n^17-\ 282715640743428796399454674436885*n^16+3449468494623608911150610295733855*n^15-\ 34577011468658775570037212215150165*n^14+278048535176479652395481285435632630*n ^13-1706871139022652367878026479826400140*n^12+ 6962133105941407127971264356193441720*n^11-\ 6540054111200236499915514611802050960*n^10-\ 161593581353803163352207384219905677664*n^9+ 1530513526649987294051413103802729438912*n^8-\ 8177639469005668790707674842651899752576*n^7+ 30202150694382721146880291829983346078208*n^6-\ 79979590285170431426385375332955360245760*n^5+ 150412318955242173168385490847399225139200*n^4-\ 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536870912*(3811207897565*n^30-1709579595829563*n^29+365948411512639321*n^28-\ 49741173687795086427*n^27+4818899522686092273549*n^26-354026655174853749525255* n^25+20485688509907803901300205*n^24-957164449383917316108982935*n^23+ 36723035421987218596500717795*n^22-1170069926781601824121258900785*n^21+ 31182933577607538969119862433995*n^20-697699181462807398093008772138065*n^19+ 13111806499625456792100583011707055*n^18-206312186720473774234175138163403725*n ^17+2696356073514428013078845741314469775*n^16-\ 28805570714373038039271759035164400925*n^15+ 243481235142612496261417062513909884100*n^14-\ 1503778168836483073964308422823462485120*n^13+ 4944098614125367956461919381768441463840*n^12+ 20262967274764828427831197428521298373920*n^11-\ 455507978723664618228895090477641847703680*n^10+ 3941087414340220532458742488593802510238208*n^9-\ 23018861150390931148737757351055936821959936*n^8+ 99283869998292850105487237568175535055090432*n^7-\ 322375632184554261107143401126257197953936384*n^6+ 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-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n -21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-63)/(2*n-45), 310155/1073741824*(26106323801846*n^33-13824790308586917*n^ 32+3493658680603915472*n^31-560460529548127439880*n^30+64042633403789071307144* n^29-5543445335681463024458028*n^28+377297314399986944372123088*n^27-\ 20681878548436720623456762600*n^26+927229459027505832974749399140*n^25-\ 34305940113893581229840006779230*n^24+1050649852530024539912310123251280*n^23-\ 26526789819952704307552339816403400*n^22+543539625805815033682173417668423880*n ^21-8666187263607447578471887908081944460*n^20+ 94091352410136662236724030274995968560*n^19-\ 225310580382465735125853445626976435800*n^18-\ 18384693256823526529622342682076810186410*n^17+ 533867829426385489866207294043856616912795*n^16-\ 9490469884403558524073402670573548197356320*n^15+ 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*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 13485/1073741824*(566523447310438*n^33-\ 297108147016306791*n^32+74248053942125927776*n^31-11758359941760430926840*n^30+ 1323604569732748737098632*n^29-112569852832566531387983844*n^28+ 7502630060821924614183308304*n^27-400892530063727273502093391800*n^26+ 17406612900506541892147340364420*n^25-617584018631420347581129687493290*n^24+ 17843237116738931483676382003848240*n^23-412191523507188878573749742787874200*n ^22+7211459379789619781188330607164517640*n^21-\ 78179482672900394820739476775303352580*n^20-\ 218471947541257946502858128198192069520*n^19+ 35105342886646519880591788647398755332600*n^18-\ 1008498070912610319280582095647343132908730*n^17+ 19588985745731019372069540339069210280117785*n^16-\ 296435846655869303807470302641947509137420560*n^15+ 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2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 13485/1073741824*(1046956970043401*n^34-\ 577334083046561353*n^33+151741026271167363899*n^32-25276947641019971834872*n^31 +2992704777467592756168644*n^30-267592070004940774575730732*n^29+ 18731346772396230152708204556*n^28-1049035813141610630135372439768*n^27+ 47549578752304936742474236244190*n^26-1747437231452509858461413913598470*n^25+ 51448100670958556364455003973658410*n^24-1164609069140235809786200655019843480* n^23+17563701298678446851759432256366606180*n^22-\ 36725285367705618774389596153229418540*n^21-\ 8009407862825817650700534203967900258180*n^20+ 320120058450184741348209198945967884302040*n^19-\ 8104747795284445071541115178973552338797535*n^18+ 157998100028602182872951532221831236263342855*n^17-\ 2511824854566306350486392972348874314103229965*n^16+ 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/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 13485/1073741824*(941025367610927*n^ 34-511020899923832629*n^33+131942303844935540825*n^32-21523340121937465405528*n ^31+2485099648248529756748348*n^30-215448865085617027682570236*n^29+ 14500274819163297238907013060*n^28-770598231160385279224104022392*n^27+ 32409108006091836380037663974130*n^26-1057806971783901701219418399176910*n^25+ 24866949350365217552157599556576750*n^24-291006352408751988242629363106606520*n ^23-7055982537837834900218212607290012740*n^22+ 560633818754636439410278429957371037380*n^21-\ 20522055773093683501643456712272033045100*n^20+ 546735366713717707633350043042940001853560*n^19-\ 11654924097901513749810054101942059457888745*n^18+ 206078631888181642866047616259769514646548315*n^17-\ 3073804905260677561387012288653220269019138175*n^16+ 39002548907922547313439883464488463630140475280*n^15-\ 422593740251170893078116961033594098918231243312*n^14+ 3912856650557776625883084878409358347161087638624*n^13-\ 30912372541957129360987225536029553442004325965600*n^12+ 207628115178201866191467530048269109124237925139968*n^11-\ 1179048727892196130951089885294120579601187007633408*n^10+ 5616827046651974060163062031578553041317567130662656*n^9-\ 22215431705859979895395002942511191576324650388821760*n^8+ 71958564838285509873327485564091637881715902524485632*n^7-\ 187462769703315532221288530414934238735691642715955200*n^6+ 383324454614932268700436452108132092021630877887692800*n^5-\ 594725568103241018625383889339719238083923200860160000*n^4+ 666265457104175928638385723663816477670408816558080000*n^3-\ 498364303354647853843415565873299268613759736217600000*n^2+ 216116074638843178526649131647736946956134514688000000*n-\ 38933403206920758859163223258548991689207316480000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/2147483648*(16290371771042911*n ^35-9225137374364033360*n^34+2480552870617266444685*n^33-\ 420504283307493462675290*n^32+50277461044898069400821788*n^31-\ 4487300785896866017961120240*n^30+307717597558678137603490495140*n^29-\ 16341399063764050869205202334360*n^28+658661629805328802763282905999266*n^27-\ 18367886757736565993959206603812400*n^26+197027937530911933424112636950270550*n ^25+13124476606702751464044757032687804900*n^24-\ 1029132904234819589357853991410151140660*n^23+ 44461757078485953383362045423662957673200*n^22-\ 1438562851945485533557493833177921124990700*n^21+ 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3574210060181465799970928205153000*n^26-417018855396300636814953820159694100*n^ 25+24391258461640269019109419708343943450*n^24-\ 1045394483967141312583262073205315109280*n^23+ 35895935419269585303861618119992180584600*n^22-\ 1024360627772468586690729975512325221826600*n^21+ 24758157900847099650142009303636854431136900*n^20-\ 512358741100459490727443857120224208156313820*n^19+ 9138015989729631074092064789588140239837388800*n^18-\ 140989146623124702983707750091665598395216327550*n^17+ 1885207166437000229655295677001743511374791240575*n^16-\ 21851656610721067786779518212986116048005816044848*n^15+ 219326381365429126007986535259587168999002020096880*n^14-\ 1901765851055218598977092510431195440629456118239680*n^13+ 14193866507309591701412541446852238548623607213506720*n^12-\ 90726017700901099269106464042223361594146204245717504*n^11+ 493360629015270199367949862518604789624552930768753920*n^10-\ 2262962095464762518237325448818428044639042786693854720*n^9+ 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n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69), -37/137438953472*(971439638225266041413*n^39-\ 787040162553956436230274*n^38+306283076779833583055459782*n^37-\ 76312010335557846562446237036*n^36+13688386454918851765509658643769*n^35-\ 1884362136526323931432825149525882*n^34+207223659188100526121444469297669976*n^ 33-18708675931667479409219213782342425648*n^32+ 1414208919665220287748584632184092959578*n^31-\ 90829253850757404585583649139376018488804*n^30+ 5012582975195778576180482618284932516227172*n^29-\ 239785403907318020717097383373705336486552456*n^28+ 10011464942538899278210748326382916302416749930*n^27-\ 366809746306022864630637410289138159921533135460*n^26+ 11843977210104954596727144088810221557902822475880*n^25-\ 338135633233993401361557631958606271718189504044240*n^24+ 8556274456460223219080094874152485304436891395207105*n^23-\ 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71)/(2*n-69)/(2*n-77), -37/34359738368*(305535646807799122997*n^39-\ 242262625883065485836931*n^38+92479500878553221400777433*n^37-\ 22645862549243236397255428059*n^36+3998931008671389454629939978186*n^35-\ 542723432972010098033571772347558*n^34+58914918169510112380035602375425144*n^33 -5256369755960018078602198269806470812*n^32+ 393047693882184676049726792614810930382*n^31-\ 24993864909185757108431537759686038250626*n^30+ 1366767037761525886363423340157347142512718*n^29-\ 64833250857725474422201214306539155188581914*n^28+ 2685986596617698180759403971150973322664968920*n^27-\ 97710920358261915618150062853694520508604995240*n^26+ 3134293222323330381555380988337055498262801092220*n^25-\ 88940200643330050655109265423002808250423339418060*n^24+ 2238029939360347722529874778896335968849141132400245*n^23-\ 50023825368471200426805037875110818840597947177992835*n^22+ 994218946872039697686265445020917774857696523749075905*n^21-\ 17577981751198782401310902957775645905361599400389985315*n^20+ 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23290058508324103450088180724411074076074880883557149794061568757727232*n^8+ 66125432849785284030934197468218200801359073442216250533887530652139520*n^7-\ 153348901084229183054031225841503452095476994554476293941798981048729600*n^6+ 283287259828877534505385158896851108515892337085615070736095712378880000*n^5-\ 402807942612284101776558059146767502476653139806151146699857204346880000*n^4+ 419488388712516125193828653951523624802479860203131724388708961484800000*n^3-\ 295962240758265457695718240131144858953562448349389882830612332544000000*n^2+ 122998292480756135004067855232859274188368331260710815096058675200000000*n-\ 21661764475161338308638613267717066835189518834212902566625280000000000)/(2*n-5 )/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-\ 59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), -41/137438953472*( 6808774300471954201124*n^41-5756051302312505157146347*n^40+ 2353151475762318768593597140*n^39-619763917848193598814443093270*n^38+ 118203534102229293373536734178468*n^37-17397822522314117359130771474389239*n^36 +2056507149370195345285079621943358900*n^35-\ 200599142994565641255206037138437506460*n^34+ 16465646059482939731291650651528092854552*n^33-\ 1154046544957381967526155635646196032682886*n^32+ 69844449136653167437943137044748872047656680*n^31-\ 3682223834885926646195956179369221515656282420*n^30+ 170281220698748877950820227623609080518605261416*n^29-\ 6945322604922051165876329148154023776355046802358*n^28+ 250948590896338052125243090887971032945786539508680*n^27-\ 8060025781161956177512328883784706603132554963603200*n^26+ 230727171292691156699685297243437175269551397938454580*n^25-\ 5898310138497266982253588048347590368064682510302796015*n^24+ 134839304250032193708534494433438013166710635952075020100*n^23-\ 2758795491812962866893919711699080807612439564086074098950*n^22+ 50531977933735687727104320783962039520216641733569578247156*n^21-\ 828457870679713573225709391340351240861572812956835614045843*n^20+ 12148748173977367366947598805278808659873789252736321086297860*n^19-\ 159159180208720924208758113995367001378767381171854818420274580*n^18+ 1859627169434630417217579597602879930087243447277105506196839312*n^17-\ 19334181960484539268544591020805071109240499988208859793572167776*n^16+ 178351251223976640590767698469471390792234334260324074792110992000*n^15-\ 1454526539645645091716459201701862434533463166150918378404726567040*n^14+ 10441480173841684705291302185309612167864272698780917900146380348928*n^13-\ 65628393851657487025946150674877720358856712824186260920769441218304*n^12+ 358853930357259858181398862253172321722916116632538675251260888069120*n^11-\ 1693741333654407406831574949263588123123592339279580293352813930644480*n^10+ 6834879735644772314142772001219942956450813728440330329439264780734464*n^9-\ 23304521350789907914392553879414571393214918511244849668590829051871232*n^8+ 66151058108410823129699512627352568790713165960593131595294196830699520*n^7-\ 153379789169418166908587262075768256100422910269442801111363309803929600*n^6+ 283304347799703706409196397191458497928792683875881588245500220538880000*n^5-\ 402792727185392962554816408694379964448692290270968896562198877306880000*n^4+ 419448852403337384896874818843919368644902251923815500484810951884800000*n^3-\ 295929865194689403043799427663270263654047830962580174652827500544000000*n^2+ 122988326634015630155747723371768127212459613226650172428333875200000000*n-\ 21661764475161338308638613267717066835189518834212902566625280000000000)/(2*n-5 )/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65) /(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-\ 59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), (603413916820341406642800*(n-41)!* (n-40)!*(2*n-87)!*(n+1)+894472394345447261611680*(n-42)!*(85/84*(n-40)!*(n+1)*( 2*n-86)!+(n-41)!*((2*n-85)!*(n+1)-1/894472394345447261611680*binomial(2*n,n)*(n -44)!*(n-40)!)))/(n-42)!/(n-41)!/(n-44)!/(n-40)!/binomial(2*n,n), ( 603413916820341406642800*(n-41)!*(2*n-87)!*(n+1)+905120875230512109964200*(n-42 )!*((2*n-86)!*(n+1)-1/905120875230512109964200*binomial(2*n,n)*(n-44)!*(n-41)!) )/(n-42)!/(n-44)!/(n-41)!/binomial(2*n,n), ((603413916820341406642800*n+ 603413916820341406642800)*(2*n-87)!-(n-44)!*(n-42)!*binomial(2*n,n))/(n-44)!/(n -42)!/binomial(2*n,n)] The limits, as n goes to infinity are 8796093022163 8796093021217 35184372044233 8796092936797 70368740170301 [-------------, -------------, --------------, -------------, --------------, 8796093022208 8796093022208 35184372088832 8796093022208 70368744177664 70368724993061 140737332236035 17592116658245 1125885827850605 --------------, ---------------, --------------, ----------------, 70368744177664 140737488355328 17592186044416 1125899906842624 1125859489015355 4503174275055195 281410252616835 2250623926322505 ----------------, ----------------, ---------------, ----------------, 1125899906842624 4503599627370496 281474976710656 2251799813685248 2249292396897405 35948019148270605 17937241413967665 571967025227067375 ----------------, -----------------, -----------------, ------------------, 2251799813685248 36028797018963968 18014398509481984 576460752303423488 568640438987478525 2253625383822977925 555499090400980575 ------------------, -------------------, ------------------, 576460752303423488 2305843009213693952 576460752303423488 4352025258601667475 4223095632029960115 4048503429380773065 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 3819784343490628215 14118214741035262485 12689727082233350595 -------------------, --------------------, --------------------, 4611686018427387904 18446744073709551616 18446744073709551616 43935132666502730967 4502577129329837109 27101083544760182257 --------------------, -------------------, --------------------, 73786976294838206464 9223372036854775808 73786976294838206464 17331209133939902557 55442909339366505011 -15313441411139818601 --------------------, ---------------------, ---------------------, 73786976294838206464 590295810358705651712 295147905179352825856 -3746589284080145818739 -6442181675955876646289 -35943266614334843532281 -----------------------, -----------------------, ------------------------, 18889465931478580854784 18889465931478580854784 75557863725914323419136 -11304818931888567550889 -106723100787204867230017 ------------------------, -------------------------, 18889465931478580854784 151115727451828646838272 -120415814940318895789817 -262721955796471403027209 -------------------------, -------------------------, 151115727451828646838272 302231454903657293676544 -69789936579837530561521 -2324233744781396557646447 ------------------------, --------------------------, 75557863725914323419136 2417851639229258349412352 -2380138269427987011497177 -9633693187115762059734233 --------------------------, --------------------------] 2417851639229258349412352 9671406556917033397649408 and in Maple notation [8796093022163/8796093022208, 8796093021217/8796093022208, 35184372044233/ 35184372088832, 8796092936797/8796093022208, 70368740170301/70368744177664, 70368724993061/70368744177664, 140737332236035/140737488355328, 17592116658245/ 17592186044416, 1125885827850605/1125899906842624, 1125859489015355/ 1125899906842624, 4503174275055195/4503599627370496, 281410252616835/ 281474976710656, 2250623926322505/2251799813685248, 2249292396897405/ 2251799813685248, 35948019148270605/36028797018963968, 17937241413967665/ 18014398509481984, 571967025227067375/576460752303423488, 568640438987478525/ 576460752303423488, 2253625383822977925/2305843009213693952, 555499090400980575 /576460752303423488, 4352025258601667475/4611686018427387904, 4223095632029960115/4611686018427387904, 4048503429380773065/ 4611686018427387904, 3819784343490628215/4611686018427387904, 14118214741035262485/18446744073709551616, 12689727082233350595/ 18446744073709551616, 43935132666502730967/73786976294838206464, 4502577129329837109/9223372036854775808, 27101083544760182257/ 73786976294838206464, 17331209133939902557/73786976294838206464, 55442909339366505011/590295810358705651712, -15313441411139818601/ 295147905179352825856, -3746589284080145818739/18889465931478580854784, -\ 6442181675955876646289/18889465931478580854784, -35943266614334843532281/ 75557863725914323419136, -11304818931888567550889/18889465931478580854784, -\ 106723100787204867230017/151115727451828646838272, -120415814940318895789817/ 151115727451828646838272, -262721955796471403027209/302231454903657293676544, -\ 69789936579837530561521/75557863725914323419136, -2324233744781396557646447/ 2417851639229258349412352, -2380138269427987011497177/2417851639229258349412352 , -9633693187115762059734233/9671406556917033397649408] and in floating point [1.000000000, .9999999999, .9999999987, .9999999903, .9999999431, .9999997274, .9999988907, .9999960559, .9999874953, .9999641018, .9999055528, .9997700538, .\ 9994778011, .9988864833, .9977579637, .9957169208, .9922046261, .9864339189, .9\ 773542148, .9636373130, .9436950480, .9157378918, .8778792427, .8282836967, .76\ 53499547, .6879114835, .5954320786, .4881703905, .3672881707, .2348816824, .\ 9392394180e-1, -.5188395764e-1, -.1983427852, -.3410462582, -.4757051727, -.598\ 4721311, -.7062342391, -.7968450205, -.8692740333, -.9236621198, -.9612805464, -.9844021158, -.9961005290] The cut off is at j=, 32 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 45], vs. those in the, 2, -th row from j=1 to j=, 44, are as follws 22 21 20 [15 (586406201479 n - 141910300757159 n + 16131008391480059 n 19 18 - 1144861351279176685 n + 56882639982877147824 n 17 16 - 2102591450119720126182 n + 59980471142049116578150 n 15 14 - 1351972815418107221262506 n + 24452199776665469633217239 n 13 12 - 358365446033202377604789955 n + 4279962132735194450639718351 n 11 10 - 41745813471253308101493705897 n + 332241900081900752481659897314 n 9 8 - 2149044622333095868033560821408 n + 11215663394430278697168260194112 n 7 - 46698194466950083637289784276048 n 6 + 152547011650718422381259196629664 n 5 - 380869372813056630310686201434496 n 4 + 691107987574987860244424037539328 n 3 - 778364304310783537847280214308864 n 2 + 11046002004571997435125259796480 n + 2074376232745277527329262859059200 n - 3547513149806085207163070054400000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) 23 22 21 (2 n - 29)), 14835 (1185856828 n - 313659130973 n + 39077836501517 n 20 19 18 - 3049158823448353 n + 167122386425982603 n - 6840143203224812712 n 17 16 + 216964156550716784770 n - 5463161097068858672786 n 15 14 + 110963746071515775988418 n - 1837288447651773646409341 n 13 12 + 24960870274361191704141153 n - 279158209780413201980417949 n 11 10 + 2571255723475457630772017143 n - 19461205927783024057490909390 n 9 8 + 120424549948452132850847294816 n - 604082798636966607278926853632 n 7 6 + 2423834759944982692996497433008 n - 7604615833654442843672769151584 n 5 + 17790259163949245165570767407744 n 4 3 - 26876046430455086640053656097280 n + 7674202371431893813842062592000 n 2 + 76354640174364432483307106304000 n - 151761775887214964054186803200000 n + 3586969817801906175089049600000)/( 2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 989 ( 23 22 21 17787852409 n - 4704886956752 n + 586167544864484 n 20 19 - 45737381782109710 n + 2506835709820822149 n 18 17 - 102602138126195257242 n + 3254461457999698124464 n 16 15 - 81947352394149870448700 n + 1664452434093268899833699 n 14 13 - 27559145243846570648383012 n + 374405788676595775643966604 n 12 11 - 4187131344672154707300170790 n + 38562147990086065427418587119 n 10 9 - 291764895448079604500864112242 n + 1803482549867642378017401450464 n 8 7 - 9017054092930921240277216314480 n + 35817143336749910535297463359024 n 6 - 108933064536554680196096901022752 n 5 + 230523959297310378087065115993984 n 4 - 226032689218520502440125889656320 n 3 - 387852723388864960781429879654400 n 2 + 1554283272721170449486680952832000 n - 1145335489094690046934721740800000 n + 53804547267028592626335744000000) /(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 40549 ( 24 23 22 1735400227 n - 499795261428 n + 67973889601258 n 21 20 19 - 5806079493402672 n + 349430588306519797 n - 15757336826230458228 n 18 17 + 552741948590221180648 n - 15456264487048526248272 n 16 15 + 350262111647561864759437 n - 6504623653213887057177948 n 14 13 + 99706343349443716416804658 n - 1266712686160980299761399152 n 12 11 + 13356870205038861886572058987 n - 116759014331841225110457440508 n 10 9 + 842657069505859442555475583708 n - 4980186534066050335522581906192 n 8 + 23741166389513619696905944157392 n 7 - 88603981731068422947431831585088 n 6 + 241990012567730819973699581905728 n 5 - 394869471158219102621020354643712 n 4 - 15725029393604066708552392995840 n 3 + 1706104128928043106141487651123200 n 2 - 3148433979794635007193935486976000 n + 1806020637943250431131203174400000 n - 123356766904894822118916096000000 )/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 943 ( 24 23 22 74622207787 n - 21491194875396 n + 2922876804373138 n 21 20 - 249661325325820584 n + 15025501683752187397 n 19 18 - 677563982860437019116 n + 23767774671196511284648 n 17 16 - 664610492738044501116864 n + 15060774910612431096030517 n 15 14 - 279676107056999978987526876 n + 4286514942989591252617079578 n 13 12 - 54441864156058081026757045704 n + 573655218891546831361383314347 n 11 - 5006023863507442559949311996916 n 10 + 35982258060875567699011457326348 n 9 - 210657667996231673654018386787664 n 8 + 982840471370114381745367062417232 n 7 - 3494194697220448756662575522940096 n 6 + 8505560853570900946467073077108288 n 5 - 9394651403813344873139114772029184 n 4 - 17289659668913834756487553826657280 n 3 + 82373414181322701149569060576358400 n 2 - 116830885668973831275282113052672000 n + 60743933427411912985646808268800000 n - 5304340976910477351113392128000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 23575 (5969775886 n 24 23 22 - 1865554605107 n + 275952717342808 n - 25701709381220450 n 21 20 + 1691371145954915194 n - 83654526432759903005 n 19 18 + 3229397495232954912988 n - 99750217885674701830760 n 17 16 + 2507303480432362333918834 n - 51884173580652765285408365 n 15 14 + 890740676848990708249864768 n - 12745959290940791328720825770 n 13 12 + 152310801507756779057293033174 n - 1518617313160842145539561574355 n 11 + 12579941534510976789052209899788 n 10 - 85792538691512588429716071596780 n 9 + 473409323195899115552512064792944 n 8 - 2045419754518161391143668848978640 n 7 + 6466059825802229920459698108280128 n 6 - 12462825816471897640265498483019840 n 5 + 2333072681389655754993736565683968 n 4 + 62448790510466745054511074908009472 n 3 - 173964615510264293078206574863380480 n 2 + 206755145906174198680475946604953600 n - 100537708311234589420602495959040000 n + 10396508314744535608182248570880000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 1725 (81586897190 n 24 23 22 - 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(n - 41)! (2 n - 89)! (n + 1) + /87 3459573123103290731418720 (n - 43)! |-- (n - 41)! (n + 1) (2 n - 88)! + \86 (n - 42)! ((2 n - 87)! (n + 1) \\ - 1/3459573123103290731418720 binomial(2 n, n) (n - 45)! (n - 41)!)||/( // (n - 43)! (n - 42)! (n - 45)! (n - 41)! binomial(2 n, n)), ( 2333200478371986772352160 (n - 42)! (2 n - 89)! (n + 1) + 3499800717557980158528240 (n - 43)! ((2 n - 88)! (n + 1) - 1/3499800717557980158528240 binomial(2 n, n) (n - 45)! (n - 42)!))/( (n - 43)! (n - 45)! (n - 42)! binomial(2 n, n)), ( (2333200478371986772352160 n + 2333200478371986772352160) (2 n - 89)! - (n - 45)! (n - 43)! binomial(2 n, n))/((n - 45)! 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(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29), 14835/ 2097152*(1185856828*n^23-313659130973*n^22+39077836501517*n^21-3049158823448353 *n^20+167122386425982603*n^19-6840143203224812712*n^18+216964156550716784770*n^ 17-5463161097068858672786*n^16+110963746071515775988418*n^15-\ 1837288447651773646409341*n^14+24960870274361191704141153*n^13-\ 279158209780413201980417949*n^12+2571255723475457630772017143*n^11-\ 19461205927783024057490909390*n^10+120424549948452132850847294816*n^9-\ 604082798636966607278926853632*n^8+2423834759944982692996497433008*n^7-\ 7604615833654442843672769151584*n^6+17790259163949245165570767407744*n^5-\ 26876046430455086640053656097280*n^4+7674202371431893813842062592000*n^3+ 76354640174364432483307106304000*n^2-151761775887214964054186803200000*n+ 3586969817801906175089049600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n -9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 989/2097152*(17787852409*n^23-4704886956752*n^22+586167544864484*n^21-\ 45737381782109710*n^20+2506835709820822149*n^19-102602138126195257242*n^18+ 3254461457999698124464*n^17-81947352394149870448700*n^16+ 1664452434093268899833699*n^15-27559145243846570648383012*n^14+ 374405788676595775643966604*n^13-4187131344672154707300170790*n^12+ 38562147990086065427418587119*n^11-291764895448079604500864112242*n^10+ 1803482549867642378017401450464*n^9-9017054092930921240277216314480*n^8+ 35817143336749910535297463359024*n^7-108933064536554680196096901022752*n^6+ 230523959297310378087065115993984*n^5-226032689218520502440125889656320*n^4-\ 387852723388864960781429879654400*n^3+1554283272721170449486680952832000*n^2-\ 1145335489094690046934721740800000*n+53804547267028592626335744000000)/(2*n-5)/ (-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/( 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/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/ (2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17 )/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 943/4194304*(74622207787*n^24-\ 21491194875396*n^23+2922876804373138*n^22-249661325325820584*n^21+ 15025501683752187397*n^20-677563982860437019116*n^19+23767774671196511284648*n^ 18-664610492738044501116864*n^17+15060774910612431096030517*n^16-\ 279676107056999978987526876*n^15+4286514942989591252617079578*n^14-\ 54441864156058081026757045704*n^13+573655218891546831361383314347*n^12-\ 5006023863507442559949311996916*n^11+35982258060875567699011457326348*n^10-\ 210657667996231673654018386787664*n^9+982840471370114381745367062417232*n^8-\ 3494194697220448756662575522940096*n^7+8505560853570900946467073077108288*n^6-\ 9394651403813344873139114772029184*n^5-17289659668913834756487553826657280*n^4+ 82373414181322701149569060576358400*n^3-116830885668973831275282113052672000*n^ 2+60743933427411912985646808268800000*n-5304340976910477351113392128000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-\ 41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 23575/4194304*(5969775886*n ^25-1865554605107*n^24+275952717342808*n^23-25701709381220450*n^22+ 1691371145954915194*n^21-83654526432759903005*n^20+3229397495232954912988*n^19-\ 99750217885674701830760*n^18+2507303480432362333918834*n^17-\ 51884173580652765285408365*n^16+890740676848990708249864768*n^15-\ 12745959290940791328720825770*n^14+152310801507756779057293033174*n^13-\ 1518617313160842145539561574355*n^12+12579941534510976789052209899788*n^11-\ 85792538691512588429716071596780*n^10+473409323195899115552512064792944*n^9-\ 2045419754518161391143668848978640*n^8+6466059825802229920459698108280128*n^7-\ 12462825816471897640265498483019840*n^6+2333072681389655754993736565683968*n^5+ 62448790510466745054511074908009472*n^4-173964615510264293078206574863380480*n^ 3+206755145906174198680475946604953600*n^2-100537708311234589420602495959040000 *n+10396508314744535608182248570880000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-45), 1725/4194304*(81586897190*n^25-25495886031281*n^24+ 3771345213187976*n^23-351254967370432350*n^22+23115160297273576378*n^21-\ 1143252365588214293335*n^20+44132926371738747099876*n^19-\ 1363109076977504187162920*n^18+34258762041555533064316778*n^17-\ 708745078799735174243616735*n^16+12161239472259812399463871936*n^15-\ 173831669404002652026947439190*n^14+2072702053691921342599943191398*n^13-\ 20576685989032343772149607146825*n^12+169037569183749358578966144353876*n^11-\ 1134905102544534854108601291275460*n^10+6085475999279382092173782504763408*n^9-\ 24949347239310280075556409415596400*n^8+71136677826093138107103120711077056*n^7 -102179872052445642842102978667430080*n^6-135171389182163983011514324469521152* n^5+1051563511886146993090372133447064576*n^4-\ 2357027361278870354561343486232350720*n^3+2556369044589194374838563998351360000 *n^2-1209827526719944204180496674750464000*n+ 142085613634841986645157397135360000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29)/(2*n-45), 12765/4194304*(22050475417*n^26-7453042731823*n^25+ 1194989219506485*n^24-120923822501323790*n^23+8667914307597369305*n^22-\ 468268693826312123745*n^21+19804835079885351875665*n^20-\ 672422460524470490202940*n^19+18645632633988922500645135*n^18-\ 427307582821910090531868065*n^17+8158405919993460374048116415*n^16-\ 130398723313849753548472167390*n^15+1748216007771538627956865818595*n^14-\ 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)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 12765/16777216*(176400313304*n^27-64296700322553*n^26+11139213644142363*n^25-\ 1220578865684129250*n^24+94957978621024617930*n^23-5581489386553820124255*n^22+ 257523886086317887275525*n^21-9565526593566042637334940*n^20+ 291045095169843239129773440*n^19-7341493376822597591841131175*n^18+ 154764025227946291059598361445*n^17-2739618245542845984505335829890*n^16+ 40795666600263098236906825923810*n^15-510336925717887685350790698565665*n^14+ 5336440759890702177223199268874635*n^13-46192024610339483337524465312089440*n^ 12+325492085419995804792025407694793820*n^11-\ 1814262447382749870092100932790303360*n^10+ 7572605595562167354795107467211302480*n^9-\ 20608408222144002457582692429442045440*n^8+ 15350925173087832469916825718688677696*n^7+ 155700970764636968028908746076297035008*n^6-\ 835831988600847773425159652646721256448*n^5+ 2212993588324871994930000474006629928960*n^4-\ 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10829805862618035364125618481408636860*n^11-\ 57218758787080233858453079585722941520*n^10+ 217136027840056813737874247775478989520*n^9-\ 457554541301097873917762261125172001600*n^8-\ 513812943955075381750091326241017639872*n^7+ 8415231745520026351954639710614494865664*n^6-\ 34911389502728410074245306023758586113024*n^5+ 84166714276018228031221624393446053990400*n^4-\ 125968726676084376928184591672451339878400*n^3+ 111412402442619289906675687901929783296000*n^2-\ 50138267597731324887921838087944929280000*n+ 7681148273099557798037208889137561600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1035/33554432*(8701327463091*n^28-\ 3410567262271162*n^27+636538116886991487*n^26-75282853798310430090*n^25+ 6334287794199643309395*n^24-403526367666932504448090*n^23+ 20223207575980768060645515*n^22-817768101236229179151365250*n^21+ 27148904140649854027682502285*n^20-748874302093932706256195901270*n^19+ 17299854332419776567182246832045*n^18-336247170193271772367167964187550*n^17+ 5507528700875326153798778625859665*n^16-75912531149473363015831762782924030*n^ 15+876210695050285093242286106304526905*n^14-\ 8390903084240356800836690477929648950*n^13+ 65622690463601649229697129154288397980*n^12-\ 407810008418504843142049869122888535960*n^11+ 1908290291435590128891866991361764340560*n^10-\ 5825251226507500598053535329337047748000*n^9+ 4045822198804274994597970062921508920384*n^8+ 69067153400686974709925904744254395696512*n^7-\ 442859343021298823673469835397766721536512*n^6+ 1512428582668809134119419746829147958179840*n^5-\ 3310207608330500047381595441753678645452800*n^4+ 4673169343776339124218826091587082870784000*n^3-\ 3995551124848809630765683140110535065600000*n^2+ 1780936776688440776438485315239247872000000*n-\ 281642103346983785928030992601710592000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 345/8388608*( 6524734514245*n^28-2557120875702638*n^27+477160827912559917*n^26-\ 56416080916279530810*n^25+4744552323132561692025*n^24-302026338587858949767430* n^23+15119115177632045286703425*n^22-610319911810306247872492170*n^21+ 20210054241479470230339251775*n^20-555392221317431736031121778450*n^19+ 12761504744234371056463421999655*n^18-246166609293265343486285284254270*n^17+ 3989894240219560619075260140817875*n^16-54207478197538858037979292787352690*n^ 15+613512901594145891329169285161530315*n^14-\ 5718871253077925785187385210068423070*n^13+ 43047377811900325959420337296735250800*n^12-\ 252256218168605939303305565974270932360*n^11+ 1059318498352225947833093431916546657120*n^10-\ 2342879564739660789089065307496996308320*n^9-\ 5428891565422750750230231138026224657920*n^8+ 77951291645377842751515248364412891709568*n^7-\ 388326002702073849237719376906838614450432*n^6+ 1207354274623830155259101320867291885808640*n^5-\ 2512374644996720759203032323505067377868800*n^4+ 3443512304407032923991108543033717786624000*n^3-\ 2901921538590790026686728508301047193600000*n^2+ 1295803284542174545989059595810860236800000*n-\ 211231577510237839446023244451282944000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 310155/8388608*( 14509749634*n^29-6098747667335*n^28+1222513044951156*n^27-155538638436743415*n^ 26+14101552239365682000*n^25-969588734191272390195*n^24+52532122640649509890500 *n^23-2300146582671194011366755*n^22+82809985461033799627118460*n^21-\ 2480564164847528071134667845*n^20+62308635475091098260387123540*n^19-\ 1318370393604943484346704677605*n^18+23534181584473883155693936376880*n^17-\ 353944931219556756116484822432945*n^16+4463800870285977724330637604893820*n^15-\ 46781734091991049857832153260353505*n^14+401045968926184706483585991075765090*n ^13-2732985685952661310148507083184751120*n^12+ 13937004640769903441721016926170963960*n^11-\ 44298236039931301964931574494123746080*n^10-\ 4682075716674530854167201659502202464*n^9+ 1065010743055514171957163840245631997440*n^8-\ 7384656322452428300835894668597669102976*n^7+ 30479376056594694796955414614102296687360*n^6-\ 85964195962784343256185022567205310489600*n^5+ 168173745718350798633094480370162829312000*n^4-\ 221254176776151059624129630406901585920000*n^3+ 181908569730186656346157581854433484800000*n^2-\ 80614099000702462385548590217342156800000*n+ 13392880887745891933730061105364992000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 930465/8388608* (4832828444*n^29-2030542876157*n^28+406794776141166*n^27-51712711841934633*n^26 +4682787994557786750*n^25-321428870379634799145*n^24+17373407253341788002510*n^ 23-758196621542944143650805*n^22+27174086820751325560655550*n^21-\ 809088914265382661433097575*n^20+20160895697161498988606256450*n^19-\ 422112798985955298977022451275*n^18+7432592031390120445213418023770*n^17-\ 109810110009200548872703804534035*n^16+1352899229239891614484634742975930*n^15-\ 13739389397886020835901611872553015*n^14+112611410667600119566586534880325230*n ^13-714235430339333787789985426206500640*n^12+ 3148974846881938549441539473856446920*n^11-\ 5603598681889367044273769786473113760*n^10-\ 45956369476526953015740359742660948384*n^9+ 521238681121536886885397805074680196352*n^8-\ 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21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-63)/(2*n-45), 7753875/1073741824*(569705385313*n^32-288125634034064*n^31+ 69602785955342504*n^30-10685657707749372160*n^29+1170121088788536988572*n^28-\ 97228030799300878233792*n^27+6366612405002073966869928*n^26-\ 336763009949057733170243712*n^25+14630543685624462362118687270*n^24-\ 527833098115703912723624515680*n^23+15918673595323180861610690805000*n^22-\ 402381989142758647332035087456640*n^21+8510262145701832959203017604446140*n^20-\ 149435325689018221045628678629222080*n^19+2138792942846242162813618846441391640 *n^18-23894282119476714044476708729958171520*n^17+ 183243279717712291974794798551512398145*n^16-\ 372448042915062865352352025376989011600*n^15-\ 15509743400142063457777685536919593323760*n^14+ 314522311575428078139625009487329540449920*n^13-\ 3773898500138432487597153817378383524052128*n^12+ 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3004230854753557990949618095968536984097185747349504*n^7-\ 7970202145034598635982935422701286217087355105484800*n^6+ 16525586303644277261298552409837527417382486464921600*n^5-\ 25894487760355716019276732317576797983397649653760000*n^4+ 29185285251985456326945030189877826092413020405760000*n^3-\ 21877388921940215714571850823429857956161794867200000*n^2+ 9467161379544120215128895185833263059279282176000000*n-\ 1692756661170467776485357532980390943009013760000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-25 )/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-\ 29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 40455/1073741824*(370667563611757*n^34 -206017568087423991*n^33+54632712177629552923*n^32-9193931454421205377864*n^31+ 1101464045838128608662708*n^30-99869942470514358433200404*n^29+ 7109791136782092226242160012*n^28-406668752739267211693629378216*n^27+ 18948786345022455596528137399830*n^26-723631469480668698454797871812090*n^25+ 22589953417926342521662832637990570*n^24-567010596646756578431803699140902760*n ^23+10893683674285037028476638648192270260*n^22-\ 134508245089129779570781215757924519380*n^21-\ 108734335214598253256889835295471737860*n^20+ 60323435920683487934900554365649972461480*n^19-\ 1974938325962666463967431011097045228907995*n^18+ 42824988810204931719297511083722460371920185*n^17-\ 722249794947230816352062709668033494045773805*n^16+ 9959448945346884743481975216664405629044712240*n^15-\ 114642725561538581921384993755578718725485093392*n^14+ 1111602693701382953228323502158713114913575966496*n^13-\ 9106400129833596832930092580506176871874690958688*n^12+ 62973782279215771829199123840254939274550312400384*n^11-\ 366188532807466167446758953585011624109052364627968*n^10+ 1778649031164351416364548779402686898231050750614784*n^9-\ 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n^32+12286283977359240214759624*n^31-1168337314775456453659371340*n^30+ 87121724886481856235680287140*n^29-5207260830710925895374367832700*n^28+ 252418975761143506381904323444728*n^27-9944074640456635634061379879255650*n^26+ 314764564446445920311916428377664850*n^25-7692571240409134525961367042943184250 *n^24+126443725164841867845790244177427378120*n^23-\ 356374264807733617922757957557329980300*n^22-\ 64264949799129498794368656474443723252700*n^21+ 2835917221114930386546971816630612695024500*n^20-\ 78752569151564495878359508493100045891951120*n^19+ 1686628427350951836309907807216880071751025225*n^18-\ 29559948489958880826785152091612907677684038725*n^17+ 434581297408318645620994273470464517126400068125*n^16-\ 5422665437536438330905099603613142528284872661328*n^15+ 57736600646913235229182830460192983954780495590160*n^14-\ 525364721740041018199171999329069946496463760848160*n^13+ 4080856967919857181773810936277661680949578221538400*n^12-\ 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2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*( 9129127500423089*n^35-5242291320719062120*n^34+1432907604966000520805*n^33-\ 247755942174343605627625*n^32+30359704741267665075456452*n^31-\ 2797194257372301025082822440*n^30+200346012576134960782750811220*n^29-\ 11346209218219470938550355602300*n^28+509126327327344035926036188847694*n^27-\ 17739410979172232190228783253698840*n^26+444224215556280983194874882030256150*n ^25-5393692861618823295788931459467889750*n^24-\ 158599418994648659143078767263738467740*n^23+ 13067252160872192092924623438123714135000*n^22-\ 514698902412926628720122982957998379179100*n^21+ 14871503358987207186329634761290409075254500*n^20-\ 345572111357376135201289570153646039495718135*n^19+ 6691350635213884750811780358609949696246646400*n^18-\ 109812611008356852435251896679965562790598022675*n^17+ 1540837896859588096272046597942851256956971992375*n^16-\ 18564703132332636369743926662511806627920246237744*n^15+ 192335051193929484651746360014776924589750454330320*n^14-\ 1712200067810925017587094084767668440061387880074080*n^13+ 13063980400520541130124502379545419774593073125108000*n^12-\ 85068666613330523780974738817017078029878957271197952*n^11+ 469888263969144231388482059080422735777219330040323840*n^10-\ 2183776814149124277789103312397448313436520691461861120*n^9+ 8448421545983141449242554471965138139595079495923244800*n^8-\ 26832368368112290470311567075796916046967925505327345664*n^7+ 68694922372269600063785468443382805892962187153808547840*n^6-\ 138334726169983769789189977371382707133491422541894451200*n^5+ 211797856864404567813118884404240826031668497083883520000*n^4-\ 234618525589785753192133937219024320157239371786485760000*n^3+ 173883151252734730335890505668692103010025830521241600000*n^2-\ 74877992801205386537118641412324725808220647456768000000*n+ 13432024106387661806411312024199402132776524185600000000)/(2*n-69)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2* n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/( 2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 899/2147483648*( 94025117858851295*n^36-56177806255790552622*n^35+15951648128253258236025*n^34-\ 2858034476702752041619890*n^33+361449145514267380922816310*n^32-\ 34141356900762297526493218296*n^31+2478009524102514874310544870340*n^30-\ 139126968946922922927400124865960*n^29+5901718023273971093827149952634970*n^28-\ 169956440427897663134485497412032612*n^27+1493744484803161944711392210035715310 *n^26+174673930653025417713277945147903949700*n^25-\ 13521921715793485715917028535158423494200*n^24+ 616072263831038077004645750352723468500520*n^23-\ 21332486911437179351488985517661432092381900*n^22+ 602360985634391574629348861902258785838375800*n^21-\ 14293522154588215168706591440075671247363369425*n^20+ 289483348461147805726220850552987598956296381730*n^19-\ 5047522173719981495296500859300781314398117118175*n^18+ 76139093683847327490278809145746272365411577765150*n^17-\ 995954990135541745592843127191211582890353929548070*n^16+ 11303493609097930859468641700356658491560135718677312*n^15-\ 111203299924016820079431330580929783601612883274268800*n^14+ 946146603967441016019099917376751169155096512674859840*n^13-\ 6936771135294879210761836026006360387938731368689450560*n^12+ 43603084626603665407253026962880318764330653386193719296*n^11-\ 233420194728497745838079698339135735838739952900845431040*n^10+ 1055074219629050565837428258457720150330487449513802268160*n^9-\ 3982583216636776685015141056142974830447826870502686840320*n^8+ 12377394076275792774992627628255969838355574484541359644672*n^7-\ 31092247068572463472766588955037491396649013169832575221760*n^6+ 61592180843217145457031103951664883802291625507551081267200*n^5-\ 92992586580801846877354112022594218031264416238252441600000*n^4+ 101828367965639581327633502279558777847386195405608058880000*n^3-\ 74784318452389815658510625998824578541470099552364134400000*n^2+ 31997261186625986073919053533220572196041927675609088000000*n-\ 5722042269321143929531218922308945308562799303065600000000)/(2*n-5)/(-1+2*n)/(2 *n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2* n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53) /(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 899/1073741824 *(38083103822761951*n^36-22070450386003944870*n^35+6034069403907074995605*n^34-\ 1029657091811908458276630*n^33+121895533158783343794593913*n^32-\ 10458951666171091283720880360*n^31+649296466680986681813238027980*n^30-\ 26696120840265088473169449170520*n^29+362490767066902857744567664861326*n^28+ 46668653938673187489761426725663980*n^27-4952838839579992499850513179102407410* n^26+299089665425674795788295177040749406700*n^25-\ 13552662959276158223239598626251583531110*n^24+ 496955886846881107392180304801408778272200*n^23-\ 15238726456516015552271059752071625445265700*n^22+ 397687421731085371124604167965710179405142600*n^21-\ 8925167246087952797444478053588780298017050365*n^20+ 173364421714951490022137467750747847465293399050*n^19-\ 2925884874401314041844691876389164838860993818875*n^18+ 42993898228795735778781656386903278582808577005050*n^17-\ 550414896118746764645737833488139379756607246355171*n^16+ 6135633396772241564451620494139306979877903456877520*n^15-\ 59453226742028135967620870875990090400103792608502480*n^14+ 499356266842606427461138354504800438525481778642729280*n^13-\ 3620930573035266119553099021017862795144473851269497888*n^12+ 22546772763231488578486030648074639427196115983070735360*n^11-\ 119732321482137326583319857154384822498255693364127175680*n^10+ 537518259948159569668238530855152897523578255252116881920*n^9-\ 2017390669879332997660586027624901151731466132881198502656*n^8+ 6240302635599741312646481935032140538053682965393548677120*n^7-\ 15616531318925815078329066720944418452362147767440328253440*n^6+ 30845754016532448223795290963607686150439128508562512281600*n^5-\ 46475163189769252815667407578883472498729725863470817280000*n^4+ 50827895690579968341785328837895262236185015168889651200000*n^3-\ 37313603807084686331295379407077334451353158918248857600000*n^2+ 15972896540779009056324068903732177084728485639880704000000*n-\ 2861021134660571964765609461154472654281399651532800000000)/(2*n-5)/(-1+2*n)/(2 *n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2* n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/( 2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53) /(2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 1147/ 1073741824*(44037773012583002*n^37-25798187475203399239*n^36+ 7031257764689682245244*n^35-1166979619270470658228485*n^34+ 127910675564883694705338816*n^33-8988539413363804399805840217*n^32+ 268960388208319947982986182512*n^31+24104024339020707703066066257460*n^30-\ 4473508329785809841944219650778788*n^29+405854852121843924003470158472479746*n^ 28-26663660424234472163434415624710216296*n^27+ 1393477071955903753090019944196178373650*n^26-\ 60295243430830939449693593959633189418520*n^25+ 2207359026761641592785378584987628334160390*n^24-\ 69286283755610809824551025862892080522662240*n^23+ 1881066810095830439070671927628406102013633700*n^22-\ 44433592211381674168958928925716759444124533030*n^21+ 916831929955964878865342205289228429562807393685*n^20-\ 16566330347238003805912836465944819022202852071860*n^19+ 262481106544397856715698206445576166826276323926475*n^18-\ 3647897698526125278092113442766798270299990534033592*n^17+ 44442277352840160146528260585972728117737754168856419*n^16-\ 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n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), -41/137438953472*( 5921400893383216838414*n^41-5057688430933668852693577*n^40+ 2087746963436148310780663240*n^39-554880390023432834423274805970*n^38+ 106735250742052222814179708161798*n^37-15836143163051845797771987824768349*n^36 +1886024092878040418283205994573915500*n^35-\ 185270243005128746225983103683061489060*n^34+ 15308159443089400461541384543268444531372*n^33-\ 1079576327713995950081731148635316659171426*n^32+ 65716666901608741602566252117639735889703680*n^31-\ 3483406514937563101015370352017463849195106620*n^30+ 161903498780717583760077337424114035519214906076*n^29-\ 6634840646000758927528264594842857577380653196178*n^28+ 240786423018582228416802426791107870716107628261480*n^27-\ 7765339585395343102161532591105614996704208503287200*n^26+ 223138165234809195742945249839858439333839865723764630*n^25-\ 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/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-\ 59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), -41/137438953472*( 12878070761703293933323*n^42-11467161654190675576122861*n^41+ 4940769958417770267411891781*n^40-1372365325440363620598771602910*n^39+ 276235925537927005776821060986621*n^38-42941976602655765866086466353712697*n^37 +5365511174639028059433423771215160287*n^36-\ 553715665940075177622995240151994746120*n^35+ 48131033640854709952714787735909884591634*n^34-\ 3576029455629897508003739442765268764133338*n^33+ 229676719429524792765336345653529887848502498*n^32-\ 12865142744047356417303769854634345377548577780*n^31+ 632909702962972076779567191769145349899252767242*n^30-\ 27499957034668191353673825204990952978294974377994*n^29+ 1060058556070568910380165570137153141170120406711374*n^28-\ 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10252053134698498919357709286167332638127106861689884938329495961600000000*n+ 1797926451438391079617004901220516547320730063239670913029898240000000000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83), -41/549755813888*( 54618089971623756502777*n^42-48437148943581052600855539*n^41+ 20789741582350316269970067694*n^40-5753691770264509150796600032590*n^39+ 1154159733927038562891556850562829*n^38-178837806562678450501914741994701303*n^ 37+22277113960224356381685688187990719688*n^36-\ 2292348489556914866274886850342943450780*n^35+ 198717994180898912218304981322738699683666*n^34-\ 14726524860580408532087057579328367611081462*n^33+ 943559583970187063295264672949433082840863252*n^32-\ 52733091146187051792360457692253353193162164420*n^31+ 2588731343889757857069216658322140361480925884658*n^30-\ 112256513796119572050113713073156980989421685638806*n^29+ 4319155765116385151822753210501270233724286039307176*n^28-\ 147975822751149304718949511145359600538183180065702560*n^27+ 4526820248369183871718708229370382128134162657523151365*n^26-\ 123914838630645609017715452923201170583107378577333654455*n^25+ 3039785429047717984957895940898165097497084615521194656430*n^24-\ 66893403359430217409493417427148786306221237193103525143550*n^23+ 1321178149268703874712243635502063123272093848155547777733313*n^22-\ 23420172148780280605423244880946022536498460395841911712159091*n^21+ 372462629529661679069946350799358119042643877939619631841517536*n^20-\ 5309483544403099343342119366392482795757028668537890957846609460*n^19+ 67749431938595981478873411129827817352023598346128590529887229936*n^18-\ 772381958068596584624649499745396379628604656160448609525752403552*n^17+ 7848484339401596097575507966605701588615125090907973666096162619392*n^16-\ 70869728687106593501928037299473794957100676749912592751014737339520*n^15+ 566574633103085886999167099516096359121698354263454371970641260410624*n^14-\ 3992411729426652812217649432062907968199066922874409654381118698061568*n^13+ 24663519526699708033620302232290899461959622302431980655730590633648128*n^12-\ 132706679778724265848654483722778785084790044593412655832904919716602880*n^11+ 617062487522327955627101198904371461089308242594619668962473960132677632*n^10-\ 2455795217895712822318237418882048542213533432966488072826391004453044224*n^9+ 8266725354809551143920359048756892144111351936787879977421900738967240704*n^8-\ 23189920488571250949004748838457620585118145234092290026907727429785354240*n^7+ 53189397667129912117205918627782532084823438194793913387177423393338163200*n^6-\ 97279858242817085493066129312183649048815971264565753769287722886758400000*n^5+ 137082333725208716941534518031360299417095417762576116743236456137359360000*n^4 -141623336424996461776997865704176264388641164411302759895111772510617600000*n^ 3+99231710164384727395409439521080986859898982800474913503901382606848000000*n^ 2-41005139160244528342252004657658697190889075771437762538082035302400000000*n+ 7191705805753564318468019604882066189282920252958683652119592960000000000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83), ( 2333200478371986772352160*(n-42)!*(n-41)!*(2*n-89)!*(n+1)+ 3459573123103290731418720*(n-43)!*(87/86*(n-41)!*(n+1)*(2*n-88)!+(n-42)!*((2*n-\ 87)!*(n+1)-1/3459573123103290731418720*binomial(2*n,n)*(n-45)!*(n-41)!)))/(n-43 )!/(n-42)!/(n-45)!/(n-41)!/binomial(2*n,n), (2333200478371986772352160*(n-42)!* (2*n-89)!*(n+1)+3499800717557980158528240*(n-43)!*((2*n-88)!*(n+1)-1/ 3499800717557980158528240*binomial(2*n,n)*(n-45)!*(n-42)!))/(n-43)!/(n-45)!/(n-\ 42)!/binomial(2*n,n), ((2333200478371986772352160*n+2333200478371986772352160)* (2*n-89)!-(n-45)!*(n-43)!*binomial(2*n,n))/(n-45)!/(n-43)!/binomial(2*n,n)] The limits, as n goes to infinity are 8796093022185 4398046510845 17592186032501 70368743804623 70368741943141 [-------------, -------------, --------------, --------------, --------------, 8796093022208 4398046511104 17592186044416 70368744177664 70368744177664 70368733256225 70368698826375 281474318698005 1125891396632915 --------------, --------------, ---------------, ----------------, 70368744177664 70368744177664 281474976710656 1125899906842624 281468749915695 2251666241829465 9005873924299185 2251033407414525 ---------------, ----------------, ----------------, ----------------, 281474976710656 2251799813685248 9007199254740992 2251799813685248 2250135698866635 1124194429536615 71845379942532465 573316825240332645 ----------------, ----------------, -----------------, ------------------, 2251799813685248 1125899906842624 72057594037927936 576460752303423488 285447562608266595 1134024108977433645 4488273354314522475 ------------------, -------------------, -------------------, 288230376151711744 1152921504606846976 4611686018427387904 4417424344543837875 4316211473442859875 4176537711323510235 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 1995026498159979135 14995356285913629435 3446225238495370761 -------------------, --------------------, -------------------, 2305843009213693952 18446744073709551616 4611686018427387904 24621256868641071033 84528580955107314205 34236710336662993949 --------------------, ---------------------, --------------------, 36893488147419103232 147573952589676412928 73786976294838206464 25255662822716351647 3870542484973016109 82195887958184295951 --------------------, --------------------, ----------------------, 73786976294838206464 18446744073709551616 1180591620717411303424 -1410409211121179101529 -2067976314786653469137 -13562666012612509089053 -----------------------, -----------------------, ------------------------, 18889465931478580854784 9444732965739290427392 37778931862957161709568 -74154002123012213584207 -92247945825341465609657 ------------------------, ------------------------, 151115727451828646838272 151115727451828646838272 -108092372202516270085997 -121388718314355945187487 -------------------------, -------------------------, 151115727451828646838272 151115727451828646838272 -528000901229835051266243 -2239341688836574016613857 -------------------------, --------------------------, 604462909807314587353088 2417851639229258349412352 -581834887926551784603983 -4762790763509392112188699 -------------------------, --------------------------, 604462909807314587353088 4835703278458516698824704 -19269900598884942208662811 ---------------------------] 19342813113834066795298816 and in Maple notation [8796093022185/8796093022208, 4398046510845/4398046511104, 17592186032501/ 17592186044416, 70368743804623/70368744177664, 70368741943141/70368744177664, 70368733256225/70368744177664, 70368698826375/70368744177664, 281474318698005/ 281474976710656, 1125891396632915/1125899906842624, 281468749915695/ 281474976710656, 2251666241829465/2251799813685248, 9005873924299185/ 9007199254740992, 2251033407414525/2251799813685248, 2250135698866635/ 2251799813685248, 1124194429536615/1125899906842624, 71845379942532465/ 72057594037927936, 573316825240332645/576460752303423488, 285447562608266595/ 288230376151711744, 1134024108977433645/1152921504606846976, 4488273354314522475/4611686018427387904, 4417424344543837875/ 4611686018427387904, 4316211473442859875/4611686018427387904, 4176537711323510235/4611686018427387904, 1995026498159979135/ 2305843009213693952, 14995356285913629435/18446744073709551616, 3446225238495370761/4611686018427387904, 24621256868641071033/ 36893488147419103232, 84528580955107314205/147573952589676412928, 34236710336662993949/73786976294838206464, 25255662822716351647/ 73786976294838206464, 3870542484973016109/18446744073709551616, 82195887958184295951/1180591620717411303424, -1410409211121179101529/ 18889465931478580854784, -2067976314786653469137/9444732965739290427392, -\ 13562666012612509089053/37778931862957161709568, -74154002123012213584207/ 151115727451828646838272, -92247945825341465609657/151115727451828646838272, -\ 108092372202516270085997/151115727451828646838272, -121388718314355945187487/ 151115727451828646838272, -528000901229835051266243/604462909807314587353088, -\ 2239341688836574016613857/2417851639229258349412352, -581834887926551784603983/ 604462909807314587353088, -4762790763509392112188699/4835703278458516698824704, -19269900598884942208662811/19342813113834066795298816] and in floating point [1.000000000, .9999999999, .9999999993, .9999999947, .9999999682, .9999998448, .9999993555, .9999976623, .9999924414, .9999778780, .9999406822, .9998528588, .\ 9996596472, .9992609846, .9984852319, .9970549378, .9945461559, .9903451760, .9\ 836091221, .9732391443, .9578762142, .9359291713, .9056422520, .8652048254, .81\ 28998931, .7472809781, .6673605047, .5727879444, .4639939466, .3422780562, .209\ 8225285, .6962262523e-1, -.7466644193e-1, -.2189555091, -.3590007802, -.4907100\ 232, -.6104456987, -.7152953172, -.8032831550, -.8735042178, -.9261700149, -.96\ 25650780, -.9849220453, -.9962305113] The cut off is at j=, 33 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 46], vs. those in the, 2, -th row from j=1 to j=, 45, are as follws 23 22 21 [15 (2345624805919 n - 620417761163954 n + 77295960634838726 n 20 19 - 6031236160521936364 n + 330568081573817240769 n 18 17 - 13529803402186895269446 n + 429155115384008856700660 n 16 15 - 10806133687512636807136088 n + 219486353920765013072435489 n 14 13 - 3634159837091281175245325758 n + 49372741793729958936903577734 n 12 11 - 552179957351306265223692265452 n + 5086094153460990410547300920239 n 10 - 38498033227875010959341966209130 n 9 + 238278417537464775278346186647648 n 8 - 1196232635813402814296497673621776 n 7 + 4813450103025135886202345677019184 n 6 - 15257498958472529912619057391511712 n 5 + 37084801477956490015108556014415232 n 4 - 65589366013981857032463674064760320 n 3 + 71345094102927363678546086702438400 n 2 + 4036862170698648133068830822400000 n - 198627075383603593728947402096640000 n + 326371209782159839059002445004800000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) 23 22 (2 n - 29) (2 n - 45)), 15 (1172812402925 n - 310208880555826 n 21 20 + 38647980307975126 n - 3015618078099267872 n 19 18 + 165284040435003545055 n - 6764901657755870572482 n 17 16 + 214577553499442061078788 n - 5403066517186750256314624 n 15 14 + 109743156135667505672523115 n - 1817078819140394238442109558 n 13 12 + 24686322497659684895150052390 n - 276088194878930786725851646896 n 11 + 2542991974103364993936009964505 n 10 - 19247592242983408061943007972534 n 9 + 119108536997096578233723898488352 n 8 - 597570451511830708205578717712048 n 7 + 2398793711772862372952485955653200 n 6 - 7536373578389029859769352599669600 n 5 + 17703556097631588366938978151625344 n 4 - 27111733941434998078495102735618560 n 3 + 9098683422227173085856982359091200 n 2 + 74287848122129311839279505059840000 n - 153485669800096202691224989286400000 n + 3547513149806085207163070054400000)/(2097152 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) 24 23 (2 n - 29) (2 n - 45)), 645 (109098828143 n - 31420462486572 n 22 21 20 + 4273291990222514 n - 365008892189865264 n + 21967540587463190489 n 19 18 - 990611732176909074540 n + 34749087938227880726168 n 17 16 - 971687009680527093400656 n + 22019998626778416261844385 n 15 14 - 408933082043685229674395268 n + 6268570305346763159829777722 n 13 12 - 79646206999990864990874081328 n + 840041418432057939150944292071 n 11 - 7348083250555522678116970667556 n 10 + 53123928433965203743365939775196 n 9 - 315394047193645019807384638435920 n 8 + 1521168182366861243597924831890832 n 7 - 5848353581990574426288688566166464 n 6 + 17244693102433514378345304610478400 n 5 - 35281292691513328412460906928856832 n 4 + 32330912703333484043046559580974080 n 3 + 62578064392731300136106512944230400 n 2 - 233311419475863335155688393318400000 n + 168955108652288175296450737766400000 n - 7755028746087721150542525235200000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), 43 (1636482418009 n 23 22 21 - 471306934258620 n + 64099378790948206 n - 5475133148059708320 n 20 19 + 329513071999904496319 n - 14859171628470267478380 n 18 17 + 521235915756944572317496 n - 14575275171696844507238160 n 16 15 + 330298163064116296645399639 n - 6133905525043089823861054740 n 14 13 + 94024797420598401687657833926 n - 1194563625747532566400965636000 n 12 + 12596909003201895218559882068929 n 11 - 110132978020449735518762082288900 n 10 + 795129608399778230991561460982836 n 9 - 4703296607561502447944700007896240 n 8 + 22463979272069285904625482413121904 n 7 - 84185107806828509699582203655175360 n 6 + 231996601225053211860002723626777536 n 5 - 387531383014180766850751715384321280 n 4 + 18398264091809084869874695260595200 n 3 + 1590834320304310953920452671853056000 n 2 - 3006076793870731759851995389992960000 n + 1736907369588780867662714449920000000 n - 116325431191315817258137878528000000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 8815 (15965681891 n 24 23 22 - 4989275470030 n + 738013584715910 n - 68737234568822020 n 21 20 + 4523449913726705885 n - 223728936633064506490 n 19 18 + 8636909382573464043260 n - 266784147793481385378520 n 17 16 + 6706160340510327638246525 n - 138786916709668116261437890 n 15 14 + 2383243468992777372483448310 n - 34120685045680431340657278820 n 13 12 + 408199996745096113817943511955 n - 4079978121159417002439560165590 n 11 + 33973119888530802591298329253160 n 10 - 234168804236573229540184121998720 n 9 + 1320047077677934315526247881893520 n 8 - 5948492435843690828205489512019040 n 7 + 20460339694765875837491703245756160 n 6 - 48108826916299665781458953146801920 n 5 + 50032354493791167284389999487860224 n 4 + 101547741959643451422454929475799040 n 3 - 455819875251614563955946760487116800 n 2 + 633677213749616026265740781998080000 n - 325328659427609187330792075264000000 n + 27804615260363292905603688038400000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 1025 (137304854590 n 24 23 22 - 42907762126349 n + 6346914481306984 n - 591139714423358150 n 21 20 + 38901592979450033602 n - 1924060145413363395515 n 19 18 + 74276643789537571470484 n - 2294288219245914312382280 n 17 16 + 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(2 n - 27) (2 n - 75) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 79) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 73) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69) (2 n - 77) (2 n - 81) (2 n - 83) / (2 n - 85)), | \ 9028471416308992293014880 (n - 43)! (n - 42)! (2 n - 91)! (n + 1) + /89 13390541875874011041325440 (n - 44)! |-- (n - 42)! (n + 1) (2 n - 90)! + \88 (n - 43)! ((2 n - 89)! (n + 1) \\ - 1/13390541875874011041325440 binomial(2 n, n) (n - 46)! (n - 42)!)||/( // (n - 44)! (n - 43)! (n - 46)! (n - 42)! binomial(2 n, n)), ( 9028471416308992293014880 (n - 43)! (2 n - 91)! (n + 1) + 13542707124463488439522320 ((2 n - 90)! (n + 1) - 1/13542707124463488439522320 binomial(2 n, n) (n - 46)! (n - 43)!) (n - 44)!)/((n - 44)! (n - 46)! (n - 43)! binomial(2 n, n)), ( (9028471416308992293014880 n + 9028471416308992293014880) (2 n - 91)! - (n - 46)! (n - 44)! binomial(2 n, n))/((n - 46)! (n - 44)! binomial(2 n, n))] and in Maple notation [15/4194304*(2345624805919*n^23-620417761163954*n^22+77295960634838726*n^21-\ 6031236160521936364*n^20+330568081573817240769*n^19-13529803402186895269446*n^ 18+429155115384008856700660*n^17-10806133687512636807136088*n^16+ 219486353920765013072435489*n^15-3634159837091281175245325758*n^14+ 49372741793729958936903577734*n^13-552179957351306265223692265452*n^12+ 5086094153460990410547300920239*n^11-38498033227875010959341966209130*n^10+ 238278417537464775278346186647648*n^9-1196232635813402814296497673621776*n^8+ 4813450103025135886202345677019184*n^7-15257498958472529912619057391511712*n^6+ 37084801477956490015108556014415232*n^5-65589366013981857032463674064760320*n^4 +71345094102927363678546086702438400*n^3+4036862170698648133068830822400000*n^2 -198627075383603593728947402096640000*n+326371209782159839059002445004800000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/( 2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 15/2097152*(1172812402925*n^23-\ 310208880555826*n^22+38647980307975126*n^21-3015618078099267872*n^20+ 165284040435003545055*n^19-6764901657755870572482*n^18+214577553499442061078788 *n^17-5403066517186750256314624*n^16+109743156135667505672523115*n^15-\ 1817078819140394238442109558*n^14+24686322497659684895150052390*n^13-\ 276088194878930786725851646896*n^12+2542991974103364993936009964505*n^11-\ 19247592242983408061943007972534*n^10+119108536997096578233723898488352*n^9-\ 597570451511830708205578717712048*n^8+2398793711772862372952485955653200*n^7-\ 7536373578389029859769352599669600*n^6+17703556097631588366938978151625344*n^5-\ 27111733941434998078495102735618560*n^4+9098683422227173085856982359091200*n^3+ 74287848122129311839279505059840000*n^2-153485669800096202691224989286400000*n+ 3547513149806085207163070054400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/( 2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/ (2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45 ), 645/4194304*(109098828143*n^24-31420462486572*n^23+4273291990222514*n^22-\ 365008892189865264*n^21+21967540587463190489*n^20-990611732176909074540*n^19+ 34749087938227880726168*n^18-971687009680527093400656*n^17+ 22019998626778416261844385*n^16-408933082043685229674395268*n^15+ 6268570305346763159829777722*n^14-79646206999990864990874081328*n^13+ 840041418432057939150944292071*n^12-7348083250555522678116970667556*n^11+ 53123928433965203743365939775196*n^10-315394047193645019807384638435920*n^9+ 1521168182366861243597924831890832*n^8-5848353581990574426288688566166464*n^7+ 17244693102433514378345304610478400*n^6-35281292691513328412460906928856832*n^5 +32330912703333484043046559580974080*n^4+62578064392731300136106512944230400*n^ 3-233311419475863335155688393318400000*n^2+168955108652288175296450737766400000 *n-7755028746087721150542525235200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-\ 13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2* n-29)/(2*n-45), 43/4194304*(1636482418009*n^24-471306934258620*n^23+ 64099378790948206*n^22-5475133148059708320*n^21+329513071999904496319*n^20-\ 14859171628470267478380*n^19+521235915756944572317496*n^18-\ 14575275171696844507238160*n^17+330298163064116296645399639*n^16-\ 6133905525043089823861054740*n^15+94024797420598401687657833926*n^14-\ 1194563625747532566400965636000*n^13+12596909003201895218559882068929*n^12-\ 110132978020449735518762082288900*n^11+795129608399778230991561460982836*n^10-\ 4703296607561502447944700007896240*n^9+22463979272069285904625482413121904*n^8-\ 84185107806828509699582203655175360*n^7+231996601225053211860002723626777536*n^ 6-387531383014180766850751715384321280*n^5+18398264091809084869874695260595200* n^4+1590834320304310953920452671853056000*n^3-\ 3006076793870731759851995389992960000*n^2+1736907369588780867662714449920000000 *n-116325431191315817258137878528000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n -13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2 *n-29)/(2*n-45), 8815/4194304*(15965681891*n^25-4989275470030*n^24+ 738013584715910*n^23-68737234568822020*n^22+4523449913726705885*n^21-\ 223728936633064506490*n^20+8636909382573464043260*n^19-266784147793481385378520 *n^18+6706160340510327638246525*n^17-138786916709668116261437890*n^16+ 2383243468992777372483448310*n^15-34120685045680431340657278820*n^14+ 408199996745096113817943511955*n^13-4079978121159417002439560165590*n^12+ 33973119888530802591298329253160*n^11-234168804236573229540184121998720*n^10+ 1320047077677934315526247881893520*n^9-5948492435843690828205489512019040*n^8+ 20460339694765875837491703245756160*n^7-48108826916299665781458953146801920*n^6 +50032354493791167284389999487860224*n^5+101547741959643451422454929475799040*n ^4-455819875251614563955946760487116800*n^3+ 633677213749616026265740781998080000*n^2-325328659427609187330792075264000000*n +27804615260363292905603688038400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29)/(2*n-45), 1025/4194304*(137304854590*n^25-42907762126349*n^24+ 6346914481306984*n^23-591139714423358150*n^22+38901592979450033602*n^21-\ 1924060145413363395515*n^20+74276643789537571470484*n^19-\ 2294288219245914312382280*n^18+57669761324795122641496402*n^17-\ 1193414137001748821100614915*n^16+20489854747067492585120185024*n^15-\ 293240857746758545821864495710*n^14+3505197411658580065513830900382*n^13-\ 34969217577506869089769393372325*n^12+290005185706344925580065610220484*n^11-\ 1981903680575436577213666189929140*n^10+10977133991443657923948673424254672*n^9 -47738051909095694827859860031226800*n^8+152696177484190673741152922667847104*n ^7-302370858572546789935689272940342720*n^6+92212221478382685056339492749316352 *n^5+1390607556761557872250973005260155904*n^4-\ 4005914399204096044838938845757870080*n^3+4817510959563972648218933065433088000 *n^2-2350256788127686753112566046294016000*n+ 239119691239124318988191717130240000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47 )/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43 )/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 35)/(2*n-29)/(2*n-45), 3075/4194304*(91536543626*n^26-30939338512550*n^25+ 4960701868522569*n^24-501989306848952728*n^23+35983559205901461388*n^22-\ 1944009518756126415594*n^21+82224594591241250668091*n^20-\ 2792070406543725693991748*n^19+77440126327520377970930238*n^18-\ 1775546844813457584379407754*n^17+33929992375917316687697868751*n^16-\ 543224573983705373594548725648*n^15+7305403679328944551444757172248*n^14-\ 82523701589606987469394582245974*n^13+780695016164878539371673205479261*n^12-\ 6141794659552449586005242855975188*n^11+39652980256079757316315509773509108*n^ 10-205160275750509467623912606549283664*n^9+ 813668541601799524427753360455220016*n^8-2245816782628203744713974974173754048* n^7+3087502257738149624596599240121363392*n^6+ 4404311310468162026714911673372833536*n^5-\ 32126156062268266657066637715945458688*n^4+ 70533297134294504373341365875936399360*n^3-\ 75433988622042326114468979241697280000*n^2+ 35282867662078796802970399095816192000*n-4065034751065113422799259191214080000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 75/ 4194304*(3752994507798*n^26-1268510239968686*n^25+203387903315877293*n^24-\ 20581379411410734384*n^23+1475299060314179027512*n^22-79701412079573330096050*n ^21+3370950896430767726142495*n^20-114457104769188748612792604*n^19+ 3174048647246231772338019842*n^18-72751636745052302167595772450*n^17+ 1389394862623912380310133764315*n^16-22217851532888197224863266385944*n^15+ 298113933290212242919619802892572*n^14-3353448281920861320740352564166190*n^13+ 31484551225783739126089826856178505*n^12-244409400227946056742403709740943004*n ^11+1542286789262106701700025547129442292*n^10-\ 7675817388567144141102673867470138640*n^9+ 28435376378075541141103304437784271600*n^8-\ 68032587847750477987975271826193237824*n^7+ 45997715398977773828873312007420609984*n^6+ 347362871142404491653352791412194230016*n^5-\ 1477020351704348349621282502858771934208*n^4+ 2834865360733956790429457449011983093760*n^3-\ 2847346658601473830389526817748172800000*n^2+ 1312022200088230330142158713927401472000*n-\ 166666424793669650334769626839777280000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-51)/(2*n-45), 555/33554432*(8114556994333*n^27-\ 2957743174681872*n^26+512433787461467625*n^25-56152581829380780120*n^24+ 4368902978676569933925*n^23-256836665736404079300000*n^22+ 11853344478981824378301885*n^21-440489696286343314205263240*n^20+ 13413384252836148024288553995*n^19-338813046351857813623681931280*n^18+ 7158896221609484236126728094635*n^17-127206504201046351487495218520040*n^16+ 1905796958188300339193995568307735*n^15-24070075213301463989724826870549440*n^ 14+255427593845201402129601107008181535*n^13-\ 2260664324999850333098770924522125240*n^12+ 16468362344423184949364941561567487580*n^11-\ 96550882083682701119123167019142817920*n^10+ 437738632864169641942956643105572737360*n^9-\ 1411696918035988263127348446226093395840*n^8+ 2467542421067682943334707304828810260032*n^7+ 2540359055392746112084061594155539520512*n^6-\ 30993055470587867085711145251702498263040*n^5+ 96896567637186963778487486082194328084480*n^4-\ 164589535564883186168125220123197104537600*n^3+ 155046211818472082565423173767161692160000*n^2-\ 70041737179930888113554877833977528320000*n+ 9549535690880531316478692132441292800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 2775/16777216*(811448377637*n^27-\ 295769309359518*n^26+51241756053041775*n^25-5614927017798557160*n^24+ 436842552956577904305*n^23-25678497111247089853146*n^22+ 1184898238808000479895847*n^21-44019587608073625252940092*n^20+ 1339733315446010900896447395*n^19-33809446350835356624355284066*n^18+ 713236578860677974075628845957*n^17-12639622074410816523225128861712*n^16+ 188532041985962443465739509229475*n^15-2364294748228535894838867790361046*n^14+ 24811036480418807651763730014550197*n^13-215852220325028228455420494953582412*n ^12+1531932201444032399562726064061729700*n^11-\ 8628720165562978297708083003050203536*n^10+ 36635425145178637786044486592353111472*n^9-\ 103511722476668988933817452568695799872*n^8+ 101522746520756177451246366244039309248*n^7+ 640387441882231237650551200356923373312*n^6-\ 3730952692578269973255323005722634933248*n^5+ 10114457606065897913283384841710476341248*n^4-\ 16080214776071184034592355928449841397760*n^3+ 14654774208781266673691696481786544128000*n^2-\ 6598779781357278816459079536471048192000*n+ 954953569088053131647869213244129280000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 555/33554432*(16228586048617*n^28-\ 6361418794037594*n^27+1187421602766749469*n^26-140463574331590585530*n^25+ 11822435114190590663865*n^24-753544088063421155922330*n^23+ 37796254548323002831690305*n^22-1530372995276960126225183250*n^21+ 50909557159567624782605636295*n^20-1408628978331974002150180623990*n^19+ 32691735992313894626658732437415*n^18-639744656431324879384912271386350*n^17+ 10581963412622358059021419500194355*n^16-147899409439644497830025331950314110*n ^15+1740651744395909083179670471241108235*n^14-\ 17125887341589340846494915798711834150*n^13+ 139112546623753482648718600697114428260*n^12-\ 913761098120021345112480224175587594520*n^11+ 4678720980339850066162114497418062572720*n^10-\ 17281938294525415605291472168665423652000*n^9+ 35697149782086419686075691202745132214208*n^8+ 37639789094425728517190814957315906524544*n^7-\ 620791875407871786144444100310892505158144*n^6+ 2539744618641368389681317509830659958241280*n^5-\ 6041896572477502950063201126047542732185600*n^4+ 8935057840265356128383533370797370400768000*n^3-\ 7816726251301489129432016674012016640000000*n^2+ 3479442689120074140128632533762932736000000*n-\ 525224462998429222406328067284271104000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 15/33554432*( 600423695249689*n^28-235349621085819758*n^27+43927309404361634013*n^26-\ 5195689334000412557310*n^25+437222747463848826639705*n^24-\ 27859061405877917352301710*n^23+1396630535580762870044413185*n^22-\ 56502519526456393199397738150*n^21+1877133902243223353484486747015*n^20-\ 51831761523649082388503611851330*n^19+1199121592702649054344530820031655*n^18-\ 23354259755924186477056897973321850*n^17+383602265807883097244540331729910035*n ^16-5307377162593528046518797832381330170*n^15+ 61569504641578673252691621560350734795*n^14-\ 593591012524280592741662292208606488450*n^13+ 4685011024717991530448983973401780674420*n^12-\ 29503287006392866363610978609753011844040*n^11+ 141138555714769415409359359426628780182640*n^10-\ 453514741850787550778994313457130936498400*n^9+ 494590122632701152711808882692571368006336*n^8+ 4099973634959989320317630195922445431931008*n^7-\ 29127220239187080213911894996956467374796288*n^6+ 102498554519218509771742669530418422131804160*n^5-\ 227647842897561157575992448407081431058227200*n^4+ 324010695353209399331866553699858392866816000*n^3-\ 278104637197552302249060270949346842214400000*n^2+ 123900256085390021771995962149071847424000000*n-\ 19433305130941881229034138489518030848000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 435/67108864*( 82806545799409*n^29-34815120787189217*n^28+6981780534616311711*n^27-\ 888851411461961275293*n^26+80662341784594145436315*n^25-\ 5553897212166658020530865*n^24+301518893099096970293786595*n^23-\ 13240466397681434842999996785*n^22+478636030914822131551289954865*n^21-\ 14419163385450740816494640140095*n^20+365018904944427117754803433467285*n^19-\ 7804807767668224204010812948965255*n^18+141284933474973134614252657714564185*n^ 17-2164451975776343704708290032588898555*n^16+ 27968654558113746851943756080635962465*n^15-\ 302740402584375210209131691816668145595*n^14+ 2712689457377927122785041554215790928970*n^13-\ 19724335782212705621965099945640539591860*n^12+ 112231637118120018976817812285724607262280*n^11-\ 460930232890946317548045209717646058499440*n^10+ 1020181629673084401410809021715342204402016*n^9+ 2015035420620451047676135423288938654979392*n^8-\ 30415903824542752494065736668687341708734336*n^7+ 150328096961010861652327322404447465909882368*n^6-\ 461977352649043139125411532073696725495285760*n^5+ 950222546853318649902246669834317892017971200*n^4-\ 1288240275260802540575921788269152376569856000*n^3+ 1074445454911887694401372166038265685606400000*n^2-\ 474765017521423362151366310568750415872000000*n+ 76392992583702567589996268545001914368000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 435/8388608 *(10348053542804*n^29-4350051584787439*n^28+872143534568063226*n^27-\ 110991565377825083091*n^26+10066693233732657590250*n^25-\ 692541557903474220536595*n^24+37550552546207278250892570*n^23-\ 1645908115312346855180208855*n^22+59340984108435238234465530090*n^21-\ 1780953954030156314777989826205*n^20+44847851170353077958701308901910*n^19-\ 952017481865799627032212656925545*n^18+17065481137989992230442177249302110*n^17 -258025608834885597405177935834613825*n^16+ 3276268855686718082930270022137446590*n^15-\ 34640560130104538626513735650094469325*n^14+ 300547687073671480060860706903456860570*n^13-\ 2084973334314153961841730786165300113760*n^12+ 10974292811997605434221653122922872262680*n^11-\ 37934240935145817693213683860426948463840*n^10+ 27427884570692421313574230935093958104096*n^9+ 644151735427180315715981986275123229511424*n^8-\ 4944050029409866028780941598541435531374976*n^7+ 21107466440844092236430374639515966403750656*n^6-\ 60569058816022818659890153168590168773329920*n^5+ 119711730579480949057608222964624542846566400*n^4-\ 158455943045077660145666314275205006497792000*n^3+ 130642977191243114590223236245909727027200000*n^2-\ 57852578526595940739308612585943785472000000*n+ 9549124072962820948749533568125239296000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 13485/8388608 *(667262622181*n^30-300103693011495*n^29+64469546445286310*n^28-\ 8805046966518738705*n^27+858469839566759258454*n^26-63597624364337213318805*n^ 25+3720251572445655440436750*n^24-176270161871249737350371325*n^23+ 6884431316398258979143770840*n^22-224346094103449228296305494575*n^21+ 6150309614855758676612936425050*n^20-142561767749408214066673433167275*n^19+ 2800547137681768792040390628104730*n^18-46609219581752932425089552525160375*n^ 17+655088944925381196738146619666359850*n^16-\ 7723170763658869919566353597209786175*n^15+ 75471802672173346231541108462447818515*n^14-\ 598660485475318515275656440246717685950*n^13+ 3701040476692238049410731106915374733000*n^12-\ 16125598417985456580201059076591387549000*n^11+ 30940721949041567921875723223674153440064*n^10+ 193230284729195392838606367103303208044320*n^9-\ 2293346945331386675626760012936632058847360*n^8+ 12821295247380739078250462615413772687364480*n^7-\ 47997638142259223640388800866701583353814784*n^6+ 127452085620171478906637168072661338853626880*n^5-\ 239165975210133917992521920219864684970393600*n^4+ 305585186936335906101762567206493319477248000*n^3-\ 246560850721990122119487330126521169715200000*n^2+ 108430202692888836882831334171655036928000000*n-\ 18174139364671175354071692919980294144000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 13485/8388608*(666598028363*n^30-299665061091615*n^29+64332360989374750*n^28-\ 8778002676866617425*n^27+854699067558288773982*n^26-63202088695551413856885*n^ 25+3687835832821981060773150*n^24-174142260104812327014198525*n^23+ 6770620746427668012136417020*n^22-219327178232478868116928538175*n^21+ 5966316529759371759512143837050*n^20-136923716609317070595038728399275*n^19+ 2655664646970113960987926848273090*n^18-43482468880684691298385011719162775*n^ 17+598426877409320120244968924868409050*n^16-\ 6862397850256836898007632881877139775*n^15+ 64544838745383233713460839678006668345*n^14-\ 483305878245092657452231743493153874550*n^13+ 2695074778314675502668115916731754402800*n^12-\ 8944262990856387641320852436836106843400*n^11-\ 10514384483299400711666278037491281284128*n^10+ 383495721822762858462505244355729564174240*n^9-\ 2970944435356469283423673055389023614995200*n^8+ 14624001163790542215855171806279371640150400*n^7-\ 51343710919256830111983579934805187045756672*n^6+ 131120969274331935888155138197133830092149760*n^5-\ 239915140688818363384508828561215353147801600*n^4+ 301782364214251439578117937102669733378048000*n^3-\ 241745542920381208213115109785623277445120000*n^2+ 106563872853369809351647160323143602995200000*n-\ 18174139364671175354071692919980294144000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 13485/1073741824*(170345043589193*n^31-81714122218162354*n^30+ 18743297426756895820*n^29-2736224324521933081400*n^28+285443259326799391549452* n^27-22648115662372523987742456*n^26+1420204274900888516702789880*n^25-\ 72193902899071234207607163600*n^24+3027284131408280679824733683670*n^23-\ 105987653703991247150947791094860*n^22+3123560646672432001169564833603200*n^21-\ 77879586809712288428297279592400800*n^20+1646555071857031002076649201187434940* n^19-29508480798251061271421937902856400920*n^18+ 446758663508231993528330837037058545000*n^17-\ 5672566758155920442460753520687525042800*n^16+ 59592744626776992350184243348678988064745*n^15-\ 504930468995777008509952642211022447768210*n^14+ 3264515874268908913576281495662292066519700*n^13-\ 13604203938720384409586642470857452591229800*n^12+ 2182958401324348417873459544602794918844592*n^11+ 532837613001680032143861895892548360032965024*n^10-\ 5283547347714163445819861275832886430461011520*n^9+ 32105628329561830010624345690613594963787497600*n^8-\ 140405350772287496187177751717891021631665262592*n^7+ 457568773400234383621344538772574907345370443776*n^6-\ 1109052747121717117174606188368306312839245742080*n^5+ 1953572317232762379139351854958104668158729420800*n^4-\ 2391643475918305743332393174381813982562811904000*n^3+ 1883133658201914096256193619429726331644149760000*n^2-\ 824749456294347920971068747690904473147801600000*n+ 141903680159352537164591778319206136676352000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n -21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-45), 13485/536870912*(84913183603999*n^31-40686416579668022*n^30+ 9318387910499058140*n^29-1357621619170763423920*n^28+141258117864318563077116*n ^27-11170192802240536642862928*n^26+697437624781860205387644240*n^25-\ 35259958188836913581432643000*n^24+1468441097246657545372062896610*n^23-\ 50974273244834570811764436747180*n^22+1486466192672061487234684198644600*n^21-\ 36581027312299840641557648053637400*n^20+760983171829557252258370229684907420*n ^19-13363677833791593275263005876979906560*n^18+ 197119926067875436803085929806783636400*n^17-\ 2416997464015501419037071264024088881000*n^16+ 24146213207132601162955509250518370198935*n^15-\ 188415386012795436699184104183687833667630*n^14+ 1024431654486093129680456305424999586718700*n^13-\ 2005618561676614723187562182154909132923800*n^12-\ 30096204863258743633853136206739776226068144*n^11+ 431452572091693628826906121381574138436046432*n^10-\ 3340352228057198137396632656099060497827695040*n^9+ 18359847255414456689526107441938731752584398720*n^8-\ 75917115134518517922880192299982821807057063936*n^7+ 238659038824920318041250827531256437570837925888*n^6-\ 564421404015649707712902211297736789522357207040*n^5+ 977685655733449274581567360716691871400171110400*n^4-\ 1184415399913082655306242514101079325363863552000*n^3+ 928315752048926434503832696856320037655674880000*n^2-\ 407441216249050690531362584877353238056140800000*n+ 70951840079676268582295889159603068338176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 121365/1073741824*(37550932396751*n^32-19144748930565136*n^31+ 4670470167860859448*n^30-725610845476549901600*n^29+80603841512714571587844*n^ 28-6813450752298918387547104*n^27+455374365009171052129192632*n^26-\ 24680528198186058243178467360*n^25+1103745825704387691875585558490*n^24-\ 41222360714393635679824752080640*n^23+1296162245421817444640813189575320*n^22-\ 34481711918046478683852894976408800*n^21+777731400112547715302133075563458980*n ^20-14860193855373825243505416680313499680*n^19+ 239486835631432954378252154825281810440*n^18-\ 3224531409756062963452889675829046159200*n^17+ 35598089571749930657251142283503050320015*n^16-\ 309666159200740580743993431078598335437040*n^15+ 1907954375623582393764815286988507801588720*n^14-\ 4691747818768086995445204113102755975952000*n^13-\ 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121365/1073741824*(37255729287599*n^32-18950210081633968*n^ 31+4609578622536076408*n^30-713564545337943884960*n^29+78912862367964095253636* n^28-6634195482691840527388512*n^27+440461727937600582216347832*n^26-\ 23681623113925303161378550560*n^25+1048897095610875476160270909210*n^24-\ 38721612095093848933591746697920*n^23+1200599725033546102858855413616920*n^22-\ 31400379615513589569970871144191200*n^21+693502287098762308889820473429789220*n ^20-12902337826231653236323280069624295840*n^19+ 200725295999616759827937180451261176840*n^18-\ 2570689153827952597450036339260708962400*n^17+ 26209236784926801388666466839736024404335*n^16-\ 195143378797077529871759241032337475858320*n^15+ 725469388913213489837486624072716533513520*n^14+ 5591602539061204950962319567738548254854400*n^13-\ 138447588470057246363963786097327097423464544*n^12+ 1545996572798664639389784671697026393192350208*n^11-\ 12172312682483828032196398272889622735748142848*n^10+ 73528298632639770703284758246414123270454005760*n^9-\ 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2886745198029082210744888536*n^26+136692484422129553873317673740*n^25-\ 5403806877905692638190211904930*n^24+179754638560264975425195784138800*n^23-\ 5053985121364607260211566922574840*n^22+120258217851354330187317084093425880*n^ 21-2415933950617484065219369905846610740*n^20+ 40670889000723479111496009589319745360*n^19-\ 564389512753928700722539314938550238440*n^18+ 6226456268813663307483776115087075874290*n^17-\ 49530314469050875405672009624645391445435*n^16+ 173825864026890552912819420785970289096160*n^15+ 2371826034840607425849381267581461910746800*n^14-\ 56987646563979852488461043297754853866242176*n^13+ 700597266633793058203068430944171921930660960*n^12-\ 6241410989248226466803832913387989992724564736*n^11+ 43425533992774943734225737427046589339610996992*n^10-\ 241560673017858602952313295534646480723202189824*n^9+ 1079405294217820672690605215663141939837077941504*n^8-\ 3853756075379863808749527764067540086644195602432*n^7+ 10849018806323237882429979743639359897632745168896*n^6-\ 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595774428018205861695630219395974212922095990988800000*n^4+ 672516974929322105512564658993618209957606146048000000*n^3-\ 504398678115197475117042457686909832551234443673600000*n^2+ 218158496691134212626661010004449145927270137856000000*n-\ 38933403206920758859163223258548991689207316480000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 202275/2147483648*(310844109463617*n ^35-184632218470345324*n^34+52460975132263308163*n^33-9487755125866603843026*n^ 32+1225805880294141107531108*n^31-120357401916249299420755136*n^30+ 9325741073656790580481146012*n^29-584327633338717986259057478424*n^28+ 30083372222167003058203511395230*n^27-1285045090781984228202696927336600*n^26+ 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1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2* n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/( 2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55) /(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-\ 71)/(2*n-69)/(2*n-77), -37/137438953472*(975082460207456068631*n^40-\ 907765411028702208212500*n^39+399732229807571394668610910*n^38-\ 111463620770543567941102126400*n^37+22196360567565557020540180362867*n^36-\ 3371828042009055958383038929084140*n^35+407322166813168291842272517473921980*n^ 34-40257686951424369719211377376599766800*n^33+ 3322842473811004325929363026819739083038*n^32-\ 232591285004015217035846301196031715302920*n^31+ 13970938327408582518534758185218593390073060*n^30-\ 726796579176214545873161700264643552027612800*n^29+ 32985344687032223307790731893464612310314507854*n^28-\ 1313597646559274860447658223157860888617555510040*n^27+ 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2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59) /(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2* n-45)/(2*n-71)/(2*n-69)/(2*n-77), -37/137438953472*(1536398692571136742511*n^40 -1342785491110554730469500*n^39+562188373778267855302960510*n^38-\ 150405995278453082048000759600*n^37+28930672839491831398434126918027*n^36-\ 4267037778456749498262529922593140*n^35+502506594169757158271596628777104780*n^ 34-48574196749359728883102738194117680400*n^33+ 3931625364929154385200092331407856387278*n^32-\ 270469573942389740978054226630781957208920*n^31+ 15996292593625572163230630128352520863522660*n^30-\ 820659952027482030339920415890741631948084000*n^29+ 36780835029798019081595095051503927128741811774*n^28-\ 1448207218479019868493434837367185306638929368040*n^27+ 50317375453731596746514694538605810681233551233600*n^26-\ 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2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59) /(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2* n-45)/(2*n-71)/(2*n-69)/(2*n-77), -1517/137438953472*(100616227337174993867*n^ 41-89304044787464138031076*n^40+38182443228945478189200970*n^39-\ 10481043419839221919302830360*n^38+2076918126372667278673713488319*n^37-\ 316714407807001529723488020274812*n^36+38687406594796478381091669288144400*n^35 -3890558310770271372033427242624508280*n^34+ 328522099683447739457614589140456873766*n^33-\ 23639817158835982581157677694225596670888*n^32+ 1466168313306413288556199213503410297443740*n^31-\ 79076335675758415344537958545164859271336560*n^30+ 3735011233799420776374702264210863276213650278*n^29-\ 155366281245671235445412967728626300345628343864*n^28+ 5717159146153706993197098176338665540210434424840*n^27-\ 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(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/( 2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59) /(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2* n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), -41/137438953472*( 4583910540669757625054*n^41-3995721090879182237285737*n^40+ 1680695149491334133786680840*n^39-454544525368671024020842192370*n^38+ 88859148853343328697227442569078*n^37-13383160121058038398056739758759069*n^36+ 1616255731762705972174515807406297900*n^35-\ 160841004751765134804471904402441908260*n^34+ 13450828875034994436198866760842614220492*n^33-\ 959289423225791849648428011504962154991906*n^32+ 59006919084333819366353794496084096404990080*n^31-\ 3158250767043501871270223291681193461277985020*n^30+ 148121636299769353918856165225988970862540616636*n^29-\ 6121195945843298576507967109691592062278367107218*n^28+ 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2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83), -1763/1099511627776*( 4813822738908479507431*n^43-4490552059517895729790472*n^42+ 2028579013240134670623591367*n^41-591268512569962858428384651680*n^40+ 124995291849374172354119056788037*n^39-20426440955673619402936223810912744*n^38 +2685574015930080048849453705628076009*n^37-\ 291922474043044457604583188354141882160*n^36+ 26756111707592168827265558402262925526498*n^35-\ 2098473611277650474407250114115960110292976*n^34+ 142442299866054118317495039942986891799169286*n^33-\ 8443085362080051161940189462277219310293335040*n^32+ 440119717323243734089423480696771047879007129674*n^31-\ 20291667137779006802480426123123926930527285827088*n^30+ 831239692020424154005609856235439231359765625646418*n^29-\ 30365808456014610726404435892566158425461767568764320*n^28+ 992089242674217212629386304880257414640846668820947995*n^27-\ 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28432325278560603119524728670463982608792940534952935368844902400000000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), -43/ 549755813888*(104422357204339033498123*n^43-97037761457345659181848226*n^42+ 43677508190944240565507931061*n^41-12687001252230304390617379803440*n^40+ 2673350851426562119929646747378021*n^39-435531314917788427629283655130615602*n^ 38+57095410571605450398705495054473945447*n^37-\ 6189262038263271463564620527333459257780*n^36+ 565807222811501424080124523498499130583634*n^35-\ 44267721876348187836881486445551732045369908*n^34+ 2997934275164979822838358508925122084675733538*n^33-\ 177313770096222064577331049610434160349783554320*n^32+ 9224151874448538384064385879118918270026605702842*n^31-\ 424464592978247123744179653760397878542584735798004*n^30+ 17356798270997974155686508312365993298545652728132094*n^29-\ 632990322045139397433159045672715300102789685894557560*n^28+ 20648047661927201450249605183778348050795260753693544335*n^27-\ 603791572544173976217136485922185848203025486424626582970*n^26+ 15854196422375954954392441379067011895737772791502369171545*n^25-\ 374239986691889350833531232068890669435263089139239681296800*n^24+ 7946922011653412056157193452632222426394819587058003276303537*n^23-\ 151841545996059394753373089794761019822207270250633585964327194*n^22+ 2609983271799510270619707448001721708831059825236264314168882059*n^21-\ 40333588083915011545182495428403268453269103975168229837552735860*n^20+ 559777793414729543360814658798492038416262047568892285826383296964*n^19-\ 6966668352935013418681744923208729469909867247753634381126602129968*n^18+ 77593753881961249011813666243553904309928364358841363206000180436448*n^17-\ 771478445622438894970210374009688629791958621633623397465994919611520*n^16+ 6825986287534999409406728002659023130452846278080192551684832235528576*n^15-\ 53544144082748987096778222500925487837086354208447053444178295401306112*n^14+ 370671197247332057752957654010592404385814571876289849610472625960306432*n^13-\ 2252291820949509345710013996005393090278839525254041183306583248575636480*n^12+ 11933521140452692400991954792557130021570304058827490867668123951861306368*n^11 -54698756381835849725289016763037443150118886678788713146702700341066838016*n^ 10+214812047739049867038737112373071531976114276420751624822286902216814821376* n^9-714243891576357171495125025528282531185894879758094868701083996998407946240 *n^8+ 1980948599513553859973481090739030084148593577860344033958133714653288857600*n^ 7-4496390797399886587470183142463570840078197439863784211560953481585885184000* n^6+ 8145640050966494666007847878625605129419777862343169881399754907467120640000*n^ 5-11380082677749394546850678895623784198847506782891854111221296608680345600000 *n^4+ 11667225576044790329127211016360032661746112866720077863286216792735744000000*n ^3-8120512057457169319081545507387078851234264691707504680165652677263360000000 *n^2+ 3337017300343041883868154154621395299273170024721224014834023006208000000000*n-\ 582862668210492363950256937744511643480255280966535175061320499200000000000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n -65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2 *n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63 )/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), ( 9028471416308992293014880*(n-43)!*(n-42)!*(2*n-91)!*(n+1)+ 13390541875874011041325440*(n-44)!*(89/88*(n-42)!*(n+1)*(2*n-90)!+(n-43)!*((2*n -89)!*(n+1)-1/13390541875874011041325440*binomial(2*n,n)*(n-46)!*(n-42)!)))/(n-\ 44)!/(n-43)!/(n-46)!/(n-42)!/binomial(2*n,n), (9028471416308992293014880*(n-43) !*(2*n-91)!*(n+1)+13542707124463488439522320*((2*n-90)!*(n+1)-1/ 13542707124463488439522320*binomial(2*n,n)*(n-46)!*(n-43)!)*(n-44)!)/(n-44)!/(n -46)!/(n-43)!/binomial(2*n,n), ((9028471416308992293014880*n+ 9028471416308992293014880)*(2*n-91)!-(n-46)!*(n-44)!*binomial(2*n,n))/(n-46)!/( n-44)!/binomial(2*n,n)] The limits, as n goes to infinity are 35184372088785 17592186043875 70368744152235 70368743974387 [--------------, --------------, --------------, --------------, 35184372088832 17592186044416 70368744177664 70368744177664 140737485869165 70368737977375 140737435824975 140737294042425 ---------------, --------------, ---------------, ---------------, 140737488355328 70368744177664 140737488355328 140737488355328 4503579131854815 2251769247942675 9006865256982435 9006355428745335 ----------------, ----------------, ----------------, ----------------, 4503599627370496 2251799813685248 9007199254740992 9007199254740992 36020847422742915 1125350822779935 8998036460110785 8989074412475055 -----------------, ----------------, ----------------, ----------------, 36028797018963968 1125899906842624 9007199254740992 9007199254740992 2297102912800267605 1145054280899926515 4557368910331685115 -------------------, -------------------, -------------------, 2305843009213693952 1152921504606846976 4611686018427387904 4521541584989452635 4467480710142691125 4388820707058778875 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 4278256646779993125 4127889524800844505 62875992241753128675 -------------------, -------------------, --------------------, 4611686018427387904 4611686018427387904 73786976294838206464 29411799152207536683 107587057813340568867 95451017361638855887 --------------------, ---------------------, ---------------------, 36893488147419103232 147573952589676412928 147573952589676412928 324851365326783813349 8119887216635749679 93798765247038759519 ---------------------, --------------------, ---------------------, 590295810358705651712 18446744073709551616 295147905179352825856 54732208843042660569 3483008512941972438639 -3650331636553875370763 ---------------------, -----------------------, -----------------------, 295147905179352825856 75557863725914323419136 37778931862957161709568 -36078051027675874539347 -56846751625132059472907 ------------------------, ------------------------, 151115727451828646838272 151115727451828646838272 -152634816870494465696239 -93970166083730031313607 -------------------------, ------------------------, 302231454903657293676544 151115727451828646838272 -218786203426914215735329 -244622916105773249962639 -------------------------, -------------------------, 302231454903657293676544 302231454903657293676544 -8486769488695649371600853 -4490161359786578440419289 --------------------------, --------------------------, 9671406556917033397649408 4835703278458516698824704 -18642218948453347941100681 -19060673382074410786142101 ---------------------------, ---------------------------, 19342813113834066795298816 19342813113834066795298816 -77089112723576611172038549 ---------------------------] 77371252455336267181195264 and in Maple notation [35184372088785/35184372088832, 17592186043875/17592186044416, 70368744152235/ 70368744177664, 70368743974387/70368744177664, 140737485869165/140737488355328, 70368737977375/70368744177664, 140737435824975/140737488355328, 140737294042425 /140737488355328, 4503579131854815/4503599627370496, 2251769247942675/ 2251799813685248, 9006865256982435/9007199254740992, 9006355428745335/ 9007199254740992, 36020847422742915/36028797018963968, 1125350822779935/ 1125899906842624, 8998036460110785/9007199254740992, 8989074412475055/ 9007199254740992, 2297102912800267605/2305843009213693952, 1145054280899926515/ 1152921504606846976, 4557368910331685115/4611686018427387904, 4521541584989452635/4611686018427387904, 4467480710142691125/ 4611686018427387904, 4388820707058778875/4611686018427387904, 4278256646779993125/4611686018427387904, 4127889524800844505/ 4611686018427387904, 62875992241753128675/73786976294838206464, 29411799152207536683/36893488147419103232, 107587057813340568867/ 147573952589676412928, 95451017361638855887/147573952589676412928, 324851365326783813349/590295810358705651712, 8119887216635749679/ 18446744073709551616, 93798765247038759519/295147905179352825856, 54732208843042660569/295147905179352825856, 3483008512941972438639/ 75557863725914323419136, -3650331636553875370763/37778931862957161709568, -\ 36078051027675874539347/151115727451828646838272, -56846751625132059472907/ 151115727451828646838272, -152634816870494465696239/302231454903657293676544, -\ 93970166083730031313607/151115727451828646838272, -218786203426914215735329/ 302231454903657293676544, -244622916105773249962639/302231454903657293676544, -\ 8486769488695649371600853/9671406556917033397649408, -4490161359786578440419289 /4835703278458516698824704, -18642218948453347941100681/ 19342813113834066795298816, -19060673382074410786142101/ 19342813113834066795298816, -77089112723576611172038549/ 77371252455336267181195264] and in floating point [1.000000000, 1.000000000, .9999999996, .9999999971, .9999999823, .9999999119, .9999996267, .9999986193, .9999954491, .9999864261, .9999629188, .9999063165, .\ 9997793544, .9995123154, .9989827254, .9979877383, .9962095874, .9931762712, .9\ 882218547, .9804530419, .9687304583, .9516737890, .9276990302, .8950933581, .85\ 21285923, .7972084134, .7290382613, .6468012524, .5503196188, .4401799680, .317\ 8025783, .1854399367, .4609723384e-1, -.9662347389e-1, -.2387445148, -.37618024\ 66, -.5050262453, -.6218423963, -.7239028231, -.8093893344, -.8775113980, -.928\ 5436060, -.9637801306, -.9854137177, -.9963534294] The cut off is at j=, 34 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 47], vs. those in the, 2, -th row from j=1 to j=, 46, are as follws 23 22 21 [47 (93575457683 n - 24750708557119 n + 3083615450874673 n 20 19 - 240607825556569625 n + 13187556446387930838 n 18 17 - 539752795376848795434 n + 17120549823285038374178 n 16 15 - 431095759389368610474850 n + 8756104581634623749988463 n 14 13 - 144979782684324597149291459 n + 1969657338223633635910695453 n 12 11 - 22028458908158002328757217125 n + 202902789992020238384134594328 n 10 9 - 1535828319145641567381067756804 n + 9505842316249278817674181550368 n 8 - 47723014489709989398251538609200 n 7 + 192039997620222097534931674850688 n 6 - 608840607210069429309342140789184 n 5 + 1480934180665545535426520940035328 n 4 - 2626673199255581052603137235379200 n 3 + 2893330639973585445958693283328000 n 2 + 32907769155921268122878730240000 n - 7827902595785082363607385210880000 n + 13303174311772819526861512704000000)/(524288 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) 24 23 (2 n - 29) (2 n - 45)), 705 (99813821527 n - 28746380598948 n 22 21 20 + 3909607574958226 n - 333944307780215616 n + 20097963016705436401 n 19 18 - 906304394012531269140 n + 31791722940679676865352 n 17 16 - 888990569166674277897264 n + 20145976995907543585341625 n 15 14 - 374131364938825855007376972 n + 5735123314643540151931105978 n 13 12 - 72869588550833579123819159232 n + 768603583410366262779773950159 n 11 - 6724137516586343330068359974844 n 10 + 48633386433448899778051845383884 n 9 - 289097362054556225205908543977680 n 8 + 1399631063163466961775142916026768 n 7 - 5442911808976588355099817923877696 n 6 + 16615128298761456884410758652906560 n 5 - 37956321427111088503866336718910208 n 4 + 56128626626431309132527653373803520 n 3 - 14950291133497808473594223421542400 n 2 - 159333109329020240680929167155200000 n + 314066365899804575796776118681600000 n - 7095026299612170414326140108800000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), 15 (4691249610941 n 23 22 21 - 1351079887504716 n + 183751555783004390 n - 15695382409123662432 n 20 19 + 944604252310031877179 n - 42596305317386884200060 n 18 17 + 1494210858574612972228952 n - 41782547155871166229929168 n 16 15 + 946860288760791889164423347 n - 17584139897038245675813759684 n 14 13 + 269549242586005016114980431134 n - 3424811694898989613249272949344 n 12 + 36122491853419145954093552235845 n 11 - 315984482681141686688326170609588 n 10 + 2284660068681510110463253468222916 n 9 - 13567231835775428524962105965888880 n 8 + 65477931653921899320329714805075248 n 7 - 252157113883196210123024951399113152 n 6 + 746629551443003789212179834628732608 n 5 - 1544227331825002322459764891492690176 n 4 + 1479571561948100209212330004239037440 n 3 + 2565974424425316691663012082150707200 n 2 - 10126909132874994994829706754990080000 n + 7417766977192218683620922565427200000 n - 333466236081772009473328585113600000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 645 (218197656022 n 24 23 22 - 68186767345175 n + 10086186564967000 n - 939409073591355650 n 21 20 + 61820513808187100890 n - 3057632560077376610945 n 19 18 + 118038111391116303854500 n - 3646076153694756288089000 n 17 16 + 91652451044523810308166970 n - 1896834626194026087790702745 n 15 14 + 32574338115859616095963795600 n - 466432343212640381308611631850 n 13 + 5582105059226870568172840957030 n 12 - 55840950974799789211683255624095 n 11 + 465917473319663447236449035187700 n 10 - 3226689267158917090149049875946700 n 9 + 18387710567844726179556794887861360 n 8 - 84902214549024704379562444658763920 n 7 + 308283601368792958717740500289124800 n 6 - 823082080745702169048108081077236800 n 5 + 1320512542587056316685118649910057728 n 4 + 41826677653918077794477321437946880 n 3 - 5541186921939380347698504222840729600 n 2 + 10175475111497034866723723710095360000 n - 5800254397438452649226004123648000000 n + 379996408558298336376583736524800000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 215 (654592962676 n 24 23 22 - 204560297949905 n + 30258558221889460 n - 2818226884804953470 n 21 20 + 185461487036518152460 n - 9172891034738086979015 n 19 18 + 354113697930508950934060 n - 10938179555159346163217720 n 17 16 + 274954285452196675214609500 n - 5690345152490752861797544415 n 15 14 + 97716194508634932281522162860 n - 1399052892003798303960452741270 n 13 + 16739031015670880301280621916980 n 12 - 167342185023102116850666022135865 n 11 + 1394048790607965021511901723505460 n 10 - 9617914609794764749887930522830420 n 9 + 54321542353192422462946938881054320 n 8 - 245706797869144540070631107105619440 n 7 + 851247727427313600008051176291776960 n 6 - 2031698633838302433095592464190237120 n 5 + 2231148889122151344611806203920504064 n 4 + 3888299454436538854055832634162928640 n 3 - 18642942825720340316439669996158668800 n 2 + 26316996124735706639483902732001280000 n - 13576544016824398094356222852300800000 n + 1139989225674895009129751209574400000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 8815 (31931362753 n 25 24 23 - 10792799928287 n + 1730479931173435 n - 175113095726426390 n 22 21 + 12552487947389535925 n - 678152478457725816545 n 20 19 + 28683864630052922313595 n - 974040559873918927787540 n 18 17 + 27017669064842142021973495 n - 619554240095503854183410225 n 16 15 + 11843145246769943702558445145 n - 189733060728515572780321581590 n 14 + 2554919688660319985326676699935 n 13 - 28936661944161747585067061517935 n 12 + 275151511953778047796710817590145 n 11 - 2185829068751503347748070445647840 n 10 + 14369445649570004377175839328032180 n 9 - 76819620115700810156426501199155840 n 8 + 323269232033734641414383524410372720 n 7 - 1001634872804146327220747993991997440 n 6 + 1913435761989358004463914435731115712 n 5 - 459334982339816541097490725894311168 n 4 - 8906320254634411585109082999319895040 n 3 + 24912777001009024320459361489716940800 n 2 - 29491031741812365893052601644318720000 n + 14219067262346096718985375996477440000 n - 1418035378278527938185788089958400000)/(4194304 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 1025 ( 26 25 24 274609674194 n - 92818045772218 n + 14882115610567439 n 23 22 - 1505970007593951952 n + 107950985424405638216 n 21 20 - 5832062676278190982630 n + 246676733647162156028405 n 19 18 - 8376414940242843258548212 n + 232331795881537772327342006 n 17 16 - 5327165063484720071540383030 n + 101809852370830318683712840265 n 15 - 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695965164974169168470024912896000000 n + 142643124257368699763797652502\ 2 9737203116443887772786660741613458439536640000000 n - 58300678487747901\ 0798416723708676180633858225025172360336551540948992000000000 n + 101418\ 104268625671327344707167545025965564418888177120460669766860800000000000)/( 1099511627776 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 65) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 75) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 87) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 79) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 73) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69) (2 n - 77) (2 n - 81) 44 (2 n - 83) (2 n - 85)), - 43 (418701165842552127331567 n 43 42 - 407283709630924214214865706 n + 192036931649559719659536888343 n 41 - 58478499338887515238030840033874 n 40 + 12928823860907479560522541552705669 n 39 - 2211876094630341739338651729651850862 n 38 + 304769769201011771995858462288551451961 n 37 - 34757591009031556145335257725060839070398 n 36 + 3346199300993856917311344284420562195232106 n 35 - 275993484008622255353384925056613153429999148 n 34 + 19726198073081048017385233200866596666296793094 n 33 - 1232771016424226085589500953076522536751119115892 n 32 + 67846028151151771305599209143544180397054294713098 n 31 - 3307273037983282816991684307164452706783302008548124 n 30 + 143462309010791743958505597423972704118878219782343922 n 29 - 5558454460987257477507510435498180055400947051690369596 n 28 + 192938213797738112470511432917926106004053840321111711155 n 27 - 6013830841819975312614343049389957627334164071465091699170 n 26 + 168628335864974110644718618565187695832717553581534509662835 n 25 - 4259090956032030328834639239037542782692207399463597116253530 n 24 + 96977883477069472327338906709629826249092319420180779668042273 n 23 - 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11 0847466496 n + 1037765494908234094693302510285418672658932094623075130\ 10 9670450217041247346688 n - 4023660963726452511450221267555654498295561\ 9 7009987271861222342183066851344384 n + 13220752202442733962232307180713\ 8 0445908258254993822130197602324879352893276160 n - 36267967518852263424\ 7 2141600621182862741868337468742204512283141129927353958400 n + 81496504\ 6 0067906100275199930459474984430644321373300113025284288117507096576000 n - 146286697133531284060529003769710043159731633631747175250547001927979\ 5 2373760000 n + 20267974606308422364281793478104823306763978442799030709\ 4 77250962367211110400000 n - 2062565651700929167202165850085086017132145\ 3 866487870194238473245500112896000000 n + 142630634002463977983304880518\ 2 7359361910537161027118166697890449866096640000000 n - 58296919198539337\ 5958425862745236509410652070225376905847538022612992000000000 n + 101418\ 104268625671327344707167545025965564418888177120460669766860800000000000)/( 1099511627776 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 65) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 75) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 87) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 79) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 73) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69) (2 n - 77) (2 n - 81) / (2 n - 83) (2 n - 85)), | \ 34961314846132693560185280 (n - 44)! (n - 43)! (2 n - 93)! (n + 1) + /91 51865686859647402534340800 (n - 45)! |-- (n - 43)! (n + 1) (2 n - 92)! + \90 (n - 44)! ((2 n - 91)! (n + 1) \\ - 1/51865686859647402534340800 binomial(2 n, n) (n - 47)! (n - 43)!)||/( // (n - 45)! (n - 44)! (n - 47)! (n - 43)! binomial(2 n, n)), ( 34961314846132693560185280 (n - 44)! (2 n - 93)! (n + 1) + 52441972269199040340277920 ((2 n - 92)! (n + 1) - 1/52441972269199040340277920 binomial(2 n, n) (n - 47)! (n - 44)!) (n - 45)!)/((n - 45)! (n - 47)! (n - 44)! binomial(2 n, n)), ( (34961314846132693560185280 n + 34961314846132693560185280) (2 n - 93)! - (n - 47)! (n - 45)! binomial(2 n, n))/((n - 47)! (n - 45)! binomial(2 n, n))] and in Maple notation [47/524288*(93575457683*n^23-24750708557119*n^22+3083615450874673*n^21-\ 240607825556569625*n^20+13187556446387930838*n^19-539752795376848795434*n^18+ 17120549823285038374178*n^17-431095759389368610474850*n^16+ 8756104581634623749988463*n^15-144979782684324597149291459*n^14+ 1969657338223633635910695453*n^13-22028458908158002328757217125*n^12+ 202902789992020238384134594328*n^11-1535828319145641567381067756804*n^10+ 9505842316249278817674181550368*n^9-47723014489709989398251538609200*n^8+ 192039997620222097534931674850688*n^7-608840607210069429309342140789184*n^6+ 1480934180665545535426520940035328*n^5-2626673199255581052603137235379200*n^4+ 2893330639973585445958693283328000*n^3+32907769155921268122878730240000*n^2-\ 7827902595785082363607385210880000*n+13303174311772819526861512704000000)/(2*n-\ 5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21 )/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 705/4194304*(99813821527*n^24-\ 28746380598948*n^23+3909607574958226*n^22-333944307780215616*n^21+ 20097963016705436401*n^20-906304394012531269140*n^19+31791722940679676865352*n^ 18-888990569166674277897264*n^17+20145976995907543585341625*n^16-\ 374131364938825855007376972*n^15+5735123314643540151931105978*n^14-\ 72869588550833579123819159232*n^13+768603583410366262779773950159*n^12-\ 6724137516586343330068359974844*n^11+48633386433448899778051845383884*n^10-\ 289097362054556225205908543977680*n^9+1399631063163466961775142916026768*n^8-\ 5442911808976588355099817923877696*n^7+16615128298761456884410758652906560*n^6-\ 37956321427111088503866336718910208*n^5+56128626626431309132527653373803520*n^4 -14950291133497808473594223421542400*n^3-159333109329020240680929167155200000*n ^2+314066365899804575796776118681600000*n-7095026299612170414326140108800000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2* n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 15/4194304*(4691249610941 *n^24-1351079887504716*n^23+183751555783004390*n^22-15695382409123662432*n^21+ 944604252310031877179*n^20-42596305317386884200060*n^19+ 1494210858574612972228952*n^18-41782547155871166229929168*n^17+ 946860288760791889164423347*n^16-17584139897038245675813759684*n^15+ 269549242586005016114980431134*n^14-3424811694898989613249272949344*n^13+ 36122491853419145954093552235845*n^12-315984482681141686688326170609588*n^11+ 2284660068681510110463253468222916*n^10-13567231835775428524962105965888880*n^9 +65477931653921899320329714805075248*n^8-252157113883196210123024951399113152*n ^7+746629551443003789212179834628732608*n^6-\ 1544227331825002322459764891492690176*n^5+1479571561948100209212330004239037440 *n^4+2565974424425316691663012082150707200*n^3-\ 10126909132874994994829706754990080000*n^2+ 7417766977192218683620922565427200000*n-333466236081772009473328585113600000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2* n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 645/4194304*(218197656022 *n^25-68186767345175*n^24+10086186564967000*n^23-939409073591355650*n^22+ 61820513808187100890*n^21-3057632560077376610945*n^20+118038111391116303854500* n^19-3646076153694756288089000*n^18+91652451044523810308166970*n^17-\ 1896834626194026087790702745*n^16+32574338115859616095963795600*n^15-\ 466432343212640381308611631850*n^14+5582105059226870568172840957030*n^13-\ 55840950974799789211683255624095*n^12+465917473319663447236449035187700*n^11-\ 3226689267158917090149049875946700*n^10+18387710567844726179556794887861360*n^9 -84902214549024704379562444658763920*n^8+308283601368792958717740500289124800*n ^7-823082080745702169048108081077236800*n^6+ 1320512542587056316685118649910057728*n^5+41826677653918077794477321437946880*n ^4-5541186921939380347698504222840729600*n^3+ 10175475111497034866723723710095360000*n^2-\ 5800254397438452649226004123648000000*n+379996408558298336376583736524800000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2* n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/( 2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 215/4194304*( 654592962676*n^25-204560297949905*n^24+30258558221889460*n^23-\ 2818226884804953470*n^22+185461487036518152460*n^21-9172891034738086979015*n^20 +354113697930508950934060*n^19-10938179555159346163217720*n^18+ 274954285452196675214609500*n^17-5690345152490752861797544415*n^16+ 97716194508634932281522162860*n^15-1399052892003798303960452741270*n^14+ 16739031015670880301280621916980*n^13-167342185023102116850666022135865*n^12+ 1394048790607965021511901723505460*n^11-9617914609794764749887930522830420*n^10 +54321542353192422462946938881054320*n^9-245706797869144540070631107105619440*n ^8+851247727427313600008051176291776960*n^7-\ 2031698633838302433095592464190237120*n^6+2231148889122151344611806203920504064 *n^5+3888299454436538854055832634162928640*n^4-\ 18642942825720340316439669996158668800*n^3+ 26316996124735706639483902732001280000*n^2-\ 13576544016824398094356222852300800000*n+1139989225674895009129751209574400000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 8815/4194304*( 31931362753*n^26-10792799928287*n^25+1730479931173435*n^24-175113095726426390*n ^23+12552487947389535925*n^22-678152478457725816545*n^21+ 28683864630052922313595*n^20-974040559873918927787540*n^19+ 27017669064842142021973495*n^18-619554240095503854183410225*n^17+ 11843145246769943702558445145*n^16-189733060728515572780321581590*n^15+ 2554919688660319985326676699935*n^14-28936661944161747585067061517935*n^13+ 275151511953778047796710817590145*n^12-2185829068751503347748070445647840*n^11+ 14369445649570004377175839328032180*n^10-76819620115700810156426501199155840*n^ 9+323269232033734641414383524410372720*n^8-\ 1001634872804146327220747993991997440*n^7+1913435761989358004463914435731115712 *n^6-459334982339816541097490725894311168*n^5-\ 8906320254634411585109082999319895040*n^4+ 24912777001009024320459361489716940800*n^3-\ 29491031741812365893052601644318720000*n^2+ 14219067262346096718985375996477440000*n-1418035378278527938185788089958400000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 1025/ 4194304*(274609674194*n^26-92818045772218*n^25+14882115610567439*n^24-\ 1505970007593951952*n^23+107950985424405638216*n^22-5832062676278190982630*n^21 +246676733647162156028405*n^20-8376414940242843258548212*n^19+ 232331795881537772327342006*n^18-5327165063484720071540383030*n^17+ 101809852370830318683712840265*n^16-1630296458453744699584943426632*n^15+ 21932337589436786822627735818196*n^14-247915072662063476579270125061050*n^13+ 2348087685252138018961241256338435*n^12-18510210927423192676160865607988212*n^ 11+119915401993542289032223734680741756*n^10-\ 623910086532095028436949277925025200*n^9+2497429907609306389196585913421565520* n^8-7012935293351126993253355224983643072*n^7+ 10185795055011510290249880859255885632*n^6+ 11302127463495371612213000339877328128*n^5-\ 94547149574596214182704246215700440064*n^4+ 212252917184233472567066274163504558080*n^3-\ 229113959522945031996640831862784000000*n^2+ 107390383049082548136337373298917376000*n-\ 12195104253195340268397777573642240000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 3075/4194304*(183073006808*n^27-\ 66730053423834*n^26+11561167121551164*n^25-1266890952007168137*n^24+ 98571774608716858956*n^23-5795054012718642197916*n^22+267472856956348695889380* n^21-9941375636315991863667243*n^20+302818729460696175573717156*n^19-\ 7653322510561006771667558526*n^18+161875427716723014712198902420*n^17-\ 2881625941460290316776582140783*n^16+43310465405628490763127905238996*n^15-\ 550007687246337976568458462920936*n^14+5890142194251552632495631272374620*n^13-\ 52912058453004158170290095400584973*n^12+394695396950067102450251306587138596*n ^11-2402131054375939658337441023838691716*n^10+ 11566363795014714172736599072351074160*n^9-\ 41550689821157897993708864878398480048*n^8+ 96373026124800932513068649368284005568*n^7-\ 60402409861900776903888531588317691072*n^6-\ 489159782771933618395082003378356095744*n^5+ 2022794178233688018065781133736642141184*n^4-\ 3819634056663299561658301741781723566080*n^3+ 3788923307954595303739269076713160704000*n^2-\ 1726366846549306075374938917331828736000*n+ 215446841806451011408360737134346240000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 75/8388608*(15011957779415*n^27-\ 5471843573652345*n^26+948008566008819900*n^25-103883512969077547680*n^24+ 8082648792081322040205*n^23-475167134604901319523891*n^22+ 21930316109421203474698422*n^21-815015651844807555373596222*n^20+ 24820754416183033094272922865*n^19-627072457642500775236178924791*n^18+ 13253864586556215281382345179232*n^17-235634596062771179374834898300292*n^16+ 3533370399194488864202877723795435*n^15-44689130456933934837915385335405381*n^ 14+475273333074681825099123001378753422*n^13-\ 4220369304158338419465856696770390942*n^12+ 30895526885931811512454850402999338080*n^11-\ 182451633801883820759587284818406675576*n^10+ 836471702581422907480599187395199574272*n^9-\ 2752255758300114168913254249038737169952*n^8+ 5107254063557024647569854810186846242560*n^7+ 3257011175281072053300270232481145273984*n^6-\ 54978555300615866598946925346778481653248*n^5+ 177672425706839843214270388422747206105088*n^4-\ 305904867517655322478683553987896814018560*n^3+ 290008856625302560145826147364145922048000*n^2-\ 131090971993076071820806729255318781952000*n+ 17666641028128982935485580445016391680000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 2775/33554432*(3245810646667*n^28 -1272348417172490*n^27+237505539802930623*n^26-28097106686715625650*n^25+ 2365130127731843929275*n^24-150779661959254792436154*n^23+ 7565345342295269788119003*n^22-306494090451409400638956858*n^21+ 10205394670166394521750492565*n^20-282806682186139607607305648454*n^19+ 6579540096018816512609813566413*n^18-129253826783606312271012920774838*n^17+ 2150796103912571407170149255697705*n^16-30334008726563642893648912507015614*n^ 15+361828839711507026387610866031535113*n^14-\ 3630228840845211993554905913941768638*n^13+ 30331642197779733850964952231550486140*n^12-\ 207580955454011942323399893270026514744*n^11+ 1131625005456575625624138464024144339408*n^10-\ 4665735191839134821794672899645330826528*n^9+ 12837369341966817529769415568250067643968*n^8-\ 12229407352167358649141137932942829398144*n^7-\ 76654290271210750281723401319829399042560*n^6+ 438041827018185258034199634920371882432512*n^5-\ 1169843887783881324745536189998836173496320*n^4+ 1836523005049855991488592202998070081945600*n^3-\ 1655281104843027336443138916136021229568000*n^2+ 737275944783940388624512337722142883840000*n-\ 105044892599685844481265613456854220800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 555/33554432*( 16228813327669*n^28-6361571752839590*n^27+1187470349805101481*n^26-\ 140473356750406895670*n^25+11823821660922039482805*n^24-\ 753691667624001897311430*n^23+37808497532991130845069885*n^22-\ 1531183994843344591988268030*n^21+50953154022344868793664079915*n^20-\ 1410551542113720818321356176330*n^19+32761765797263812873017556195275*n^18-\ 641859682481405836626269971536930*n^17+10634993312349157852536175784419335*n^16 -149002090809496474793620185413890530*n^15+ 1759606582640145916659620650348265055*n^14-\ 17393811359984340970729046811344688330*n^13+ 142202932765984340158195589618880984420*n^12-\ 942554174025508978827409516900317698280*n^11+ 4892529689237026605630216410339785907120*n^10-\ 18525152109427045448929582639558060202080*n^9+ 41221920975004024629785666041589499243456*n^8+ 19536253639636299327526820752136985068160*n^7-\ 579614289182914907486546169526705337058816*n^6+ 2482718454553607494308142349749490328791040*n^5-\ 6014495741550555209844238163022286896537600*n^4+ 8979100935768515126317616040229642395648000*n^3-\ 7891046357749288978027255461781059502080000*n^2+ 3511801569079089298511853115891639910400000*n-\ 525224462998429222406328067284271104000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 16095/33554432*( 1119187894366*n^29-470597899503233*n^28+94388318324695164*n^27-\ 12019662524230914357*n^26+1091210622435614559060*n^25-75180894384461440232385*n ^24+4085475917216702406021780*n^23-179666607393903679824525465*n^22+ 6509123169959251394282489760*n^21-196726847613638208206724038655*n^20+ 5003579947734236188873358327340*n^19-107706475034614541919905085600495*n^18+ 1968174204388034715370802951144940*n^17-30546362286791889258555575772021195*n^ 16+401768023272614786066484779221209660*n^15-\ 4454438627931420083497950909965227155*n^14+ 41240550022642595825533966892365090530*n^13-\ 313959977684298668657655201958193857140*n^12+ 1915155419579564503756815403019416691720*n^11-\ 8914419559188780232081068949315624520560*n^10+ 28048696092139401587848083847721789263584*n^9-\ 31029765775870116713014485436941923136192*n^8-\ 234530839408641106776998455178229491201664*n^7+ 1659109992955192221774680058396546887788032*n^6-\ 5768237701944104082785970694372059871242240*n^5+ 12658193018137562261201263504685289386188800*n^4-\ 17817498633606695478886054881595349458944000*n^3+ 15135529547224918767228182659481422233600000*n^2-\ 6672757174752403409965726504084144128000000*n+ 1032337737617602264729679304662188032000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 435/33554432* (41406517078597*n^29-17409717772558121*n^28+3491569724891222163*n^27-\ 444560487884400714309*n^26+40350062620602508741695*n^25-\ 2778938277510531858939945*n^24+150922916920887823847868135*n^23-\ 6630970359127019903998076505*n^22+239891262366353619831861096045*n^21-\ 7234741895082017115440478772935*n^20+183424160238051448886674056457905*n^19-\ 3930077610558010029287052451455615*n^18+71340820505645562964984302446149605*n^ 17-1096930189312818148642805325253170915*n^16+ 14242263832648782193144097942351848845*n^15-\ 155127533433486831424567432629881506035*n^14+ 1401567205206949587648375365773144304010*n^13-\ 10308741019411723744860764161609680853380*n^12+ 59700270220269978815902551282309576767240*n^11-\ 253560738289912165247608278994727240793520*n^10+ 627545728332329753594027303983634575928928*n^9+ 547211069248771261261171253731838293509696*n^8-\ 13867814624429674132280219322597028411836288*n^7+ 72445836447168050526744695375720018989545984*n^6-\ 227677367931546575487935324244412449067898880*n^5+ 474015386743535318225084747935657110066585600*n^4-\ 647140670370434304792093398873685829705728000*n^3+ 541520678780503474487666017130125583155200000*n^2-\ 239134506547764680910560112568690507776000000*n+ 38196496291851283794998134272500957184000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 435/ 67108864*(165596621152133*n^30-74503903173143625*n^29+16013557423817841535*n^28 -2188740072162116844525*n^27+213630632555168886416667*n^26-\ 15851112381607697091944865*n^25+929291812097109514029441075*n^24-\ 44167070087833333337451158625*n^23+1732333056358612731565904335545*n^22-\ 56778406685725142150695846058175*n^21+1568588100359518480727241834365925*n^20-\ 36731243388292263209655403444308375*n^19+731218691353036709399500533941303865*n ^18-12380879113711657378830523059081259275*n^17+ 177925977772828798713875873574824838225*n^16-\ 2159293819404181355722273529149164916875*n^15+ 21932006143510432799987885114414069491470*n^14-\ 183684261085939106266996889688275812385100*n^13+ 1236235272212868716408319088260434382391000*n^12-\ 6352591224175235678688579139217862441330000*n^11+ 21720910403529888484403076229731448858917152*n^10-\ 18641439812343945874164513538109203908283200*n^9-\ 324402214488065596650878051148888498486264960*n^8+ 2510417334420494515424415184899191999783654400*n^7-\ 10624332734594043153805235246525657976913016832*n^6+ 30154536854102552613671054276672478668344074240*n^5-\ 58952536013578382495686051896238976753425612800*n^4+ 77228481051971927455087310678001255084539904000*n^3-\ 63045515389322629413629710216705134688665600000*n^2+ 27639131297178738782340537809482910859264000000*n-\ 4507186562438451487809779844155112947712000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 435/8388608*(20692140733141*n^30-9307834117406145*n^29+2000000765350178210*n^28 -273241284119055368655*n^27+26652278476445981476194*n^26-\ 1975692103295495543386155*n^25+115669198556745506269145250*n^24-\ 5486785898314837987997799075*n^23+214616014045270337474066260740*n^22-\ 7007587712018209657798057425825*n^21+192597397573777700335956185156550*n^20-\ 4478794275004954976885156829465525*n^19+88342851329885239816464228003516030*n^ 18-1477816479311548976618600547406967625*n^17+ 20904517365462353968331969867635573350*n^16-\ 248483880740105953584675403089879115425*n^15+ 2454707736959950450123513363363507253415*n^14-\ 19773379955670231412323971453882590533450*n^13+ 125326999898669878694234675359927511690000*n^12-\ 575526764498589348129758411941968417195000*n^11+ 1395764033271539232676089773641917696822304*n^10+ 3986279435938761529404670907011951656090720*n^9-\ 63957356208200308415504625166696576371221760*n^8+ 378474207378954984763757297471902028167255680*n^7-\ 1452686228694848375332065260086657938702061824*n^6+ 3912374283825115288534445929154333777588628480*n^5-\ 7406255084451411022256828045348388622426521600*n^4+ 9513191800290919704669620262154328544626688000*n^3-\ 7694100784334808942803133809852889381888000000*n^2+ 3380992310512786254600455743470522310656000000*n-\ 563398320304806435976222480519389118464000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 13485/8388608*(1334058614660*n^31-640573965960889*n^30+147130309949384455*n^29-\ 21517872098513347160*n^28+2250241679594369107185*n^27-179122691046013071882336* n^26+11280257535182964534900525*n^25-576593251743495284921500800*n^24+ 24350295377903007066065330325*n^23-860244096356281158522397121010*n^22+ 25641585236906088294078976771575*n^21-648453405666251013817632993858600*n^20+ 13954159573215990105272712478723275*n^19-255651150159161566864858690058347320*n ^18+3979638212225342656250774995052490975*n^17-\ 52375570400700077165857427946530240400*n^16+ 577518177581529966375898857208922909475*n^15-\ 5252107733459042005844073772108121640285*n^14+ 38273440941827752002103308987157476750550*n^13-\ 209887154851872057970713498718604634066800*n^12+ 708861102138215822579149198897171045108040*n^11+ 408180544873027402547234287521123265863584*n^10-\ 24998529387658139853944818425771848664619680*n^9+ 195823840174061486498161825546669100282876160*n^8-\ 957653167467804269980883636389810739228184960*n^7+ 3328647707719931704296361045997101101497408256*n^6-\ 8410572689349059679113980068102447570075878400*n^5+ 15229711392728351473125859580542024266826137600*n^4-\ 18967563055687935936212301394524741013444608000*n^3+ 15049523289015624663436169438332838991175680000*n^2-\ 6570656133060416668214604941437438766284800000*n+ 1108622501244941696598373268118797942784000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 13485/67108864*(10659099269303*n^31-5115380040230959*n^30+ 1174030231057617655*n^29-171521037543124948415*n^28+17910936532403663084427*n^ 27-1422943662340542219151491*n^26+89374854321469123261633155*n^25-\ 4552571558520291899190550875*n^24+191389861145619654807712416045*n^23-\ 6721835175187445867557392188085*n^22+198862539211965502323235527794325*n^21-\ 4981475434849238494739549586440925*n^20+105920433909640281091551025685819865*n^ 19-1911487173141882175846490023939002945*n^18+ 29191556857805289681680700292722427425*n^17-\ 374799882879784993607196947992389611625*n^16+ 3997623744694183506790770046723996939320*n^15-\ 34653740349620880218400148934759053998360*n^14+ 233409025978535202126557734816754514000400*n^13-\ 1082081047330079540346618843081864965669600*n^12+ 1643763357684232130404532469791040285165632*n^11+ 25326122413267157738827032799923755436111104*n^10-\ 296458787546197820789778383703033088263982080*n^9+ 1895100813264170092136807415235522341082832640*n^8-\ 8499150477248755039574937090179389980704750592*n^7+ 28120797634124035667134207908209019152746100736*n^6-\ 68837561088937571511858343970482591485216890880*n^5+ 122054796332853327272762426984383005794397388800*n^4-\ 150028995688610085115803534928424927462866944000*n^3+ 118336289955156327033985981210364152930959360000*n^2-\ 51785282028146466563429189686127987038617600000*n+ 8868980009959533572786986144950383542272000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 13485/1073741824*(340347981480151*n^32-173913323974016432*n^31+ 42550143972185467832*n^30-6635011485490412521120*n^29+740466229973940916133604* n^28-62954558601017539939283808*n^27+4237746292094773395595248888*n^26-\ 231704054507391872004438846240*n^25+10473597853529260407922766938890*n^24-\ 396273626698642196985653513714880*n^23+12657155125829319978747665280661080*n^22 -343172845591432934405891502677119200*n^21+ 7921320050350184405372248617913198980*n^20-\ 155738512714532628097454148174444473760*n^19+ 2602346925120296292954186909423788105160*n^18-\ 36755504522770284712441213100669350754400*n^17+ 434258457808401875296257208736809183249815*n^16-\ 4210148273412306745595029092482473060916880*n^15+ 32214886663208536934479561690495099579221680*n^14-\ 176031388141469665533156200161175801452422400*n^13+ 421671177657455243375767808523691430617680544*n^12+ 3737748579066905456190327271690512594168904192*n^11-\ 58772752798752922727747884813441083566569934592*n^10+ 458174396678412431563353611929177397399968542720*n^9-\ 2502732658996626656404448646918665090721806563584*n^8+ 10248607200537339426598882172446322767558165229568*n^7-\ 31878637858621563430205130944415024051384786690048*n^6+ 74600994464360776539328413158106556453980798320640*n^5-\ 127918050230700887896744608225904024558874134118400*n^4+ 153474386657019299317204540533720190223889334272000*n^3-\ 119179162257970662213453946673890916849960878080000*n^2+ 51833016777627102979740663433305932630379724800000*n-\ 8939931850039209841369282034109986610610176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 13485/1073741824*(339111986699431*n^32-173062959564881072*n ^31+42272183577938908472*n^30-6577572158057544257440*n^29+ 732042584314839034078884*n^28-62021550767523598118756448*n^27+ 4156644909383499248751226488*n^26-226028276488835347238760491040*n^25+ 10148050376598533417370613088490*n^24-380773675431203589892709194015680*n^23+ 12038899561519318060787593419320280*n^22-322376642004940449301832048628472800*n ^21+7328733248322119217387171040182138180*n^20-\ 141392657266146957517487454183032499360*n^19+ 2306853920237015193639897413853673663560*n^18-\ 31575868382956260370846022603377399797600*n^17+ 357072578383618483908098267569368809472615*n^16-\ 3234527716511759843934246118824333484379280*n^15+ 21792505888962690239372402074149059495925680*n^14-\ 82411021483550654324046167146602524572038400*n^13-\ 280525494794822599924349424882706760775223136*n^12+ 8094466981794657751123425975565889693850860032*n^11-\ 80851314511057599290310533215194891957967986432*n^10+ 547974487914527039324997075230566899986453360640*n^9-\ 2788503459143141503990581652901343060964697336064*n^8+ 10932042973627437067482913493696934933454903799808*n^7-\ 33018502698614972701292571461968779879918558978048*n^6+ 75691241248697325247021542756617163861659286896640*n^5-\ 127950823283808134476591936738374557406673634918400*n^4+ 152147249567477509722549430707819828368461135872000*n^3-\ 117697962306475994679630593315688678064670638080000*n^2+ 51292219898922186239232300523441123260943564800000*n-\ 8939931850039209841369282034109986610610176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 121365/1073741824*(74919718194238*n^33-40598434329051951*n^ 32+10539975387598794016*n^31-1744927288737315443640*n^30+ 206845884343978585608232*n^29-18687727830446413591398084*n^28+ 1337232334060646578302746064*n^27-77745952498064570339479774200*n^26+ 3737820680960159008176133878420*n^25-150445400819020038517432240569690*n^24+ 5112517847095392124888831208025840*n^23-147480055156482468640389091981258200*n^ 22+3621221826649727762807370699367817640*n^21-\ 75686764827082220507277329724763441380*n^20+ 1342447053473746169254623164764648597680*n^19-\ 20057687398791678018057563597099055251400*n^18+ 248774350921423554179273429936307731466270*n^17-\ 2485992222802081395771504627120162839997615*n^16+ 18621496212582418512313587009793148532541040*n^15-\ 79630288344003181705841003660444383671870000*n^14-\ 280912827750430000550472833889903326881851328*n^13+ 9605527189090072672897021178693283840317660256*n^12-\ 110536130587697641368259460867277678114183632896*n^11+ 870717150932881122321321903731892541786457859840*n^10-\ 5220246861553304309608658001371052602313012367872*n^9+ 24534538755311075647272020240270845375073631838464*n^8-\ 90817415609651703335681588153909868297931364511744*n^7+ 262548773353623448510709792766255238239782448537600*n^6-\ 581679592450644284502302248476613543761140750745600*n^5+ 957626603495553196015524688393058788302405672960000*n^4-\ 1116466640943914445260243213056398530925737410560000*n^3+ 852313851329672306672284314141180737182118707200000*n^2-\ 369168813671926233027561202238043235801890816000000*n+ 64566174472505404409889259135238792187740160000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 121365/1073741824*(74246089822822*n^33-\ 40133630752774911*n^32+10387395866959584352*n^31-1713213457804017009720*n^30+ 202159447800408136196488*n^29-18163619468168600120800644*n^28+ 1291124664962618850913571088*n^27-74471520113338995487567841400*n^26+ 3546657849427131053680451451780*n^25-141149673387708305468290024658490*n^24+ 4732363009498679253288707158232880*n^23-134312653966718125107854147173671000*n^ 22+3232978495840591616834855057983908360*n^21-\ 65908151269018838231509426424169769380*n^20+ 1131615836368404997792154128904225224560*n^19-\ 16163047798651160146511312378593767798600*n^18+ 187147059640922267537088817745168720092230*n^17-\ 1651803312074989677564368872496763169330815*n^16+ 8986662744370390009503398550490947110512880*n^15+ 14961329393162720774112216240079230166971600*n^14-\ 1066100611437242512711697697363115630368471232*n^13+ 15076734449031380425659339439540596478809976416*n^12-\ 142232383640714993673378112580301625998533312512*n^11+ 1021420267325162558062151528840606886091239601920*n^10-\ 5797860004956365154046151039081361583884725365248*n^9+ 26273163065568501429878151749280956666389389037824*n^8-\ 94759594346505119170359078753282183980213795893248*n^7+ 268772834899035368647427048983844481462175434547200*n^6-\ 587198651249371731963499686124456829612841251635200*n^5+ 957223001365481335966255967048750664845148856320000*n^4-\ 1109128335404085560227281720437887521085487185920000*n^3+ 844598497850711518775343084249162202467414835200000*n^2-\ 366435777395262510422081007394468738215444480000000*n+ 64566174472505404409889259135238792187740160000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 3034125/1073741824*(5859859932560*n^34-\ 3353072458381732*n^33+919422072310181693*n^32-160793558060703493552*n^31+ 20137403480394352914344*n^30-1922216439204361011921616*n^29+ 145324493906161535215758156*n^28-8926089939330307335429478080*n^27+ 453290259980355606270501608712*n^26-19265026119493266939591856960440*n^25+ 690890593084188683082200562759150*n^24-21011420170049319315068136369191520*n^23 +542940782308369223783867189095967880*n^22-\ 11904080553281285783217348645646183440*n^21+ 220161715593280563341573571094289403180*n^20-\ 3389546115209937646259771268570926825280*n^19+ 42216656484170176184520591487642607421480*n^18-\ 396086332471111303014081093815549602661700*n^17+ 2125775115715322220910662945144906016599805*n^16+ 10015528851576067777010217938495797298241040*n^15-\ 431410975161250352315117465818438893638299760*n^14+ 6479546102824924356747592150193839746267386752*n^13-\ 67859129539007804593579040682733885848763865888*n^12+ 552121403577423241271621541664432895968479703552*n^11-\ 3609002044866030782975162384741756171482755273984*n^10+ 19146684074094725362777612139022799084348922981376*n^9-\ 82374631147151403497450875662439160178796783792896*n^8+ 285071182296643754458446089820297373906776429219840*n^7-\ 781954993674087448039835737096748993158978915295232*n^6+ 1662648914791547665264982131326188052064110939340800*n^5-\ 2652309560046136529858216892645726562428980630323200*n^4+ 3022594853913890159690909048999228424463939928064000*n^3-\ 2275143345367541996145447352747779124337740087296000*n^2+ 981059569575402199005316302525367517634845736960000*n-\ 173037347586314483818503214482439963063143628800000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-25) /(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-\ 29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 337125/1073741824*(51710661131284*n^34 -29463137733515168*n^33+8038172045752827253*n^32-1397435055437385895568*n^31+ 173798193656006560591192*n^30-16455637142922216781666784*n^29+ 1232345043030048042518535948*n^28-74858667717519696281091664800*n^27+ 3752414587390610857072798323168*n^26-157045793783776691648334317254560*n^25+ 5529215737168970061726387145378590*n^24-164407892331310795461154789732889280*n^ 23+4129356550699054840990458239281984440*n^22-\ 87209499833735881423428916332673990560*n^21+ 1530020431657652941583583001435228809900*n^20-\ 21690801213606152311826129945997443284320*n^19+ 231525240744518224364357794882269017139180*n^18-\ 1411283220325126299830008098072130565428800*n^17-\ 7746160756176054235256850147998930018028715*n^16+ 377908716425615066497874895708812364790318160*n^15-\ 6514789338794624490118629893481938488970798704*n^14+ 78768330215554633808664893419625799422470139648*n^13-\ 744744560925348598524003271249465936156897542688*n^12+ 5702283076081050179486496671527893937296611399168*n^11-\ 35786282514993293965504825373796142348087954159872*n^10+ 184374799005460486300590127268145345111500836737024*n^9-\ 776000875065881019118895533792149772494696883231488*n^8+ 2640702082802109473808142204218635222328246124400640*n^7-\ 7150652025590735276757503699964297529421656589426688*n^6+ 15057418437827622690648975096693392029425138627379200*n^5-\ 23854674365323753968143944967666118172615081176268800*n^4+ 27068523704181917161798478950598541816125486923776000*n^3-\ 20341273570510036837165717676492943471044121329664000*n^2+ 8782917885256636751058059313465975809035091312640000*n-\ 1557336128276830354366528930341959667568292659200000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-25 )/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-\ 29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 337125/1073741824*(100574336306936*n^ 35-60455869140856420*n^34+17412475159603668410*n^33-3198093172083149294185*n^32 +420530004355200671357888*n^31-42132684414749241540675448*n^30+ 3341743399013122557017238840*n^29-215189442902642147801819519100*n^28+ 11445532459572254119173332913216*n^27-508732702763744353495831438240032*n^26+ 19036944842519207195855127103266300*n^25-601889477510829044065534315547587350*n ^24+16070141220159303685615439164682793840*n^23-\ 360132629685221654463228228181582984920*n^22+ 6668523850091624875340728274412914911800*n^21-\ 98289245734537000415896176098596101212700*n^20+ 1036060835653195911777124671544743222141960*n^19-\ 4202821551685578040592581395592646786885340*n^18-\ 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189987395563033606211020183527691*n^38-30438061084878706019597352576392787*n^37 +3911092382171204276165700749672550527*n^36-\ 414253660239118933677854245372526982720*n^35+ 36889848674407077373521965288822051391614*n^34-\ 2803207620548493852267947663727298893629598*n^33+ 183851784453384810024626861995590076531040858*n^32-\ 10501089499311345079109546840929834462942095180*n^31+ 526072421607986141971374061300034374733803230582*n^30-\ 23247360872667038976069109031375383229980055006574*n^29+ 910328545429750011981858870724364569045326978604254*n^28-\ 31702614024233574009581489154672925057118766722948040*n^27+ 984694935030582409495338652292308698680424284105885085*n^26-\ 27337815684860491767030199478006822760230445523498158195*n^25+ 679469394435575579719052055007257811435059377462074782845*n^24-\ 15134766907016933558800977856528062900285506569497665898450*n^23+ 302288144403329346175743779022330831431038108467029632032527*n^22-\ 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2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83), -41/137438953472*( 9327893065041923932558*n^42-8512930174275776406689406*n^41+ 3752829078605772942145180651*n^40-1064848515714068905419843792360*n^39+ 218635142140520416860585785009716*n^38-34622616843315857921433866009783862*n^37 +4401356618321958819344552768780053427*n^36-\ 461591768625492009927371554280103393420*n^35+ 40731005981228848244043633522359727332264*n^34-\ 3068979440989783663925439743877065276749548*n^33+ 199707986651084068089874484376623296501971158*n^32-\ 11323927110094472584617631005289465417541206080*n^31+ 563469228361222727942501186180509425554886030432*n^30-\ 24744007385229122220967893787356210571886124344124*n^29+ 963298448296051867200028365458519460091022625604254*n^28-\ 33366082866135411946043069666048234876464301296784040*n^27+ 1031165503702056488788866929609381202784117276762510710*n^26-\ 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-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83), -1763/137438953472*( 502007784995104273112*n^43-474363117414443626067494*n^42+ 216897633116218314604704584*n^41-63941388773978718764288530885*n^40+ 13662430538345401520932712063324*n^39-2255198051244570807801951933732688*n^38+ 299311926899930852736798441791890168*n^37-\ 32824777180056253951143316844450575445*n^36+ 3033689062071169502077459216621720747996*n^35-\ 239797565175282075383500849060702739661152*n^34+ 16397014582790414353899899181810470898945472*n^33-\ 978619828621572183055946258323326784038073130*n^32+ 51343268020327361984148791624338370894475373848*n^31-\ 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71165759511670137223762311344745872367196005259211745701185780187136000000*n^3-\ 49534615789699056139727145139713517260633810356705392638393204080640000000*n^2+ 20353520338943984093399209445204829490289195120634644762179141632000000000*n-\ 3554040659820075389940591083807997826099117566869116921105612800000000000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), -1763/ 274877906944*(1117981349839872163729*n^43-1048446292197342917451863*n^42+ 475992691742692478881280488*n^41-139387389393932287905290926520*n^40+ 29596415346171753499824567212833*n^39-4856557202141054618641210273901951*n^38+ 640993498205346255231423301675766426*n^37-\ 69929641398705751467752911636376205190*n^36+ 6431265097874566160330721443619424783682*n^35-\ 506016083848998848855578456734900988231654*n^34+ 34450851784862268342548289069454732269900104*n^33-\ 2047763593253912555798025749747478973485267160*n^32+ 107025921489844352997635802783031522566456640266*n^31-\ 4946538238537427466464114131516084636318145638902*n^30+ 203097812170946564054592796363590821904423955737852*n^29-\ 7435199473837952377786052017554809038991475604650180*n^28+ 243402039796766558179835296355954784820625862311516205*n^27-\ 7141294393164944307492836835238026665270570446751156235*n^26+ 188095674398877070876767417025775541441935485834153931360*n^25-\ 4452810668280309594473004338166390301222969916380797222400*n^24+ 94807301803628812013321242650010986217521008296234411011701*n^23-\ 1815958478013480445327516089539237109108319707393010057082347*n^22+ 31285513727075603754176062213585624938206568882045837759681722*n^21-\ 484489604276512204729363926983893495749025003343783831150998630*n^20+ 6737066926508527955287630646728671688436336864329510273984735072*n^19-\ 83993775506740163324605102601866155189798424546688165540177706584*n^18+ 937018468606445878517754698385977228161569390314744866758521917984*n^17-\ 9329977199130410816532649372299810444448257988135639943724384608960*n^16+ 82660258431770075272780543312681570564728971563729406094595369147648*n^15-\ 649172491935335348468013379876908878001325924068467226481493639353856*n^14+ 4498822898552796271834461779210114444780375394982359141154088007814656*n^13-\ 27361795707855929955814594695758723779979912871361012233723552449994240*n^12+ 145093868067161261357113385417368858047525898635542448672500821377368064*n^11-\ 665536102040511952583212132137404111661565431568871020057033164304324608*n^10+ 2615303860794775886383238738724454773825490954403323507837990925928849408*n^9-\ 8700362705627029390269955699306365105629670561089519457347290118743326720*n^8+ 24140788545698577910229513860668894359432392581543932488852242475273420800*n^7-\ 54814043050474126566014482043886485552194759881771568269956684015665152000*n^6+ 99326800999691875373512624489522309686925198894286168534104211155845120000*n^5-\ 138792336276468952172898195583524562383801760629622602401782858304716800000*n^4 +142309168244776996762368932987715639129451639798727423653617390845952000000*n^ 3-99051345324382005368721786974152275869612193342798950850875180974080000000*n^ 2+40701611701429977891603746956402750964259199008761251593436790784000000000*n-\ 7108081319640150779881182167615995652198235133738233842211225600000000000)/(2*n -5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), -43/ 1099511627776*(396442951288240273871917*n^44-387028734386500427566584206*n^43+ 183113413110805108800882066493*n^42-55942758733062856963745830767374*n^41+ 12406384613731667103287055760181119*n^40-2128702614379640382354116984386764362* n^39+294121276594690428959389068170810691011*n^38-\ 33631000938645630653636516339090546250898*n^37+ 3245752581689543276545141005670467240505406*n^36-\ 268334440287657728824060500207348938664294148*n^35+ 19221086459085362128340211135624109582999137794*n^34-\ 1203702756594174434563417095094150807356464106892*n^33+ 66376015944107874708823200188431268980074049447998*n^32-\ 3241578281551317292096501262987462834575662933071124*n^31+ 140856172312694431549275308321115364998987754784798022*n^30-\ 5466352400484802528516924061283739376023428617242594596*n^29+ 190030441673214001324882724528967697668193969199295646905*n^28-\ 5931645502938125140240563411105726137865483464246182391670*n^27+ 166545571955766796908340734399628178564389840229322806009585*n^26-\ 4211717529661597949087334055470917139874521365411563992601030*n^25+ 96010272831882381878772130509013885944052923947447461091283923*n^24-\ 1973727478429951314986325965494056716252112804958830206905229314*n^23+ 36590546793616591282502512141979565052869765413414007765771842567*n^22-\ 611490964328085764487783298873859283886724324196252609351979617706*n^21+ 9204562288975679046767987702409968477803131327399856683895594312396*n^20-\ 124646824500863777440919923089563627325859576230201624356268050470808*n^19+ 1516032992419328184204673545995647942713137805627763707798225174418624*n^18-\ 16525977135379258429688345481555042140138055817093236975453593227476032*n^17+ 161033431841978262766410102971216411975556969819713877807236066233696384*n^16-\ 1398188517713343148950633456813398086702506581171395066186846831268617472*n^15+ 10775619677174214811920279994276535478608069041639869972822797089849986816*n^14 -73373733345591426052130539855918486496124696525304108262823989377302041088*n^ 13+438998183281761408185781343764400262337311548358162233895729423549670177792* n^12-\ 2292631446678890829337403018521947642252378892753368775956150627613472618496*n^ 11+ 10367916848504915605532010011413337102229735271516691458047732666520992473088*n ^10-\ 40209106213116654657878340838207892999552092582683298593246470208907040784384*n ^9+ 132145614986733861728288666668926273350197394829143362460126986799437680476160* n^8-\ 362574291475104872881194052722725336515828534679641368074434915843455673958400* n^7+ 814843870519059429471077196194697185479583804760083918006232549951243288576000* n^6-146280702314125202359965902366807409683601100095972737051110339710865637376\ 0000*n^5+2026866091727355986899303583866714991932848006933835497314494343996139\ 110400000*n^4-20627238768953206801055197331591929946234786959651649741691684700\ 24912896000000*n^3+142643124257368699763797652502973720311644388777278666074161\ 3458439536640000000*n^2-\ 583006784877479010798416723708676180633858225025172360336551540948992000000000* n+ 101418104268625671327344707167545025965564418888177120460669766860800000000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/( 2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13) /(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2* n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), -43/1099511627776*(418701165842552127331567*n^44-407283709630924214214865706*n^ 43+192036931649559719659536888343*n^42-58478499338887515238030840033874*n^41+ 12928823860907479560522541552705669*n^40-2211876094630341739338651729651850862* n^39+304769769201011771995858462288551451961*n^38-\ 34757591009031556145335257725060839070398*n^37+ 3346199300993856917311344284420562195232106*n^36-\ 275993484008622255353384925056613153429999148*n^35+ 19726198073081048017385233200866596666296793094*n^34-\ 1232771016424226085589500953076522536751119115892*n^33+ 67846028151151771305599209143544180397054294713098*n^32-\ 3307273037983282816991684307164452706783302008548124*n^31+ 143462309010791743958505597423972704118878219782343922*n^30-\ 5558454460987257477507510435498180055400947051690369596*n^29+ 192938213797738112470511432917926106004053840321111711155*n^28-\ 6013830841819975312614343049389957627334164071465091699170*n^27+ 168628335864974110644718618565187695832717553581534509662835*n^26-\ 4259090956032030328834639239037542782692207399463597116253530*n^25+ 96977883477069472327338906709629826249092319420180779668042273*n^24-\ 1991475655599481845304110089401520744793636782547322817762337814*n^23+ 36882765557236818427352936743920493373498697011292081620657507717*n^22-\ 615805890208495456499367532151920446449613448488505720912274471206*n^21+ 9261624296740284150065883398114977904949737492411497022706941414596*n^20-\ 125321353256874834719261236198861227893890743295463164896306084256808*n^19+ 1523142781823522073823483250357549582075368315252538007578127656223424*n^18-\ 16592590230833691809078852734548287129887731157654944963209463189444032*n^17+ 161586055787929374506550923778978293215429187060954519502662130427725184*n^16-\ 1402228721181781620670023884646932156014692259671248653307910138659017472*n^15+ 10801498568811647144829494328324879024762871779260526394255918952025206016*n^14 -73517920499417307517080543793067897316645099521461912970372864933706137088*n^ 13+439690703042427660816498967515273072849870780486383011839884792673677448192* n^12-\ 2295465906880338497585324128382067594614384514300132400853320219980847466496*n^ 11+ 10377654949082340946933025102854186726589320946230751309670450217041247346688*n ^10-\ 40236609637264525114502212675556544982955617009987271861222342183066851344384*n ^9+ 132207522024427339622323071807130445908258254993822130197602324879352893276160* n^8-\ 362679675188522634242141600621182862741868337468742204512283141129927353958400* n^7+ 814965040067906100275199930459474984430644321373300113025284288117507096576000* n^6-146286697133531284060529003769710043159731633631747175250547001927979237376\ 0000*n^5+2026797460630842236428179347810482330676397844279903070977250962367211\ 110400000*n^4-20625656517009291672021658500850860171321458664878701942384732455\ 00112896000000*n^3+142630634002463977983304880518735936191053716102711816669789\ 0449866096640000000*n^2-\ 582969191985393375958425862745236509410652070225376905847538022612992000000000* n+ 101418104268625671327344707167545025965564418888177120460669766860800000000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/( 2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13) /(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2* n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), (34961314846132693560185280*(n-44)!*(n-43)!*(2*n-93)!*(n+1)+ 51865686859647402534340800*(n-45)!*(91/90*(n-43)!*(n+1)*(2*n-92)!+(n-44)!*((2*n -91)!*(n+1)-1/51865686859647402534340800*binomial(2*n,n)*(n-47)!*(n-43)!)))/(n-\ 45)!/(n-44)!/(n-47)!/(n-43)!/binomial(2*n,n), (34961314846132693560185280*(n-44 )!*(2*n-93)!*(n+1)+52441972269199040340277920*((2*n-92)!*(n+1)-1/ 52441972269199040340277920*binomial(2*n,n)*(n-47)!*(n-44)!)*(n-45)!)/(n-45)!/(n -47)!/(n-44)!/binomial(2*n,n), ((34961314846132693560185280*n+ 34961314846132693560185280)*(2*n-93)!-(n-47)!*(n-45)!*binomial(2*n,n))/(n-47)!/ (n-45)!/binomial(2*n,n)] The limits, as n goes to infinity are 4398046511101 70368744176535 70368744164115 70368744067095 35184371743835 [-------------, --------------, --------------, --------------, --------------, 4398046511104 70368744177664 70368744177664 70368744177664 35184372088832 281474962667695 140737458024425 70368686991825 1125896833456125 ---------------, ---------------, --------------, ----------------, 281474976710656 140737488355328 70368744177664 1125899906842624 9007124544500925 9006991396856295 9006664579910385 18011834929189695 ----------------, ----------------, ----------------, -----------------, 9007199254740992 9007199254740992 9007199254740992 18014398509481984 72034530201177855 9001081218916335 4497445104672525 143737953646550955 -----------------, ----------------, ----------------, ------------------, 72057594037927936 9007199254740992 4503599627370496 144115188075855872 4589592530259836235 4572925140641827035 4546315799321847435 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 4505438345673396015 1111221719867413125 4358239158471029625 -------------------, -------------------, -------------------, 4611686018427387904 1152921504606846976 4611686018427387904 4238265390934474875 16309202344625055675 123768898088638698369 -------------------, --------------------, ---------------------, 4611686018427387904 18446744073709551616 147573952589676412928 115292696039673011487 104875814733705079797 184843236412663710689 ---------------------, ---------------------, ---------------------, 147573952589676412928 147573952589676412928 295147905179352825856 623420081430283442681 123001708828465630649 43366768480604838897 ----------------------, ---------------------, ---------------------, 1180591620717411303424 295147905179352825856 147573952589676412928 1527421386150196017633 3524419647357437479353 -17799987639198961299947 ----------------------, ------------------------, ------------------------, 9444732965739290427392 151115727451828646838272 151115727451828646838272 -38950318131579389476967 -3708272961757842866327 ------------------------, -----------------------, 151115727451828646838272 9444732965739290427392 -313532626682831943440153 -191221807833359440617439 -------------------------, -------------------------, 604462909807314587353088 302231454903657293676544 -110629965618296104187057 -1971001119767694624654227 -------------------------, --------------------------, 151115727451828646838272 2417851639229258349412352 -17047046905394331776492431 -18004150131229741475257381 ---------------------------, ---------------------------, 19342813113834066795298816 19342813113834066795298816 -18664477163007659794560331 -38139355683197310253719737 ---------------------------, ---------------------------, 19342813113834066795298816 38685626227668133590597632 -154196234366201711025512633 ----------------------------] 154742504910672534362390528 and in Maple notation [4398046511101/4398046511104, 70368744176535/70368744177664, 70368744164115/ 70368744177664, 70368744067095/70368744177664, 35184371743835/35184372088832, 281474962667695/281474976710656, 140737458024425/140737488355328, 70368686991825/70368744177664, 1125896833456125/1125899906842624, 9007124544500925/9007199254740992, 9006991396856295/9007199254740992, 9006664579910385/9007199254740992, 18011834929189695/18014398509481984, 72034530201177855/72057594037927936, 9001081218916335/9007199254740992, 4497445104672525/4503599627370496, 143737953646550955/144115188075855872, 4589592530259836235/4611686018427387904, 4572925140641827035/ 4611686018427387904, 4546315799321847435/4611686018427387904, 4505438345673396015/4611686018427387904, 1111221719867413125/ 1152921504606846976, 4358239158471029625/4611686018427387904, 4238265390934474875/4611686018427387904, 16309202344625055675/ 18446744073709551616, 123768898088638698369/147573952589676412928, 115292696039673011487/147573952589676412928, 104875814733705079797/ 147573952589676412928, 184843236412663710689/295147905179352825856, 623420081430283442681/1180591620717411303424, 123001708828465630649/ 295147905179352825856, 43366768480604838897/147573952589676412928, 1527421386150196017633/9444732965739290427392, 3524419647357437479353/ 151115727451828646838272, -17799987639198961299947/151115727451828646838272, -\ 38950318131579389476967/151115727451828646838272, -3708272961757842866327/ 9444732965739290427392, -313532626682831943440153/604462909807314587353088, -\ 191221807833359440617439/302231454903657293676544, -110629965618296104187057/ 151115727451828646838272, -1971001119767694624654227/2417851639229258349412352, -17047046905394331776492431/19342813113834066795298816, -\ 18004150131229741475257381/19342813113834066795298816, -\ 18664477163007659794560331/19342813113834066795298816, -\ 38139355683197310253719737/38685626227668133590597632, -\ 154196234366201711025512633/154742504910672534362390528] and in floating point [1.000000000, 1.000000000, .9999999998, .9999999984, .9999999902, .9999999501, .9999997845, .9999991873, .9999972703, .9999917055, .9999769231, .9999406392, .\ 9998576927, .9996799250, .9993207616, .9986334214, .9973824103, .9952092384, .9\ 915950744, .9858250933, .9769612085, .9638312022, .9450424728, .9190273089, .88\ 41236307, .8386906762, .7812536970, .7106661636, .6262732453, .5280573490, .416\ 7459998, .2938646538, .1617220298, .2332265282e-1, -.1177904374, -.2577515841, -.3926286720, -.5186962204, -.6326998885, -.7320877018, -.8151869568, -.8813116\ 688, -.9307927459, -.9649308533, -.9858792374, -.9964698093] The cut off is at j=, 35 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 48], vs. those in the, 2, -th row from j=1 to j=, 47, are as follws 24 23 22 [47 (2994414645857 n - 862391418006228 n + 117288227263251638 n 21 20 - 10018329236943232392 n + 602938891116036559007 n 19 18 - 27189131901468253865148 n + 953751696647818505100488 n 17 16 - 26669717782626012132917472 n + 604379358712154173500989087 n 15 14 - 11223943750920729571345766988 n + 172053834270517550989050219758 n 13 - 2186093119641755116442693475432 n 12 + 23058294354252693151172615074577 n 11 - 201729519042943867888868004541668 n 10 + 1459132638315630519694714619276468 n 9 - 8675586603400035822405855010111152 n 8 + 42033944535874345459371420263004272 n 7 - 163910230323125577963106112129419968 n 6 + 505385512608496319309968551266931648 n 5 - 1198999228919784840263349660979383552 n 4 + 2076192348656882390433724511829427200 n 3 - 2212258176530673133923900415065600000 n 2 - 177310096026892129455927351828480000 n + 6244682011594350746822518574776320000 n - 10216837871441525396629641756672000000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 24 (2 n - 35) (2 n - 29) (2 n - 45)), 47 (2994414645833 n 23 22 21 - 862391417986380 n + 117288227255414342 n - 10018329234977203800 n 20 19 + 602938890764360970743 n - 27189131853742662670020 n 18 17 + 953751691543649523750632 n - 26669717341526965137785760 n 16 15 + 604379327378528048097655223 n - 11223941899694958136597224660 n 14 13 + 172053742583812909308649333502 n - 2186089294496957057202636652920 n 12 + 23058159629168434227741991377593 n 11 - 201725513847546312397186542251100 n 10 + 1459032401161538887205588229983732 n 9 - 8673485977769706591169356197137680 n 8 + 41997396570314037961118231987697008 n 7 - 163389013745372746279846384689195840 n 6 + 499406112257254392208854032422137792 n 5 - 1145348525459144289483266571964419840 n 4 + 1716038629056666255087398204112153600 n 3 - 539304236996592967816769496617472000 n 2 - 4706708474470300451460485128581120000 n + 9625972983107987173315160088576000000 n - 212850788988365112429784203264000000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 705 (798510572147 n 24 23 22 - 249534553753675 n + 36911151174419750 n - 3437837541241199650 n 21 20 + 226236787438074023765 n - 11189635764584253729445 n 19 18 + 431969429877207374718500 n - 13343100961607295471043000 n 17 16 + 335409875840592975810804845 n - 6941656983426582064659476245 n 15 14 + 119210464736701249047486693350 n - 1707029621918467853949589755850 n 13 + 20430870860797369104715332426155 n 12 - 204427574783209884040705732707595 n 11 + 1706701532800764972540760490889200 n 10 - 11838698489756806255156298748960700 n 9 + 67751665370216752767128702389487360 n 8 - 316321697138352665612171171500259920 n 7 + 1181931008083231858498035058787308800 n 6 - 3400256365758328559823512871793300800 n 5 + 6813072772999732624493676266826985728 n 4 - 6117991590105676693536760840711173120 n 3 - 11911201168441342672039725918610329600 n 2 + 43902740731189949119807009380495360000 n - 31628413028022659005936878747648000000 n + 1390625154723985401207923461324800000)/(16777216 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 15 (18764998431367 n 24 23 22 - 5864062001899100 n + 867412048269690850 n - 80789181168779369600 n 21 20 + 5316564323474198552065 n - 262956416780389495675820 n 19 18 + 10151279170243102033060600 n - 313562671482561347528277200 n 17 16 + 7882118459032484575777087945 n - 163128174667914557053936845620 n 15 + 2801410127561286774268809481450 n 14 - 40113791860372379892454055724800 n 13 + 480079245498535197370959074689855 n 12 - 4802773453586024999165719345513220 n 11 + 40078161778868619762828230481286300 n 10 - 277650659954871836054216337093940400 n 9 + 1583447082210500841631192974812401360 n 8 - 7324090065660948509312510807830377920 n 7 + 26696397409413857640088608836303726400 n 6 - 71878927965127396723259124420246048000 n 5 + 117890050509084387248529078256789137408 n 4 - 5809954275450006861355952030959288320 n 3 - 471493620137031211718710969845211545600 n 2 + 885614432613133493439958410063912960000 n - 508382426118434327466740828430336000000 n + 32679691136013656928386201341132800000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 645 (3491162482877 n 25 24 23 - 1180012910278501 n + 189199181912858000 n - 19145708268236471350 n 22 21 + 1372406944150096875215 n - 74144867046767754151915 n 20 19 + 3136121428861840153501910 n - 106496564520917150849183200 n 18 17 + 2954024382710976996812471795 n - 67742766100666537153435803355 n 16 + 1295060650166912722637347502660 n 15 - 20751767569122313077365483276350 n 14 + 279568739025012214060998024051305 n 13 - 3169573679351875938327659466963685 n 12 + 30205493340039966378647411948596310 n 11 - 241093761667504665288283670396855500 n 10 + 1600664084771046120093112017246976760 n 9 - 8730819087851645719007291524338432400 n 8 + 38241153129519082969805230687951961760 n 7 - 128470761352820928967976807584206350400 n 6 + 296832397708194322583767819285761522048 n 5 - 308068277802894581717083177754940870144 n 4 - 590532206621619198622414293436983920640 n 3 + 2670559505548361652178231190558482636800 n 2 - 3698135143667264828389964145119354880000 n + 1884995791816724072967706858605772800000 n - 155038534691785721241646164502118400000)/(33554432 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 215 ( 26 25 24 10473487210393 n - 3540038544533387 n + 567597476412507190 n 23 22 - 57437108476534430930 n + 4117218101359489637095 n 21 20 - 222434256126425323025045 n + 9408330109513823398201720 n 19 18 - 319486973012009705916436280 n + 8861896243648745828515456615 n 17 16 - 203218797758008329544343278325 n + 3884757624697497520230618947470 n 15 - 62239483238010554003574589636730 n 14 + 838213516545794780443058376584905 n 13 - 9495926217973690971163529197848635 n 12 + 90340772508590965637773507044251620 n 11 - 718391529869691778633729807295407580 n 10 + 4731487326022760178771117717296230000 n 9 - 25380528630264525425849485555261690640 n 8 + 107445735546469020018753357989140451520 n 7 - 336541861343582652432630729646112834880 n 6 + 658955692422994268204862851359964500992 n 5 - 226923915765548794808234061006176083968 n 4 - 2831509066596385627161948561669178859520 n 3 + 8181375681453180660055091741867462246400 n 2 - 9795602978597662586339311050728325120000 n + 4738322039084989142061887628198543360000 n - 465115604075357163724938493506355200000)/(33554432 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 26445 ( 27 26 25 340601177431 n - 124149111094584 n + 21509209835073675 n 24 23 - 2357020214447965560 n + 183391534523730428415 n 22 21 - 10781757242575127148600 n + 497648389959292623838215 n 20 19 - 18497420324221546136234520 n + 563496238597251596464475505 n 18 17 - 14244423139252886881332362760 n + 301401841709079772218996334065 n 16 - 5369486542171568709738183510120 n 15 + 80820028225497762834694196818485 n 14 - 1029149069449624812112536022562280 n 13 + 11076575358940067892397914895266765 n 12 - 100395375837834822055812151250038920 n 11 + 760623136486801791498589579111440900 n 10 - 4752941658246375092662915340611719840 n 9 + 23923754566388929463132204203196173040 n 8 - 92848510747864584824541480092439745920 n 7 + 252911351632404550623719358046508144064 n 6 - 352681142583529279635139751104951911936 n 5 - 424234525430536330974434741971995125760 n 4 + 3310756807968973497810023307972545495040 n 3 - 7284167048780004691945476166062945484800 n 2 + 7757876129478940700839630185965568000000 n - 3595848414315925507636262691101736960000 n + 400831333593397230527182766761574400000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 45)), 3075 ( 27 26 25 732292273205 n - 266920391533815 n + 46244729494395900 n 24 23 - 5067577013973204510 n + 394289121732589319535 n 22 21 - 23180449336669674267777 n + 1069912437748466489439834 n 20 19 - 39767016650675110851652884 n + 1211363647753844986933944555 n 18 17 - 30617563445863700420350241577 n + 647671971432832844532709838904 n 16 - 11532133946499046964549446830174 n 15 + 173396457896591291072798890236345 n 14 - 2203547842124375660467835634795807 n 13 + 23626537382595488228833112807070834 n 12 - 212660366196529474993108510885054224 n 11 + 1591341379800043545599857848033563160 n 10 - 9732653385511820438683805797374989472 n 9 + 47222464826296323824995985490580266784 n 8 - 171800202079801108783133516530601938944 n 7 + 409383848761397335338614549934780827520 n 6 - 311780654695098285046486859880819447552 n 5 - 1829566882255962594173916038558268128256 n 4 + 7990293780527252707024865257541931380736 n 3 - 15340973193565465268661388068308129464320 n 2 + 15332469767091665443681810639556345856000 n - 6994964047940532219106065600659718144000 n + 861787367225804045633442948537384960000)/(16777216 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 45)), 3075 ( 28 27 26 11716662842765 n - 4592924447794690 n + 857356829022226365 n 25 24 - 101428029167981844510 n + 8538214415128320371085 n 23 22 - 544356342005712218300682 n + 27316238122560829029579849 n 21 20 - 1106896152571373782440076254 n + 36870511582762447229317793115 n 19 - 1022416616126338131832127776782 n 18 + 23814068379621754917299361791319 n 17 - 468740027075290558674893691169194 n 16 + 7825425450730153752560278318334655 n 15 - 110958135457353979226404192699828062 n 14 + 1334865689584466424238900550412126099 n 13 - 13572046180889997241890389448798211194 n 12 + 115724872429320335881242661969468838220 n 11 - 816618494720346161749472025225907692552 n 10 + 4664417497002041715059585112907325836624 n 9 - 20744958467589024257553788659369624554464 n 8 + 66358794935360995989797405298065153613120 n 7 - 119497819173753695685844114272029676124032 n 6 - 80470630150774643112642213060618673376256 n 5 + 1266884453193602358102600116602642842095616 n 4 - 4018533181622249808318873193937399063592960 n 3 + 6824225452534030480787211185436281838796800 n 2 - 6396316462210059644003908434752993132544000 n + 2860223908362996074124492778972611870720000 n - 379186441579353780078714897356449382400000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 28 27 (2 n - 51) (2 n - 45)), 225 (160127178364515 n - 62769557635392370 n 26 25 + 11717064220231939275 n - 1386150312691508535990 n 24 23 + 116683715753201524198635 n - 7438911048029825900733834 n 22 21 + 373263200106617948754175263 n - 15123162453332552322469323558 n 20 + 503626078939930854672382462845 n 19 - 13959340370286337951343284799934 n 18 + 324885169883060532283819077822753 n 17 - 6386075932770600475615692461622738 n 16 + 106363746741983204317575599351570025 n 15 - 1502258328294336818715032500775343294 n 14 + 17957554260255356601558742363809455013 n 13 - 180732177310887516175640571064249588338 n 12 + 1516847451240114583023634181757378783180 n 11 - 10447253442761115742331839625129240290824 n 10 + 57485623501623742496186862029716481622288 n 9 - 240583904471896013305298723742951607624928 n 8 + 683067559962494801343257560266268488217920 n 7 - 778131225241281393932966577261690369769344 n 6 - 3356202171421531175729507945435207278766592 n 5 + 20976293274061330120723369609918479213975552 n 4 - 57354199808303990084193911411715021496197120 n 3 + 91042748776334552714208796126552991406489600 n 2 - 82495479366265351956936471286991648391168000 n + 36750532611495719902132477235183710371840000 n - 5182214701584501661075770263871474892800000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 28 (2 n - 51) (2 n - 45)), 48285 (2984635544213 n - 1255024432166691 n 27 26 + 251735191921157147 n - 32059549073162988279 n 25 24 + 2910981766067270437815 n - 200607457903420619326275 n 23 22 + 10905864437537261978723535 n - 479924123364691128629809875 n 21 20 + 17405536111903242264905428485 n - 526929335797192232488927708365 n 19 + 13436653278515135786939837789865 n 18 - 290375084221663499994684694503285 n 17 + 5337417181327232515493773337126925 n 16 - 83552981165764222810767200967950145 n 15 + 1112613517804390039767474013291462725 n 14 - 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(n - 44)! (2 n - 95)! (n + 1) + /93 201027560365262987971065360 (n - 46)! |-- (n - 44)! (n + 1) (2 n - 94)! + \92 (n - 45)! ((2 n - 93)! (n + 1) \\ - 1/201027560365262987971065360 binomial(2 n, n) (n - 48)! (n - 44)!)||/( // (n - 48)! (n - 46)! (n - 45)! (n - 44)! binomial(2 n, n)), ( 135475095028764187545717960 (n - 45)! (2 n - 95)! (n + 1) + 203212642543146281318576940 (n - 46)! ((2 n - 94)! (n + 1) - 1/203212642543146281318576940 binomial(2 n, n) (n - 48)! (n - 45)!))/( (n - 46)! (n - 48)! (n - 45)! binomial(2 n, n)), ( (135475095028764187545717960 n + 135475095028764187545717960) (2 n - 95)! - (n - 48)! (n - 46)! binomial(2 n, n))/((n - 48)! (n - 46)! binomial(2 n, n))] and in Maple notation [47/8388608*(2994414645857*n^24-862391418006228*n^23+117288227263251638*n^22-\ 10018329236943232392*n^21+602938891116036559007*n^20-27189131901468253865148*n^ 19+953751696647818505100488*n^18-26669717782626012132917472*n^17+ 604379358712154173500989087*n^16-11223943750920729571345766988*n^15+ 172053834270517550989050219758*n^14-2186093119641755116442693475432*n^13+ 23058294354252693151172615074577*n^12-201729519042943867888868004541668*n^11+ 1459132638315630519694714619276468*n^10-8675586603400035822405855010111152*n^9+ 42033944535874345459371420263004272*n^8-163910230323125577963106112129419968*n^ 7+505385512608496319309968551266931648*n^6-\ 1198999228919784840263349660979383552*n^5+2076192348656882390433724511829427200 *n^4-2212258176530673133923900415065600000*n^3-\ 177310096026892129455927351828480000*n^2+6244682011594350746822518574776320000* n-10216837871441525396629641756672000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2* n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/( 2*n-29)/(2*n-45), 47/8388608*(2994414645833*n^24-862391417986380*n^23+ 117288227255414342*n^22-10018329234977203800*n^21+602938890764360970743*n^20-\ 27189131853742662670020*n^19+953751691543649523750632*n^18-\ 26669717341526965137785760*n^17+604379327378528048097655223*n^16-\ 11223941899694958136597224660*n^15+172053742583812909308649333502*n^14-\ 2186089294496957057202636652920*n^13+23058159629168434227741991377593*n^12-\ 201725513847546312397186542251100*n^11+1459032401161538887205588229983732*n^10-\ 8673485977769706591169356197137680*n^9+41997396570314037961118231987697008*n^8-\ 163389013745372746279846384689195840*n^7+499406112257254392208854032422137792*n ^6-1145348525459144289483266571964419840*n^5+ 1716038629056666255087398204112153600*n^4-539304236996592967816769496617472000* n^3-4706708474470300451460485128581120000*n^2+ 9625972983107987173315160088576000000*n-212850788988365112429784203264000000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2* n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 705/16777216*( 798510572147*n^25-249534553753675*n^24+36911151174419750*n^23-\ 3437837541241199650*n^22+226236787438074023765*n^21-11189635764584253729445*n^ 20+431969429877207374718500*n^19-13343100961607295471043000*n^18+ 335409875840592975810804845*n^17-6941656983426582064659476245*n^16+ 119210464736701249047486693350*n^15-1707029621918467853949589755850*n^14+ 20430870860797369104715332426155*n^13-204427574783209884040705732707595*n^12+ 1706701532800764972540760490889200*n^11-11838698489756806255156298748960700*n^ 10+67751665370216752767128702389487360*n^9-316321697138352665612171171500259920 *n^8+1181931008083231858498035058787308800*n^7-\ 3400256365758328559823512871793300800*n^6+6813072772999732624493676266826985728 *n^5-6117991590105676693536760840711173120*n^4-\ 11911201168441342672039725918610329600*n^3+ 43902740731189949119807009380495360000*n^2-\ 31628413028022659005936878747648000000*n+1390625154723985401207923461324800000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 15/8388608*( 18764998431367*n^25-5864062001899100*n^24+867412048269690850*n^23-\ 80789181168779369600*n^22+5316564323474198552065*n^21-262956416780389495675820* n^20+10151279170243102033060600*n^19-313562671482561347528277200*n^18+ 7882118459032484575777087945*n^17-163128174667914557053936845620*n^16+ 2801410127561286774268809481450*n^15-40113791860372379892454055724800*n^14+ 480079245498535197370959074689855*n^13-4802773453586024999165719345513220*n^12+ 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434690094934115838646739656842512098275*n^16+ 4841040208895420971858178887362391330680*n^15-\ 44629582406335755856813133843418686359800*n^14+ 332012157119039917719825056022600404328400*n^13-\ 1890941693858004404333771880501633626583200*n^12+ 7099868818835934288778847537475742839338048*n^11-\ 4562538176406500899071827102969802982808320*n^10-\ 165756748749837385878641831300837973561561600*n^9+ 1450022624276735111286370703744841030423717120*n^8-\ 7363897065143417073492019155967842535255415808*n^7+ 26099898574791376138874226595636347078158776320*n^6-\ 66733524246547444050059855458293996038971269120*n^5+ 121760653815978693386125874910771196448877772800*n^4-\ 152347814004401146396564334978956089626640384000*n^3+ 121127117277500977081958294773091232864337920000*n^2-\ 52842011561248028010399553992760373162803200000*n+ 8868980009959533572786986144950383542272000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 13485/1073741824*(341371939225213*n^32-174634190226540080*n^31+ 42791429375231877512*n^30-6686105256719994616960*n^29+748150020733933528030572* n^28-63827880431009709154512960*n^27+4315695802884847096756722888*n^26-\ 237308976470069591806169293440*n^25+10804085180537691794125045843470*n^24-\ 412457113241147137763123367136800*n^23+13321321150354571864654572434650280*n^22 -366164936406167229374578630644931200*n^21+ 8595688639935804261367819986896761740*n^20-\ 172543094955184857029339746439787201600*n^19+ 2958587854478127249109679387925529472760*n^18-\ 43180331736262177136699892613322285577600*n^17+ 532719671270788247953597681909045221788445*n^16-\ 5489244305645473424857474794542729640375600*n^15+ 46247790914595005405823860110327246702478480*n^14-\ 305365459712276210801897523934273812759251200*n^13+ 1415967858120209519174151797609042277772341472*n^12-\ 2578296306178869400514764261683351932829739520*n^11-\ 26040417405608912891576455851163427693761135872*n^10+ 322186747921561802984453430598578143401215528960*n^9-\ 2061157328215032242618288098041284609061338272512*n^8+ 9171738851624566921437523109914452515215189954560*n^7-\ 30046094367691139924872426351418064746736649986048*n^6+ 72800267285433117096570593249521876916411681341440*n^5-\ 127799716114273085685859033561636154860269487718400*n^4+ 155598372014601732085337971478554744718405271552000*n^3-\ 121605355959906469918869206680090009809078190080000*n^2+ 52729061261136456406686416461547452982152396800000*n-\ 8939931850039209841369282034109986610610176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 13485/1073741824*(340843444905181*n^32-174254202810437072*n ^31+42661567750913614472*n^30-6658036733125139885440*n^29+ 743842950001080894715884*n^28-63328566051512956272512448*n^27+ 4270256760164378980321516488*n^26-233979258626899444432059911040*n^25+ 10604097488678215623963069990990*n^24-402486969232572328277221710835680*n^23+ 12904990330488070747585107880250280*n^22-351509254787742421969156714860412800*n ^21+8158865621852813985038680577951943180*n^20-\ 161489222010652746045458038438243239360*n^19+ 2720798862422646043972415026182497773560*n^18-\ 38831824182264699642477216187945865577600*n^17+ 465199349129543493395820922222033126186365*n^16-\ 4601237548376750115657325715543710046339280*n^15+ 36392823783919156514845793778082089785025680*n^14-\ 213560242017273407095428756498914227098438400*n^13+ 703155188597376965043487216742635533201904864*n^12+ 1991305426249315096583524774016331256796396032*n^11-\ 49922294870975617036720616273082875892431922432*n^10+ 422176946312211317245108602933361777828662128640*n^9-\ 2388177984799963376984145329865264007822716728064*n^8+ 9974643722531395113658300651427758407953007095808*n^7-\ 31421709280520757127570941426602527102877714178048*n^6+ 74163955882708625548744529310427735381075197296640*n^5-\ 127904912756826000259133394641163681650057954918400*n^4+ 154006385576533896008165425075999214588349775872000*n^3-\ 119772919135078438423133135735584055673374638080000*n^2+ 52049801733314849776582377875622429489679564800000*n-\ 8939931850039209841369282034109986610610176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 13485/1073741824*(679897716331087*n^33-369348187032443784*n ^32+96189440325472606024*n^31-15987017551155677674560*n^30+ 1904356680250396398327268*n^29-173083236966145354244405856*n^28+ 12475994025577774568066292696*n^27-731791442902225896522011908800*n^26+ 35559445406450577202176586902330*n^25-1449654516891144409947873046806960*n^24+ 50022130897539156662624370788300760*n^23-1469676039502064981952977334511588800* n^22+36893155458109084998956253142804033860*n^21-\ 792221496931275010423919064161958157920*n^20+ 14534630887521070906047562976116286252520*n^19-\ 226913827403900533264659903668263447521600*n^18+ 2990338687015756753123125851762252566263855*n^17-\ 32776809131352912727142098864611820532369160*n^16+ 290402552219962361199411021891770146257510560*n^15-\ 1948044396339392764346581042488094693238976000*n^14+ 7901775591557784147892931320024570437993780128*n^13+ 12358215724480473746969544596300020651247771904*n^12-\ 557649642290560238602234458719333254837295321344*n^11+ 5721413239413545592808958984479462695144651663360*n^10-\ 38741094407200746675153584188770587362218579280128*n^9+ 195612642887126670785184071769894732414996752971776*n^8-\ 759342970685843990293192473272094151847518545801216*n^7+ 2269868143549563857108233663962036485084089071206400*n^6-\ 5150689558042445381891021974330514644558873888358400*n^5+ 8622140714397831861526101502168393903249549885440000*n^4-\ 10157524739848799098237316147762646311981232291840000*n^3+ 7787782776042063760258783355107368955449625804800000*n^2-\ 3364323751941873296029957702579405784208113664000000*n+ 581095570252548639689003332217149129689661440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 13485/1073741824*(676990689133012*n^33-\ 367258034477027859*n^32+95474334890963740624*n^31-15832081862117179907160*n^30+ 1880488884062966025049168*n^29-170300542488747804356933556*n^28+ 12220802451523986440042224896*n^27-712902258476614698431074363800*n^26+ 34410347533427238112526148013080*n^25-1391449731747297049215935201436210*n^24+ 47543839830845289301277758295009760*n^23-1380355862311559712547323133306327800* n^22+34154754300051014897655635405439993360*n^21-\ 720566965830384163176227800261373262420*n^20+ 12931204772382450797237664321764098187520*n^19-\ 196205929423024410256246028670648905058600*n^18+ 2487190192617053281033371112974412823452980*n^17-\ 25733857562313963093611171766870158677942035*n^16+ 206400434049095770302696653271438870298260560*n^15-\ 1097666610981835742563491724209997342119942000*n^14+ 634346456762458169028455750646864058217779328*n^13+ 64412937682602608940780519835382323102264891104*n^12-\ 867159711603514436640940578405819312937910930944*n^11+ 7229455695149240984713555588010266767072749832960*n^10-\ 44655724370828521496382463943784062245042156182528*n^9+ 213808056437040604756617930251146882341196442270976*n^8-\ 801478520574539071798127734026605876516633598652416*n^7+ 2337869823402479691601276964019896025623403620966400*n^6-\ 5212886789672035230468675001319041992503139739238400*n^5+ 8620265051150545978227054545396547925313379901440000*n^4-\ 10077756832475096349836061596109922790188200099840000*n^3+ 7701900391257309489079905447669034621684403404800000*n^2-\ 3333527385828342757742346049373453070829092864000000*n+ 581095570252548639689003332217149129689661440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 121365/2147483648*(298864905161491*n^34-\ 171816806069331719*n^33+47378095106852997124*n^32-8341392394151348549336*n^31+ 1052973517680592033128724*n^30-101455556127619869259831076*n^29+ 7754940492532880914972623336*n^28-482483536914113691807878193144*n^27+ 24873159348665471682459838764690*n^26-1075941012975711728914904434637610*n^25+ 39397834736074279886270956611797760*n^24-1228310300305313291298219981960172440* n^23+32712076484851850694170023149573398580*n^22-\ 744743426373309501511052530659084986820*n^21+ 14465400057416833652734128252468321264120*n^20-\ 238344087958992327317915649216298468507080*n^19+ 3293057673865130590426902247127387460776115*n^18-\ 37282510857633262685868173187801650311924935*n^17+ 328432308367438246288554291550231522967160860*n^16-\ 1919483381565649201103433025889919347867682640*n^15+ 940059035835085039246443683796578334087291104*n^14+ 146242964101658520596991741926562830004721204064*n^13-\ 2210524625880671311160639734768345036444163964544*n^12+ 21042885521540878314520788679749372647951858998016*n^11-\ 150455708017423104051705069592217433152377844431104*n^10+ 846514669578954270428216035896155412653654656010496*n^9-\ 3796417448075246256527567096036451228193788238918656*n^8+ 13546401746379619234805137178059380697553772670746624*n^7-\ 38018812511183888271861669329691462552125781371289600*n^6+ 82221897918375019677190509567387390385102217689497600*n^5-\ 132746289357205425141217739061148492457774591016960000*n^4+ 152412365587020833788726082980095649858346762895360000*n^3-\ 115058806755295291574558187265093072001416613068800000*n^2+ 49505623668029902967397424028894363399101612032000000*n-\ 8651867379315724190925160724121998153157181440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-25 )/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-\ 29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 121365/2147483648*(295820211847829*n^ 34-169618537496867755*n^33+46621826779365806606*n^32-8176372062584774930200*n^ 31+1027325877605459838721436*n^30-98432676687489889491557140*n^29+ 7474057051152942640451948304*n^28-461363886741761308550351408280*n^27+ 23564306818207401711769799862510*n^26-1008184769526735801701571935342850*n^25+ 36438728555828436390646927416455340*n^24-1118481953800081931283167521831620600* n^23+29229092963128198665120312226898141820*n^22-\ 650003896815002438381067291347198738100*n^21+ 12249532053539176520427771185351253660480*n^20-\ 193718868774525920200396044974183811157800*n^19+ 2519000660144295552790930277775045107622885*n^18-\ 25726424496321472575770039493013973744032875*n^17+ 180203631979204485160028393056578782290414390*n^16-\ 290588188515130366537642172296673155129857200*n^15-\ 14332299711618246487728798362852976454793210624*n^14+ 267733163093387852693054056323018572621694275680*n^13-\ 3024422390496124747429024295098326642763630732736*n^12+ 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29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 3034125/4294967296*(46621859895041*n^ 35-28245538060445020*n^34+8209342552474482335*n^33-1523600405729082271360*n^32+ 202761195268394022950228*n^31-20596264121979361298571088*n^30+ 1659669476467275669793530540*n^29-108845568711966525359739957600*n^28+ 5913660020036046821925524526846*n^27-269499299158146434556405597162792*n^26+ 10390216691225287357043733722824050*n^25-340721649641691801973886377767021600*n ^24+9527413764196453276004248486594527540*n^23-\ 227042338103095643388344351506586405520*n^22+ 4590216571516435687487587439851374258300*n^21-\ 77889970069841820809762032765175332471200*n^20+ 1084081099316414167316950972022907557765385*n^19-\ 11721788890096659711854501697809428248456540*n^18+ 82458726922814207722440031167555343865988775*n^17+ 29000397824377933725495591500288066969034400*n^16-\ 11837610909045515729503908668876889077058517136*n^15+ 221422463990060226000061677878660963738225982400*n^14-\ 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14350592017543312502457542257784840061602732801862244866145383676979454528*n^17 -137088914112003115458918039350865519796774282992177251205424608649458495360*n^ 16+1168369839280404291412166290639335693724009673271730103311275412890763051520 *n^15-\ 8849076253605320681864176419898283028873287789804248750105961532848354976000*n^ 14+ 59281848447095642533602591847494398685099789648331097121864357389351743216128*n ^13-\ 349324023148860577408786852950906582496180291542850225954639256627652087383040* n^12+17985459321797062326281201022173968945347106363091043246882589160082144363\ 41760*n^11-80263457223168282324797111656758039514237127523658149456947265382043\ 32203417600*n^10+30746141460611462844017816175279446754878718526030656851105401\ 178609644004179968*n^9-99895417495902347692892313384570845707469500756910765653\ 386718788160332504760320*n^8+27120148116529648570019726425779434180552329436038\ 3052501416181966532949691596800*n^7-6035863995575453297660028214935068737924798\ 18204125037577086863124487107510272000*n^6+107395460249541318538354779087481782\ 6472748412925816792927292478568837435883520000*n^5-1476124928548621125698353488\ 069556212702628809882376686163357086224567028940800000*n^4+14914657254616557756\ 86767013877476992110856317089632993621125568336276488192000000*n^3-102493414036\ 4121303801622302485754128714076057386935186884969852235866439680000000*n^2+4167\ 19617105768174678824504865221008498079847846754793129219509771567104000000000*n -722096902392614779850694315032920584874818662483821097679968740048896000000000\ 00)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9 )/(2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-\ 13)/(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/( 2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83) /(2*n-85), 1/4398046511104*(-144363364565726890961710523*n^45+ 146843916421581120978118299600*n^44-72453548206790868093004154176215*n^43+ 23105250636611678582476893562276200*n^42-5353667038793976860452445832464620653* n^41+960694208140896483107439076509239963280*n^40-\ 138963120332006484870962048770971839062505*n^39+ 16652150895242801264883129714332192411896200*n^38-\ 1686068141638092477762096343758654959497919598*n^37+ 146405048119490618443649362465077526080713493360*n^36-\ 11027808446482071478844478148561653406925832174870*n^35+ 727104820223248099889974912099802014629706879866000*n^34-\ 42268396571064695327670712685901681421313416610751098*n^33+ 2179096238602517279717755904969961646653160884718470240*n^32-\ 100099215481611841708962231440326426718959257773586048210*n^31+ 4112835453694446918645012692234838606244046041712320539600*n^30-\ 151617040759471967730014012374103454516613464761268585496263*n^29+ 5027080216485263236884324749108196426649131488245508928324720*n^28-\ 150200530759937175387125399330267734788590068997372845946619475*n^27+ 4049768400735877481035454923726733735139122586674898426212013000*n^26-\ 98631429118073843468484208676024290180422638856713810224280127177*n^25+ 2171062265549577825466453547326825068918134993524677243438301626000*n^24-\ 43199391111830661994166045962400168867303633285812662430987243268085*n^23+ 776862413263917245717926118963913262853764668534603538502390714245800*n^22-\ 12618909424648238755217676993590866875294730834406598765943291037810272*n^21+ 184967619757624558060120956939121821112598011596749244863908786408666320*n^20-\ 2443314037644907720800554113354536956316989022666787818537980786745595920*n^19+ 29033830648186094208321142407502268888332070980223708197719397770841396800*n^18 -309674820661419951815249791295612563512717833064970512960914240063506628352*n^ 17+2956662780228955818048489954849161382845872210030753275807821497631095626240 *n^16-\ 25186361185799926303074819468181323058165904385828235933077657689667178919680*n ^15+ 190673292413218150576794304000989022577883766486907414081166716729725029888000* n^14-12768524168807303888773700507352433694527345540469269225135891638333326186\ 98752*n^13+75213134025054206234221449174118278698024668760546763461916014788794\ 62272143360*n^12-38712509574752389533115740275551277799449740603480596451084727\ 357949957885603840*n^11+1727148229363216900458697599101622162218849331127660114\ 42632839680930309181030400*n^10-66145585813874303628816009528302937381585945304\ 8314076200960999626962446620557312*n^9+2148671553640085618578088916588004822911\ 472237211261326226075027574674812376186880*n^8-58323785786063422491926432705585\ 84948782116469974079326826806393440747625460531200*n^7+129788594242441415355083\ 83114788166483897986381353381549522470994952815179726848000*n^6-230908577366806\ 65353660392156353428389863119464851631830963239744074989887815680000*n^5+317356\ 52081236979248548786703855279766441117877863501316419624744488363307827200000*n ^4-3206420990632220731137861619846962320890991738182933101263151204296978910412\ 8000000*n^3+2203428632481151298859603363393284044456141076511759196584535763744\ 3644293120000000*n^2-8958934493293068869597814135617852188225922964819451399675\ 779007157108736000000000*n+1552508340144121776678992777320779257480860124340215\ 360011932791105126400000000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n -49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2 *n-89)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/ (2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69 )/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), (135475095028764187545717960*(n-45)!*(n-\ 44)!*(2*n-95)!*(n+1)+201027560365262987971065360*(n-46)!*(93/92*(n-44)!*(n+1)*( 2*n-94)!+(n-45)!*((2*n-93)!*(n+1)-1/201027560365262987971065360*binomial(2*n,n) *(n-48)!*(n-44)!)))/(n-48)!/(n-46)!/(n-45)!/(n-44)!/binomial(2*n,n), ( 135475095028764187545717960*(n-45)!*(2*n-95)!*(n+1)+203212642543146281318576940 *(n-46)!*((2*n-94)!*(n+1)-1/203212642543146281318576940*binomial(2*n,n)*(n-48)! *(n-45)!))/(n-46)!/(n-48)!/(n-45)!/binomial(2*n,n), (( 135475095028764187545717960*n+135475095028764187545717960)*(2*n-95)!-(n-48)!*(n -46)!*binomial(2*n,n))/(n-48)!/(n-46)!/binomial(2*n,n)] The limits, as n goes to infinity are 140737488355279 140737488354151 562949953363635 281474976470505 [---------------, ---------------, ---------------, ---------------, 140737488355328 140737488355328 562949953421312 281474976710656 2251799801455665 2251799750234495 9007198137162795 2251798740105375 ----------------, ----------------, ----------------, ----------------, 2251799813685248 2251799813685248 9007199254740992 2251799813685248 36028738241502375 36028615132015875 144113127252324705 72054896254521645 -----------------, -----------------, ------------------, -----------------, 36028797018963968 36028797018963968 144115188075855872 72057594037927936 576408104269814715 576340289187327465 2304803027856140985 ------------------, ------------------, -------------------, 576460752303423488 576460752303423488 2305843009213693952 143982217115090745 4603400600451997305 4596273854546365785 ------------------, -------------------, -------------------, 144115188075855872 4611686018427387904 4611686018427387904 9168420704724708195 2282304860739666705 36271739214924355215 -------------------, -------------------, --------------------, 9223372036854775808 2305843009213693952 36893488147419103232 35902220010911766585 141456550654041274125 34605970104772368375 --------------------, ---------------------, --------------------, 36893488147419103232 147573952589676412928 36893488147419103232 537132476547005752125 515189754359640034569 1947810261563850707523 ---------------------, ---------------------, ----------------------, 590295810358705651712 590295810358705651712 2361183241434822606848 903243952589776217131 6537693391573899964673 5721728791504619741923 ----------------------, ----------------------, ----------------------, 1180591620717411303424 9444732965739290427392 9444732965739290427392 19117158713698404338587 1859222825653682231819 81742848334163799254331 -----------------------, ----------------------, ------------------------, 37778931862957161709568 4722366482869645213696 302231454903657293676544 41906043513124372963331 1539848070664001360699 ------------------------, -------------------------, 302231454903657293676544 1208925819614629174706176 -83537643344017761636383 -1334732393655573547594759 ------------------------, --------------------------, 604462909807314587353088 4835703278458516698824704 -1974838913550345547554649 -10285717680124874295332221 --------------------------, ---------------------------, 4835703278458516698824704 19342813113834066795298816 -3109612464339138996140149 -57245307406192476216757129 --------------------------, ---------------------------, 4835703278458516698824704 77371252455336267181195264 -63498381814983819582021469 -273869370916450619739104671 ---------------------------, ----------------------------, 77371252455336267181195264 309485009821345068724781056 -144363364565726890961710523 -1195877207361126877959526309 ----------------------------, -----------------------------, 154742504910672534362390528 1237940039285380274899124224 -1221005652406784751455909479 -4934825770262925576153282151 -----------------------------, -----------------------------] 1237940039285380274899124224 4951760157141521099596496896 and in Maple notation [140737488355279/140737488355328, 140737488354151/140737488355328, 562949953363635/562949953421312, 281474976470505/281474976710656, 2251799801455665/2251799813685248, 2251799750234495/2251799813685248, 9007198137162795/9007199254740992, 2251798740105375/2251799813685248, 36028738241502375/36028797018963968, 36028615132015875/36028797018963968, 144113127252324705/144115188075855872, 72054896254521645/72057594037927936, 576408104269814715/576460752303423488, 576340289187327465/576460752303423488, 2304803027856140985/2305843009213693952, 143982217115090745/144115188075855872, 4603400600451997305/4611686018427387904, 4596273854546365785/ 4611686018427387904, 9168420704724708195/9223372036854775808, 2282304860739666705/2305843009213693952, 36271739214924355215/ 36893488147419103232, 35902220010911766585/36893488147419103232, 141456550654041274125/147573952589676412928, 34605970104772368375/ 36893488147419103232, 537132476547005752125/590295810358705651712, 515189754359640034569/590295810358705651712, 1947810261563850707523/ 2361183241434822606848, 903243952589776217131/1180591620717411303424, 6537693391573899964673/9444732965739290427392, 5721728791504619741923/ 9444732965739290427392, 19117158713698404338587/37778931862957161709568, 1859222825653682231819/4722366482869645213696, 81742848334163799254331/ 302231454903657293676544, 41906043513124372963331/302231454903657293676544, 1539848070664001360699/1208925819614629174706176, -83537643344017761636383/ 604462909807314587353088, -1334732393655573547594759/4835703278458516698824704, -1974838913550345547554649/4835703278458516698824704, -\ 10285717680124874295332221/19342813113834066795298816, -\ 3109612464339138996140149/4835703278458516698824704, -\ 57245307406192476216757129/77371252455336267181195264, -\ 63498381814983819582021469/77371252455336267181195264, -\ 273869370916450619739104671/309485009821345068724781056, -\ 144363364565726890961710523/154742504910672534362390528, -\ 1195877207361126877959526309/1237940039285380274899124224, -\ 1221005652406784751455909479/1237940039285380274899124224, -\ 4934825770262925576153282151/4951760157141521099596496896] and in floating point [1.000000000, 1.000000000, .9999999999, .9999999991, .9999999946, .9999999718, .9999998759, .9999995232, .9999983686, .9999949516, .9999857002, .9999625607, .\ 9999086702, .9997910298, .9995489800, .9990773286, .9982033864, .9966580197, .9\ 940421646, .9897919553, .9831474614, .9731316233, .9585468721, .9379966992, .90\ 99378094, .8727653921, .8249297333, .7650773873, .6922052127, .6058116002, .506\ 0269778, .3937057474, .2704643974, .1386554670, .1273732470e-2, -.1382014380, -\ .2760161897, -.4083871156, -.5317591407, -.6430527858, -.7398782570, -.82069734\ 95, -.8849196640, -.9329263776, -.9660219150, -.9863205112, -.9965801278] The cut off is at j=, 36 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 49], vs. those in the, 2, -th row from j=1 to j=, 48, are as follws 24 23 22 [7 (20105355479329 n - 5790342378044652 n + 787506668768666326 n 21 20 - 67265924876899707288 n + 4048303983257913408799 n 19 18 - 182555599916676217551012 n + 6403761392507376960153256 n 17 16 - 179068105174931659606036128 n + 4057975698686410325707117279 n 15 14 - 75360765449214294469714227252 n + 1155218614628718505452251602126 n 13 - 14678054349758184076006664309688 n 12 + 154819995624994405289791933171729 n 11 - 1354470200030537049466925382249852 n 10 + 9797047748284103722591530499326676 n 9 - 58250667283633144697758812041171088 n 8 + 282233134450236363441244277233786864 n 7 - 1100614577394950713707035285360423232 n 6 + 3394156927564366990667090918341511616 n 5 - 8058087780384361149022502450720855808 n 4 + 13991599158067669783675911194814336000 n 3 - 15092726890925102494361492918075904000 n 2 - 543453733545788823203432537026560000 n + 41445537653345835524880818785812480000 n - 70027909577172121989399002873856000000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) 25 (2 n - 35) (2 n - 29) (2 n - 45)), 1645 (171109408334 n 24 23 22 - 53471690103955 n + 7909532399999960 n - 736679474498163970 n 21 20 + 48479311839836586170 n - 2397779125790417260045 n 19 18 + 92564881400710242848060 n - 2859236228851194398128120 n 17 16 + 71873566737871528952861330 n - 1487499219776101773277945645 n 15 14 + 25545163712553811685456101760 n - 365794737164158420850547070570 n 13 + 4378137983250629233766965784150 n 12 - 43808710134849813069868917836995 n 11 + 365791863371626820410355526273260 n 10 - 2538333149659575653904258127898620 n 9 + 14543775829054400182508231597441360 n 8 - 68147762171825319552507024691879120 n 7 + 257452951382543293692322726670276160 n 6 - 766149921570982695656584680617958720 n 5 + 1711837392148903395148605278348518656 n 4 - 2481070149637254645743519483233674240 n 3 + 615482544066738938345865510109900800 n 2 + 7042839149821344572620041427537920000 n - 13774353280934893200042922008576000000 n + 297991104583711157401697884569600000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 235 (1197765858278 n 24 23 22 - 374301830676685 n + 55366726779299720 n - 5156756316149100790 n 21 20 + 339355181897194876490 n - 16784453743559402908315 n 19 18 + 647954154741816250946420 n - 20014652263289351761834840 n 17 16 + 503114869365874975372439210 n - 10412488595313791960113952515 n 15 14 + 178815843201995516748157092320 n - 2560550164529585922456471586990 n 13 + 30646494943156065806419330220150 n 12 - 306646564435302202696939682216965 n 11 + 2560171989984614222451465801862820 n 10 - 17760328365469548611068965198978340 n 9 + 101663077867527548940065835344804720 n 8 - 474929361287489566306311907137841840 n 7 + 1777300517351550200133237139385853120 n 6 - 5133148208352567633337691463065039040 n 5 + 10391924096872698240524483028867601152 n 4 - 9740101050840284001271500149743303680 n 3 - 17084840966218505196117640262577254400 n 2 + 66450437523600557601651356153917440000 n - 48401334970769083036141137803673600000 n + 2085937732085978101811885191987200000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 705 (1597021143719 n 25 24 23 - 539793146195797 n + 86548564390028000 n - 8758143740493044950 n 22 21 + 627803279892087150605 n - 33917346344237300699755 n 20 19 + 1434610135179818420736770 n - 48716629575905435177004400 n 18 17 + 1351316926025314540894609865 n - 30989158074441606319744327435 n 16 15 + 592442383370023131429719697020 n - 9493661119448599973767334669950 n 14 + 127915442777375565086032080451835 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(2 n - 35) (2 n - 53) (2 n - 29) (2 n - 73) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69) (2 n - 77) (2 n - 81) (2 n - 91) / (2 n - 83) (2 n - 85)), | \ 525311592968677461911967600 (n - 46)! (n - 45)! (2 n - 97)! (n + 1) + /95 779672995879826548732499280 (n - 47)! |-- (n - 45)! (n + 1) (2 n - 96)! + \94 (n - 46)! ((2 n - 95)! (n + 1) \\ - 1/779672995879826548732499280 binomial(2 n, n) (n - 49)! (n - 45)!)||/( // (n - 49)! (n - 47)! (n - 46)! (n - 45)! binomial(2 n, n)), ( 525311592968677461911967600 (n - 46)! (2 n - 97)! (n + 1) + 787967389453016192867951400 ((2 n - 96)! (n + 1) - 1/787967389453016192867951400 binomial(2 n, n) (n - 49)! (n - 46)!) (n - 47)!)/((n - 47)! (n - 49)! (n - 46)! binomial(2 n, n)), ( (525311592968677461911967600 n + 525311592968677461911967600) (2 n - 97)! - (n - 49)! (n - 47)! binomial(2 n, n))/((n - 49)! (n - 47)! binomial(2 n, n))] and in Maple notation [7/8388608*(20105355479329*n^24-5790342378044652*n^23+787506668768666326*n^22-\ 67265924876899707288*n^21+4048303983257913408799*n^20-182555599916676217551012* n^19+6403761392507376960153256*n^18-179068105174931659606036128*n^17+ 4057975698686410325707117279*n^16-75360765449214294469714227252*n^15+ 1155218614628718505452251602126*n^14-14678054349758184076006664309688*n^13+ 154819995624994405289791933171729*n^12-1354470200030537049466925382249852*n^11+ 9797047748284103722591530499326676*n^10-58250667283633144697758812041171088*n^9 +282233134450236363441244277233786864*n^8-1100614577394950713707035285360423232 *n^7+3394156927564366990667090918341511616*n^6-\ 8058087780384361149022502450720855808*n^5+ 13991599158067669783675911194814336000*n^4-\ 15092726890925102494361492918075904000*n^3-543453733545788823203432537026560000 *n^2+41445537653345835524880818785812480000*n-\ 70027909577172121989399002873856000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-\ 13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2* n-29)/(2*n-45), 1645/8388608*(171109408334*n^25-53471690103955*n^24+ 7909532399999960*n^23-736679474498163970*n^22+48479311839836586170*n^21-\ 2397779125790417260045*n^20+92564881400710242848060*n^19-\ 2859236228851194398128120*n^18+71873566737871528952861330*n^17-\ 1487499219776101773277945645*n^16+25545163712553811685456101760*n^15-\ 365794737164158420850547070570*n^14+4378137983250629233766965784150*n^13-\ 43808710134849813069868917836995*n^12+365791863371626820410355526273260*n^11-\ 2538333149659575653904258127898620*n^10+14543775829054400182508231597441360*n^9 -68147762171825319552507024691879120*n^8+257452951382543293692322726670276160*n ^7-766149921570982695656584680617958720*n^6+ 1711837392148903395148605278348518656*n^5-2481070149637254645743519483233674240 *n^4+615482544066738938345865510109900800*n^3+ 7042839149821344572620041427537920000*n^2-\ 13774353280934893200042922008576000000*n+297991104583711157401697884569600000)/ (2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2 *n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 235/8388608*( 1197765858278*n^25-374301830676685*n^24+55366726779299720*n^23-\ 5156756316149100790*n^22+339355181897194876490*n^21-16784453743559402908315*n^ 20+647954154741816250946420*n^19-20014652263289351761834840*n^18+ 503114869365874975372439210*n^17-10412488595313791960113952515*n^16+ 178815843201995516748157092320*n^15-2560550164529585922456471586990*n^14+ 30646494943156065806419330220150*n^13-306646564435302202696939682216965*n^12+ 2560171989984614222451465801862820*n^11-17760328365469548611068965198978340*n^ 10+101663077867527548940065835344804720*n^9-\ 474929361287489566306311907137841840*n^8+1777300517351550200133237139385853120* n^7-5133148208352567633337691463065039040*n^6+ 10391924096872698240524483028867601152*n^5-\ 9740101050840284001271500149743303680*n^4-\ 17084840966218505196117640262577254400*n^3+ 66450437523600557601651356153917440000*n^2-\ 48401334970769083036141137803673600000*n+2085937732085978101811885191987200000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-35)/(2*n-29)/(2*n-45), 705/16777216*( 1597021143719*n^26-539793146195797*n^25+86548564390028000*n^24-\ 8758143740493044950*n^23+627803279892087150605*n^22-33917346344237300699755*n^ 21+1434610135179818420736770*n^20-48716629575905435177004400*n^19+ 1351316926025314540894609865*n^18-30989158074441606319744327435*n^17+ 592442383370023131429719697020*n^16-9493661119448599973767334669950*n^15+ 127915442777375565086032080451835*n^14-1450681328362605629896453481325445*n^13+ 13835359318862004310454910285233570*n^12-110635326544441184369980428358097500*n ^11+737758033870493420202897114846215720*n^10-\ 4065364468134919950887235968097278800*n^9+ 18224352572175535916446827909438474720*n^8-\ 64505314880992503650068471073494612800*n^7+ 168606604371255084024911523467546288256*n^6-\ 266190939886099382696763421803000672768*n^5-\ 6812408603630432185967168339370670080*n^4+ 1088113477870921529384625413600875929600*n^3-\ 1988582971383408477659843576659599360000*n^2+ 1126885446083705371214928625129881600000*n-\ 70921882890923255461604096527564800000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 15/33554432*(150119987128361*n^26-\ 50740555427131843*n^25+8135564928364220750*n^24-823265480526272834050*n^23+ 59013502778695187555495*n^22-3188229811093479662354845*n^21+ 134853273752903560879618880*n^20-4579356438507715670133928600*n^19+ 127023319228741682029143366935*n^18-2912953483977627345428296888765*n^17+ 55688257435447513370877509288630*n^16-892350218945764133594861485259050*n^15+ 12022209911538650503204449262117865*n^14-136311252120419344190670146675931955*n ^13+1299258214914875085146382686445055580*n^12-\ 10374516070783550939892811914659417500*n^11+ 68936471351721888543863334600463929680*n^10-\ 376667968685718334893597804451277501200*n^9+ 1655508957797006632330189716635070223680*n^8-\ 5599044405592034845323971644141035643200*n^7+ 13118193357628256941836364096375480241664*n^6-\ 14314202656202734794971845223450181691392*n^5-\ 23770569915242592839802530411629756907520*n^4+ 114574318180199812721949046035557588582400*n^3-\ 161007774115045650731223429747218595840000*n^2+ 82457878236433190867470424941913702400000*n-\ 6666656991746786013390785073591091200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 645/33554432*(6982324892824*n^27-\ 2545057371252933*n^26+440939023825554423*n^25-48318967054653439950*n^24+ 3759535325552213948130*n^23-221027152206442563560475*n^22+ 10201904751159342342729465*n^21-379206177444305189752367880*n^20+ 11552268391048880009028296640*n^19-292043045970223091210026097595*n^18+ 6180203866536056427908571396585*n^17-110130022432725243447682636571430*n^16+ 1658573669701732446063715349781210*n^15-21144379930662535288407793473483285*n^ 14+228106401999238055584395611392353975*n^13-\ 2077108592833498254981391246402998180*n^12+ 15878380614349444900029245665255420620*n^11-\ 100904378597340024981181776878076004080*n^10+ 523814715443803944972670702365129642320*n^9-\ 2150684948872533990332650209262593958080*n^8+ 6538837421220927729493166336103189350976*n^7-\ 12378157576539883354362319670527708741632*n^6+ 3509064183223129413513276045035910623232*n^5+ 53887812750281035125877011550270393835520*n^4-\ 151291581412752270810313075768066969190400*n^3+ 178418135224391083933809009429049098240000*n^2-\ 85342426639540956573266037603336192000000*n+ 8217042338664643225807246718612275200000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 3225/33554432*(1396464901259*n^27-\ 509011411864806*n^26+88187780795519925*n^25-9663787578405207780*n^24+ 751906056545417928375*n^23-44205298618805692481178*n^22+ 2040367427594246229683061*n^21-75840119628280114398919056*n^20+ 2310378368314516101351400365*n^19-58404405243373980891895355418*n^18+ 1235845261933423827737351335191*n^17-22018398285523381927465880310516*n^16+ 331466859378320513057044387791045*n^15-4222117658226981068396859639280518*n^14+ 45467701725690180570209280847094511*n^13-412532381465070025407940567068739416*n ^12+3131125820637767432607677202087203340*n^11-\ 19625966850158105311501227806490967968*n^10+ 99291281576891717804767252532044739536*n^9-\ 388628635788561297377657401427043962496*n^8+ 1075329558049507928557929959380743871296*n^7-\ 1573559485123752808266662580957396314112*n^6-\ 1477229367813681961972644915422757556224*n^5+ 13324912443131660001954434092572876939264*n^4-\ 29956509460372746328125419620494420295680*n^3+ 32192138962660244605763453551107293184000*n^2-\ 14953295653792068187683505372148269056000*n+ 1643408467732928645161449343722455040000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 132225/67108864*(136240449889*n^28 -53406240122270*n^27+9969317421083181*n^26-1179407729646987750*n^25+ 99283543753450259385*n^24-6329970919892672536302*n^23+317654630053538194912209* n^22-12872759596145171260311774*n^21+428846665716985590636737175*n^20-\ 11894891072826552242207758002*n^19+277185796526773744881322206519*n^18-\ 5460678428936855804672810786514*n^17+91307382083264375969723064086835*n^16-\ 1298289259848496162412547176109882*n^15+15694955965913481278246137392253179*n^ 14-160895567282925568037304385876200714*n^13+ 1390685620185481155980640635501522460*n^12-\ 10030800047895834817242916953570772872*n^11+ 59325082730455607592257711757150003344*n^10-\ 279111112034950519474863931275058185184*n^9+ 986604212855456039365805764511921687616*n^8-\ 2283330839233578149629248230987736089472*n^7+ 1630282696626016431606773192464940525568*n^6+ 10181299852744793629930284466751763111936*n^5-\ 43221720639852852226264850716027302543360*n^4+ 81682360087684034250160611767179377868800*n^3-\ 80667293103163546802568103775120031744000*n^2+ 36409681225106097090319603101762846720000*n-\ 4409144669527369535799010434377318400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 3075/134217728*( 11716670573345*n^28-4592930237999110*n^27+857358888657464025*n^26-\ 101428491920632064370*n^25+8538288090610219410105*n^24-544365181306236817585422 *n^23+27317067774233388577226229*n^22-1106958577510598279045893314*n^21+ 36874339234599815074144553535*n^20-1022609992030517957033481235722*n^19+ 23822174944372039162626598525899*n^18-469023135294011648842037634433254*n^17+ 7833672875788175844721555930565475*n^16-111158332798963882015120712629342602*n^ 15+1338900998024954902680589854064255479*n^14-\ 13639192310327174659742434602053994054*n^13+ 116639365753724370522027810770769086340*n^12-\ 826695094195624868522154132724531685592*n^11+ 4752883337844486689135954360348695920304*n^10-\ 21351512237237981657760111251859262469024*n^9+ 69521312576470806499461083177602983284160*n^8-\ 131577961608918800903995019860890644468352*n^7-\ 48590046109568207510270816981402396007936*n^6+ 1215142142795268164947355984503520011094016*n^5-\ 3984140376386458310148506161204062943272960*n^4+ 6855614764425199671853063321495443131596800*n^3-\ 6465841753505898213234760076655180447744000*n^2+ 2893273747021060497983394375043220766720000*n-\ 379186441579353780078714897356449382400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 267525/134217728*( 269348186050*n^29-113260593991815*n^28+22718390762094580*n^27-\ 2893366478147616135*n^26+262728654396614242200*n^25-18107292002231069404767*n^ 24+984540336493907131484676*n^23-43337269495351951639911963*n^22+ 1572437912767925157875484564*n^21-47639873565688045848934424697*n^20+ 1216356225090357871201659118596*n^19-26340837103615818563016089389173*n^18+ 485785814629996141903518084989544*n^17-7644377997739682224143338233285317*n^16+ 102613908279138087383017693635614556*n^15-1171767228993097597849907930324558553 *n^14+11316189411409619259356858906443941834*n^13-\ 91494363245876787545297565857812343772*n^12+ 609239277536621216735505690106727894696*n^11-\ 3251520136870479336502835173206704752848*n^10+ 13237439061017563807670380992612196322144*n^9-\ 36643193200267189781125187395435229692992*n^8+ 40557356937985469249512801041293384781696*n^7+ 174934663925886557138899606685706929244672*n^6-\ 1071021452333603175056354530320535278606336*n^5+ 2885791319531590807538837075718566663823360*n^4-\ 4525506292177690797313559613615154023628800*n^3+ 4057408313162172253866622649878002991104000*n^2-\ 1788825580589742196460974988123119288320000*n+ 248432496207162821430882174130087526400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 6525/134217728 *(11043211979608*n^29-4643634962473423*n^28+931435801815706838*n^27-\ 118623776035243203963*n^26+10771171601390982422550*n^25-\ 742311156473423969839407*n^24+40357561211170784600191926*n^23-\ 1776152797880084459244752343*n^22+64426617296101885231351198254*n^21-\ 1950937299247903090529524776657*n^20+49769228482738362404058933310266*n^19-\ 1076234278500948593113094445219873*n^18+19801614362030989687305021415223874*n^ 17-310426035833178165389279546719894917*n^16+ 4142377220066573319389813722493156466*n^15-\ 46875127358120377290646795482250168893*n^14+ 446570106139553631940389095783102928834*n^13-\ 3538789431253001880707107079963846299692*n^12+ 22875099789904820460815370162206130533576*n^11-\ 116679146073251614823131651501614574379408*n^10+ 439703819070999122120996984196996501347296*n^9-\ 1015003507135830558393284762960077435001664*n^8-\ 34343612209494377478371485146575510278272*n^7+ 11271323694851151047902175183048393573460480*n^6-\ 49974284318318368352470410037434288196540416*n^5+ 121695131649068408192384190919933143506165760*n^4-\ 181284121811883876642394531148579418950860800*n^3+ 158372836574084449901304772392618919133184000*n^2-\ 69738732600685068590337792232467301662720000*n+ 10185732344493675678666169139333588582400000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 48285/ 268435456*(5969216119317*n^30-2686098606031185*n^29+577500260838904785*n^28-\ 78967506855784934065*n^27+7712802118042111279413*n^26-572876142568127917494885* n^25+33638829833287060224502725*n^24-1602585661111881364213006725*n^23+ 63079584023711626058284180155*n^22-2078202898524107329024223047875*n^21+ 57843739184217724216825886214675*n^20-1368958587637081027161300226485075*n^19+ 27659589479835289417209431353802535*n^18-477998046760795096391897531424679575*n ^17+7062827897242040052072782739614685975*n^16-\ 88981375599794520212562354019240514775*n^15+ 950435533986190250791844159493316322580*n^14-\ 8525642773630685554832313818502975812400*n^13+ 63263594428086720014323926545916708960000*n^12-\ 378854409274213935482833269856563061212000*n^11+ 1750190285692759182747298173782590645563648*n^10-\ 5611961479065817799572305624479793829533440*n^9+ 7788732949795308956242552602971682298871040*n^8+ 33320819568877827882990912858271601918096640*n^7-\ 265898351803970998181262647901941287107867648*n^6+ 942459530553007567379942414287379719315599360*n^5-\ 2074530734561434873889346231596553666489139200*n^4+ 2912542639110858423810093027700391899693056000*n^3-\ 2459221160536264170898648186820263585382400000*n^2+ 1072897582323605398999053809053998514176000000*n-\ 162421137385169422984136210600184250368000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 16095/536870912*(35814049339967*n^30-16115637393596835*n^29+3464654626127747485 *n^28-473725295362099908015*n^27+46263808765209455881713*n^26-\ 3435656621977380669085035*n^25+201678649962741325388811225*n^24-\ 9603555900025339386273458475*n^23+377720336947615225067464258155*n^22-\ 12429577859982058128981570067125*n^21+345332907256477649616270290120175*n^20-\ 8150539324282596867970292126867325*n^19+164016854031016172275119287134657035*n^ 18-2817900124414188963173920827576696825*n^17+ 41291255165633000841325360068550247475*n^16-\ 514176180418421391256444459923177548025*n^15+ 5404146705217535895752667797736471558330*n^14-\ 47408046337775406775835155258464072408900*n^13+ 340929547754638429140214517124864041017000*n^12-\ 1948599974544605903378861874704690827542000*n^11+ 8313659786547673696948580587141005680481248*n^10-\ 22042354160648160745315308863882653744367040*n^9-\ 850378010068863496020558057799344379148160*n^8+ 344365452756927001530339435032369990301987840*n^7-\ 1900821565858997934192233063318131795414376448*n^6+ 6045273437592071272371826794835161215871221760*n^5-\ 12598831227726985885216170879645877191295795200*n^4+ 17140181894090430186245078938360961338638336000*n^3-\ 14248439478614619627325167244647087754444800000*n^2+ 6225544938131290827467703640869576376320000000*n-\ 974526824311016537904817263601105502208000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 498945/536870912*(2310404956924*n^31-1110052544034407*n^30+255186264810759755*n ^29-37368485826605179765*n^28+3914940454006948420551*n^27-\ 312440519784575978116713*n^26+19747356732785417473848435*n^25-\ 1014477587869300309891873425*n^24+43138934842513159824217288635*n^23-\ 1538285818078529270600973935955*n^22+46427206212048322977684561463725*n^21-\ 1193574651322972465346581935091575*n^20+26241890447040611686841144877943845*n^ 19-494299103859801616463999085594778035*n^18+ 7974154951184411366901205576313987025*n^17-\ 109883934463687110495502843333595605275*n^16+ 1286527667140843038881258195742827178285*n^15-\ 12684612147483122364857919361965792321930*n^14+ 103831058802741800167077531000394132040900*n^13-\ 689072054846749761555614376536472858649000*n^12+ 3545124429380905158711268658056966757747056*n^11-\ 12663096628172912928209173075289816286538208*n^10+ 18151509045392648954262446145866388587651520*n^9+ 119659089977582134698134987223763195393875840*n^8-\ 1071891949031253598110533525318653984299319296*n^7+ 4668124797188937212898695646815348760246325248*n^6-\ 13346104353999223047632846092968624610052751360*n^5+ 26088222420449340338725491526711343067849523200*n^4-\ 34045483111968212777559647152817364836745216000*n^3+ 27608968114719728415277686529407203731046400000*n^2-\ 11978985451346022675548173387109629427712000000*n+ 1917617299450709961683672679989272117248000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 13485/536870912*(85471120867963*n^31-41061450187836734*n^30+ 9438115711898312060*n^29-1381774710796465331680*n^28+144714023954656839483612*n ^27-11543376332705463787899456*n^26+729033587158391906491014720*n^25-\ 37411158307692213273665958600*n^24+1588302839008557561652700177370*n^23-\ 56507937630301252978852684460460*n^22+1700010640746587971275277252012200*n^21-\ 43511208118578835737955700408051400*n^20+950870458463533146174476911507666140*n ^19-17766100570982074334570081709188491920*n^18+ 283539967784039000389688334624240556800*n^17-\ 3852261475929902111036404741701220075800*n^16+ 44270075196767074348551726375889422093795*n^15-\ 425769907725254640001590476879627704175910*n^14+ 3367282467654930656097545883173941398158300*n^13-\ 21222849347232026197145727237306255859243000*n^12+ 99664209402164635032590159408329865223286672*n^11-\ 280755083421584591401493228450316908501412896*n^10-\ 217553882126156582608090338635281837180781760*n^9+ 7691086692303634147453504352923705543555102080*n^8-\ 48601912287535974486747905665365084055058583552*n^7+ 189820858024257994885063881982368257873525477376*n^6-\ 513259326393419494330690840454088568865945272320*n^5+ 970118156045904334485969319662637726616147558400*n^4-\ 1240390949542932194839078770058383317326036992000*n^3+ 996121856780738436756809357316495476608204800000*n^2-\ 433197868909895673889463529884493845889024000000*n+ 70951840079676268582295889159603068338176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 13485/1073741824*(341774507335267*n^32-174930480355539824*n^31+ 42895166346375472664*n^30-6709095148554407253280*n^29+751770159311291855486388* n^28-64258905241869454869824736*n^27+4356013463363819516808324696*n^26-\ 240348133648257692223752994720*n^25+10991996434828907827566237774930*n^24-\ 422107716336397432598699960256960*n^23+13736717029089263357061135557058360*n^22 -381245785288343515368434094291784800*n^21+ 9059451417542163826363369678803432660*n^20-\ 184654349756806042643708900350276876320*n^19+ 3227490082660814838961515394623328105320*n^18-\ 48255336984715710164874541979171695596000*n^17+ 614022553986950763078441002205596869265955*n^16-\ 6591943833951074174750881473235082288836560*n^15+ 58859471196244946529945491117941371962126960*n^14-\ 426337976145610150293575298608333777583596800*n^13+ 2382123231965780579662398389914443511507740448*n^12-\ 8941737065368301764366691334000846135858515456*n^11+ 8084599639503401061113443168750684152493737216*n^10+ 175770164257859092082354905283988255582404014080*n^9-\ 1571020023873673106497222795294674064335627349248*n^8+ 7940995291065367342982480083870753070528273657856*n^7-\ 27888401047789377446488157422618463619500639625216*n^6+ 70595956287373313623513874010184894444932895211520*n^5-\ 127544003802124148533104541040643403796282115686400*n^4+ 158082520765850042563611887365414872536280072192000*n^3-\ 124544002761970017343789975084395781252330291200000*n^2+ 53832869173233134356488133937187563573870592000000*n-\ 8939931850039209841369282034109986610610176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 13485/1073741824*(341567123763421*n^32-174774527909511632*n ^31+42839390119960347272*n^30-6696471577810655251840*n^29+ 749740716376547305048044*n^28-64012288177486126775844288*n^27+ 4332477437490628935677164488*n^26-238538714264683451369615879040*n^25+ 10877945019335036186133517504590*n^24-416139305992174437382340502874080*n^23+ 13475081249067611447719726888250280*n^22-371577600914354403143138008363900800*n ^21+8757016956686665054666468577033722380*n^20-\ 176625532560110804535565973028902516160*n^19+ 3046408542129954079973409017246030317560*n^18-\ 44786330905315886731340393913280265577600*n^17+ 557656406439477061190538275842168187971965*n^16-\ 5817204668270722783949827416872189806441680*n^15+ 49887446319670154337909900408147874109121680*n^14-\ 339271139190154138964721077937721296337734400*n^13+ 1679225203480518316321816596730868600158131424*n^12-\ 4265950402223774932359491153735703657240708608*n^11-\ 17220316511553138681289507861237943533488383232*n^10+ 285258175639037525499116003554592877493514711040*n^9-\ 1940381430687615279893307250738287609255457223424*n^8+ 8875208481263322276384293221310789180240920424448*n^7-\ 29538049623716541339807339480332822230742484738048*n^6+ 72296627191245124837406947344210365001549118832640*n^5-\ 127760864688024115919165869265642117645607650918400*n^4+ 156186327506212038891983000136773080730887913472000*n^3-\ 122282115050576993730000171656961530724239278080000*n^2+ 52979926695703883213459534069647478116633804800000*n-\ 8939931850039209841369282034109986610610176000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 13485/1073741824*(682387953704282*n^33-371178511501742109*n ^32+96829989143555316704*n^31-16129065372568982081160*n^30+ 1926767488924154455173848*n^29-175760695041452408986808556*n^28+ 12727744287789488905408769616*n^27-750906599812143653613788281800*n^26+ 36752808145814342163144772732380*n^25-1511710613190273934021268737333710*n^24+ 52735520458241105577574380405414960*n^23-1570120500605347680145726414642597800* n^22+40056351275491664329971177295245255960*n^21-\ 877239355976450586859953090063831537420*n^20+ 16488444346270775147372238465848822047920*n^19-\ 265330952845420504193513019360656540868600*n^18+ 3636329898582584007231172587722736556021530*n^17-\ 42051485325875083625269491379793631520068285*n^16+ 403786494152192426047044888734972997427659760*n^15-\ 3123603110929660662911704697733028295572602000*n^14+ 18181812841415219919314756175079806184625238208*n^13-\ 62913278220852869445012271610066043505325980896*n^12-\ 100611935571133293995929709703095964086327911424*n^11+ 3449782335577152216613944752585331664902100296960*n^10-\ 29661813622663131383780715170873867233201654199808*n^9+ 167174931978340568665268862819703729207243995230976*n^8-\ 692329218993397983620788320643747551098282542657536*n^7+ 2159733315648945853365156138653066229375939871334400*n^6-\ 5047383816605376574706585133776224407058038285926400*n^5+ 8621711174024704302509203134347178177187070730240000*n^4-\ 10286180784198010557529718755901946413754350960640000*n^3+ 7929132942494270180455232380408521106894474444800000*n^2-\ 3415533913282573673432189046991066143768182784000000*n+ 581095570252548639689003332217149129689661440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 13485/1073741824*(681102669898762*n^33-\ 370214548647602109*n^32+96485849263492114624*n^31-16051237903614280901160*n^30+ 1914249806185932281410168*n^29-174236655014834416090796556*n^28+ 12581770124886487592278942896*n^27-739620950028675576800380813800*n^26+ 36035742835874044284375115055580*n^25-1473780216011289445465896544993710*n^24+ 51049374696471378953851080681719760*n^23-1506698970089825434929705219458737800* n^22+38028214066199305470220196736968298360*n^21-\ 821922073158174876031864281922299117420*n^20+ 15199245934541934554245332369088445837520*n^19-\ 239642153871695069824797674515269053088600*n^18+ 3198892144581440317557232495777761087386730*n^17-\ 35696090253213532948218605592669412954788285*n^16+ 325221164185226556863144141608431034040760560*n^15-\ 2300523295407373824044145861957341237791482000*n^14+ 10914101953552468878048579532806744299843227328*n^13-\ 9218301681102874338290599257197147641256860896*n^12-\ 429358698711862122736789064985807365870889554944*n^11+ 5096334417155752077093401349377672689257494856960*n^10-\ 36289499017632302361498545359985973213039770838528*n^9+ 188070701591631028906806490738608654789006874718976*n^8-\ 741877918691163200000154564345699073853071841980416*n^7+ 2241681662597158912339477503698785767717687755366400*n^6-\ 5124909002497175557251868115853215131582679246438400*n^5+ 8622918170874011583738021739704366655539770941440000*n^4-\ 10190588284847469695569011384161980692231827619840000*n^3+ 7823380752564386255490563149253166930528947404800000*n^2-\ 3377088748170493624707933257638030800520740864000000*n+ 581095570252548639689003332217149129689661440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 13485/1073741824*(1357956293006249*n^34-\ 782803957289513347*n^33+216584198665233767951*n^32-38290395352124788437928*n^31 +4858176126062033375655956*n^30-470986189029871003262243668*n^29+ 36269165025331733929997041644*n^28-2276707187222169524150892923832*n^27+ 118621267076825432936347545438510*n^26-5196260761285319462402923645060530*n^25+ 193136045818391501925593380771299090*n^24-6129276970803971299871334565187726520 *n^23+166736065890928368325581111661933244820*n^22-\ 3894877273445364269551788472128046913460*n^21+ 78096414912991699896078078487687184229180*n^20-\ 1340254845117629678882282248572735514220040*n^19+ 19563946515549185661034424121288596183356785*n^18-\ 240123230262020414644760582195410182120632355*n^17+ 2425172212726472677022610402217844488053512215*n^16-\ 19254404469233117260324077847825828409158663520*n^15+ 105679159138542617392960358676255218056727735056*n^14-\ 163719947600274299616731064554983983619081921568*n^13-\ 4345595037217664359817658717265359933958837084256*n^12+ 62878857208025493018405077831690712013464613619968*n^11-\ 528481318904813136812261347234377374865704142091776*n^10+ 3249097881556748680169063036515618899309592595152128*n^9-\ 15422652932966742033098230350003677801243887322605824*n^8+ 57244768163749860271105972810397534162206425115471872*n^7-\ 165304332555028597221624682659766833248658420878745600*n^6+ 364975696195884244169210162659894413407539954429132800*n^5-\ 597868506779185035677650791709861591787620897628160000*n^4+ 692694206845555544840459566591673364913420391546880000*n^3-\ 524850248589570302857450535401458553107110205849600000*n^2+ 225266493695757487133316409396259306501563219968000000*n-\ 38933403206920758859163223258548991689207316480000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 13485/2147483648*(2702397875895733*n ^34-1555458367281336179*n^33+429535967604123906262*n^32-75756187206709370509016 *n^31+9583016168008018817845852*n^30-925623362829117310949900276*n^29+ 70958124404227101086339263728*n^28-4429847525798197385480657561304*n^27+ 229280823192744106449282995762670*n^26-9964173553927162972975088837388210*n^25+ 366839666799972727744023873315455580*n^24-\ 11509795852555206684603482886745552440*n^23+ 308838191525093214653593296454386650940*n^22-\ 7095183174101343346566554481652436483220*n^21+ 139368625104244653851304919264557028592160*n^20-\ 2329972699680863203919536346235446939867880*n^19+ 32844326693058206184048289811463334038619845*n^18-\ 383417812644133923200362969711478084302877235*n^17+ 3569981006058198369843738774854359346650968830*n^16-\ 24023630519585849410274887912181294072151563440*n^15+ 71731731847679567393543995918860217703576269952*n^14+ 812870138234905166689238946268034536343602397024*n^13-\ 16522859465233448993048164434405181530388256212672*n^12+ 170548767242037903640486153021173916410757701360896*n^11-\ 1267495283630757505244495423788614894506719040961792*n^10+ 7296859345761489393815252748244888074525863184266496*n^9-\ 33226981191455601706238422276860709696099123764453888*n^8+ 119843128049549100201567921422849935838458587098333184*n^7-\ 338987473218967970549190082758898521835224636504883200*n^6+ 737290934838268181214659661282656254273440334887321600*n^5-\ 1195074978958208004193051705109180067810841762283520000*n^4+ 1375488914977512744078037177072993725667457712783360000*n^3-\ 1039383543116407585782717959167125238451851572019200000*n^2+ 446894742408698533857666975158261081019279998976000000*n-\ 77866806413841517718326446517097983378414632960000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 4247775/2147483648*(17026509970688*n ^35-10366569690509281*n^34+3030691209843367547*n^33-566380706498174715994*n^32+ 75988142957247919091672*n^31-7792281942991673421097804*n^30+ 634875219811071025164659028*n^29-42173949155473665637848603456*n^28+ 2325731376733337299612348772760*n^27-107845146388332833406134075297430*n^26+ 4243352796873253931538699978840930*n^25-142549443240387288960657410363251860*n^ 24+4103747858519412096603721538303627640*n^23-\ 101380460638897087932462946225644372780*n^22+ 2146794611916259260884448021044188035060*n^21-\ 38797174916617402290901884682489834851120*n^20+ 592896965855460268099189104967405093633320*n^19-\ 7523906396765383933496216724267859513891665*n^18+ 76277653064386786119046645796089795139021755*n^17-\ 556673739902286606524285853427721050471975010*n^16+ 1679056217851348968238151034491521441011817072*n^15+ 26377070849106926832114420447072828957058687936*n^14-\ 570136325551202209800685490855909125001351668832*n^13+ 6586855035202267120793938399279008363794703648064*n^12-\ 55740680313675462024416846207406911179569560493312*n^11+ 370689408715376725866309912921470632227631511549184*n^10-\ 1980242940701623708462831367071167212162074229599488*n^9+ 8531866418701295884387482328952526559593391266725376*n^8-\ 29475497580091727269470666033571085037181187158179840*n^7+ 80556922765398285533068935870949511736735809743011840*n^6-\ 170440777446134832628007192764039317138789547155456000*n^5+ 270292756717397565304679506132423335573508509057024000*n^4-\ 305965246885216348088984288734363452204657039114240000*n^3+ 228568133815387215157947646267907833932841793617920000*n^2-\ 97712642639245853789798722824967635005804327731200000*n+ 17056538547793856262109602570411939216224157696000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-69), 4247775/2147483648*( 16831627003487*n^35-10219433050272526*n^34+2977679145105351527*n^33-\ 554246864217712613794*n^32+74006465864793929092604*n^31-\ 7546382426603760940544344*n^30+610770007646352634922613948*n^29-\ 40257523507990947804046692456*n^28+2199847144184387042045714476914*n^27-\ 100919663638580206384624736496420*n^26+3921005349592238913894938452785330*n^25-\ 129758824978122047471400836050984860*n^24+3668625751705224116125482816888594300 *n^23-88637363416394513294036830120228304280*n^22+ 1824597454931947895734666136681378492460*n^21-\ 31751682798949182058509276327545740578120*n^20+ 459553606544246855420075906560069772858535*n^19-\ 5340070553281483876662905672106029036931390*n^18+ 45362675242231301459366972749856853450335455*n^17-\ 179182306435268844480436691518148274500509010*n^16-\ 2284287695519094837312042864998002790697742832*n^15+ 62000245904468594974438451699866925297086812256*n^14-\ 842649876590323967346323012018325015076550335712*n^13+ 8347633946807600531022818949767092535531906380864*n^12-\ 65253616347533842576992249625715466064139988831744*n^11+ 413091410791514717741981243836743242392411476762624*n^10-\ 2133310855873122848054779548792017146035883412375808*n^9+ 8967476904278089715204038363238154413957905465253376*n^8-\ 30411404381923723491107938988587854718143626919051264*n^7+ 81953784701701011710531502683742364841862532161454080*n^6-\ 171581409456471504273354537750520446811004561871667200*n^5+ 270073499976710722009012946013336788926696149663744000*n^4-\ 304295456824956777254060436706306886115592824422400000*n^3+ 226910719350349412199866942519242828886151156203520000*n^2-\ 97143036528187802931323819585139406980030608179200000*n+ 17056538547793856262109602570411939216224157696000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-69), 3034125/4294967296*( 92692963578427*n^36-59361173960325606*n^35+18256347500802426645*n^34-\ 3589422712138452043890*n^33+506670240047090490574896*n^32-\ 54664608735866625608025144*n^31+4685616511365950414860111748*n^30-\ 327418929164711297449966490280*n^29+18988972507187788663191040957242*n^28-\ 925677688139748972901567499175540*n^27+38265995384902526157134988581266182*n^26 -1349155707491909849038916291953515900*n^25+ 40692199846791373028767846331418465780*n^24-\ 1050075942981683637281247830040412594680*n^23+ 23105049861230426204622025343420125026420*n^22-\ 429686034945589491937124446917664309989000*n^21+ 6628006960026592021259830863923337169123395*n^20-\ 81261262365764851258854690458589980817944790*n^19+ 699429923139306916396654552877131166528373965*n^18-\ 1781089378393982576386131141280400794664585250*n^17-\ 72606071940202657973356254244285981983491814492*n^16+ 1755245472229508289674006748473330772150678800736*n^15-\ 25235914366080993470382035063547624276082353228000*n^14+ 274329276830913922153032008045257069351336133624640*n^13-\ 2398056989761953223481045083945611837842411325332096*n^12+ 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n^3-396765889810291259875009378675442326704673527671363335767286999941120000000 *n^2+ 162976871026705601811961922503987883871737929521022662537910419456000000000*n-\ 28432325278560603119524728670463982608792940534952935368844902400000000000)/(2* n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), -1763/ 1099511627776*(2323188669657521849071*n^43-2302800490744479357488702*n^42+ 1099297797376303952145666997*n^41-336970091514678075386384584580*n^40+ 74602398981569136570161585139817*n^39-12719850899420298252514002567987254*n^38+ 1739032966075934870260530413414367119*n^37-\ 195982350379716742096269414856640276560*n^36+ 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1556290262742724864002085222666040344556438041725535869924869919324435*n^23+ 27241104826685701820197140054574998084362951782805710102258543687490884*n^22-\ 431434797332178777058392712144947526032656306697121297919232183351541112*n^21+ 6175751502393986544191931339974075044105219174150681926545580011340676560*n^20-\ 79783277572438842987739149622657174970780344649842791415808929804014245520*n^19 +928471668188054246429861776615076226635669047283950429720815091472988318144*n^ 18-9710817250965877047769506184712369588932901529645612414320320589591659794432 *n^17+ 91024032939849488880303132552365700905605481111152820400308994808696262483200*n ^16-\ 762099694551688349045920827378866172900702634995281799690486199463070244455680* n^15+56766001962661091291392303318116682992844519949214742582430502330885290461\ 50144*n^14-37439206805274047555060390638034045324982454427033078104604717510560\ 881739241472*n^13+2174106255098550153667689535967543844039422976160967185790366\ 96429072849526435840*n^12-11041652049585271058513870724784382794015390694672784\ 17686877163742976187585392640*n^11+48650336211372313190385558057945575103575083\ 34463587402239791994775251910905380864*n^10-18415954222060749813342416459589251\ 211057701416458705389214751768596640055305961472*n^9+59176922854283761667749662\ 729044804086871604906746995967047026474146786051586785280*n^8-15902293453686410\ 8225153158957767442714412516197289151086027844536591236411241267200*n^7+3506056\ 37597811843123376815700565932749944230509346896431759552418903077127979008000*n ^6-6184789266262111230319102771398752934554270759541897801587844552234555750010\ 06080000*n^5+843473622685619734808438415635778582945559003432782869036553910013\ 534386991923200000*n^4-84631276045821556871524740886314854048246083497414027799\ 8696125467231380307968000000*n^3+5780511835414948139176250013393307478160435288\ 86894711992403317240510933893120000000*n^2-233831417994134676512719503993328362\ 346622463894383167712431460160620199936000000000*n+4036521684374716619365381221\ 0340260694502363232845599360310252568733286400000000000)/(2*n-5)/(-1+2*n)/(2*n-\ 3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-87)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-91)/(2*n-83)/(2*n-85), 1/ 17592186044416*(-1157415301678997479750776559*n^46+ 1229910825019036384626442189547*n^45-634393240128740755558240199267745*n^44+ 211640799242247704945611495316447235*n^43-\ 51339571610829460396879621506982131399*n^42+ 9652401125077493191455226865140565859057*n^41-\ 1464037110888643855371651905533055601539335*n^40+ 184117626463118608480407693516782292375614045*n^39-\ 19582191797583174655667282422588742244468730584*n^38+ 1787771450244422933178628713291454138862558357802*n^37-\ 141724578750529460750450559855244357011461004647250*n^36+ 9844795618631270869953414533442482144615688240902430*n^35-\ 603612708969286269548507273949977240195167234912045534*n^34+ 32859345521353174910365125030810253553797824675860847842*n^33-\ 1595843755566470032731632555499428377746910496852308403790*n^32+ 69414263726249142594873889583077214530564958090117779370490*n^31-\ 2712751571704739746160582796678891365421864207353144353404879*n^30+ 95494745977003838030180170733199638516134876715443514422568367*n^29-\ 3034087498148957971144180858751956330493497093127906816504262005*n^28+ 87140524248206895367804368598623914408012825491633796775539109775*n^27-\ 2264819409009857750496365644443609443662348246882078145089908453291*n^26+ 53305817942843498285827738646181541861235422223473813229974347793053*n^25-\ 1136548809054912831782860968414474523415431931491809474722803885602555*n^24+ 21951469612696753717054948456956908846350492433882469407211610442061465*n^23-\ 383917737842190296392808659897448356820595219223310797327386508840929626*n^22+ 6075690947541712734104377789107200771140545727435029368853223490646005068*n^21-\ 86908609094486508514856842336160062040006129920191346407523229298948096840*n^20 +1122022778341421049045868651639173542094096188813923841677871808802208957280*n ^19-\ 13049639947067035549606600195810246164747053521356553940323271049256870670016*n ^18+ 136410547209413257793512095973865251392281376572481385197404064740749303314048* n^17-12780026729378890509661545067757150883607655652467006877310632068522976942\ 84800*n^16+10695257576946720131804619917948329095627418469579020442342266231033\ 458158859520*n^15-7963259322106789744362133365003280234593174256545404542957388\ 4332949224077718016*n^14+525013770649908122095328957097056660684341183213783686\ 491549503387834750836292608*n^13-3047781788929205645710311063806942222236028112\ 580631708737059152991571390039541760*n^12+1547436781359422080949727598343446452\ 5779118988417426572989040351978622711428136960*n^11-681642026057203505650706411\ 51336911588020256097747013675007849269465971139905028096*n^10+25797137886657276\ 4410589560223135621783353221696102975301626872306613282172051652608*n^9-8288021\ 37479293888183344941311319753921768723367799562193148054272602501376489553920*n ^8+2226861191414349229645522116214824608048863554795744481614514876853972382114\ 958540800*n^7-49090816873410893952057119831982030713813413773503272660662447399\ 72222817756250112000*n^6+865898070390565610026484358585720456005990301772955461\ 0983905859874793126488965120000*n^5-1180826086796487078533805720702984863166710\ 9028048771422310555893354203357957324800000*n^4+1184758071017080157828566256156\ 4028931959027256730438822036684290237739986583552000000*n^3-8092100511873925378\ 363701207002724831691004152668371941757000995809586241863680000000*n^2+32734569\ 83221825003798760217735906917639661780704241821851900709905423663104000000000*n -565113035812460326711153370944763649723033085259838391044343535962266009600000\ 000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2 *n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/( 2*n-13)/(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37) /(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2* n-91)/(2*n-83)/(2*n-85), (525311592968677461911967600*(n-46)!*(n-45)!*(2*n-97)! *(n+1)+779672995879826548732499280*(n-47)!*(95/94*(n-45)!*(n+1)*(2*n-96)!+(n-46 )!*((2*n-95)!*(n+1)-1/779672995879826548732499280*binomial(2*n,n)*(n-49)!*(n-45 )!)))/(n-49)!/(n-47)!/(n-46)!/(n-45)!/binomial(2*n,n), ( 525311592968677461911967600*(n-46)!*(2*n-97)!*(n+1)+787967389453016192867951400 *((2*n-96)!*(n+1)-1/787967389453016192867951400*binomial(2*n,n)*(n-49)!*(n-46)! )*(n-47)!)/(n-47)!/(n-49)!/(n-46)!/binomial(2*n,n), (( 525311592968677461911967600*n+525311592968677461911967600)*(2*n-97)!-(n-49)!*(n -47)!*binomial(2*n,n))/(n-49)!/(n-47)!/binomial(2*n,n)] The limits, as n goes to infinity are 140737488355303 140737488354715 140737488347665 1125899906321895 [---------------, ---------------, ---------------, ----------------, 140737488355328 140737488355328 140737488355328 1125899906842624 2251799806925415 562949944483935 4503599306560275 18014393486573025 ----------------, ---------------, ----------------, -----------------, 2251799813685248 562949953421312 4503599627370496 18014398509481984 36028762013035875 36028686736513125 9007119770867775 288223600321221345 -----------------, -----------------, ----------------, ------------------, 36028797018963968 36028797018963968 9007199254740992 288230376151711744 576427124126768865 288191250308111295 1152578064904481055 ------------------, ------------------, -------------------, 576460752303423488 288230376151711744 1152921504606846976 4608829231416075495 4606032663949732185 4601000777851121385 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 4592334751792402785 18312040611189267765 36441835356453959505 -------------------, --------------------, --------------------, 4611686018427387904 18446744073709551616 36893488147419103232 282518685120075075 71496964394736991425 281242038117394821375 ------------------, --------------------, ---------------------, 288230376151711744 73786976294838206464 295147905179352825856 549301244397142019625 531534843335943069075 254137262204063606139 ---------------------, ---------------------, ---------------------, 590295810358705651712 590295810358705651712 295147905179352825856 3829285539518745082699 7071444671318466726923 3181421917208098529899 ----------------------, ----------------------, ----------------------, 4722366482869645213696 9444732965739290427392 4722366482869645213696 11058788557262293093121 36588926691459870244589 112148899360834708300931 -----------------------, -----------------------, ------------------------, 18889465931478580854784 75557863725914323419136 302231454903657293676544 74832382191738592652831 35127091432233210086081 ------------------------, ------------------------, 302231454903657293676544 302231454903657293676544 -48537181846028994615377 -763505610595489443734449 -------------------------, -------------------------, 2417851639229258349412352 4835703278458516698824704 -354911120757342458325931 -4095781624606211019912173 -------------------------, --------------------------, 1208925819614629174706176 9671406556917033397649408 -21054701767383391418858567 -50518239018892168048294909 ---------------------------, ---------------------------, 38685626227668133590597632 77371252455336267181195264 -57819569341693722036016099 -7988001409670210461556303 ---------------------------, --------------------------, 77371252455336267181195264 9671406556917033397649408 -549861221577901691218825967 -1157415301678997479750776559 ----------------------------, -----------------------------, 618970019642690137449562112 1237940039285380274899124224 -598579635441932262283242317 -2443048104010218208428750473 ----------------------------, -----------------------------, 618970019642690137449562112 2475880078570760549798248448 -9870688339722499857823495817 -----------------------------] 9903520314283042199192993792 and in Maple notation [140737488355303/140737488355328, 140737488354715/140737488355328, 140737488347665/140737488355328, 1125899906321895/1125899906842624, 2251799806925415/2251799813685248, 562949944483935/562949953421312, 4503599306560275/4503599627370496, 18014393486573025/18014398509481984, 36028762013035875/36028797018963968, 36028686736513125/36028797018963968, 9007119770867775/9007199254740992, 288223600321221345/288230376151711744, 576427124126768865/576460752303423488, 288191250308111295/288230376151711744, 1152578064904481055/1152921504606846976, 4608829231416075495/ 4611686018427387904, 4606032663949732185/4611686018427387904, 4601000777851121385/4611686018427387904, 4592334751792402785/ 4611686018427387904, 18312040611189267765/18446744073709551616, 36441835356453959505/36893488147419103232, 282518685120075075/ 288230376151711744, 71496964394736991425/73786976294838206464, 281242038117394821375/295147905179352825856, 549301244397142019625/ 590295810358705651712, 531534843335943069075/590295810358705651712, 254137262204063606139/295147905179352825856, 3829285539518745082699/ 4722366482869645213696, 7071444671318466726923/9444732965739290427392, 3181421917208098529899/4722366482869645213696, 11058788557262293093121/ 18889465931478580854784, 36588926691459870244589/75557863725914323419136, 112148899360834708300931/302231454903657293676544, 74832382191738592652831/ 302231454903657293676544, 35127091432233210086081/302231454903657293676544, -\ 48537181846028994615377/2417851639229258349412352, -763505610595489443734449/ 4835703278458516698824704, -354911120757342458325931/1208925819614629174706176, -4095781624606211019912173/9671406556917033397649408, -\ 21054701767383391418858567/38685626227668133590597632, -\ 50518239018892168048294909/77371252455336267181195264, -\ 57819569341693722036016099/77371252455336267181195264, -\ 7988001409670210461556303/9671406556917033397649408, -\ 549861221577901691218825967/618970019642690137449562112, -\ 1157415301678997479750776559/1237940039285380274899124224, -\ 598579635441932262283242317/618970019642690137449562112, -\ 2443048104010218208428750473/2475880078570760549798248448, -\ 9870688339722499857823495817/9903520314283042199192993792] and in floating point [1.000000000, 1.000000000, .9999999999, .9999999995, .9999999970, .9999999841, .9999999288, .9999997212, .9999990284, .9999969390, .9999911755, .9999764916, .\ 9999416644, .9998642550, .9997021135, .9993805331, .9987741242, .9976830078, .9\ 958038629, .9926977107, .9877579266, .9801835910, .9689645515, .9528850898, .93\ 05525039, .9004550498, .8610505368, .8108827541, .7487183277, .6736922957, .585\ 4473915, .4842504127, .3710695811, .2475995830, .1162257960, -.2007450791e-1, -\ .1578892597, -.2935755983, -.4234938941, -.5442512845, -.6529329359, -.74730041\ 85, -.8259399874, -.8883487150, -.9349526350, -.9670575576, -.9867392711, -.996\ 6848178] The cut off is at j=, 36 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 50], vs. those in the, 2, -th row from j=1 to j=, 49, are as follws 25 24 23 [7 (80421421917323 n - 25131694349158975 n + 3717480228125604950 n 22 21 - 346239353047611276250 n + 22785276571097921396285 n 20 19 - 1126956190044823806600625 n + 43505494364020651390237700 n 18 17 - 1343841037365561834964975000 n + 33780577117203304888727826005 n 16 - 699124681246737005555602734625 n 15 + 12006229525196991200417597196950 n 14 - 171923644031928247589151276561250 n 13 + 2057729401623123075623182229129795 n 12 - 20590243425704770494880413471421375 n 11 + 171926355904701221574484345126737200 n 10 - 1193115364469007539152296985369247500 n 9 + 6837538000179106666402975518931840640 n 8 - 32061969178722655352953242366710506000 n 7 + 121445897318820289778261282890847059200 n 6 - 364959089749605483488191380631808040000 n 5 + 846515153793270435822639393369194909952 n 4 - 1437077722027765848054315083590031078400 n 3 + 1501655850057000005574030404699825664000 n 2 + 154642829335971452974335501122457600000 n - 4297354084301104964507910896431595520000 n + 7002790957717212198939900287385600000000)/(16777216 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 25 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 7 (40210710958574 n 24 23 - 12565847174500825 n + 1858740114028970600 n 22 21 - 173119676514541798750 n + 11392638283735923467330 n 20 19 - 563478094752510981187375 n + 21752747150255324697880100 n 18 17 - 671920515654167296195195000 n + 16890288320318725322139038690 n 16 15 - 349562324967750405354281560375 n + 6003113896350379377152233781600 n 14 - 85961781427444561132774862653750 n 13 + 1028863085379335544297166161651710 n 12 - 10295067050801811155728195531392625 n 11 + 85961607367395785725022980709086100 n 10 - 596519495325006752020265838431012500 n 9 + 3417988013937978596161922925570918320 n 8 - 16017671073860352833624882974579030000 n 7 + 60536261774154428015694071493743409600 n 6 - 180367093308882969210059161766985240000 n 5 + 404509099862429773595644295899302725376 n 4 - 593729831160023539676233469285962828800 n 3 + 174588904140243599381632573242511872000 n 2 + 1631056411343224370735074242845798400000 n - 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(n - 46)! (2 n - 99)! (n + 1) + /97 3025794775499582180612933376 (n - 48)! |-- (n - 46)! (n + 1) (2 n - 98)! + \96 (n - 47)! ((2 n - 97)! (n + 1) \\ - 1/3025794775499582180612933376 binomial(2 n, n) (n - 50)! (n - 46)!)||/( // (n - 50)! (n - 48)! (n - 47)! (n - 46)! binomial(2 n, n)), ( 2038208980718468552218434288 (n - 47)! (2 n - 99)! (n + 1) + 3057313471077702828327651432 ((2 n - 98)! (n + 1) - 1/3057313471077702828327651432 binomial(2 n, n) (n - 50)! (n - 47)!) (n - 48)!)/((n - 48)! (n - 50)! (n - 47)! binomial(2 n, n)), ( (2038208980718468552218434288 n + 2038208980718468552218434288) (2 n - 99)! - (n - 50)! (n - 48)! binomial(2 n, n))/((n - 50)! (n - 48)! binomial(2 n, n))] and in Maple notation [7/16777216*(80421421917323*n^25-25131694349158975*n^24+3717480228125604950*n^ 23-346239353047611276250*n^22+22785276571097921396285*n^21-\ 1126956190044823806600625*n^20+43505494364020651390237700*n^19-\ 1343841037365561834964975000*n^18+33780577117203304888727826005*n^17-\ 699124681246737005555602734625*n^16+12006229525196991200417597196950*n^15-\ 171923644031928247589151276561250*n^14+2057729401623123075623182229129795*n^13-\ 20590243425704770494880413471421375*n^12+171926355904701221574484345126737200*n ^11-1193115364469007539152296985369247500*n^10+ 6837538000179106666402975518931840640*n^9-\ 32061969178722655352953242366710506000*n^8+ 121445897318820289778261282890847059200*n^7-\ 364959089749605483488191380631808040000*n^6+ 846515153793270435822639393369194909952*n^5-\ 1437077722027765848054315083590031078400*n^4+ 1501655850057000005574030404699825664000*n^3+ 154642829335971452974335501122457600000*n^2-\ 4297354084301104964507910896431595520000*n+ 7002790957717212198939900287385600000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-45), 7/8388608*(40210710958574*n^25-12565847174500825*n^ 24+1858740114028970600*n^23-173119676514541798750*n^22+11392638283735923467330* n^21-563478094752510981187375*n^20+21752747150255324697880100*n^19-\ 671920515654167296195195000*n^18+16890288320318725322139038690*n^17-\ 349562324967750405354281560375*n^16+6003113896350379377152233781600*n^15-\ 85961781427444561132774862653750*n^14+1028863085379335544297166161651710*n^13-\ 10295067050801811155728195531392625*n^12+85961607367395785725022980709086100*n^ 11-596519495325006752020265838431012500*n^10+ 3417988013937978596161922925570918320*n^9-\ 16017671073860352833624882974579030000*n^8+ 60536261774154428015694071493743409600*n^7-\ 180367093308882969210059161766985240000*n^6+ 404509099862429773595644295899302725376*n^5-\ 593729831160023539676233469285962828800*n^4+ 174588904140243599381632573242511872000*n^3+ 1631056411343224370735074242845798400000*n^2-\ 3304199814213785139119909714774261760000*n+ 70027909577172121989399002873856000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-45), 329/8388608*(1711094083298*n^26-578349800126878*n^25+ 92730604960197425*n^24-9383725502506006600*n^23+672646383517666537460*n^22-\ 36340015646135733742210*n^21+1537082475649495426040675*n^20-\ 52196405570870100061050100*n^19+1447840789485828229748257430*n^18-\ 33202744064302654834252962130*n^17+634763514048981793339321711175*n^16-\ 10171944211953182479562354485600*n^15+137058244208030880534738882368720*n^14-\ 1554485389977695346925570872879070*n^13+14828365668212549422745917381411925*n^ 12-118641407464738544388270134402064100*n^11+ 792326632945973446438911839944123940*n^10-4383549276576462764223346463881552240 *n^9+19859652199236742584777232814338042800*n^8-\ 72263206867360122033772377677390481600*n^7+ 203165473205968972341629192142358249152*n^6-\ 399193589457052931996163930519462857472*n^5+ 351554637576535552486966070628084096000*n^4+ 689223647449869514557354003860633088000*n^3-\ 2512329235468935077144833324681605120000*n^2+ 1801156771235061331284474500680581120000*n-\ 75987731668846345137432960565248000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-\ 47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-\ 43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-35)/(2*n-29)/(2*n-51)/(2*n-45), 47/8388608*(11977658580434*n^26-\ 4048448598570298*n^25+649114233753866075*n^24-65686078260898326400*n^23+ 4708524636065060326880*n^22-254380102550211932604310*n^21+ 10759576540208633959990025*n^20-365374766763473398737562900*n^19+ 10134880090640778288809240390*n^18-232418868058444605055600255030*n^17+ 4443326719284071225613005342525*n^16-71202817904413861562850129849400*n^15+ 959378098893881088911124378724460*n^14-10880461914357312791020710980398570*n^13 +103773631965186054461987686341697775*n^12-829932979912601685380755535025742900 *n^11+5535937146242730604556454989313820220*n^10-\ 30526742119804993901586285604395369040*n^9+ 137066512184566477665456623366469536400*n^8-\ 486886229562252180694063297296325982400*n^7+ 1282755049139544603230205220070422967616*n^6-\ 2067817020433016084749218439696050922752*n^5+ 103108292182780640034057014706246067200*n^4+ 8078906742296356877319395554292734464000*n^3-\ 15087165786675528556907505934672220160000*n^2+ 8608103203597357710956850460609413120000*n-\ 531914121681924415962030723956736000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n -47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n -43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2 *n-35)/(2*n-29)/(2*n-51)/(2*n-45), 705/67108864*(12776169134503*n^27-\ 4656913638615684*n^26+806824656473417595*n^25-88413444067914728100*n^24+ 6879152424985026960735*n^23-404433024554366487634500*n^22+ 18667353768295447888201815*n^21-693869948217790511332419540*n^20+ 21138427713338875290694305105*n^19-534390357658711871474811199260*n^18+ 11309129479646265816792249136065*n^17-201542104783908051810614312648940*n^16+ 3035791707442024570980380837351445*n^15-38717508361664377893075529126894380*n^ 14+418064773100631254978687726302627965*n^13-\ 3814487613535232534663747246916398940*n^12+ 29286808341386187501173729458759477140*n^11-\ 187833465548803874005476913336610957040*n^10+ 993739142866303589328655125952327630640*n^9-\ 4238957066566816486686455033391664856640*n^8+ 13930101481980590352454770260991262510272*n^7-\ 31656463617139440159923117180205937019136*n^6+ 32752350977411885167182540588824431825920*n^5+ 59676835931133175711255640449021897052160*n^4-\ 271610260743809202798831086811310674739200*n^3+ 374707565940745973794676995727440773120000*n^2-\ 189712059798550483004584025760729661440000*n+ 15035439172875730157860068463843737600000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 3/33554432*(1501199862393443*n^27 -547187343241460976*n^26+94801893358388980491*n^25-10388578704141429417600*n^24 +808300231150004355116835*n^23-47520855520420331896675920*n^22+ 2193411347055337605933317295*n^21-81529478791704040476218954400*n^20+ 2483747870911369288502408737605*n^19-62789822416938500652420442636560*n^18+ 1328770225340713438809331847722785*n^17-23678981660173852354131304208366400*n^ 16+356626802051745982684718122295261745*n^15-\ 4546954059359691095561109969280085040*n^14+ 49063609550847435074445485042591902965*n^13-\ 446968407967922124275424705075245738400*n^12+ 3419866122886467062405041028710039339140*n^11-\ 21769354095784941828393324250618112311680*n^10+ 113358838124520172779742730880535333604080*n^9-\ 468000891795697450514517601448896802918400*n^8+ 1437229165598780940182025708820571400311232*n^7-\ 2783083603030945142639681911838457013069824*n^6+ 1048619496994844382936519866480762640912384*n^5+ 11241987505745236400198388192668096921395200*n^4-\ 32568433924735728781798832687105459804160000*n^3+ 38831249964343509290203714052430397194240000*n^2-\ 18634172333506982285393107874767342141440000*n+ 1766664102812898293548558044501639168000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 903/67108864*(19949499196817*n^ 28-7820203302488554*n^27+1459794399666257349*n^26-172699354629819928170*n^25+ 14538011224310998379865*n^24-926897006424629357241930*n^23+ 46514584851056039427500505*n^22-1885012497109926497506157250*n^21+ 62800373530731567604832739495*n^20-1742033280672914127826692897990*n^19+ 40601221403516223668452505067615*n^18-800126629364748134549341371095550*n^17+ 13387572289887676950624066439735155*n^16-190597928950348655800004802285248910*n ^15+2309699871420842046033203658813715635*n^14-\ 23784366509733988845223193993094948150*n^13+ 207260209345490592182328085494146961660*n^12-\ 1516566287527935554145166340529400857720*n^11+ 9193439759690997530834920091331109327120*n^10-\ 45100222042899907259421367716142624647200*n^9+ 171494369644333806041075643987235004563008*n^8-\ 461149517787455820224712304353129160864896*n^7+ 649640396693545788601399212149992969291776*n^6+ 665075879350800974944523234916157665976320*n^5-\ 5565034045610480571491363961536246808576000*n^4+ 12267806233273133569601269605926677401600000*n^3-\ 13014315195340172658053675316138803036160000*n^2+ 5981083007869059177223207542660385996800000*n-\ 645624755180793396313426527890964480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 129/67108864*( 139646477370443*n^28-54741408950358970*n^27+10218555175721639607*n^26-\ 1208894068546180208010*n^25+101765825691659521663635*n^24-\ 6488244828499055390782650*n^23+325598456625618442155461715*n^22-\ 13194776081564573440501406850*n^21+439580780394032121843885291405*n^20-\ 12192964630112347119855723856950*n^19+284147048553804094297689484151445*n^18-\ 5598385808849393100509279242590750*n^17+93627590602091106346315395850810545*n^ 16-1331735578440722142377577544544843550*n^15+ 16109060295148527284699446750398544305*n^14-\ 165313609417007555621168480689704266550*n^13+ 1431371955854572202664771219153784211940*n^12-\ 10353477468940064539195463327975151756600*n^11+ 61506357352666388909660572252611162162160*n^10-\ 291392976631359808495891903232526122727200*n^9+ 1041922617635039266121947333765564293508032*n^8-\ 2469991091344755443298220092770538810161280*n^7+ 2048628480080478083653849145343338601120768*n^6+ 9756731638144981645131838537694820422799360*n^5-\ 43768118853130947494542013984676555257856000*n^4+ 84062555496107389653736125642322829045760000*n^3-\ 83627066337474969096204500065508317102080000*n^2+ 37795840932409877221145776162723685990400000*n-\ 4519373286265553774193985695236751360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 153381/268435456*( 939589029679*n^29-395096955642413*n^28+79250906132567841*n^27-\ 10093308450203051577*n^26+916524258146133547365*n^25-63168708801863355175485*n^ 24+3434821645822986002881245*n^23-151207394116730152798733565*n^22+ 5487308972488417964761286415*n^21-166299467239554788731017981555*n^20+ 4248353392078598562734053496235*n^19-92089610815530513491715493133595*n^18+ 1701194983838985849013621841613735*n^17-26846412180829773782866164961703295*n^ 16+362075371622039068305784233903080415*n^15-\ 4166317887200071618235488755607637055*n^14+ 40724350028803199351824431567732016470*n^13-\ 335462664569863900686758038155022504740*n^12+ 2298004881947859298088826441203847556280*n^11-\ 12807791216733160586802263859975767782960*n^10+ 55922795095463733242083817201566774075296*n^9-\ 177306988137897813787756002048185383192512*n^8+ 326182896645605678642233018874433826137984*n^7+ 132568184908482805324918619795383966858752*n^6-\ 2975834343143114282159529560827576656168960*n^5+ 9577735804794892390737778452516388531200000*n^4-\ 16262568209981889075033803497326491934720000*n^3+ 15171360594925650485141741390018457108480000*n^2-\ 6719081314887606474837528439294682726400000*n+ 866624986769172632898426188825886720000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 17835/ 134217728*(4040227686784*n^29-1698912709199609*n^28+340777262926047234*n^27-\ 43400824062122631069*n^26+3940983910477690672050*n^25-271616135081008339206585* n^24+14768766015491638340783970*n^23-650110984847921240598036225*n^22+ 23589902316702188617481781450*n^21-714774835558036594601461496295*n^20+ 18253141587583545784579432206030*n^19-395400252150183429827592511586535*n^18+ 7295679837742161523225018972485270*n^17-114895936279491295346811405871956915*n^ 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)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 10701/ 134217728*(13467377478279*n^30-6060291700789965*n^29+1302968347487436390*n^28-\ 178175288023875740675*n^27+17403671298383477956926*n^26-\ 1292824712285863595129655*n^25+75928385674714008716122050*n^24-\ 3618469364121595292114622375*n^23+142502779663183169481119214060*n^22-\ 4698912395886638641549129417125*n^21+130968159641038055745991481322150*n^20-\ 3106306651809412649889300888230625*n^19+62973892179388364355123352081615170*n^ 18-1093845926314087879757750434555692525*n^17+ 16285449319746501746987731752671359350*n^16-\ 207448817574431642904086384159503342125*n^15+ 2251033097068579126424452907859198026685*n^14-\ 20646744851225952228540556098049790248650*n^13+ 158096150246026578698241265533434181272700*n^12-\ 990827889062470084475729643868822184181000*n^11+ 4915669767335552594510661189164568098250576*n^10-\ 18074278686270434671245732596995312906718560*n^9+ 40850641478743764513612079648955204911959360*n^8+ 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35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2* n-69), 698523/8589934592*(2869868020012163*n^38-2012501882620899683*n^37+ 677690206464434320494*n^36-145862491286633207428515*n^35+ 22532029231907711742629739*n^34-2658882211915959599283577374*n^33+ 249059143052623690059862887232*n^32-18993208580820832468903458257020*n^31+ 1199624862321111300920933848670118*n^30-63478478442305229681326869566504738*n^ 29+2833541310637626794944659583695067284*n^28-\ 106958803538657629709854123503762442890*n^27+ 3404104896959351144541405835466245457070*n^26-\ 90299467730488492013006021937374016244920*n^25+ 1938569364200333748631729461575389590841560*n^24-\ 31033012462373185665985035712348971652709100*n^23+ 255040352696555651872626801422980677683976855*n^22+ 4393569462547994185879617347612535408841568445*n^21-\ 256442365341711825817800958607943015298047301010*n^20+ 6994758147564770008533224362100508448411333728725*n^19-\ 140833637497817536447462544275879672382567647732473*n^18+ 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59)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2* n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/( 2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83), 1517/274877906944*( 75243277959539397706*n^42-35342620718786946117882*n^41+ 1744236545191548342328597*n^40+3213068336239690097240489880*n^39-\ 1380482469839664020919414567788*n^38+326879862797515653663006483560046*n^37-\ 54629706027520901847476021133331651*n^36+7033506827010290178655445665473493980* n^35-730182083478855011748150636675678347752*n^34+ 62877479265012463238489042140253709980124*n^33-\ 4578801494638546091956068941655017134576534*n^32+ 285937363140863178278357965240071923864004000*n^31-\ 15474299102732199668050568081802184958959640976*n^30+ 731606630311241590837512092985208625887676533932*n^29-\ 30409106397112827050078294454883663138993258717182*n^28+ 1116683894074437895093660620472456887562399882013640*n^27-\ 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0130347251181616422799680155126284366643200000000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-87)/(2*n-67 )/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2* n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-91)/(2*n-83)/(2*n-85), -329/ 35184372088832*(13419585936199719990876239*n^47-14929241383385731750925217778*n ^46+8065956123726962342845761370848*n^45-2820092722787416550739569581846440*n^ 44+717350956227634531114771240878891894*n^43-\ 141512597785505439901849952862227307948*n^42+ 22535852908074548449393534796907208085228*n^41-\ 2977689194868920258113015028860747008354160*n^40+ 332986381320731474594306961861296282318352969*n^39-\ 31988795012319244502739154454437299883466753958*n^38+ 2670626561231860660945144245832623671662419825948*n^37-\ 195542708050684995992803552875321632465500070171880*n^36+ 12649409265651365086350545911813142652963180600411284*n^35-\ 727251964291713655929489315926411337397861438340224648*n^34+ 37341953654170391004105757161009006532965398017850284248*n^33-\ 1719231936559917133082516592377181848311360506791824706400*n^32+ 71204580367869254742810670427838939856779446110058677989969*n^31-\ 2659870372240707628899416681222688570027565587727613125846878*n^30+ 89805838991743004314388132163942492168841849904808634937162488*n^29-\ 2745049013770638990671640383428177706281578441397380208696817720*n^28+ 76054562116445404142134806428636072306528910201175898636090176086*n^27-\ 1911578366102450482899179584218974993859657523936159788228891792972*n^26+ 43607126802327700095736264371072091306892713234595308612282434417852*n^25-\ 902984505465612089772608457673053034169698999933120654872360973206960*n^24+ 16969752707236461610999374065896494637434120429699641207555119166453431*n^23-\ 289277675978283324246326258531932742532991372357231969643524414747907402*n^22+ 4469151621892692041633436695909445110781985353129576218558612679560866572*n^21-\ 62500317930850549376250301000705486007773066664782665795259184687535117240*n^20 +789957568204394169500225016111208212042581819902390550543044065407952397056*n^ 19-9006089764369666857649437067155828552244560067021072636270746386031189417792 *n^18+ 92392995798681201349412487278650838986017786814695746549175351776777517242752*n ^17-\ 850476821106128210092649120931387077715525058786080729111006738764579503435520* n^16+70003510990697979433111681745637186066093697072203923340639324344503515324\ 18816*n^15-51315732575086413825062901990684393844238107932434990260703245929837\ 014386954752*n^14+3334052892193919371950365393751709120631682519280570747989621\ 43960283987557604352*n^13-19090578596886100429381551415023924915958693589045715\ 03293983142192436581607168000*n^12+95687574230296235776926659516349609940982845\ 53046883171437645505232338967692845056*n^11-41645232858176545465458908922149736\ 248991762690457588504327793240458202920758607872*n^10+1558449786306461426856870\ 28215710721128486155559657058057918960636778610307497459712*n^9-495472982799495\ 260237648418494649063110172918199880306052783472977223804104859975680*n^8+13183\ 7042681524060268762750796230311654577581565447805717325455799231007278073118720\ 0*n^7-2880327274456511795777956716208028276523649398172379376277354447315098777\ 019219968000*n^6+50387863960060874219519629383529593925120151640825572983783582\ 13618234084221255680000*n^5-682002727584316469735629910563305760562154790776668\ 3965411943858721133375468339200000*n^4+6796803476460667766920984975122582945852\ 243739169987695536448739133251930554368000000*n^3-46149657994036437929669392007\ 86141032080077421532427240948942107403981915422720000000*n^2+185760699908550493\ 7311273238834168202151213178280289518866651604169865035776000000000*n-319486397\ 146254166468919534941416531454359130268480063022030084161037926400000000000)/(2 *n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-93)/(2*n-49)/(2*n-7)/(2*n-61)/(2* n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2 *n-13)/(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/ (2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73 )/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-\ 91)/(2*n-83)/(2*n-85), -47/17592186044416*(49353188928991042657009321*n^47-\ 54737137796538348770116331072*n^46+29487102427329633256767332942382*n^45-\ 10280936470132671368727936856060410*n^44+2608271278503927606853340384130974016* n^43-513244637383071280463785088073352373322*n^42+ 81539079927401301879061224526187568003142*n^41-\ 10749439186864379592212351752078819665772430*n^40+ 1199492873872322686455409232505130835734673241*n^39-\ 114995811675209449350655725329315315161016444982*n^38+ 9582008839545181239063106167234296719182373708512*n^37-\ 700307563290979698717064393736003058331866970572300*n^36+ 45223484963243904047635781624922600084627083752835176*n^35-\ 2595771252129806172884401930150984993588115499844803812*n^34+ 133077865647705636119882161933281922236914837812525563852*n^33-\ 6117986831138656042270966559654935775277797126892100443420*n^32+ 253036859656660041798423626104159520526120761876856224934891*n^31-\ 9440026161967808865669792518706137024776246333954255626692652*n^30+ 318337543993903236186631241190114338625208860654269403874728302*n^29-\ 9719329496888989751541874899668866903887211163378852406276784690*n^28+ 268995769037680497363987765210248991082523995992357834001492709304*n^27-\ 6754233556791470397264283109905993083223153961400801896554361662578*n^26+ 153933743709448653256421899233653027132695026598975357142052405872718*n^25-\ 3184758337794330379209295752281661738265000517563161719312059085487990*n^24+ 59802291924236924195612831298945950469379576122586024712002302188088659*n^23-\ 1018658409222893535208257768769456964354079511093608981065328233595876478*n^22+ 15726578063620494028771044213593073187294411585225598692853197741846035108*n^21 -219791043208859112752527373550749550485337594234243169628753893346870871720*n^ 20+2776337636405623702700157225544579445152725572051477677962379490759032485984 *n^19-\ 31634918132966952518186258283758889370806688117673771998243641997893831375168*n ^18+ 324378204775268390273494292568396508574764590043919965720528167570218963305088* n^17-29845346052301931416928218383318136317855956858580544611963649987325684637\ 37600*n^16+24555792626083117839782926047523234689046514253958821729181638198924\ 032768020224*n^15-1799377570953480471542424009807012286849729220511092257983836\ 23890820453711922688*n^14+11686883503287455038607010448149235716205242259696310\ 41804698422214870918483022848*n^13-66898414198849139266176485324964341040230632\ 48404101326645229559067653955681044480*n^12+33522628734787139447833936042375891\ 953152967724621125828523022720491154329688899584*n^11-1458640176342015255795554\ 87746813605157241783910408112932852255335176286212550197248*n^10+54574518989579\ 3484177435105757523843728221198186056827306047936210843983302372098048*n^9-1734\ 7804597373571283832378174923916278309611020626392821080537716719073886576548249\ 60*n^8+461532229808178306929944466216641170437330560800368413318227733485020463\ 9349545369600*n^7-1008227379459008301049410892930921180578444077550964579607789\ 5719548172404203192320000*n^6+1763624796606225533117980225388135274634023779286\ 6209680500605120070355680724254720000*n^5-2386937704833620155759699996183045743\ 7410093009591858160209453491998052732790374400000*n^4+2378730806469319514329486\ 7077865060554050691341947865742420346036751381282422784000000*n^3-1615123117888\ 1190317097959695544588049177760335496939932706745986714308496588800000000*n^2+ 6501285647590172768594716572776469023329218773388304322027045128141629554688000\ 000000*n-1118202390011889582641218372294957860090256955939680220577105294563632\ 742400000000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-93)/(2*n-49)/( 2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89) /(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-\ 53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2* n-77)/(2*n-81)/(2*n-91)/(2*n-83)/(2*n-85), (2038208980718468552218434288*(n-47) !*(n-46)!*(2*n-99)!*(n+1)+3025794775499582180612933376*(n-48)!*(97/96*(n-46)!*( n+1)*(2*n-98)!+(n-47)!*((2*n-97)!*(n+1)-1/3025794775499582180612933376*binomial (2*n,n)*(n-50)!*(n-46)!)))/(n-50)!/(n-48)!/(n-47)!/(n-46)!/binomial(2*n,n), ( 2038208980718468552218434288*(n-47)!*(2*n-99)!*(n+1)+ 3057313471077702828327651432*((2*n-98)!*(n+1)-1/3057313471077702828327651432* binomial(2*n,n)*(n-50)!*(n-47)!)*(n-48)!)/(n-48)!/(n-50)!/(n-47)!/binomial(2*n, n), ((2038208980718468552218434288*n+2038208980718468552218434288)*(2*n-99)!-(n -50)!*(n-48)!*binomial(2*n,n))/(n-50)!/(n-48)!/binomial(2*n,n)] The limits, as n goes to infinity are 562949953421261 140737488355009 281474976702521 281474976640199 [---------------, ---------------, ---------------, ---------------, 562949953421312 140737488355328 281474976710656 281474976710656 9007199239824615 4503599587180329 18014397774725751 18014395580787147 ----------------, ----------------, -----------------, -----------------, 9007199254740992 4503599627370496 18014398509481984 18014398509481984 144115104961194699 562948912451505 144114406395063579 144113069864574909 ------------------, ---------------, ------------------, ------------------, 144115188075855872 562949953421312 144115188075855872 144115188075855872 2305757483205319593 1152820334385522561 4610783339164393815 -------------------, -------------------, -------------------, 2305843009213693952 1152921504606846976 4611686018427387904 4609778369465258679 9215700865605710673 4604321822676204513 -------------------, -------------------, -------------------, 4611686018427387904 9223372036854775808 4611686018427387904 4598140989251744247 4587750880113975315 146270340015767667849 -------------------, -------------------, ---------------------, 4611686018427387904 4611686018427387904 147573952589676412928 72716004516257309067 288338863527594292581 284659028129713085289 --------------------, ---------------------, ---------------------, 73786976294838206464 295147905179352825856 295147905179352825856 2235698846554499138025 272340990060615146241 2102879054415956419503 ----------------------, ---------------------, ----------------------, 2361183241434822606848 295147905179352825856 2361183241434822606848 2004668818942956135249 30094128021048459098195 13831104974399934800311 ----------------------, -----------------------, -----------------------, 2361183241434822606848 37778931862957161709568 18889465931478580854784 49502555421610691616145 42705908721531311946905 559425653435554296678581 -----------------------, -----------------------, -------------------------, 75557863725914323419136 75557863725914323419136 1208925819614629174706176 105431926270397407484273 136164965525245355232967 ------------------------, ------------------------, 302231454903657293676544 604462909807314587353088 57072026332310633160001 -788156202910290829625233 ------------------------, --------------------------, 604462909807314587353088 19342813113834066795298816 -1710730105472465396548295 -12010532046679643770148249 --------------------------, ---------------------------, 9671406556917033397649408 38685626227668133590597632 -16943715530699869760353877 -172137501752082300843522757 ---------------------------, ----------------------------, 38685626227668133590597632 309485009821345068724781056 -12812093012793080861766757 -233468676463615354340380753 ---------------------------, ----------------------------, 19342813113834066795298816 309485009821345068724781056 -257161210363807063636970599 -4415043773009707876998282631 ----------------------------, -----------------------------, 309485009821345068724781056 4951760157141521099596496896 -2319599879662579004879438087 -9587020079519414028391033313 -----------------------------, -----------------------------, 2475880078570760549798248448 9903520314283042199192993792 -9776132252988137914679341649 -39486693195837264512258323025 -----------------------------, ------------------------------] 9903520314283042199192993792 39614081257132168796771975168 and in Maple notation [562949953421261/562949953421312, 140737488355009/140737488355328, 281474976702521/281474976710656, 281474976640199/281474976710656, 9007199239824615/9007199254740992, 4503599587180329/4503599627370496, 18014397774725751/18014398509481984, 18014395580787147/18014398509481984, 144115104961194699/144115188075855872, 562948912451505/562949953421312, 144114406395063579/144115188075855872, 144113069864574909/144115188075855872, 2305757483205319593/2305843009213693952, 1152820334385522561/ 1152921504606846976, 4610783339164393815/4611686018427387904, 4609778369465258679/4611686018427387904, 9215700865605710673/ 9223372036854775808, 4604321822676204513/4611686018427387904, 4598140989251744247/4611686018427387904, 4587750880113975315/ 4611686018427387904, 146270340015767667849/147573952589676412928, 72716004516257309067/73786976294838206464, 288338863527594292581/ 295147905179352825856, 284659028129713085289/295147905179352825856, 2235698846554499138025/2361183241434822606848, 272340990060615146241/ 295147905179352825856, 2102879054415956419503/2361183241434822606848, 2004668818942956135249/2361183241434822606848, 30094128021048459098195/ 37778931862957161709568, 13831104974399934800311/18889465931478580854784, 49502555421610691616145/75557863725914323419136, 42705908721531311946905/ 75557863725914323419136, 559425653435554296678581/1208925819614629174706176, 105431926270397407484273/302231454903657293676544, 136164965525245355232967/ 604462909807314587353088, 57072026332310633160001/604462909807314587353088, -\ 788156202910290829625233/19342813113834066795298816, -1710730105472465396548295 /9671406556917033397649408, -12010532046679643770148249/ 38685626227668133590597632, -16943715530699869760353877/ 38685626227668133590597632, -172137501752082300843522757/ 309485009821345068724781056, -12812093012793080861766757/ 19342813113834066795298816, -233468676463615354340380753/ 309485009821345068724781056, -257161210363807063636970599/ 309485009821345068724781056, -4415043773009707876998282631/ 4951760157141521099596496896, -2319599879662579004879438087/ 2475880078570760549798248448, -9587020079519414028391033313/ 9903520314283042199192993792, -9776132252988137914679341649/ 9903520314283042199192993792, -39486693195837264512258323025/ 39614081257132168796771975168] and in floating point [1.000000000, 1.000000000, 1.000000000, .9999999997, .9999999983, .9999999911, .9999999592, .9999998374, .9999994233, .9999981509, .9999945760, .9999853020, .\ 9999629090, .9999122488, .9998042626, .9995863446, .9991682899, .9984031446, .9\ 970628900, .9948098942, .9911663776, .9854856259, .9769300695, .9644623022, .94\ 68552916, .9227271659, .8906039216, .8490102690, .7965849360, .7322126006, .655\ 1608659, .5652079958, .4627460547, .3488449814, .2252660392, .9441774740e-1, -.\ 4074672067e-1, -.1768853471, -.3104649767, -.4379847810, -.5562062662, -.662369\ 6842, -.7543779797, -.8309326856, -.8916109894, -.9368789303, -.9680416433, -.9\ 871370929, -.9967842732] The cut off is at j=, 37 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 51], vs. those in the, 2, -th row from j=1 to j=, 50, are as follws 25 24 23 [3 (93824992236881 n - 29320310074022550 n + 4337060266148150150 n 22 21 - 403945911889320957500 n + 26582822666367243401895 n 20 19 - 1314782221731813532563250 n + 50756410092869569526574400 n 18 17 - 1567814543737375170377580000 n + 39410673314750661756955076735 n 16 - 815645462200032130168370800250 n 15 + 14007267820646400030156556959150 n 14 - 200577586636702934695047954827500 n 13 + 2400684378818987683156304265075865 n 12 - 24021953266277021486489625774006750 n 11 + 200580823345244512077956459402825900 n 10 - 1391969743638103891751769466466675000 n 9 + 7977164856692366098281545954653576080 n 8 - 37406264685438382428105056009199220000 n 7 + 141695770056969181735278000037020622400 n 6 - 425886197639583529428320254953724560000 n 5 + 988493797379491958189063835158598572544 n 4 - 1682533962834958221984840172403086387200 n 3 + 1779371778442131263808053692583482368000 n 2 + 106429253431241045410633965962649600000 n - 4958554871110182808265612509334077440000 n + 8333321239683482516738481341988864000000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) 26 (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 45)), 21 (53614281278157 n 25 24 - 18121627071978391 n + 2905558955994210100 n 23 22 - 294023399235883291650 n + 21076253381017677267815 n 21 20 - 1138653828163636654214345 n + 48161918109762473622370050 n 19 18 - 1635487425999979848846378400 n + 45365682118188606581638597795 n 17 - 1040352913284915933040782600585 n 16 + 19889271488202763766952927972800 n 15 - 318721608104852022108076282530650 n 14 + 4294519092660379799920183514371905 n 13 - 48708137412000206108215917674227015 n 12 + 464648803273759176969602083582319050 n 11 - 3718079435531130395715057564852684900 n 10 + 24839500850401917521597978963345650760 n 9 - 137577362916039059276239715574700744880 n 8 + 625440290133628378279795829761798312800 n 7 - 2300130659110374359836368332175868766400 n 6 + 6684325623674730947479987495133431613568 n 5 - 14628155190105216611387500067963611574784 n 4 + 20803758812219962750437703139451019315200 n 3 - 4797104190094142314160679494640718848000 n 2 - 59091169223245853685908434365417185280000 n + 114723742608892546877716496400036003840000 n - 2380948925623852147639566097711104000000)/(16777216 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 7 ( 26 25 24 80421421916098 n - 27182440606917674 n + 4358338433527667875 n 23 22 - 441035098723459136000 n + 31614380045325047145760 n 21 20 - 1707980738239506981480230 n + 72242876680547746091915425 n 19 18 - 2453231091573737121212208500 n + 68048519344088052400285742230 n 17 - 1560529111183502297885724123590 n 16 + 29833892522267533068967869039925 n 15 - 478081703906200100689936805579000 n 14 + 6441749670361942298298864799859020 n 13 - 73061198664740262626808183725803610 n 12 + 696943450758114089956154360885025175 n 11 - 5576375451218239637858105013398244500 n 10 + 37243613218499198138135415020436491740 n 9 - 206091917158717968632370223655340512720 n 8 + 934207028215340727543111591940786678800 n 7 - 3404176224505036365869443128785461560000 n 6 + 9606178584393228019005301082130990265152 n 5 - 19061261453914692572911910823684243922176 n 4 + 17493916154095452001688754634686564172800 n 3 + 31053488578906755894317672140809865728000 n 2 - 119108531245348720324733301046311505920000 n + 86301424681302970879560966079580405760000 n - 3571423388435778221459349146566656000000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 987 ( 27 26 25 1140729388744 n - 415795862110374 n + 72037916392487292 n 24 23 - 7894057675212888075 n + 614210070045429180180 n 22 21 - 36110096102808414711180 n + 1666728540493481765182740 n 20 19 - 61952722231612279795524225 n + 1887363267562060168211957340 n 18 17 - 47713654569479604438602302290 n + 1009755844444990611204794749620 n 16 - 17995371986326785602308161961725 n 15 + 271073236367891400479542739648460 n 14 - 3457571291797386929718364046818560 n 13 + 37344742651867753286812037332416780 n 12 - 340976141666234713035634671527379975 n 11 + 2622392132736426260618133661329442620 n 10 - 16887807914022965512600724299908459420 n 9 + 90209834651086735236258669196910650160 n 8 - 393431063558817710199471719250039267600 n 7 + 1359749977251169279279706791207701166656 n 6 - 3484115312253978280102353257572684858176 n 5 + 5417181626796943967291509925545948593408 n 4 + 114792369464007819585110948640970521600 n 3 - 21628076588288998882454927992174262784000 n 2 + 39342244687039006322384393045359349760000 n - 22171762450429794373105646927685550080000 n + 1342449926149618764094648969986048000000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 45)), 141 ( 27 26 25 7985105715007 n - 2910571029197919 n + 504265412353204944 n 24 23 - 55258403072859859275 n + 4299470362993493872665 n 22 21 - 252770653889353635599655 n + 11667097587057954233465130 n 20 19 - 433668848422549540500703575 n + 13211526790012150301865951645 n 18 17 - 333994542495278876232688303065 n + 7068234513296194346096330778540 n 16 - 125965022272042392802993355823825 n 15 + 1897412694914169444229555357818255 n 14 - 24199724410754764128093373843268685 n 13 + 261322618965931973024833599526270410 n 12 - 2384726809996812108972884293177559325 n 11 + 18315874961311034927331005847832624860 n 10 - 117559901864625249884822969019713915220 n 9 + 622941817865661147996190633686905838320 n 8 - 2665729737152138020230854039102156449200 n 7 + 8815005015853303842959291215323189365568 n 6 - 20295393721487171383258747996253841675456 n 5 + 21995154950670582981236053365131886502656 n 4 + 34994123064890103962167754343869609395200 n 3 - 169400079475045159454641880244663694848000 n 2 + 237022778295588315664522159881689640960000 n - 120561864548834084267650155072198082560000 n + 9397149483047331348662542789902336000000)/(8388608 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) (2 n - 45)), 987 ( 28 27 26 18251670131023 n - 7154654644295426 n + 1335556806739511931 n 25 24 - 158001596410372426530 n + 13300744608126483364935 n 23 22 - 848013685958754780427170 n + 42556065587249196151693095 n 21 20 - 1724600926231127733609356250 n + 57456759028341577862172262905 n 19 - 1593841819257552242340175523310 n 18 + 37149190243338846239846143465185 n 17 - 732175936557455993343640116790950 n 16 + 12253447119135774056961233001950445 n 15 - 174536532274664905420491324431432790 n 14 + 2117224754710827174963964351481231565 n 13 - 21848149049473236510501231811401964350 n 12 + 191193461599479902369915118174969201540 n 11 - 1410626537564501770655849902609366918680 n 10 + 8686951549516251246930408665647689559280 n 9 - 43877591674362527700002888998562881040800 n 8 + 176045342043396195852216106261969802953152 n 7 - 525742337666161874333280942032455993002624 n 6 + 986177708193376210232853438642423098578944 n 5 - 315071781098134060744890483058189215621120 n 4 - 4039475727479668026995742632870409252864000 n 3 + 11378168344822163726752183173680530329600000 n 2 - 13369901903989417200262185052944716759040000 n + 6347647889194230266327151062962942771200000 n - 590677967505832256201645546793861120000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 28 27 (2 n - 51) (2 n - 45)), 21 (857828480469551 n - 336268754507355682 n 26 25 + 62771164146186267627 n - 7426073496608044519410 n 24 23 + 625134705773476577683095 n - 39856601467264959517206690 n 22 21 + 2000130358005798372297558615 n - 81055812138848820606330962250 n 20 + 2700435327398888746469522990985 n 19 - 74908550187698294356781141462670 n 18 + 1745906786178382445432997174956145 n 17 - 34407651471540502586800599858648150 n 16 + 575740975018513607972175800344495965 n 15 - 8197872642698861418420020678958555030 n 14 + 99369011063564473012545091618369888605 n 13 - 1023766250640147137664062527522459567950 n 12 + 8929350333824208942614983010929417456980 n 11 - 65443956137603126775142771221350108276760 n 10 + 397830604635981064831357918386096468531760 n 9 - 1960758292846701490672816756600431903773600 n 8 + 7514144338713968991677060492167352906591424 n 7 - 20498301794125984356375657943979876254743168 n 6 + 30139408496152771059762470833955531959477248 n 5 + 24097203562652640610390814489675617789071360 n 4 - 235049804666670443844038238945538470770688000 n 3 + 529114569327925224669448206230579992473600000 n 2 - 566209500484838812099400297841730269511680000 n + 260779281211316783167142050777836906086400000 n - 27761864472774116041477340699311472640000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 28 (2 n - 51) (2 n - 45)), 3741 (9630792198242 n - 4049747691677791 n 27 26 + 812323346205638148 n - 103456806896824399179 n 25 24 + 9394445013813054746460 n - 647489022728865320698695 n 23 22 + 35207971144154718990644460 n - 1549966779848144651687434455 n 21 20 + 56251389265576494839228499120 n - 1704951645224348967391408648185 n 19 + 43564493897967061711667079648180 n 18 - 944702089313916522342091681517265 n 17 + 17464669148855431525220162151716580 n 16 - 275984646813641917829131785686536365 n 15 + 3731353051106279027290345680908057220 n 14 - 43122539425548796413559301222619254685 n 13 + 424663788452490902545250230130254007310 n 12 - 3542032753729568200059059387040268492980 n 11 + 24762202802677375777821332334883208132440 n 10 - 142583214291966051400218003982960709555120 n 9 + 656325684862614092195330830340927726997408 n 8 - 2284136099979796968548485541869970111841984 n 7 + 5267690260153355800604649960626180318439552 n 6 - 4128162921915673522278617489361009366199296 n 5 - 20830441000403765949330493119364370047365120 n 4 + 90781329308906817877173511441029351376896000 n 3 - 171700819958146341262576890238832845762560000 n 2 + 168862042390375840542128912178302358159360000 n - 75541596205320452486463192194340342988800000 n + 8882906114384019487208868435465338880000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 (2 n - 51) (2 n - 57) (2 n - 45)), 3741 (19261579644451 n 28 27 - 8099491296607202 n + 1624645016991753519 n 26 25 - 206913178176589460898 n + 18788809409246340881235 n 24 23 - 1294966747585555329375090 n + 70414697028971684945776155 n 22 21 - 3099822882205337323347436410 n + 112494710052187281775540268685 n 20 - 3409415143327386989816970819870 n 19 + 87104283947924001155321918285565 n 18 - 1888353263797726507333212411198630 n 17 + 34891666515256984282014737411717265 n 16 - 550831128617078067183620884770829830 n 15 + 7433778702405328181849412317615206185 n 14 - 85629102516350337924331977320100536670 n 13 + 838408490408812803714606013552435231980 n 12 - 6924392948660590421210213257543519572360 n 11 + 47621754142746560050259952815740872128720 n 10 - 266990212439898717473612185174580909580640 n 9 + 1176400497183744772531266156580632500226624 n 8 - 3789454153803008908494018475390351903947648 n 7 + 7272342870346814080654742942866325781689856 n 6 + 1183528189587208304236724388053121683893248 n 5 - 58531461139589144192317082919955106351370240 n 4 + 194717598368950099140585575236722901297152000 n 3 - 334899087204805408013585556163696747683840000 n 2 + 314332136444221694228768040399684386488320000 n - 139313963861330985596316315004996917657600000 n + 17765812228768038974417736870930677760000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 30 (2 n - 51) (2 n - 57) (2 n - 45)), 153381 (1879177051351 n 29 28 27 - 845628781021755 n + 181812163775940455 n - 24862322588316592995 n 26 25 + 2428535257265627033139 n - 180409708375428115260495 n 24 23 + 10596299755112685908873475 n - 505042232692839361480943775 n 22 21 + 19893835757396249219011320165 n - 656224884795743589114240276825 n 20 + 18301858186760474983196200949925 n 19 - 434546390144186767231362303186825 n 18 + 8825093088185498162557317117160305 n 17 - 153731623884387180903811931264079525 n 16 + 2299303586391363266458929711013125825 n 15 - 29499001079254452852340302052904711925 n 14 + 323591482354737172561327548962442721440 n 13 - 3016460225447812949739651251297931094200 n 12 + 23652808032064379534427989174080704556400 n 11 - 153491852201061098122830307409259539985600 n 10 + 802753659746627309488983715339904699911744 n 9 - 3226576761829564789105269048685509143982720 n 8 + 8955570002160067372240920908216033325649920 n 7 - 10960957763687402396847887625503548243418880 n 6 - 34436791452777683341261614317225683727238144 n 5 + 227516679282851841161061577567963300775915520 n 4 - 618593040732015721498574650111657038084096000 n 3 + 969214436832411623990272372334295019683840000 n 2 - 864679947163135091444032209033342589992960000 n + 377537932560358656421950599636504857804800000 n - 51130874219381185341007145140727316480000000)/(268435456 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n 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59) (2 n - 87) (2 n - 67) (2 n - 31) (2 n - 25) (2 n - 89) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 79) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 73) (2 n - 51) (2 n - 57) (2 n - 63) (2 n - 45) (2 n - 71) (2 n - 69) (2 n - 77) (2 n - 81) / (2 n - 91) (2 n - 83) (2 n - 85) (2 n - 95)), | \ 7913046631024642614495097824 (n - 48)! (n - 47)! (n + 1) (2 n - 101)! + /99 11749675300612348124553327072 (n - 49)! |-- (n - 47)! (n + 1) (2 n - 100)! \98 + (n - 48)! ((2 n - 99)! (n + 1) \\ - 1/11749675300612348124553327072 binomial(2 n, n) (n - 51)! (n - 47)!)||/ // ((n - 49)! (n - 48)! (n - 51)! (n - 47)! binomial(2 n, n)), ( 7913046631024642614495097824 (n - 48)! (2 n - 101)! (n + 1) + 11869569946536963921742646736 (n - 49)! ((2 n - 100)! (n + 1) - 1/11869569946536963921742646736 binomial(2 n, n) (n - 51)! (n - 48)!))/( (n - 49)! (n - 51)! (n - 48)! binomial(2 n, n)), ( (7913046631024642614495097824 n + 7913046631024642614495097824) (2 n - 101)! - (n - 51)! (n - 49)! binomial(2 n, n))/((n - 51)! (n - 49)! binomial(2 n, n))] and in Maple notation [3/8388608*(93824992236881*n^25-29320310074022550*n^24+4337060266148150150*n^23 -403945911889320957500*n^22+26582822666367243401895*n^21-\ 1314782221731813532563250*n^20+50756410092869569526574400*n^19-\ 1567814543737375170377580000*n^18+39410673314750661756955076735*n^17-\ 815645462200032130168370800250*n^16+14007267820646400030156556959150*n^15-\ 200577586636702934695047954827500*n^14+2400684378818987683156304265075865*n^13-\ 24021953266277021486489625774006750*n^12+200580823345244512077956459402825900*n ^11-1391969743638103891751769466466675000*n^10+ 7977164856692366098281545954653576080*n^9-\ 37406264685438382428105056009199220000*n^8+ 141695770056969181735278000037020622400*n^7-\ 425886197639583529428320254953724560000*n^6+ 988493797379491958189063835158598572544*n^5-\ 1682533962834958221984840172403086387200*n^4+ 1779371778442131263808053692583482368000*n^3+ 106429253431241045410633965962649600000*n^2-\ 4958554871110182808265612509334077440000*n+ 8333321239683482516738481341988864000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-45), 21/16777216*(53614281278157*n^26-18121627071978391*n ^25+2905558955994210100*n^24-294023399235883291650*n^23+21076253381017677267815 *n^22-1138653828163636654214345*n^21+48161918109762473622370050*n^20-\ 1635487425999979848846378400*n^19+45365682118188606581638597795*n^18-\ 1040352913284915933040782600585*n^17+19889271488202763766952927972800*n^16-\ 318721608104852022108076282530650*n^15+4294519092660379799920183514371905*n^14-\ 48708137412000206108215917674227015*n^13+464648803273759176969602083582319050*n ^12-3718079435531130395715057564852684900*n^11+ 24839500850401917521597978963345650760*n^10-\ 137577362916039059276239715574700744880*n^9+ 625440290133628378279795829761798312800*n^8-\ 2300130659110374359836368332175868766400*n^7+ 6684325623674730947479987495133431613568*n^6-\ 14628155190105216611387500067963611574784*n^5+ 20803758812219962750437703139451019315200*n^4-\ 4797104190094142314160679494640718848000*n^3-\ 59091169223245853685908434365417185280000*n^2+ 114723742608892546877716496400036003840000*n-\ 2380948925623852147639566097711104000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 7/8388608*(80421421916098*n^26-\ 27182440606917674*n^25+4358338433527667875*n^24-441035098723459136000*n^23+ 31614380045325047145760*n^22-1707980738239506981480230*n^21+ 72242876680547746091915425*n^20-2453231091573737121212208500*n^19+ 68048519344088052400285742230*n^18-1560529111183502297885724123590*n^17+ 29833892522267533068967869039925*n^16-478081703906200100689936805579000*n^15+ 6441749670361942298298864799859020*n^14-73061198664740262626808183725803610*n^ 13+696943450758114089956154360885025175*n^12-\ 5576375451218239637858105013398244500*n^11+ 37243613218499198138135415020436491740*n^10-\ 206091917158717968632370223655340512720*n^9+ 934207028215340727543111591940786678800*n^8-\ 3404176224505036365869443128785461560000*n^7+ 9606178584393228019005301082130990265152*n^6-\ 19061261453914692572911910823684243922176*n^5+ 17493916154095452001688754634686564172800*n^4+ 31053488578906755894317672140809865728000*n^3-\ 119108531245348720324733301046311505920000*n^2+ 86301424681302970879560966079580405760000*n-\ 3571423388435778221459349146566656000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 987/8388608*(1140729388744*n^27-\ 415795862110374*n^26+72037916392487292*n^25-7894057675212888075*n^24+ 614210070045429180180*n^23-36110096102808414711180*n^22+ 1666728540493481765182740*n^21-61952722231612279795524225*n^20+ 1887363267562060168211957340*n^19-47713654569479604438602302290*n^18+ 1009755844444990611204794749620*n^17-17995371986326785602308161961725*n^16+ 271073236367891400479542739648460*n^15-3457571291797386929718364046818560*n^14+ 37344742651867753286812037332416780*n^13-340976141666234713035634671527379975*n ^12+2622392132736426260618133661329442620*n^11-\ 16887807914022965512600724299908459420*n^10+ 90209834651086735236258669196910650160*n^9-\ 393431063558817710199471719250039267600*n^8+ 1359749977251169279279706791207701166656*n^7-\ 3484115312253978280102353257572684858176*n^6+ 5417181626796943967291509925545948593408*n^5+ 114792369464007819585110948640970521600*n^4-\ 21628076588288998882454927992174262784000*n^3+ 39342244687039006322384393045359349760000*n^2-\ 22171762450429794373105646927685550080000*n+ 1342449926149618764094648969986048000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 141/8388608*(7985105715007*n^27-\ 2910571029197919*n^26+504265412353204944*n^25-55258403072859859275*n^24+ 4299470362993493872665*n^23-252770653889353635599655*n^22+ 11667097587057954233465130*n^21-433668848422549540500703575*n^20+ 13211526790012150301865951645*n^19-333994542495278876232688303065*n^18+ 7068234513296194346096330778540*n^17-125965022272042392802993355823825*n^16+ 1897412694914169444229555357818255*n^15-24199724410754764128093373843268685*n^ 14+261322618965931973024833599526270410*n^13-\ 2384726809996812108972884293177559325*n^12+ 18315874961311034927331005847832624860*n^11-\ 117559901864625249884822969019713915220*n^10+ 622941817865661147996190633686905838320*n^9-\ 2665729737152138020230854039102156449200*n^8+ 8815005015853303842959291215323189365568*n^7-\ 20295393721487171383258747996253841675456*n^6+ 21995154950670582981236053365131886502656*n^5+ 34994123064890103962167754343869609395200*n^4-\ 169400079475045159454641880244663694848000*n^3+ 237022778295588315664522159881689640960000*n^2-\ 120561864548834084267650155072198082560000*n+ 9397149483047331348662542789902336000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2* n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2* n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 987/67108864*(18251670131023*n^28-\ 7154654644295426*n^27+1335556806739511931*n^26-158001596410372426530*n^25+ 13300744608126483364935*n^24-848013685958754780427170*n^23+ 42556065587249196151693095*n^22-1724600926231127733609356250*n^21+ 57456759028341577862172262905*n^20-1593841819257552242340175523310*n^19+ 37149190243338846239846143465185*n^18-732175936557455993343640116790950*n^17+ 12253447119135774056961233001950445*n^16-174536532274664905420491324431432790*n ^15+2117224754710827174963964351481231565*n^14-\ 21848149049473236510501231811401964350*n^13+ 191193461599479902369915118174969201540*n^12-\ 1410626537564501770655849902609366918680*n^11+ 8686951549516251246930408665647689559280*n^10-\ 43877591674362527700002888998562881040800*n^9+ 176045342043396195852216106261969802953152*n^8-\ 525742337666161874333280942032455993002624*n^7+ 986177708193376210232853438642423098578944*n^6-\ 315071781098134060744890483058189215621120*n^5-\ 4039475727479668026995742632870409252864000*n^4+ 11378168344822163726752183173680530329600000*n^3-\ 13369901903989417200262185052944716759040000*n^2+ 6347647889194230266327151062962942771200000*n-\ 590677967505832256201645546793861120000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 21/67108864*( 857828480469551*n^28-336268754507355682*n^27+62771164146186267627*n^26-\ 7426073496608044519410*n^25+625134705773476577683095*n^24-\ 39856601467264959517206690*n^23+2000130358005798372297558615*n^22-\ 81055812138848820606330962250*n^21+2700435327398888746469522990985*n^20-\ 74908550187698294356781141462670*n^19+1745906786178382445432997174956145*n^18-\ 34407651471540502586800599858648150*n^17+575740975018513607972175800344495965*n ^16-8197872642698861418420020678958555030*n^15+ 99369011063564473012545091618369888605*n^14-\ 1023766250640147137664062527522459567950*n^13+ 8929350333824208942614983010929417456980*n^12-\ 65443956137603126775142771221350108276760*n^11+ 397830604635981064831357918386096468531760*n^10-\ 1960758292846701490672816756600431903773600*n^9+ 7514144338713968991677060492167352906591424*n^8-\ 20498301794125984356375657943979876254743168*n^7+ 30139408496152771059762470833955531959477248*n^6+ 24097203562652640610390814489675617789071360*n^5-\ 235049804666670443844038238945538470770688000*n^4+ 529114569327925224669448206230579992473600000*n^3-\ 566209500484838812099400297841730269511680000*n^2+ 260779281211316783167142050777836906086400000*n-\ 27761864472774116041477340699311472640000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 3741/67108864*( 9630792198242*n^29-4049747691677791*n^28+812323346205638148*n^27-\ 103456806896824399179*n^26+9394445013813054746460*n^25-647489022728865320698695 *n^24+35207971144154718990644460*n^23-1549966779848144651687434455*n^22+ 56251389265576494839228499120*n^21-1704951645224348967391408648185*n^20+ 43564493897967061711667079648180*n^19-944702089313916522342091681517265*n^18+ 17464669148855431525220162151716580*n^17-275984646813641917829131785686536365*n ^16+3731353051106279027290345680908057220*n^15-\ 43122539425548796413559301222619254685*n^14+ 424663788452490902545250230130254007310*n^13-\ 3542032753729568200059059387040268492980*n^12+ 24762202802677375777821332334883208132440*n^11-\ 142583214291966051400218003982960709555120*n^10+ 656325684862614092195330830340927726997408*n^9-\ 2284136099979796968548485541869970111841984*n^8+ 5267690260153355800604649960626180318439552*n^7-\ 4128162921915673522278617489361009366199296*n^6-\ 20830441000403765949330493119364370047365120*n^5+ 90781329308906817877173511441029351376896000*n^4-\ 171700819958146341262576890238832845762560000*n^3+ 168862042390375840542128912178302358159360000*n^2-\ 75541596205320452486463192194340342988800000*n+ 8882906114384019487208868435465338880000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 3741/ 134217728*(19261579644451*n^29-8099491296607202*n^28+1624645016991753519*n^27-\ 206913178176589460898*n^26+18788809409246340881235*n^25-\ 1294966747585555329375090*n^24+70414697028971684945776155*n^23-\ 3099822882205337323347436410*n^22+112494710052187281775540268685*n^21-\ 3409415143327386989816970819870*n^20+87104283947924001155321918285565*n^19-\ 1888353263797726507333212411198630*n^18+34891666515256984282014737411717265*n^ 17-550831128617078067183620884770829830*n^16+ 7433778702405328181849412317615206185*n^15-\ 85629102516350337924331977320100536670*n^14+ 838408490408812803714606013552435231980*n^13-\ 6924392948660590421210213257543519572360*n^12+ 47621754142746560050259952815740872128720*n^11-\ 266990212439898717473612185174580909580640*n^10+ 1176400497183744772531266156580632500226624*n^9-\ 3789454153803008908494018475390351903947648*n^8+ 7272342870346814080654742942866325781689856*n^7+ 1183528189587208304236724388053121683893248*n^6-\ 58531461139589144192317082919955106351370240*n^5+ 194717598368950099140585575236722901297152000*n^4-\ 334899087204805408013585556163696747683840000*n^3+ 314332136444221694228768040399684386488320000*n^2-\ 139313963861330985596316315004996917657600000*n+ 17765812228768038974417736870930677760000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 153381/ 268435456*(1879177051351*n^30-845628781021755*n^29+181812163775940455*n^28-\ 24862322588316592995*n^27+2428535257265627033139*n^26-180409708375428115260495* n^25+10596299755112685908873475*n^24-505042232692839361480943775*n^23+ 19893835757396249219011320165*n^22-656224884795743589114240276825*n^21+ 18301858186760474983196200949925*n^20-434546390144186767231362303186825*n^19+ 8825093088185498162557317117160305*n^18-153731623884387180903811931264079525*n^ 17+2299303586391363266458929711013125825*n^16-\ 29499001079254452852340302052904711925*n^15+ 323591482354737172561327548962442721440*n^14-\ 3016460225447812949739651251297931094200*n^13+ 23652808032064379534427989174080704556400*n^12-\ 153491852201061098122830307409259539985600*n^11+ 802753659746627309488983715339904699911744*n^10-\ 3226576761829564789105269048685509143982720*n^9+ 8955570002160067372240920908216033325649920*n^8-\ 10960957763687402396847887625503548243418880*n^7-\ 34436791452777683341261614317225683727238144*n^6+ 227516679282851841161061577567963300775915520*n^5-\ 618593040732015721498574650111657038084096000*n^4+ 969214436832411623990272372334295019683840000*n^3-\ 864679947163135091444032209033342589992960000*n^2+ 377537932560358656421950599636504857804800000*n-\ 51130874219381185341007145140727316480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 3567/134217728*(40402217395427*n^30-18180943410684255*n^29+3908931174640581370* n^28-534532192245626969625*n^27+52212102479323203924738*n^26-\ 3878615645400046034398005*n^25+227799594231140438983860150*n^24-\ 10856593268802778312512289125*n^23+427587968569215195516770132280*n^22-\ 14101167814220579404736354705775*n^21+393110268717159231651170267778450*n^20-\ 9326939764778545505034506848837875*n^19+189184629453689581205284000700233710*n^ 18-3288796780887489388746669835604232375*n^17+ 49024874661528270190581195639101296050*n^16-\ 625627326198168103396495474420456620375*n^15+ 6806407825372807553045146945341340626405*n^14-\ 62658402191221911575364356187964031749350*n^13+ 482240916534858342816836073503539754769100*n^12-\ 3044062597038756037750632533521304855451000*n^11+ 15263307152205045022806300101047466801500688*n^10-\ 57152590193560566487388962060753906764335520*n^9+ 135308036732743275687662989576534706407970880*n^8-\ 37958426384182197796985081293040336461872000*n^7-\ 1214552399709869130618168719190441780338293248*n^6+ 5588184007912109859923445999219846310887905280*n^5-\ 13682763358095719555771072497545153945056256000*n^4+ 20343778096340566110441276239916417799127040000*n^3-\ 17674185437652920891268218988614775189995520000*n^2+ 7704643250216741695886729278994190512947200000*n-\ 1099313795716695484831653620525637304320000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 331731/134217728*(868859824148*n^31-417484083060029*n^30+95986409059069915*n^29 -14058783309706655540*n^28+1473364272704484828437*n^27-117646053528455683026516 *n^26+7441671456957115204577985*n^25-382778348385819804182707800*n^24+ 16308149755028264620990068345*n^23-583210244755244697200994637110*n^22+ 17677656819816949495041590381475*n^21-457334113286545100890600471646400*n^20+ 10146566846316855327459930339737415*n^19-193598836785867073310416656711680820*n ^18+3179774973369085167371629734209214075*n^17-\ 44912367888853539414007395806756850600*n^16+ 543771674833448778259082925398934736095*n^15-\ 5610151753599496133404742325080280519285*n^14+ 48853935802506279359995923413459974565350*n^13-\ 353794424954678140287818589234276884058700*n^12+ 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23611644500278935845*n^29-3458206220378628566035*n^28+362402104174452393264717* n^27-28934770780868370064296111*n^26+1829996029084158204693579285*n^25-\ 94107288997448819359627441575*n^24+4007852869962236039158663333695*n^23-\ 143238950149135354678819287660885*n^22+4337422283792926487641374577687275*n^21-\ 112039915131253379367717289903458225*n^20+2479916211946583652314583838059667215 *n^19-47150741745781722324711421340814810645*n^18+ 770423386909752436878594866754694076775*n^17-\ 10800716570106658025638412791001800872525*n^16+ 129392860174916928509958121324066592944170*n^15-\ 1315412670741620368106843206320692225206710*n^14+ 11222676534265393975268415666591269253840500*n^13-\ 78968121212488264831831005544166425412801800*n^12+ 445133968296112662844202870791382444946957712*n^11-\ 1899467637487464525914240718190141480576676576*n^10+ 5266244044513075871302458194071931239295878080*n^9-\ 2605772700654616725708785824345777768837491840*n^8-\ 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22203389448894548561602777071148358302623989760000*n^3-\ 17893825264346111385110826790994017050089226240000*n^2+ 7686876556399571965585582557464196059391590400000*n-\ 1208098898653947275860713788393241433866240000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 99789/1073741824*(46208457933587*n^32-23656796882547184*n^ 31+5803094009762654104*n^30-908136579736107790880*n^29+101839290926340422446068 *n^28-8714798111706166732656096*n^27+591719627792512773043783896*n^26-\ 32723216502954828183014566560*n^25+1501290173195657098691750159730*n^24-\ 57901883189112043502584615757760*n^23+1895374684405267430482360850475960*n^22-\ 53015886688552294257310524731676000*n^21+1272794487326617666124010685570896660* n^20-26290630247589879392023829111908204320*n^19+ 467457807479627436768275011493391270120*n^18-\ 7143723393486090003819626299176513112800*n^17+ 93479899765251713378985426320475185779155*n^16-\ 1040622778830522292470685650524531808551760*n^15+ 9753341342548234687550832265452933956533360*n^14-\ 75706369443572957689819467271839056866476800*n^13+ 473090421659813495488455151787962433936368928*n^12-\ 2247853892868709998157829839509979271834768896*n^11+ 6898790071100102315938522611252142473874334976*n^10-\ 2135186061866297759169014664645606182700625920*n^9-\ 122438741767925457236601843205864895508325072128*n^8+ 839497405699549861906456000690475684436372566016*n^7-\ 3344535545855775418724981235340648341192833212416*n^6+ 9086939237861900028207529629611216409361783848960*n^5-\ 17157641713957320774852699453374266937745088512000*n^4+ 21846946845483083270600172016237908202190929920000*n^3-\ 17428292947303798717131131801017483692634275840000*n^2+ 7503878597823031363359751531124948326573670400000*n-\ 1208098898653947275860713788393241433866240000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 99789/1073741824*(92403346888582*n^33-50305786295696019*n^ 32+13139345986935973024*n^31-2192325500839860206040*n^30+ 262500046783351235123848*n^29-24020748592886293083286836*n^28+ 1746836018936250995865772176*n^27-103640743202825656091823787800*n^26+ 5110378222879060571897748042180*n^25-212237252449075403885240347423410*n^24+ 7496415264205445952884854734227760*n^23-226758439683230801835363594280961400*n^ 22+5901884662621635085100451434303498760*n^21-\ 132535047941114391294217062990582097620*n^20+ 2570360911023280478688354862500027636720*n^19-\ 43013476182478640103565729707027904050600*n^18+ 619337764577664412459821067854184376185830*n^17-\ 7633044358719419638504929833511618710998035*n^16+ 79847670049497228038303805439220629756906160*n^15-\ 699567510246879763865357036001444055290361200*n^14+ 5020268939785983474046887905935493390332390208*n^13-\ 28286199643855918925254155123702470478667727136*n^12+ 112734400396361995733522516423943766817703827456*n^11-\ 192508871325593300059958210201237304190501704960*n^10-\ 1253335249612112007842483238156097248856428907008*n^9+ 13580896885710269286559316287472483262104617549056*n^8-\ 71417703431812689177087375879280382820761595703296*n^7+ 253863587916401063472641422035047508932279043072000*n^6-\ 644309983933986937104799909191347595046692677222400*n^5+ 1162031072186138165027490970113826394955953479680000*n^4-\ 1433974219241085578803323037557637631910804848640000*n^3+ 1122250819283557441850504883818351276273369088000000*n^2-\ 480401994550720082082895137062645113512001536000000*n+ 78526428412506572930946396245560693201305600000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*(3417944324235424*n^33-\ 1860512853235506723*n^32+485843478671330375728*n^31-81038841401615218069080*n^ 30+9698915818268024333371936*n^29-886956180321714915420589812*n^28+ 64442746564475655558496075872*n^27-3818578286560242122018350306200*n^26+ 187959142582988908608138464705760*n^25-7787440421546582732118531449213970*n^24+ 274177182479066966089581524312982720*n^23-8258268806219543237609132139730039800 *n^22+213742070426626802833899957108265780320*n^21-\ 4765292868706990322616429409143090171540*n^20+ 91564754339897674354362459764192282055840*n^19-\ 1514306933575942075599881691523670220412200*n^18+ 21479065565010150194215679548714302576802560*n^17-\ 259666473328811133155436438971885549888882595*n^16+ 2648418787103761823021223683881526366907541520*n^15-\ 22408548886473007032823658977656523080265676400*n^14+ 152585348795160860701554239293445778375502489856*n^13-\ 782744977449171121424675130553943391278231514912*n^12+ 2438348451907550021517385211085829509958139704832*n^11+ 2150820827416236134330081861147763788708900494080*n^10-\ 86107021639135983349090048992576728534000284717056*n^9+ 635381169542438229814023668841144484804062606519552*n^8-\ 2975952756176138520874190662758899152007465918656512*n^7+ 9977455500449025662370619071242029702572483354009600*n^6-\ 24436530735794740335863758038632944579733868350668800*n^5+ 43064395192747280949845931977021359031320270479360000*n^4-\ 52384080446948895010332855598539504432690562990080000*n^3+ 40727103067559784312139019307797847925292728320000000*n^2-\ 17475780490158419472521309532676689500839084032000000*n+ 2905477851262743198445016661085745648448307200000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*(6832242692191883*n^34-\ 3946990894614409009*n^33+1095134047590611137307*n^32-194324394366802850787736*n ^31+24772553771995546658579372*n^30-2416238302551288992103208156*n^29+ 187502499081270065192597203788*n^28-11884165355878406570823610942824*n^27+ 626675534839904985495102746988570*n^26-27862136923959483596848572362354310*n^25 +1054590271949956806514819601853288330*n^24-\ 34217725773756741666106427062968684840*n^23+ 956204495901888891557415132451031682540*n^22-\ 23077384342471321535412258253882517031420*n^21+ 481499542977975839714464754882489644977660*n^20-\ 8678556003948364353750960736654525917502680*n^19+ 134760108505497504912031407061669574558655795*n^18-\ 1793526054223283129624925179493243656823952785*n^17+ 20285143281666868645171947581315482425553549555*n^16-\ 192242833605587097547456407903093207995809888640*n^15+ 1489051552424095008352092230238650132918782801552*n^14-\ 8954930523350259445541076899499724402987034758496*n^13+ 36139340185957292723737184105932017497306169291808*n^12-\ 27799761732714120752181616316629618337547754774784*n^11-\ 981995131927892531847117563343175158458118164262912*n^10+ 9753031703916591904593291773652570362111634564984576*n^9-\ 57183375680104065215452232728752039413584622962568448*n^8+ 240139684660832157193240953466773959701129624867221504*n^7-\ 752219711690884823987913088402040480907377876242636800*n^6+ 1756970021175577780726875848825455805061694161336729600*n^5-\ 2991443919358876419431830566336431977159568369786880000*n^4+ 3550650131148831028102627442139309693361056973455360000*n^3-\ 2718100945183926764506363634546836185377064812544000000*n^2+ 1159988872167622108702010016937377154653087596544000000*n-\ 194667016034603794295816116292744958446036582400000000)/(2*n-5)/(-1+2*n)/(2*n-3 )/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11 )/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 8091/1073741824*(2275249805892004*n^ 34-1313899625294658048*n^33+364359113949118697761*n^32-64605694639820723925712* n^31+8227792614782434199034936*n^30-801448930177271703714460832*n^29+ 62084053932923024817821512924*n^28-3925929635595246561527856698208*n^27+ 206406136345627323708264538150560*n^26-9141908949903647950984283863708320*n^25+ 344354491177530886658318233020583590*n^24-\ 11105474668365838920155832335864033280*n^23+ 308004051349081434301486700935258946520*n^22-\ 7364363998370183289728899593768440454240*n^21+ 151894123094677894046279345502789595581180*n^20-\ 2699027098241249259162852669804245210470560*n^19+ 41171696072229204914294073275444923978994460*n^18-\ 535667771270865364642653757778583028266551520*n^17+ 5878998505433839070528131867487268296930779265*n^16-\ 53387902969375764544063257193982434335322520880*n^15+ 386308180644604403451206751198417026782016085776*n^14-\ 2028472562961098877883603999940385952011936125312*n^13+ 5076923269230182591046107134842026475687340332384*n^12+ 32885379515341010656163017506999245183497465551872*n^11-\ 536181077510393031689159586866600489573615293747456*n^10+ 4083631675418417595576612742425042410790289448835072*n^9-\ 21663709801723873494545254075414985925431808126607104*n^8+ 86169316361685890100112045739057690449712855954976768*n^7-\ 260787615350379757818138651445734293659596037937356800*n^6+ 595064816318789617314655759813741308314722424419123200*n^5-\ 997165806101112331799796951164269591931191308410880000*n^4+ 1171808628587204666517038049766761198280669178757120000*n^3-\ 893149589565577835057472205357803438953183379456000000*n^2+ 381997734782713596719288791529844248625307189248000000*n-\ 64889005344867931431938705430914986148678860800000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 8091/1073741824*(4543101701499672*n^ 35-2778636425839296820*n^34+816931560301477603570*n^33-153734344625873135113605 *n^32+20801952538866934564074176*n^31-2155340129639751791181525080*n^30+ 177813231357197942110561597080*n^29-11990245236198628856536948919820*n^28+ 673145029432663522921729961480832*n^27-31884198315868662960271393067349600*n^26 +1286522237848602737152874302259033100*n^25-\ 44527599005796711911288189028035509950*n^24+ 1328151719622708170087643488351979229680*n^23-\ 34236069847965002512873717306344911364600*n^22+ 763451179895908647059896935043343341224600*n^21-\ 14716225681891554824058655245017756062809900*n^20+ 244502350941088383065795592642838348302250920*n^19-\ 3481875403110909639162062866268090183510580300*n^18+ 42089337495852411759973907618788284505937404050*n^17-\ 424587909288018169817286352111799075605530988325*n^16+ 3458939734387491870172357926088558143558190435088*n^15-\ 21048880486031340469894101792438986497577166016080*n^14+ 70718991489192653721986962443569143307143416278080*n^13+ 264502643512660192227213655688977347130469895406880*n^12-\ 6504542546426960017494460957067960686592651646168576*n^11+ 60144867977675588879664459099297392054974345711207680*n^10-\ 382373195038202532654581968396548330820221270757639680*n^9+ 1839285781216119715661825533382871918619812004192254720*n^8-\ 6857364981274821063818499245903900019978552247663521792*n^7+ 19800456633550881440707292016618376875932570077726924800*n^6-\ 43600527491784437154565287693581308714179052912823500800*n^5+ 71108962420238000774421614536944073614424462978375680000*n^4-\ 81914819381091878743217320114593745683267528702689280000*n^3+ 61626739992124507856932279232767447841133057343488000000*n^2-\ 26216547723456493761534548216736517736391074906112000000*n+ 4477341368795887268803770674733134044258841395200000000)/(2*n-5)/(-1+2*n)/(2*n-\ 3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2* n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-69), 2697/1073741824*( 13592819956051268*n^35-8306173881532975840*n^34+2439227064589074307310*n^33-\ 458343792033373172455375*n^32+61901554299831486670962224*n^31-\ 6398443849068228284481033880*n^30+526287399582893920438901355240*n^29-\ 35357006021576117130865524704100*n^28+1975944664770092982828079621424328*n^27-\ 93073173834863416200148585143781080*n^26+3730242275429783836877317680876477300* n^25-128060954829164208352157377020742328250*n^24+ 3782568952168113513283692460069417653120*n^23-\ 96362876523924322401697578009613828947000*n^22+ 2118451737257434245299135773662005996797800*n^21-\ 40128347577057760163621256049096412292898500*n^20+ 652290094387839414297542380202087218795855380*n^19-\ 9028916303624187841072696199605680544107544200*n^18+ 104956667054980058845717310071212458592311275150*n^17-\ 997916136935718190658152105416290771010548003375*n^16+ 7314593369531608078412684878410262264218206381872*n^15-\ 34121122880301114030934905209804724247363501723760*n^14-\ 21373664231743613208855823309366941172099467767360*n^13+ 2376205785132408904502981955782467482187253369308000*n^12-\ 28458725369881546804309754414969789278961604915316224*n^11+ 222036350739180544091381316573031636906163113620007680*n^10-\ 1303443286480996437027798618838330720750904498992151040*n^9+ 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269102872151738259011893595424247233109136578690270692305745735136356*n^26-\ 5954475412418565486386551782528995657660294164768362617800567521726960*n^25+ 119829615844374882524224230291469339305778284531844765258382372930138513*n^24-\ 2192445055259531921665908463535678483196060436383531372963261007838336136*n^23+ 36446045951933741157881006103299198410079538730956397601198412119047918936*n^22 -549925801989173863053786218267699700791446147037202586961259586343622486720*n^ 21+7521770027703278849482904904691084022334436972512309861003148567158348987888 *n^20-\ 93105395095931144873275283885208664943565812715374954990685970339572702930816*n ^19+104082707197557214416021982139674853038542648529392227750766130641459807435\ 9296*n^18-104823323247319800398959886779980635928883469205915057303998371067719\ 96369735680*n^17+94827443664336267753000711832447888468551785818288288441289301\ 479720819482087168*n^16-7678811431743422196135078315643564360611484153356266394\ 37163224410082213835667456*n^15+55431140233034718390062521536797075197706309555\ 37131301587122687188849548974012416*n^14-35498530882543288728257764090407255802\ 776404148750018942524967051004496581698109440*n^13+2005295869790963165552022127\ 47929336725112919713854593897550304294613601437010956288*n^12-99244569891506976\ 3141598016646733501576443025937809268584423086221144667794002968576*n^11+426839\ 2157791749420180959212911530366574983022816270501479343164543420363430833750016 *n^10-1579731551062212553723975536132503507601102955544767073548001766309754949\ 1766021324800*n^9+4970904295285489148196292308258704142292158276805066522743115\ 4340027995513649758208000*n^8-1310097109683488880200210112743613011935997362614\ 47325424065513920851484594609848320000*n^7+283712886897118390080265408901364750\ 720572944076097134447080137588728207727478702080000*n^6-49232658027762837227543\ 9410304149581763014427524837020383356978025491957402304512000000*n^5+6614934627\ 17675618804643991393849866775883381650828168400441663043629984797163520000000*n ^4-6549206132784001349788587663582849175113328051062199514932898915438541649477\ 63200000000*n^3+442132670621626677047689776319104709033127877945752324849357759\ 589740852124057600000000*n^2-17711037532800904684262643293565444481633269183221\ 1234832862536242379830067200000000000*n+303512077288941458145473558194345704881\ 64117375505605987092857995298603008000000000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-93)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-87)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-91)/(2*n-83)/(2*n-85)/(2*n-\ 95), -47/35184372088832*(197799014240205845232384319*n^48-\ 228763674249151752186626632968*n^47+128588905641864873867803794408148*n^46-\ 46811452335586474487388689075267280*n^45+12408346228393511469892407588541149774 *n^44-2552900199375995252028550589022284914368*n^43+ 424369239974958801543914921142333859360588*n^42-\ 58582488694223675793869612409023102992106160*n^41+ 6850697741966289986584865054658896766983219249*n^40-\ 688878551667662974298917560533371099825365428808*n^39+ 60259790302755763453139049302493582231740987363568*n^38-\ 4627827700231027284710817166466442290871767502908640*n^37+ 314338994182501644555080213034492456458023890683342164*n^36-\ 18997591287197675046395559118427583062881747586541553728*n^35+ 1026628593122146135930264354166623628659079893180881075128*n^34-\ 49807730930074503484531769682496369240968093995365752743520*n^33+ 2176648431350064656540172498788012293908629471844891471231249*n^32-\ 85913760912656945351765526780566352932620545514393650091544088*n^31+ 3069499556064178925115035434965066420143181134892865450759862228*n^30-\ 99437774930407582491166123055308966510822926023419176902722285200*n^29+ 2924727002370648597396098250522247889697466430848056027402526364206*n^28-\ 78177026325354970372404560472687534792318183455458325195804713045632*n^27+ 1900172098273639577877344745569250583998812723030049818917862004384252*n^26-\ 42009279914354530392311780886667327340117109373627837246894690901710320*n^25+ 844730079075333403348195415762937193952080686860584701606897645591483551*n^24-\ 15443953746954695413317412925100914512037046740114026654407467304350279832*n^23 +256554192509736706832345960731141259499381830718049473013754780976323095912*n^ 22-3868598960955869979071299682742104349042048514971828705584422332111829237440 *n^21+ 52882460609104589763888255131760918154344910792843879493472313971130097845776*n ^20-\ 654227240539206717665109602056351814646671349968278955677947699771907094385792* n^19+73099406052130301993857020372114572566978282585482756705828197964698353852\ 72832*n^18-73585656796486181192701290165696605934413531108005716178137597864985\ 107682344960*n^17+6654063545490778957998842737205010075397651105835377520216613\ 98790916757466439936*n^16-53861873081277855383806767860360745331805682281796817\ 24958062770263402328931510272*n^15+38867977923548289059967613593307039572464723\ 925626462720563062326986659145469724672*n^14-2488367993778398776149770547508972\ 39679208391718331589126762760546414780586254254080*n^13+14052819803721019779201\ 21084306849969923933997595948901991438325541379416294305099776*n^12-69532298135\ 25763055012438410070496300327447531251542577993433528529913107322993344512*n^11 +298987011265077699350286072491211938000727088934466675335777634998276964664954\ 86492672*n^10-11063491912532054917262475413076404966383969546985941713673767075\ 6937296513212311142400*n^9+3480786652972386695999335716200066887057904217710273\ 07534905779018546800804579901440000*n^8-917255151530223575594894105410877554608\ 584811585688694535116497132564483629105807360000*n^7+19861931833206341023266275\ 54624327522516697467169557467535051966089557417051511848960000*n^6-344637156011\ 8908953633297652872249257547320361710661080382924941191237507508862976000000*n^ 5+46303203674767881960856843622629524021088258876211516098175082317873386012803\ 07200000000*n^4-458417181590234220363604529380256461495739130873617148027264208\ 6735886119285555200000000*n^3+3094722627518392189814381109108050393317109681951\ 158102541541737228086776378163200000000*n^2-12397122526287992588387937172529809\ 31278006834066845924522873886118947219046400000000000*n+21245845410225902070183\ 1490736041993417148821628539241909650005967090221056000000000000)/(2*n-5)/(-1+2 *n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-93)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-\ 65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-17)/( 2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51) /(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-91)/(2*n-\ 83)/(2*n-85)/(2*n-95), (7913046631024642614495097824*(n-48)!*(n-47)!*(n+1)*(2*n -101)!+11749675300612348124553327072*(n-49)!*(99/98*(n-47)!*(n+1)*(2*n-100)!+(n -48)!*((2*n-99)!*(n+1)-1/11749675300612348124553327072*binomial(2*n,n)*(n-51)!* (n-47)!)))/(n-49)!/(n-48)!/(n-51)!/(n-47)!/binomial(2*n,n), ( 7913046631024642614495097824*(n-48)!*(2*n-101)!*(n+1)+ 11869569946536963921742646736*(n-49)!*((2*n-100)!*(n+1)-1/ 11869569946536963921742646736*binomial(2*n,n)*(n-51)!*(n-48)!))/(n-49)!/(n-51)! /(n-48)!/binomial(2*n,n), ((7913046631024642614495097824*n+ 7913046631024642614495097824)*(2*n-101)!-(n-51)!*(n-49)!*binomial(2*n,n))/(n-51 )!/(n-49)!/binomial(2*n,n)] The limits, as n goes to infinity are 281474976710643 1125899906841297 281474976706343 140737488336291 [---------------, ----------------, ---------------, ---------------, 281474976710656 1125899906842624 281474976710656 140737488355328 1125899905815987 18014398419319701 18014398089860571 18014396806811661 ----------------, -----------------, -----------------, -----------------, 1125899906842624 18014398509481984 18014398509481984 18014398509481984 72057569449891191 288230055313267731 144114709449488109 -----------------, ------------------, ------------------, 72057594037927936 288230376151711744 144115188075855872 72056934581110047 576447217239880131 4611425637725605767 -----------------, ------------------, -------------------, 72057594037927936 576460752303423488 4611686018427387904 4611095808734713143 4610418791332354599 288068620076966829 -------------------, -------------------, ------------------, 4611686018427387904 4611686018427387904 288230376151711744 18426558540841508451 4602261544868051091 4594779483354230769 --------------------, -------------------, -------------------, 18446744073709551616 4611686018427387904 4611686018427387904 9164958855367567449 292031106296616815283 290121325554018958713 -------------------, ---------------------, ---------------------, 9223372036854775808 295147905179352825856 295147905179352825856 287292635816431988547 1132929962530961384097 4441236614601574645881 ---------------------, ----------------------, ----------------------, 295147905179352825856 1180591620717411303424 4722366482869645213696 2159393895800675253153 1039403959135306606289 7902170976828363419479 ----------------------, ----------------------, ----------------------, 2361183241434822606848 1180591620717411303424 9444732965739290427392 59091508676180887638661 54068723638072689722105 48103245806888466390125 -----------------------, -----------------------, -----------------------, 75557863725914323419136 75557863725914323419136 75557863725914323419136 329503261330196898077995 1067552071209827639080921 ------------------------, -------------------------, 604462909807314587353088 2417851639229258349412352 197681696008639027956385 61491404496544784110403 ------------------------, ------------------------, 604462909807314587353088 302231454903657293676544 177024163259747805370489 -2350813584654299295565571 -------------------------, --------------------------, 2417851639229258349412352 38685626227668133590597632 -7552200448821792474187013 -12639271118172417670860731 --------------------------, ---------------------------, 38685626227668133590597632 38685626227668133590597632 -69927084329063205086162081 -351361588817471992322186669 ---------------------------, ----------------------------, 154742504910672534362390528 618970019642690137449562112 -207785173328113417371808567 -117779597080345546697981105 ----------------------------, ----------------------------, 309485009821345068724781056 154742504910672534362390528 -1034536300419714428249975593 -8860854053945065772218607513 -----------------------------, -----------------------------, 1237940039285380274899124224 9903520314283042199192993792 -9296553669289674725922062993 -9596290284101214218895362153 -----------------------------, -----------------------------, 9903520314283042199192993792 9903520314283042199192993792 -19559757921346564316683015777 -78980879807044817511840978529 ------------------------------, ------------------------------] 19807040628566084398385987584 79228162514264337593543950336 and in Maple notation [281474976710643/281474976710656, 1125899906841297/1125899906842624, 281474976706343/281474976710656, 140737488336291/140737488355328, 1125899905815987/1125899906842624, 18014398419319701/18014398509481984, 18014398089860571/18014398509481984, 18014396806811661/18014398509481984, 72057569449891191/72057594037927936, 288230055313267731/288230376151711744, 144114709449488109/144115188075855872, 72056934581110047/72057594037927936, 576447217239880131/576460752303423488, 4611425637725605767/4611686018427387904, 4611095808734713143/4611686018427387904, 4610418791332354599/ 4611686018427387904, 288068620076966829/288230376151711744, 18426558540841508451/18446744073709551616, 4602261544868051091/ 4611686018427387904, 4594779483354230769/4611686018427387904, 9164958855367567449/9223372036854775808, 292031106296616815283/ 295147905179352825856, 290121325554018958713/295147905179352825856, 287292635816431988547/295147905179352825856, 1132929962530961384097/ 1180591620717411303424, 4441236614601574645881/4722366482869645213696, 2159393895800675253153/2361183241434822606848, 1039403959135306606289/ 1180591620717411303424, 7902170976828363419479/9444732965739290427392, 59091508676180887638661/75557863725914323419136, 54068723638072689722105/ 75557863725914323419136, 48103245806888466390125/75557863725914323419136, 329503261330196898077995/604462909807314587353088, 1067552071209827639080921/ 2417851639229258349412352, 197681696008639027956385/604462909807314587353088, 61491404496544784110403/302231454903657293676544, 177024163259747805370489/ 2417851639229258349412352, -2350813584654299295565571/ 38685626227668133590597632, -7552200448821792474187013/ 38685626227668133590597632, -12639271118172417670860731/ 38685626227668133590597632, -69927084329063205086162081/ 154742504910672534362390528, -351361588817471992322186669/ 618970019642690137449562112, -207785173328113417371808567/ 309485009821345068724781056, -117779597080345546697981105/ 154742504910672534362390528, -1034536300419714428249975593/ 1237940039285380274899124224, -8860854053945065772218607513/ 9903520314283042199192993792, -9296553669289674725922062993/ 9903520314283042199192993792, -9596290284101214218895362153/ 9903520314283042199192993792, -19559757921346564316683015777/ 19807040628566084398385987584, -78980879807044817511840978529/ 79228162514264337593543950336] and in floating point [1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999991, .9999999950, .9999999767, .9999999055, .9999996588, .9999988869, .9999966789, .9999908482, .\ 9999765204, .9999435389, .9998720187, .9997252139, .9994387959, .9989057401, .9\ 979563931, .9963339796, .9936668302, .9894398746, .9829692858, .9733853122, .95\ 96290052, .9404684348, .9145388879, .8804093989, .8366748965, .7820696055, .715\ 5935990, .6366411573, .5451174191, .4415291881, .3270369328, .2034579906, .\ 7321547790e-1, -.6076710690e-1, -.1952198060, -.3267175008, -.4518931910, -.567\ 6552622, -.6713901053, -.7611328067, -.8356917683, -.8947176128, -.9387120311, -.9689776948, -.9875154137, -.9968788534] The cut off is at j=, 38 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 52], vs. those in the, 2, -th row from j=1 to j=, 51, are as follws 26 25 24 [3 (750599937895065 n - 253702779008520257 n + 40677825384293138600 n 23 22 - 4116327589410918774750 n + 295067547356801661361075 n 21 20 - 15941153597863476169602415 n + 674266853985037954578344550 n 19 18 - 22896824009741287059693668000 n + 635119553516145615546061465175 n 17 - 14564941059165803419309698968895 n 16 + 278449817174360536036765281070300 n 15 - 4462103344806240299196306324343750 n 14 + 60123303412583179801923369859501125 n 13 - 681915265722115394730618680844097505 n 12 + 6505125889227465279913397456379334550 n 11 - 52054268587816689319075402800749329500 n 10 + 347779669248713828251638171462622132600 n 9 - 1926601921597233801746414218918465537360 n 8 + 8764609758173471192681625702362958448800 n 7 - 32315250621642244114857556395116400264000 n 6 + 94812898216832270516973243047027809944960 n 5 - 215319383042228485100188198607580564653568 n 4 + 358796288231364308373156704326069146163200 n 3 - 368040027777013386594664048932016389120000 n 2 - 46062079419722889412437752091357388800000 n + 1067653350965934142954734039700853882880000 n - 1733330817854164363481604119133683712000000)/(33554432 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 3 ( 26 25 24 750599937894623 n - 253702779008089749 n + 40677825384092172250 n 23 22 - 4116327589351074184750 n + 295067547344037041504385 n 21 20 - 15941153595787640983117355 n + 674266853717545650357309800 n 19 18 - 22896823981719409204398201000 n + 635119551086823186143532163105 n 17 - 14564940882662357507152992588715 n 16 + 278449806330896328207337604578050 n 15 - 4462102778089210174392782669927750 n 14 + 60123278120688980184088810476346895 n 13 - 681914300179111034770050688699708485 n 12 + 6505094362768736598090258378862438300 n 11 - 52053389964146457435552480252464474500 n 10 + 347758854696193054718908989183935676240 n 9 - 1926185501981926841840846544309856148720 n 8 + 8757641158986110071255644838553802388800 n 7 - 32219023425143679014765131579717806840000 n 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(n - 50)! binomial(2 n, n))] and in Maple notation [3/33554432*(750599937895065*n^26-253702779008520257*n^25+40677825384293138600* n^24-4116327589410918774750*n^23+295067547356801661361075*n^22-\ 15941153597863476169602415*n^21+674266853985037954578344550*n^20-\ 22896824009741287059693668000*n^19+635119553516145615546061465175*n^18-\ 14564941059165803419309698968895*n^17+278449817174360536036765281070300*n^16-\ 4462103344806240299196306324343750*n^15+60123303412583179801923369859501125*n^ 14-681915265722115394730618680844097505*n^13+ 6505125889227465279913397456379334550*n^12-\ 52054268587816689319075402800749329500*n^11+ 347779669248713828251638171462622132600*n^10-\ 1926601921597233801746414218918465537360*n^9+ 8764609758173471192681625702362958448800*n^8-\ 32315250621642244114857556395116400264000*n^7+ 94812898216832270516973243047027809944960*n^6-\ 215319383042228485100188198607580564653568*n^5+ 358796288231364308373156704326069146163200*n^4-\ 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347758854696193054718908989183935676240*n^10-\ 1926185501981926841840846544309856148720*n^9+ 8757641158986110071255644838553802388800*n^8-\ 32219023425143679014765131579717806840000*n^7+ 93737597576919217063349485651201573834752*n^6-\ 205870579532486950138565193648862868606976*n^5+ 296375684362112145893104824199925905612800*n^4-\ 81450347951182068864502210804247119872000*n^3-\ 815891309433614281788876299984650321920000*n^2+ 1638183632061739819813948185531230453760000*n-\ 33333284958733930066953925367955456000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 21/67108864*(428914250222281*n^27-\ 156339244203892812*n^26+27086256585854949717*n^25-2968165692152613517500*n^24+ 230942987599551051975345*n^23-13577396327975819376580140*n^22+ 626689954627407599416267065*n^21-23294225856128495666899257900*n^20+ 709648774658220242680755283935*n^19-17940346707280791473339834708820*n^18+ 379668915183739560710357621419695*n^17-6766294528951446281931884581254900*n^16+ 101924959720910454319455792006492315*n^15-1300096496564646154152095844539924580 *n^14+14043098090460236893472457646968146355*n^13-\ 128244112328015452703111745395185452900*n^12+ 986804859052207084902653867756383776780*n^11-\ 6363708460211795211438923329355738039760*n^10+ 34121585330074133723266678542749600081360*n^9-\ 150326024368242105584031142741438434110400*n^8+ 533623985493465835786975662593172947713344*n^7-\ 1468306013972310281411271873757650156933888*n^6+ 2832390594716673530047801287065486628215808*n^5-\ 2448081849009217105773849944501885794406400*n^4-\ 4837358609868374716561846803189624766464000*n^3+ 17444616313106617712688200204825237422080000*n^2-\ 12449472640172626087141719235289827246080000*n+ 504761172232256655299588012714754048000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 21/8388608*(53614281274307*n^27-\ 19542405522189339*n^26+3385782071736090624*n^25-371020711086960547275*n^24+ 28867873360694523216165*n^23-1697174526971390635398555*n^22+ 78336242585868637337411730*n^21-2911778056454327529349312575*n^20+ 88706082235451896838986937145*n^19-2242542324491506676603746260765*n^18+ 47458555056970803305576170715340*n^17-845783873467029365223264107997825*n^16+ 12740495933723779862068563127942755*n^15-162507613977768617285958973384412985*n ^14+1755251678267014873701700726362398610*n^13-\ 16027012756133632756977591781217528325*n^12+ 123274437144374160578203936580200288860*n^11-\ 794079531554060037624824404215506600820*n^10+ 4244453703435201324612106710319437556720*n^9-\ 18538799522211901737695027417793140113200*n^8+ 64286919221138353647991649514601572648768*n^7-\ 165957565387478112706518608771091931897536*n^6+ 263182599156252995195672629023415605086976*n^5-\ 13015069046768120523769311869542055500800*n^4-\ 1006825383995801780479827639470951351808000*n^3+ 1869747849644355843317080330058872565760000*n^2-\ 1060725397341678899721244298982082805760000*n+ 63095146529032081912448501589344256000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 987/33554432*(9125835106613*n^28-\ 3577327359221266*n^27+667778419349895861*n^26-79000802585190166590*n^25+ 6650373160993402643985*n^24-424006970322302247232170*n^23+ 21278047726223978065578345*n^22-862301880145789157184460350*n^21+ 28728490235145843124903976055*n^20-796928119391596013209639160910*n^19+ 18574989607861103547760954095135*n^18-366106201315748325526444683237450*n^17+ 6127437447470265691501155915480795*n^16-87291958728215149779897461867686590*n^ 15+1059277872300107944205228942103870915*n^14-\ 10939835869312506375896539920285514650*n^13+ 95909505942567852186981120460255827240*n^12-\ 710465793102299894800863840397529329080*n^11+ 4413014165735685306861809524901013830080*n^10-\ 22693541480885291807315615170724327684000*n^9+ 94439489088505038248683888519559430909312*n^8-\ 303986905813411070439147646429921012089984*n^7+ 680039744878691799833937821962936340369664*n^6-\ 700822860204224541889740330823732090536960*n^5-\ 1220540222278881720798741763861368466944000*n^4+ 5585434318524333443309614134582374430720000*n^3-\ 7677586154136801142648370621035607408640000*n^2+ 3862636419072430682005660379903302041600000*n-\ 295338983752916128100822773396930560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 987/33554432*( 9125835085625*n^28-3577327339807366*n^27+667778410767923649*n^26-\ 79000800172723981890*n^25+6650372676894611093325*n^24-424006896534244717345470* n^23+21278038850797564546144005*n^22-862301016184812819206356050*n^21+ 28728420984597924076025328075*n^20-796923493369041071212660977210*n^19+ 18574729936103453027409958358115*n^18-366093888210343290390218116896150*n^17+ 6126942841081820070410106437830575*n^16-87275117899140643910685497560996890*n^ 15+1058792572750085900572701705101190135*n^14-\ 10928044723050842586442276633410511350*n^13+ 95669477416857062042356312348118107200*n^12-\ 706410052279595060258286713101209275880*n^11+ 4356881054490539397534911113464030601120*n^10-\ 22068909319038106556703135558681470655200*n^9+ 88997012992628421276591510185017244851200*n^8-\ 268270640725624044522138607338722298317184*n^7+ 513872115364554657451274976734337643742976*n^6-\ 207414835772483621994753789212954237199360*n^5-\ 1961796266976851902503976014895936375296000*n^4+ 5696339740204308107591353641881809950720000*n^3-\ 6764663630203251928737403295250142248960000*n^2+ 3221978008162258773356183286842268057600000*n-\ 295338983752916128100822773396930560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 28623/134217728*( 2517471721403*n^29-1058596841652919*n^28+212339917388437557*n^27-\ 27043441032165319611*n^26+2455694685305438738265*n^25-169253013274957817889255* n^24+9203363767389058830035265*n^23-405164421794057864119705095*n^22+ 14704460408361555521555534955*n^21-445699633780096525742245426665*n^20+ 11389181577235944096164061591495*n^19-247010458735855258322844928880385*n^18+ 4567763576400446067333604385522595*n^17-72222709532312642291824931295414285*n^ 16+977549700814583847321316002409013355*n^15-\ 11321688902090029609207470561262077165*n^14+ 111947979324380401366106843344737953790*n^13-\ 940733122496832108232103522510513315820*n^12+ 6664881838206457051361334740303236759960*n^11-\ 39275738294626548899649269558667912936080*n^10+ 188086293308544903870165460265138821335072*n^9-\ 701518255742919574982555318273249837365056*n^8+ 1862853191463146495413040700427952988202368*n^7-\ 2643062094264935240285002666569226886321664*n^6-\ 2319688417229766351276894174083657938606080*n^5+ 20857771370443911464056086097339694552064000*n^4-\ 46061806815621074351190006373344283607040000*n^3+ 48682969205793347612849243207472936714240000*n^2-\ 22194866294543791539887673851654779699200000*n+ 2321975458471202662309916977051729920000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 87/67108864*( 414124080973751*n^29-174139164888581713*n^28+34929909686371557669*n^27-\ 4448644204468654563597*n^26+403961414657583630061305*n^25-\ 27842067085360277177813385*n^24+1513947069711745138643717505*n^23-\ 66648954647861976876500794065*n^22+2418837667770055190058460295235*n^21-\ 73314610484682312715020040444455*n^20+1873358750932618692221726797601415*n^19-\ 40625827622602625397694072523135895*n^18+751111175724662366705940157918214115*n ^17-11871279614945007049834122238144212195*n^16+ 160548314504595024656016214816664989035*n^15-\ 1856401421072723358908080578005797856955*n^14+ 18298339741328463044977702211128866561930*n^13-\ 152860121113904860991066918892282759322140*n^12+ 1071360821731847821349565139125450987089320*n^11-\ 6193995885822348082494533557852719966934160*n^10+ 28693642084812925466989440911164125562393824*n^9-\ 100913760920442525515825868098541905146754112*n^8+ 237805811513378560074073414770569720499325056*n^7-\ 210187423450361268434575026773668119572195328*n^6-\ 837310804460040970189324715620730490379100160*n^5+ 3858060830375213103150712156890424498864128000*n^4-\ 7412567171432364009138542441356627942625280000*n^3+ 7342039951643380328580546321979417927188480000*n^2-\ 3289028274114852254760029042992463177318400000*n+ 381964962918512837949981342725009571840000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 3741/ 536870912*(154092658722527*n^30-69341681867466825*n^29+14908647815593245115*n^ 28-2038723679680828272825*n^27+199142365046280597857223*n^26-\ 14793946837408824301890765*n^25+868935739148282678295689175*n^24-\ 41416993422947143835927686125*n^23+1631556015238168626669964578105*n^22-\ 53826544485240693118862466231675*n^21+1501583870931247541461197137807025*n^20-\ 35668991480876835269813698289272875*n^19+724989061069828010545452999853496685*n ^18-12647444648210779925821651714701989775*n^17+ 189633820491168552761758511437189789725*n^16-\ 2443119180042342767903611673797847991375*n^15+ 26984861065554150438015244643075383656180*n^14-\ 254331106071305468917391096948424262605600*n^13+ 2028838553878001265046654949592143388095200*n^12-\ 13517593508480851205958374464403926929060000*n^11+ 73616707863088617768613572150368907332693888*n^10-\ 315809371726008696491908233608954453805888000*n^9+ 992041544368769505090267710579142581162645760*n^8-\ 1853311302454776029028923665159194501757996800*n^7-\ 371100966090834429924740932580909214281284608*n^6+ 14775497105922803651967820592674597772527472640*n^5-\ 48253212104461312031876572752256782817462272000*n^4+ 81927124688941658583324918185147405488865280000*n^3-\ 76093454307680519246954963044199755056414720000*n^2+ 33394121306863102812721373671182210013593600000*n-\ 4192731685989257197962585901539639951360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 3741/536870912*(154092586832797*n^30-69341618029386585*n^29+ 14908620773121069935*n^28-2038716408705945071565*n^27+199140972289541245400283* n^26-14793744613455611946516465*n^25+868912618768303558417563075*n^24-\ 41414859030115341456305083425*n^23+1631394147656341791875830754505*n^22-\ 53816338406241672585229095965775*n^21+1501044434620659702843279287620725*n^20-\ 35644963702070599521002310914918775*n^19+724084457802209794937160247698462585*n ^18-12618634029771160373101260430249503675*n^17+ 188858395965542587879082846416677017025*n^16-\ 2425534393887944082298690168120489982475*n^15+ 26650508510976172501585316982113516915430*n^14-\ 249038578991263790220219207002539185705900*n^13+ 1959756763363529750417991799766306537619800*n^12-\ 12783264361474746639833792188926801705157200*n^11+ 67361105124106087171332702949226404043995168*n^10-\ 273978379228312833074612310058353703522962240*n^9+ 778384985033816426689537675973799355852141440*n^8-\ 1050877125495518499414791787333818610400066560*n^7-\ 2459077373377655465690341596035514080862240768*n^6+ 18120540467862469994468109297874040398851440640*n^5-\ 50430003968981149513997184755198548179628032000*n^4+ 79855839432224125325163549735466749401825280000*n^3-\ 71608704456880526402086389991197238913925120000*n^2+ 31275358392896309378280878872647755209113600000*n-\ 4192731685989257197962585901539639951360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 4754811/1073741824*(484948473755*n^31-233017323044986*n^30+53575011504957700*n^ 29-7847073309175454960*n^28+822399995550377063820*n^27-65670883028331184167984* n^26+4154366035139132431472880*n^25-213721294773262421328448200*n^24+ 9107872165960072846086273450*n^23-325854842673042958857877263540*n^22+ 9883979706983824702428937914600*n^21-256000948526991093891810424770600*n^20+ 5690204855530609521465985323957900*n^19-108885170018577082409217078563441280*n^ 18+1796375029463354757762887280043918800*n^17-\ 25543538699419764832066023953808101400*n^16+ 312340896930699220502825181349211121075*n^15-\ 3268869371322564869212218834781951540290*n^14+ 29051277553661454984702976373499202311700*n^13-\ 216550649144412658186225510776698755665800*n^12+ 1327883998129193884365884899987870471748720*n^11-\ 6486257058631857247647692327892611665871584*n^10+ 23724348609072136083777340530259475411851200*n^9-\ 55009239922912785516270078448361373680711040*n^8+ 15182989461775273206075370502284661950241280*n^7+ 476764458372846561092905355413092435574769664*n^6-\ 2163188109410440351938060587199837442351226880*n^5+ 5232541801213430913625502468496360612913152000*n^4-\ 7697601740016873770236023911686338425978880000*n^3+ 6623818895217947581565778770486657152450560000*n^2-\ 2859931544468722165134731003306124194611200000*n+ 402449461597709974942120755301208555520000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21 )/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-\ 17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2* n-45), 110577/268435456*(5213177065853*n^31-2504919593124019*n^30+ 575924443278989095*n^29-84354205620424600085*n^28+8840454907888748464887*n^27-\ 705912761331276867154521*n^26+44653905364981106231211135*n^25-\ 2297002544193397242074463825*n^24+97872282966706375049046288645*n^23-\ 3500626971848952448487774882235*n^22+106132694137138638320183605343025*n^21-\ 2746751879347890223399049138289975*n^20+60974899493566000356715552603606365*n^ 19-1164401139671069763588245953348538595*n^18+ 19148479760563874994139991706206601525*n^17-\ 270940041241708885420182875298886573275*n^16+ 3288517456303127408854948903423616417370*n^15-\ 34043659992862395029485988895150658160310*n^14+ 297830074419482355138026318546557980464500*n^13-\ 2170490261718379833910179981603683127225800*n^12+ 12878929634961665025442069981228826674662032*n^11-\ 59786545725299586341054533983617620211827936*n^10+ 199229092382834781644071798691031059808454080*n^9-\ 350545074530945266047427034612289958775399040*n^8-\ 606412669838988100488671491588260326361745152*n^7+ 6837969084990143660340788127064484638804847616*n^6-\ 25574630853782554550277571936845708385899663360*n^5+ 57308523410164158968939368885484764156311552000*n^4-\ 80904449825869378539956743079436803672309760000*n^3+ 68183699417134304281319682149352041984163840000*n^2-\ 29441826479198552051222647033485184362086400000*n+ 4326331712175382230627798119487991971840000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 110577/1073741824*(41705035986559*n^32-21352740967992848*n^31+ 5238493349791125848*n^30-819928569525610001920*n^29+91973631703258579369956*n^ 28-7874043094116951303125952*n^27+535000822275382599699920472*n^26-\ 29617671110586027481309365120*n^25+1360975773599264187026241309210*n^24-\ 52615110781203557306297519591520*n^23+1728355278367043707114957428124920*n^22-\ 48590270068409282478934907645097600*n^21+1175032952012982491981826425500308420* n^20-24519634488496902352572776132939797440*n^19+ 442143001552550032878969454566482993640*n^18-\ 6887366872821258121031993957411040720000*n^17+ 92471486373887156860811863440190181765535*n^16-\ 1065292526813805878666343404032769557666320*n^15+ 10452882578138581206674993484828196632827120*n^14-\ 86374284659299810144723051447933817293974400*n^13+ 590592672817370405400729209491977444289153696*n^12-\ 3245803611727444849418436197585890000456282112*n^11+ 13555745627545784571735993004969131715211027712*n^10-\ 37032475750976447286329060971034590402299463680*n^9+ 19932610363951420840467344558315616495520890624*n^8+ 395700729939601470668508976921796386946487816192*n^7-\ 2316722661296591679865378707278713104953713139712*n^6+ 7388520878900976704759451551041465553971905822720*n^5-\ 15263827929806038128183808094183605822128758784000*n^4+ 20503990101464582283893305126307033531809136640000*n^3-\ 16787307371182902426720338926187578559370362880000*n^2+ 7193463401687395929957987186983040249023692800000*n-\ 1090235591468196322118205126110973976903680000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 2697/1073741824*(1709869943583479*n^32-875430852687832048*n ^31+214765255314400801048*n^30-33613685199375414767360*n^29+ 3770289423443980471818756*n^28-322747086955159454331707712*n^27+ 21925196929424797145094985752*n^26-1213443305759897113413611598720*n^25+ 55735123963261364908721958322410*n^24-2153244070629896379839395249602720*n^23+ 70658082782709878395607259035054520*n^22-1983318224790461809089627881992080000* n^21+47849203099280941934396781514989269220*n^20-\ 995069025259426428206602003875814751040*n^19+ 17855638764149765006753734352269671718440*n^18-\ 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44699659250196049206846410170549933053050880000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2 *n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/ (2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57 )/(2*n-63)/(2*n-45), 99789/1073741824*(92421047447194*n^33-50320601663254263*n^ 32+13145260309986117808*n^31-2193824055620756527320*n^30+ 262770614059427782506616*n^29-24057798038656915875899172*n^28+ 1750834420735090363983361392*n^27-103989648661499096868986859960*n^26+ 5135438982030205460212977321660*n^25-213737902131942783935565047106570*n^24+ 7572018790943363569709796610279920*n^23-229984029713192279565711801199469400*n^ 22+6018945359232251255504973537090916920*n^21-\ 136158723171019118620088072435147596740*n^20+ 2666169845965369703108886367223235300240*n^19-\ 45177265711012020169792137546216977051400*n^18+ 661034465220228482557241504241775448351610*n^17-\ 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*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 99789/1073741824*(92411957971150*n^33-\ 50312830161236643*n^32+13142091682279275232*n^31-2193004200879065564760*n^30+ 262619483378656477903720*n^29-24036674022469395752834292*n^28+ 1748507984968476325874893008*n^27-103782538472665718978673280920*n^26+ 5120267028712347011916018045300*n^25-212811651737595906844672096042770*n^24+ 7524460822856299000508222653818480*n^23-227917063525516484739093887456043000*n^ 22+5942567406273271966988950299894388200*n^21-\ 133752664689371465186697079882964053140*n^20+ 2601469476895488859849732182485159421360*n^19-\ 43692084403804069626484853290644275298600*n^18+ 631965735780859428431668979607068062709550*n^17-\ 7833098582401242877795004249933986556431395*n^16+ 82537296464685233439587472405107547858866480*n^15-\ 730120907984563669185748696259878402239615600*n^14+ 5311825880242427674822102088143559122787739200*n^13-\ 30605710935051504673031989049179815678074095392*n^12+ 127968021167859523826911069361087413958775492608*n^11-\ 274035456525269680161483798232828900179578307840*n^10-\ 904028722988043093038807025577625773460275806720*n^9+ 12412650865744042770415764936222702127105129573632*n^8-\ 68485855263029664813817417591583459852504128647168*n^7+ 248725102223722502471473189455308855111811713310720*n^6-\ 639061093118714641166460028764507299987297166950400*n^5+ 1161422320345609208158666652902311824248700805120000*n^4-\ 1439890400911162690059499712644170273211984773120000*n^3+ 1129250179687353846565490918046756984377298124800000*n^2-\ 483031386271319849683297867072677462515122176000000*n+ 78526428412506572930946396245560693201305600000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 99789/2147483648*(369575714068093*n^34-\ 213571852630536089*n^33+59284282161153616072*n^32-10526200220989832305256*n^31+ 1343050325192493700168012*n^30-131153317034089314370622876*n^29+ 10194141394254234467822523048*n^28-647528550000013932824418546504*n^27+ 34244545216702383595107690799470*n^26-1528328038564936443997390332471510*n^25+ 58134382774452833418825959529388680*n^24-1898263570446921802739605157439353640* n^23+53475560950793565626922402646956887340*n^22-\ 1303740534049245907297162440278496701820*n^21+ 27548126183178683895864275164223699358360*n^20-\ 504394657467444308667802577804151940940280*n^19+ 7986992457872800699934589337792061696912445*n^18-\ 108946626600740194094993747722095291316856985*n^17+ 1271810614383209183065356713157877584912631280*n^16-\ 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/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 2697/2147483648*(13669253173363951*n ^34-7897867537247668793*n^33+2191778263733961992254*n^32-\ 389021289411722983366472*n^31+49610586371984559236423284*n^30-\ 4841208134907086777918176412*n^29+375923077985355543165676434336*n^28-\ 23846439293258421694480818312648*n^27+1258826581633078617905467444441290*n^26-\ 56044714811528034837816380256030870*n^25+2124986934346229944405064481980083260* n^24-69097246552473054067122558012752344680*n^23+ 1936046725333411921636230811145082655380*n^22-\ 46877500828109643245325801435181856057340*n^21+ 981955751955000116809634442836886213833520*n^20-\ 17784068962222496123638650748556520226890360*n^19+ 277777050090191083025186343212221346846511615*n^18-\ 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n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/( 2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-\ 45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), 7585/68719476736*(13924006759478492008 *n^41-11136965388215768592797*n^40+4280479474552119541055492*n^39-\ 1052088362601150466524976186*n^38+185623498161766371474642687384*n^37-\ 25009656155712877620842565881745*n^36+2671648607194211857725589852414652*n^35-\ 231767998682960510247545036605147108*n^34+ 16573793334844800227823850917551431168*n^33-\ 984443934319391624357128212143324492234*n^32+ 48556388581072484778254111725736064209640*n^31-\ 1964742118203337588663807704968281880444076*n^30+ 62782530211432504657228211133319100549874480*n^29-\ 1397412166557611643155348220299709696400567194*n^28+ 7970078957543926384710720224869081448235535416*n^27+ 1119565164016627026348705499719943388752677896640*n^26-\ 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(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-\ 45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), 7585/1099511627776*( 394098916928229567899*n^42-327058994280040690520475*n^41+ 130357092839146685571079580*n^40-33197414108862885571541439798*n^39+ 6060601177003889557218669559067*n^38-843194252075677648536292971051711*n^37+ 92714158178739219879897998080030834*n^36-8236734059227711481227299651740012772* n^35+598135876774038499208725480729768243774*n^34-\ 35547273099189581392744955176581342182118*n^33+ 1704426114399280157706370241282547728541952*n^32-\ 62725860712951877621329846006949253711905172*n^31+ 1463197475534908856622834116607207802037205870*n^30+ 7214762358202436098713890386302559978918137114*n^29-\ 3008939987502431041091008301493668110277882401956*n^28+ 196733526457700802983939436245465674328396744844432*n^27-\ 8873969959920990562464050107296672728730059355818625*n^26+ 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2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51) /(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-91)/(2*n-\ 83)/(2*n-85), -47/8796093022208*(17911127851431728450826209*n^47-\ 20290250707886128539763431703*n^46+11150748938712450681821092496943*n^45-\ 3961631830523119257884638344246165*n^44+1023051150048601477160438022269101389*n ^43-204707544582514717694557323976292291643*n^42+ 33039038123151555672298412825846921195213*n^41-\ 4420896909732326855585373258892820702775475*n^40+ 500285266998296685272916465306455154049181364*n^39-\ 48601532167833545484384389369252487738921889548*n^38+ 4100561375838767973115452616207557375991862485098*n^37-\ 303237174586623613927348345487432095201211689490010*n^36+ 19800152840214621507043831068038373808816073684869754*n^35-\ 1148419926161419493339902401381468354993263199665472358*n^34+ 59456961510768116058095618659606075626256205601615126138*n^33-\ 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41452769191083836093629852010205946603951238630243871329337302754164521041920*n ^8+2304470968716776843240051643112239579068810003709103265373555441091931847563\ 280384000*n^7-50376231921196809972738434760948131518061026447866717959107876821\ 19090154450190336000*n^6+881657479936643294990332059318741101351994135313606224\ 3002750717238831239638548480000*n^5-1193691822327968272343452419185330426298656\ 7824047946386027643571883259793820876800000*n^4+1189833491575224351541947540322\ 8463909861772350087893250136150155289880868945920000000*n^3-8079195205715902694\ 862516569693637194262067916971786520591886074131571477053440000000*n^2+32516990\ 22092738550960115997012495078400272877371478595977081581667499376640000000000*n -559101195005944791320609186147478930045128477969840110288552647281816371200000\ 000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-93)/(2*n-49)/(2*n-7)/(2 *n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/( 2*n-43)/(2*n-13)/(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-\ 29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2* n-81)/(2*n-91)/(2*n-83)/(2*n-85), -329/35184372088832*( 23105756484648128441598181*n^48-27060662012260637680350966792*n^47+ 15393256138665249502122457696652*n^46-5667469285321171977692669290742880*n^45+ 1518478131606662959975431516289399626*n^44-\ 315606388939946380046066533058355657552*n^43+ 52971748753214791653303405947510015177332*n^42-\ 7379697398462946278971893437853988098609120*n^41+ 870497995285334142000038418083028688188530651*n^40-\ 88254981325142600419509065292392448622946022072*n^39+ 7780330569654290495640388205141723935314275641872*n^38-\ 601921935348257446434500149380893252285812336695360*n^37+ 41169959977795348238248424984033402492122586914039036*n^36-\ 2504575329994397161138047399664496576564347642129516512*n^35+ 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64117375505605987092857995298603008000000000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-93)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-\ 11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-87)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/( 2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63) /(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-91)/(2*n-83)/(2*n-85)/(2*n-\ 95), -329/35184372088832*(25292573975739455432362933*n^48-\ 29455227165005640735238370232*n^47+16666080138450009829135137009740*n^46-\ 6105089647121940963113954918512800*n^45+1627892655607005530865500185978940778*n ^44-336810691703447238164009300004629393712*n^43+ 56287280447373287096642627875854452583220*n^42-\ 7809591048279296376434304192623834794463520*n^41+ 917644492967608451829494355058327201924513163*n^40-\ 92693996215939791455326989029108389804097225352*n^39+ 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201081521266376633532476302113604244540784640*n^13+2002509558642975586569312228\ 50691443682749059169422015987569669709852549564624392192*n^12-99136471287071346\ 7760914918558292569624720210223977060493985745647423821997123207168*n^11+426486\ 1476982464405900008955595906763628059654540472020447694618354602361761357168640 *n^10-1578781287468204538743384203952316972962228157560388915292091317347560832\ 5384838840320*n^9+4968863228803295400348260361363596318336154526745473467599717\ 6888627463661282630041600*n^8-1309765954353414167318734604337611914718467121970\ 02551723810654745529222258169479168000*n^7+283676975928360516614190771030289303\ 398482192163418802852262498602000675819319459840000*n^6-49231145367734577229682\ 3249095736887457298693002941292944227745830920745625898188800000*n^5+6615171476\ 83672896632158932027403122948300528039265461374814112419904136511750144000000*n ^4-6549688207558505276555296865600148065521372783459850910636660803410475481248\ 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62295689620741521068759721525926065728680514172987723033696327047238*n^28+ 1612862374717906201223670563956914041110578276004738730942445254607264*n^27-\ 38048925488342998504092794002673776640431377759098181067753124784705612*n^26+ 817969002747675477006336289053485767390213806248179422525350962877288476*n^25-\ 16021273364194865272267774123833275024329402839928763196933690663982642283*n^24 +285767439029034074852120827881138246001836431273576828246003065081928569424*n^ 23-4638156711683480296388013705510205356870183633035691424084459324265206094392 *n^22+ 68426430349842037276219671977909411233317816055696050241122399722889414235296*n ^21-\ 916299527208701637605054861455365060242958248370802662383077784106851386573968* n^20+11117985740632423599390178274444192561605309438107566804306368344784532404\ 741504*n^19-1219739673110485496858150924988043742976280602267795241037384023072\ 91603509548032*n^18+12068626480135374676476724816731166259150968300156918983327\ 18453967060689352165376*n^17-10737247426579372028371688810525043840917263948830\ 134745051766388071955376625892608*n^16+8559252686593046237075887318315680273299\ 3151493964761681512760120189691436442189824*n^15-608809247409391875006444795016\ 982831055315421843315931574501771919055483988328814592*n^14+3845094399217237848\ 293704813198474989620425102227150794989832438322226154839678427136*n^13-2143931\ 6574387246994258785573986147073865036795480795913204982090682367683393109618688 *n^12+1048157982328574230027281446027186445167458130933954229541046898862161061\ 16354609086464*n^11-44566692887861358902062582484043996732199544653087751861908\ 9910707552231554618613235712*n^10+163186009982203527659897058487497770444057266\ 1930334385715953056016249685749401609830400*n^9-5084015387278919576605743228878\ 019392036579594678401573255621550124546126623686656000000*n^8+13275720793992609\ 634234784897221132900952074010530391506824117188434070744886773022720000*n^7-28\ 5051834602898488786613904443344551013073129815236715051917960210605702607687227\ 80160000*n^6+490788221248937180135549817275709133049426454834538308214453322677\ 66189988769693696000000*n^5-654744521483040324307532259846964856364393027963299\ 26307602844600678970540058214400000000*n^4+644110525919338454384395230397081469\ 18410868559206689528222805834994948711540326400000000*n^3-432407042460025031639\ 94892287084315483842790467774280184947821327345635145495347200000000*n^2+172402\ 9620029919247577192875659407687092361405690651303359271913236609410007040000000\ 0000*n-294406714970273214401109351448515333735191938542404378074800722554396449\ 1776000000000000)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77 )/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-29)/(2*n-\ 53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2* n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/( 2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-9)/(2*n-61)/(2*n-7)/( 2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -7/35184372088832/ (2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)*(2661098904656580792455314138*n^49-\ 3206632935467017368378979410239*n^48+1879099751458231805101960486258112*n^47-\ 713595399906878450573522423186760436*n^46+ 197445921112491555855894724284197239908*n^45-\ 42432172282441308259326096832323443332974*n^44+ 7372898395752779895958253544094592701721992*n^43-\ 1064669240229786477928277301410005031980885676*n^42+ 130337297906074023806971491787090372418164940918*n^41-\ 13731375391540812227570896381499189498161354080529*n^40+ 1259517212111842487664039983297323294023047959083432*n^39-\ 101518445457782471546680684370550111193707905180658096*n^38+ 7243750872763631535516447609624803401110223852667090008*n^37-\ 460350992884726953258538290852847881381155895914928434324*n^36+ 26186683046981821453387169488995563458470703411622480219792*n^35-\ 1338804827650945971361720612731694583797605764316672871430776*n^34+ 61725798386843662701693191351859099154425588879328491550523238*n^33-\ 2573557662106592075196144216739658382752314374276858884082926289*n^32+ 97252207048423796313094852010792895312830197102702263525495570512*n^31-\ 3336939624036178388432790388899193517505870871138616982373249330036*n^30+ 104109823016653584548602815460061516608115560895116409222221093907812*n^29-\ 2956538132352312903593356671985472877596355273357584171749136783911886*n^28+ 76477330011779135290273142019474658346752226985274781043331045777878888*n^27-\ 1802653849318287384830094710104844324265776809397126037891182090199897564*n^26+ 38722641601061732908190116792357882528079107445580580095729278010833650442*n^25 -757889797044529163066751471705901188471665455119800120623258421222404477951*n^ 24+ 13509039660321734297609111209733547005741710724283322230096755317613672590408*n ^23-\ 219119375093916939945586652087521493795114327677681916342691977864271153506024* n^22+32307469777670872884598283215387226192067020679981053803863212723056462411\ 64832*n^21-43239381101398689377787157355029832833688347883381690948625273840414\ 766503534096*n^20+5243840901940740717516420713526055683818332201282838357086589\ 82884272940923453568*n^19-57502880115490653897846147866983760896817218211128099\ 00211630080471917307125819904*n^18+56871706644217568943997525238165620505814192\ 447881873729693362437663795446908667392*n^17-5057821839757015089370553247459743\ 62724137986449565883930022974566790186922914693376*n^16+40304546986809079752399\ 42202139813797061032318223031689748279404210187270629139191808*n^15-28659054736\ 043965194290224286848917435694302956656225320700584806899115616069008001024*n^ 14+1809523334162645013480916440656026885918497726885987009389282796091201283744\ 70346432512*n^13-10086910689106021816313689633998443360459594655339635227847464\ 68900950395257477309198336*n^12+49303406226528939513013427668787110862900100890\ 26527222327496523719438870093210985791488*n^11-20959253928174374339595562589317\ 158554697555163191815817232732903953498426019042656190464*n^10+7673178422517375\ 4810547823221635276742155142125161562932051187601772450561303928229068800*n^9-\ 2390217298980111550264055345832696497207636942543835107303579472083891290835938\ 34496000000*n^8+624076008728968155235293488996557739119314155545323271499870758\ 251826787191748941578240000*n^7-13398688782848494365707247454298383913764907434\ 84112228616659801828294731735141994987520000*n^6+230675517517633332841491378400\ 1786122709492350837206071905361117789854377344407437312000000*n^5-3077213808259\ 387015516669085397836148546392434084344928798728159219033056022940876800000000* n^4+302715013072949681495227968702436518941646293571768173534158161457234313437\ 1237068800000000*n^3-2032186282295749034823947511567206273905229844153316530505\ 945995582875763357410918400000000*n^2+81025699749496042346649790373156268542499\ 8256334487918049835039933260637195468800000000000*n-138371156036028410768521395\ 180802206855540211114930057695156339600566331113472000000000000)/(2*n-91)/(2*n-\ 81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2* n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/( 2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43) /(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-9)/(2*n-61 )/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), ( 30739142682057265540923264624*(n-49)!*(n-48)!*(2*n-103)!*(n+1)+ 45652192102065245852856333600*(n-50)!*(101/100*(n-48)!*(n+1)*(2*n-102)!+(n-49)! *((2*n-101)!*(n+1)-1/45652192102065245852856333600*binomial(2*n,n)*(n-52)!*(n-\ 48)!)))/(n-52)!/(n-50)!/(n-49)!/(n-48)!/binomial(2*n,n), ( 30739142682057265540923264624*(n-49)!*(2*n-103)!*(n+1)+ 46108714023085898311384896936*(n-50)!*((2*n-102)!*(n+1)-1/ 46108714023085898311384896936*binomial(2*n,n)*(n-52)!*(n-49)!))/(n-52)!/(n-50)! /(n-49)!/binomial(2*n,n), ((30739142682057265540923264624*n+ 30739142682057265540923264624)*(2*n-103)!-(n-52)!*(n-50)!*binomial(2*n,n))/(n-\ 52)!/(n-50)!/binomial(2*n,n)] The limits, as n goes to infinity are 2251799813685195 2251799813683869 9007199254667901 1125899906760447 [----------------, ----------------, ----------------, ----------------, 2251799813685248 2251799813685248 9007199254740992 1125899906842624 9007199250227031 9007199229511875 72057593081718069 36028795044716337 ----------------, ----------------, -----------------, -----------------, 9007199254740992 9007199254740992 72057594037927936 36028797018963968 576460636280973507 576460367341493577 2305838337443485305 ------------------, ------------------, -------------------, 576460752303423488 576460752303423488 2305843009213693952 576457480410827181 4611617764285734543 4611519237844642863 ------------------, -------------------, -------------------, 576460752303423488 4611686018427387904 4611686018427387904 4611301951854021033 4610848436991543675 36879590931140932377 -------------------, -------------------, --------------------, 4611686018427387904 4611686018427387904 36893488147419103232 36865975808562575847 147365319661507018143 9199630972592450433 --------------------, ---------------------, -------------------, 36893488147419103232 147573952589676412928 9223372036854775808 73454025297186961155 73223703375621667047 1165847378916446739837 --------------------, --------------------, ----------------------, 73786976294838206464 73786976294838206464 1180591620717411303424 578610714720968511561 9157135358332882823301 9014717958043391362503 ---------------------, ----------------------, ----------------------, 590295810358705651712 9444732965739290427392 9444732965739290427392 35275576130581362415623 8556989605672638972037 65727505891863296973971 -----------------------, ----------------------, -----------------------, 37778931862957161709568 9444732965739290427392 75557863725914323419136 62265246474985518190331 115961400893198533891153 13201698908830545235085 -----------------------, ------------------------, -----------------------, 75557863725914323419136 151115727451828646838272 18889465931478580854784 2989240284900621272513915 2539694946752744433381245 -------------------------, -------------------------, 4835703278458516698824704 4835703278458516698824704 8135825765345531957427965 739012169946868024316293 --------------------------, -------------------------, 19342813113834066795298816 2417851639229258349412352 3523648331154683264010677 1017485598631039044387497 --------------------------, --------------------------, 19342813113834066795298816 19342813113834066795298816 -12404054452559981708390039 -16473996847084137885940027 ---------------------------, ---------------------------, 154742504910672534362390528 77371252455336267181195264 -423826675637890899870262549 -575952040363556218837959343 ----------------------------, ----------------------------, 1237940039285380274899124224 1237940039285380274899124224 -2865223197941799266780242207 -841823009017291237188831823 -----------------------------, ----------------------------, 4951760157141521099596496896 1237940039285380274899124224 -7601793883449234257285801549 -8321256838018280837247404957 -----------------------------, -----------------------------, 9903520314283042199192993792 9903520314283042199192993792 -4445089994796630302820477739 -9313846166298032773593599483 -----------------------------, -----------------------------, 4951760157141521099596496896 9903520314283042199192993792 -153681866604521018224976675783 -156535128610900096090780196633 -------------------------------, -------------------------------, 158456325028528675187087900672 158456325028528675187087900672 -631904103696486121652043898649 -------------------------------] 633825300114114700748351602688 and in Maple notation [2251799813685195/2251799813685248, 2251799813683869/2251799813685248, 9007199254667901/9007199254740992, 1125899906760447/1125899906842624, 9007199250227031/9007199254740992, 9007199229511875/9007199254740992, 72057593081718069/72057594037927936, 36028795044716337/36028797018963968, 576460636280973507/576460752303423488, 576460367341493577/576460752303423488, 2305838337443485305/2305843009213693952, 576457480410827181/576460752303423488, 4611617764285734543/4611686018427387904, 4611519237844642863/ 4611686018427387904, 4611301951854021033/4611686018427387904, 4610848436991543675/4611686018427387904, 36879590931140932377/ 36893488147419103232, 36865975808562575847/36893488147419103232, 147365319661507018143/147573952589676412928, 9199630972592450433/ 9223372036854775808, 73454025297186961155/73786976294838206464, 73223703375621667047/73786976294838206464, 1165847378916446739837/ 1180591620717411303424, 578610714720968511561/590295810358705651712, 9157135358332882823301/9444732965739290427392, 9014717958043391362503/ 9444732965739290427392, 35275576130581362415623/37778931862957161709568, 8556989605672638972037/9444732965739290427392, 65727505891863296973971/ 75557863725914323419136, 62265246474985518190331/75557863725914323419136, 115961400893198533891153/151115727451828646838272, 13201698908830545235085/ 18889465931478580854784, 2989240284900621272513915/4835703278458516698824704, 2539694946752744433381245/4835703278458516698824704, 8135825765345531957427965/ 19342813113834066795298816, 739012169946868024316293/2417851639229258349412352, 3523648331154683264010677/19342813113834066795298816, 1017485598631039044387497 /19342813113834066795298816, -12404054452559981708390039/ 154742504910672534362390528, -16473996847084137885940027/ 77371252455336267181195264, -423826675637890899870262549/ 1237940039285380274899124224, -575952040363556218837959343/ 1237940039285380274899124224, -2865223197941799266780242207/ 4951760157141521099596496896, -841823009017291237188831823/ 1237940039285380274899124224, -7601793883449234257285801549/ 9903520314283042199192993792, -8321256838018280837247404957/ 9903520314283042199192993792, -4445089994796630302820477739/ 4951760157141521099596496896, -9313846166298032773593599483/ 9903520314283042199192993792, -153681866604521018224976675783/ 158456325028528675187087900672, -156535128610900096090780196633/ 158456325028528675187087900672, -631904103696486121652043898649/ 633825300114114700748351602688] and in floating point [1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999995, .9999999972, .9999999867, .9999999452, .9999997987, .9999993322, .9999979739, .9999943242, .\ 9999851997, .9999638352, .9999167188, .9998183785, .9996233152, .9992542766, .9\ 985862483, .9974259887, .9954876726, .9923662285, .9875111414, .9802046780, .96\ 95494189, .9544703901, .9337367255, .9060065157, .8698962974, .8240736755, .767\ 3681810, .6988921210, .6181604025, .5251966054, .4206123338, .3056482697, .1821\ 683491, .5260277255e-1, -.8015932313e-1, -.2129214188, -.3423644621, -.46525035\ 31, -.5786272168, -.6800192112, -.7675850245, -.8402322178, -.8976787756, -.940\ 4581271, -.9698689312, -.9878755460, -.9969688865] The cut off is at j=, 39 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 53], vs. those in the, 2, -th row from j=1 to j=, 52, are as follws 26 25 24 [53 (42486788937457 n - 14360534660860115 n + 2302518417979079950 n 23 22 - 232999674872382577250 n + 16701936642851997058015 n 21 20 - 902329448937972386055725 n + 38166048339072617078486800 n 19 18 - 1296046642091928432741035000 n + 35950163409270531434241580495 n 17 - 824430626186414450940012601325 n 16 + 15761310418130800827640906782550 n 15 - 252571888070849373344966865922250 n 14 + 3403205881613358900683603132317105 n 13 - 38598978376660254707238757730514275 n 12 + 368214707965659775757387801723488300 n 11 - 2946469008437648598397745244602109500 n 10 + 19685664757199695909396532660235754960 n 9 - 109053401133267736791312644660877194000 n 8 + 496117720606031511596214086220111028800 n 7 - 1829271929966984478310574695856087688000 n 6 + 5367961276024815593584083410426464531968 n 5 - 12198376624487674001188924676964846274560 n 4 + 20378503111730113084917917647897524633600 n 3 - 21150533958327984499509465409168263168000 n 2 - 1752871054819063227145268378208583680000 n + 59799989587310583367183376114221056000000 n - 99999854876201790200861776103866368000000)/(33554432 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 29) (2 n - 51) (2 n - 45)), 159 ( 27 26 25 28324525958296 n - 10324289711792925 n + 1788715057656869199 n 24 23 - 196010941964507133750 n + 15250952017668785559570 n 22 21 - 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8349399173349596067194424682086400000000 n - 32196768269951526698729114\ 8374929225008468747888163459418823161462468163687730380800000000000 n + 547949777902672506643344724915976739147939236015123028472819104818242671\ 20934912000000000000)/(140737488355328 (2 n - 99) (2 n - 95) (2 n - 85) (2 n - 97) (2 n - 83) (2 n - 91) (2 n - 81) (2 n - 77) (2 n - 69) (2 n - 71) (2 n - 45) (2 n - 63) (2 n - 57) (2 n - 51) (2 n - 73) (2 n - 29) (2 n - 53) (2 n - 35) (2 n - 55) (2 n - 79) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 37) (2 n - 15) (2 n - 89) (2 n - 25) (2 n - 31) (2 n - 67) (2 n - 87) (2 n - 59) (2 n - 13) (2 n - 43) (2 n - 75) (2 n - 27) (2 n - 21) (2 n - 41) (2 n - 11) (2 n - 33) (2 n - 65) (2 n - 9) (2 n - 61) (2 n - 7) (2 n - 49) (2 n - 93) (2 n - 47) / (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), | \ 119476667783090503800569670048 (n - 50)! (n - 49)! (2 n - 105)! (n + 1) + / 177475050202066476519292810848 (n - 51)! | \ 103 --- (n - 49)! (n + 1) (2 n - 104)! + (n - 50)! ((2 n - 103)! (n + 1) 102 \\ - 1/177475050202066476519292810848 binomial(2 n, n) (n - 53)! (n - 49)!)|| // /((n - 53)! (n - 51)! (n - 50)! (n - 49)! binomial(2 n, n)), ( 119476667783090503800569670048 (n - 50)! (2 n - 105)! (n + 1) + 179215001674635755700854505072 (n - 51)! ((2 n - 104)! (n + 1) - 1/179215001674635755700854505072 binomial(2 n, n) (n - 53)! (n - 50)!))/ ((n - 51)! (n - 53)! (n - 50)! binomial(2 n, n)), ( (119476667783090503800569670048 n + 119476667783090503800569670048) (2 n - 105)! - (n - 53)! (n - 51)! binomial(2 n, n))/((n - 53)! (n - 51)! binomial(2 n, n))] and in Maple notation [53/33554432*(42486788937457*n^26-14360534660860115*n^25+2302518417979079950*n^ 24-232999674872382577250*n^23+16701936642851997058015*n^22-\ 902329448937972386055725*n^21+38166048339072617078486800*n^20-\ 1296046642091928432741035000*n^19+35950163409270531434241580495*n^18-\ 824430626186414450940012601325*n^17+15761310418130800827640906782550*n^16-\ 252571888070849373344966865922250*n^15+3403205881613358900683603132317105*n^14-\ 38598978376660254707238757730514275*n^13+368214707965659775757387801723488300*n ^12-2946469008437648598397745244602109500*n^11+ 19685664757199695909396532660235754960*n^10-\ 109053401133267736791312644660877194000*n^9+ 496117720606031511596214086220111028800*n^8-\ 1829271929966984478310574695856087688000*n^7+ 5367961276024815593584083410426464531968*n^6-\ 12198376624487674001188924676964846274560*n^5+ 20378503111730113084917917647897524633600*n^4-\ 21150533958327984499509465409168263168000*n^3-\ 1752871054819063227145268378208583680000*n^2+ 59799989587310583367183376114221056000000*n-\ 99999854876201790200861776103866368000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-29)/(2*n-51)/(2*n-45), 159/33554432*(28324525958296*n^27-\ 10324289711792925*n^26+1788715057656869199*n^25-196010941964507133750*n^24+ 15250952017668785559570*n^23-896620513261982648159475*n^22+ 41385185816591105227186305*n^21-1538297947896049353453382800*n^20+ 46863599540265990368352797760*n^19-1184739965083580939473016909475*n^18+ 25072480948799279474022858355665*n^17-446831053932697280539207208243550*n^16+ 6730906203245288664694899280038090*n^15-85855911439580044891648609901701725*n^ 14+927390153990360126807855515779741935*n^13-\ 8469389807145399744403992021375906300*n^12+ 65176758340465455227449596491078772780*n^11-\ 420452957027875331013008248867488915600*n^10+ 2256794009852862337386832780930927107920*n^9-\ 9975268971189618054833431404116711956800*n^8+ 35776579949554084440808956649495719741504*n^7-\ 101684578002857244143325964750379751820800*n^6+ 218232314864439336068117297960674095998976*n^5-\ 304856634847549795996107961176081820876800*n^4+ 65187076463383254313534021142435653632000*n^3+ 866944837415268507501492945743746252800000*n^2-\ 1671516917020473749880902110899185909760000*n+ 33333284958733930066953925367955456000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2 *n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2 *n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/ (2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 3/33554432*(1501199875783721*n^27 -547187354719063992*n^26+94801898052959753097*n^25-10388579923246964236200*n^24 +808300456745654038430745*n^23-47520887171051915293622040*n^22+ 2193414844070441625732806565*n^21-81529790786055614808691914600*n^20+ 2483770735382631463128243961935*n^19-62791215148038507250029056638920*n^18+ 1328841301031740604727032649526395*n^17-23682035705515947887169012574476600*n^ 16+356737563624832672274031680255056515*n^15-\ 4550345075530981315849393246714651080*n^14+ 49151066973809737535900777412526720455*n^13-\ 448860168852472949378728743256805580600*n^12+ 3453942656777433303146329539024110825580*n^11-\ 22275262694016532810517862648205524972960*n^10+ 119459768631670055684175400780499203038160*n^9-\ 526556706735146122141707451454955377769600*n^8+ 1871669502668945692114610907177025985565504*n^7-\ 5168072073301320840873610344606093540331008*n^6+ 10063259614694876740582143182621869982435328*n^5-\ 9051610027307416255198104729989038354022400*n^4-\ 16267361060818287184956562908435341850624000*n^3+ 61568899586359174066943994931640623841280000*n^2-\ 44390460746803917474906906839747918561280000*n+ 1766664102812898293548558044501639168000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 21/67108864*(857828500417787*n^ 28-336268772143155754*n^27+62771171590013286519*n^26-7426075492567463487210*n^ 25+625135087397487839596515*n^24-39856656828168550219670730*n^23+ 2000136687940701282855208755*n^22-81056397127478112291345043650*n^21+ 2700479784703315689428409500445*n^20-74911362035262698997165459134790*n^19+ 1746056013164906801373588579990165*n^18-34414331615387051025898377683783550*n^ 17+575993917818758022979558418942772705*n^16-\ 8205979021434847253037645779288413710*n^15+ 99588576452669697103111867407258976785*n^14-\ 1028774260996336073352191284607748535350*n^13+ 9024967098235782378291936234251218132260*n^12-\ 66958779588701965560269007591460169486520*n^11+ 417501451374566871939259397254465242158320*n^10-\ 2166616547862449889621816853896402826868000*n^9+ 9209903890628429733297813120781476333156288*n^8-\ 31127634399494227042629801574116690213818496*n^7+ 78278517207493042382678886610694335857939456*n^6-\ 119949719629278106311083237038736350652682240*n^5-\ 2179100517218060936946498091166912280576000*n^4+ 468822303468435364736987833283177643847680000*n^3-\ 848934839580867405628491450207122769346560000*n^2+ 475955180962695897826265424799765718630400000*n-\ 27761864472774116041477340699311472640000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 21/33554432*( 428914250107831*n^28-168134385973143002*n^27+31385585749116486447*n^26-\ 3713037732655904155230*n^25+312567540804505092756195*n^24-\ 19928327946243943258067490*n^23+1000068284160441522143718315*n^22-\ 40528192360633962138700495950*n^21+1350239361172311988511529681285*n^20-\ 37455642995848898673480581223270*n^19+873025711938917609716198690689645*n^18-\ 17207048380912326650844324805624650*n^17+287991846419124547447107292594093665*n ^16-4102799909341645132824179410708420230*n^15+ 49788303363949514900098008827290551705*n^14-\ 514226792099840773788199536730482883050*n^13+ 4508856345127087988362321680263603906880*n^12-\ 33410640511686636271922324955053659844760*n^11+ 207671149039672695229770898817556346072160*n^10-\ 1069483900161090986385344431985373589300000*n^9+ 4463814603074638140294984886682742980742144*n^8-\ 14452488062787071656860075506307111811141248*n^7+ 32730001391014808929624368143220398198761728*n^6-\ 35219522844424525645690846542953909652741120*n^5-\ 53938829485767767396073270608966032332288000*n^4+ 262002736964764543670475898678421377443840000*n^3-\ 365066662798462098183873173807870509793280000*n^2+ 184505445860045602764060790455454315315200000*n-\ 13880932236387058020738670349655736320000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 28623/33554432*( 629367937718*n^29-264649217257405*n^28+53084982386829372*n^27-\ 6760861116919909965*n^26+613923844643006723280*n^25-42313279899689241267825*n^ 24+2300844160922236708141740*n^23-101291421179715338241351825*n^22+ 3676140623634374411012476980*n^21-111426629514638412113979614775*n^20+ 2847393040154956215663124385820*n^19-61757301457016103902780446259175*n^18+ 1142131800514179411935959460575920*n^17-18062283355713011791408871526699675*n^ 16+244581422035259098594744078883263380*n^15-\ 2835242580712894392804243585973126875*n^14+ 28087805934831892265693606317653236790*n^13-\ 236943956114976250188183580805368008000*n^12+ 1691619034457517158286483755875998361960*n^11-\ 10117010042955850124642122978977974861600*n^10+ 49809040794013570571026534495122463768032*n^9-\ 195558474838978749865215499112111649592320*n^8+ 574190574614882427596685571499637791737728*n^7-\ 1067276584274583854905014292748823689890560*n^6+ 373063395372379928740393639186462730081280*n^5+ 4134723843703207872203215657702112125440000*n^4-\ 11680186628288833818618303598781368350720000*n^3+ 13677451112939461630615596309654863462400000*n^2-\ 6448257392728374794443589915906624716800000*n+ 580493864617800665577479244262932480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 28623/33554432 *(629367933956*n^29-264649213675981*n^28+53084980756758534*n^27-\ 6760860644901401289*n^26+613923747023458435590*n^25-42313264555295069538345*n^ 24+2300842256317653604552230*n^23-101291229712091099524934805*n^22+ 3676124760246661562409308310*n^21-111425533030379176670274004935*n^20+ 2847329276598859003411510436490*n^19-61754164545482800869267609217515*n^18+ 1142000838039988039133283227093490*n^17-18057638830273217723306563842711315*n^ 16+244441641404778879697905576614751810*n^15-\ 2831683526009108348173261420047555735*n^14+ 28011544696405815850907606752134284430*n^13-\ 235579607151844090121571815755656983280*n^12+ 1671464115873021886236428952508604045320*n^11-\ 9874880708592427416001996904247303540320*n^10+ 47492548833308702114219096156945336475744*n^9-\ 178416459564507198686687987169446147622144*n^8+ 480101880356836736729858032643639933855616*n^7-\ 707850699888292631579072031815675632110336*n^6-\ 484382888411059112891702448346078266731520*n^5+ 5124933945349512526954590498364683649536000*n^4-\ 11549959051858581527343039037163850086400000*n^3+ 12310429614385365279256997213954162933760000*n^2-\ 5624613267836702906680826912726011084800000*n+ 580493864617800665577479244262932480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 4089/134217728 *(35244603450697*n^30-15860070978497325*n^29+3409959050653517015*n^28-\ 466304475726452418825*n^27+45548649006628686028803*n^26-\ 3383741264397356201595165*n^25+198748231289113768939689675*n^24-\ 9473254813929167278777846125*n^23+373192293770694744702548668905*n^22-\ 12312522026591149765488686274675*n^21+343509758923835365798506791928525*n^20-\ 8161239608708549955380781271972875*n^19+165936760089005058231557909784016785*n^ 18-2896614625369162264006519837791863775*n^17+ 43483085362833862976137821556213323225*n^16-\ 561428860922063785462093979248091271375*n^15+ 6225315240180877010885417943002378912730*n^14-\ 59072720831350465218388192666091970224100*n^13+ 476678279464733697472762461963519931182200*n^12-\ 3236698600882247251944646968805847644230000*n^11+ 18175164477871255813963076730279823182830368*n^10-\ 81952493628098307781027756994307864316152000*n^9+ 280965207876305836611848248172206451498751360*n^8-\ 645079159601065809348698753838266442771340800*n^7+ 541867375688587995419812887583271143857011712*n^6+ 2280778189828753908335114114690710284610007040*n^5-\ 10203498037697196047264784403343062244923392000*n^4+ 19313476032139762047959710829469328126894080000*n^3-\ 18919653741768888388162991661290709489745920000*n^2+ 8392964543653164412697340401405350615449600000*n-\ 958975864348606699533995711522364456960000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 87 /536870912*(6625984874414711*n^30-2981692808120365875*n^29+ 641072062715815784245*n^28-87665173787498779816575*n^27+ 8563132339753828684182489*n^26-636141259304825690338408395*n^25+ 37364413457977882403519183025*n^24-1780947027169687184260874542875*n^23+ 70158145567894833363631574944515*n^22-2314619402714304583660548594333525*n^21+ 64572228557700023049628773705427575*n^20-1533950242176128925625925099771062125* n^19+31181442160406102138763131156394485955*n^18-\ 544060276485668176788666089207718708825*n^17+ 8160179694948144292859080979220060447675*n^16-\ 105188499051113785579666494611030671055625*n^15+ 1162903984018905470046997419702987837404490*n^14-\ 10976680456677774406322432554217780200425300*n^13+ 87767946970685393329545947279244503885122600*n^12-\ 586867903969157212219992287281977079000158000*n^11+ 3213320685457865750689982697304782144210044384*n^10-\ 13899454908021521584699315347788688756381201600*n^9+ 44290445021643050921755014649528866956989130880*n^8-\ 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6682611511036787617224600057105*n^22+202761301472103856448897958183975*n^21-\ 5254395464987719789678396490922525*n^20+116895938241759367445079289326908895*n^ 19-2240255114637676049003873047972510785*n^18+ 37051974600401335362744931956316799475*n^17-\ 528993864292541113829203856763370545225*n^16+ 6509849956832501957356747532396105590935*n^15-\ 68804133650381328724573253640103794603030*n^14+ 620616363500750510604287377630800758235500*n^13-\ 4728804400492421974799839681430202701498200*n^12+ 29950404878040179258336485594108873129222096*n^11-\ 153636388157893525703442269633462351929284128*n^10+ 609694146632396621843820775943700157085862720*n^9-\ 1692711209778956348138321081750111694155045760*n^8+ 2225062680675346066267951087108938165974513664*n^7+ 5263538834990318702157250901262600201222524928*n^6-\ 37835740894065185684144613952380427892750499840*n^5+ 103815578686735181071633121974139574087585792000*n^4-\ 162537314974429025422029607056304456357969920000*n^3+ 144335929375340305270748512376516838150635520000*n^2-\ 62441643058408049104670162484795014224281600000*n+ 8250213962753054486313475483674775388160000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 115971/536870912*(9941451609983*n^31-4776862025279020*n^30+ 1098290612621156170*n^29-160865763479052560570*n^28+16859343134709365834082*n^ 27-1346273537682906897676290*n^26+85166798989801657919848590*n^25-\ 4381494660618329133809410650*n^24+186726880194295021213077762720*n^23-\ 6680984216523752466080010797250*n^22+202671431723066981854140481952250*n^21-\ 5250202142557653439699737942058950*n^20+116730061037897063944372762760697390*n^ 19-2234684053996489532612316678599338950*n^18+ 36893162438634209333004151961538855450*n^17-\ 525158882868202894218477857251181387550*n^16+ 6431690992465023881706204510371448150945*n^15-\ 67467050158807181723001391749494479401450*n^14+ 601558086155952700598593561021040926191700*n^13-\ 4504636284155728121402875304948894810318600*n^12+ 27801020331240884499176229869753178250584752*n^11-\ 137094605332327476210088626455893162062035680*n^10+ 509514588252454166842789470760954767058361280*n^9-\ 1227645901477977724244941868409655062009847680*n^8+ 630720260271441974762046943197440281889200128*n^7+ 9062718085990430939592763139926464720975728640*n^6-\ 43382188489939554443546180646316591542280765440*n^5+ 106827646943052030215280634912075854329155584000*n^4-\ 158566588790060333398928514856856906731683840000*n^3+ 137042849267856414268348911435247216774348800000*n^2-\ 59169231146976439232950016653534495349145600000*n+ 8250213962753054486313475483674775388160000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 4754811/1073741824*(969895511509*n^32-496585054403600*n^31+ 121829512243658216*n^30-19069201092720389920*n^29+2139130855704835560876*n^28-\ 183147609624427815746400*n^27+12445346298811701597290664*n^26-\ 689103518898151202152513440*n^25+31674983682381160753571842110*n^24-\ 1225153542414384313347007248000*n^23+40276755582612992276096970874440*n^22-\ 1133724781368882004301825523516000*n^21+27468765566799342013728683703364620*n^ 20-574865547192783987444331127794212000*n^19+ 10411175286929433812408803871818736280*n^18-\ 163210118325552655132354488805229983200*n^17+ 2211337096328144367871423805753080880085*n^16-\ 25803671215462260053330957171407923078000*n^15+ 257734732182889443389104902890703339514640*n^14-\ 2182644067084675225217328952724223823571200*n^13+ 15444582364830142785273706712784896425456096*n^12-\ 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+5238574934866819928*n^30-819949866073877896960*n^29+91977590674262486806116*n^ 28-7874600829359942177946432*n^27+535062688840576872923830872*n^26-\ 29623213310423323162567681920*n^25+1361383841236424355324064802010*n^24-\ 52640113745979969065535274645920*n^23+1729641338935212975014995135785720*n^22-\ 48646137037052796771681469478160000*n^21+1177090486184374568943198135755172420* n^20-24584011776351912215773290708881413440*n^19+ 443855104065158752006541865163691364840*n^18-\ 6926033791840743929076481351155396201600*n^17+ 93211393323885863444320510684538918652735*n^16-\ 1077245471195078909048281675929164314615920*n^15+ 10615085721687819453454082265698456824033520*n^14-\ 88211213425882009657179422291012478369961600*n^13+ 607809918798485974320117151230074426599474336*n^12-\ 3377963851700315634278053139699320768556306432*n^11+ 14375412440326731618823114406628781202378341632*n^10-\ 41067215690944187756502390915656755620579440640*n^9+ 35308313080355897495748457616332059467368529664*n^8+ 352013691061834374681754367113979628824340099072*n^7-\ 2230059914978120751896595479973151075105775296512*n^6+ 7285485940259789712685457778314913558018016542720*n^5-\ 15232615513210801292266744591393656694868189184000*n^4+ 20600817379547155814112144154417344170504355840000*n^3-\ 16921395283816541390447675450163418118034554880000*n^2+ 7247526850224159414726790295558935184985292800000*n-\ 1090235591468196322118205126110973976903680000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 110577/1073741824*(83409382692638*n^33-45415669622826591*n^ 32+11864631442086178976*n^31-1980279408053079153240*n^30+ 237226590685757847023432*n^29-21723927043870975690786404*n^28+ 1581520087600400993841303024*n^27-93981705089347153620676082520*n^26+ 4644795237506866997598749096820*n^25-193539064762783818673511878022490*n^24+ 6867958997676410542387778271108240*n^23-209105250513952855801595009625271800*n^ 22+5491366797072906827124727159720200840*n^21-\ 124822059830851182086895242695876314180*n^20+ 2460372366187845637968351783673812587280*n^19-\ 42064168099541110725186920213954777197800*n^18+ 622862049422360239269502350598902480757470*n^17-\ 7961090979536710078355456903340712966239615*n^16+ 87332345814178588861847078869378656335173840*n^15-\ 815045537605759342064029254610763695208502000*n^14+ 6385661001062352858521300623485023155938177472*n^13-\ 41126287270347817756024771546881110024894956704*n^12+ 209858478784322840166783670935923165219073708544*n^11-\ 783249168980240996142989004107564596640849583360*n^10+ 1614074748098449093642680973858508062660368496128*n^9+ 2625104932554534822446721433289734355560850505984*n^8-\ 39132953588787275454507339508915145284791752699904*n^7+ 182572355090105804530903229717963534738144712990720*n^6-\ 530966963272992736027911943196109959245221514444800*n^5+ 1038952735042145105312905002334257593401713008640000*n^4-\ 1347050388373560898022649891008894243464774287360000*n^3+ 1079252572621093561215600973075142288109718732800000*n^2-\ 459213219827948248561588971233820051627835392000000*n+ 70865313445432760937683333197213308498739200000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 2697/1073741824*(3419679082781758*n^33-\ 1861948041326828631*n^32+486409605945118489216*n^31-81180539398041372890040*n^ 30+9724180860731869711097512*n^29-890371688741054930276060964*n^28+ 64806552150660026718204258384*n^27-3849903028031457641998427851320*n^26+ 190178706348154341504365466121620*n^25-7918526086064768347090857240534090*n^24+ 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n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-69), 8091/4294967296*( 36412997998285471*n^36-23577762527492040558*n^35+7348533956645295451065*n^34-\ 1468133496207055236765570*n^33+211242756727710137534007288*n^32-\ 23316295181856620697349804824*n^31+2053285964871315553619531403860*n^30-\ 148128520642494754554564419584680*n^29+8919774360001802238443925088356306*n^28-\ 454476349071346339636465193311133988*n^27+ 19791168250859297362907678928608653950*n^26-\ 742049828349847665209250410877678841500*n^25+ 24081182878718366587897376052032864259540*n^24-\ 678767230942901548174883073422097621016920*n^23+ 16649299659478171827797786765819897125805700*n^22-\ 355538109094163169407981584525432249464558600*n^21+ 6602348967348062812949838038989718813379866535*n^20-\ 106307611725164512818069436631199125356186526430*n^19+ 1476446199934954817386370899987348405031555798225*n^18-\ 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3643772357777153667525546205175896262008043851481088000000*n+ 627190819309975235519187659652481993280907431116800000000)/(2*n-5)/(-1+2*n)/(2* n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n -11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2 *n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/ (2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 299367 /8589934592*(7771896672509407*n^38-5587293231741525479*n^37+ 1935711685782806765319*n^36-430368736724105706535785*n^35+ 68986534290947158897554606*n^34-8491751500840680778816262292*n^33+ 834766834823218640498019368372*n^32-67286951135298642867296493570100*n^31+ 4531049282391870655480718837400042*n^30-258384645963312708263174418108324114*n^ 29+12603176521710035446800240110547844194*n^28-\ 529692508455473585282032691209467152190*n^27+ 19282418113190494737558821387350557362280*n^26-\ 610060269160314642653378145511321425237660*n^25+ 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2095569/34359738368*(4404192621141679*n^38-3159942720126759863*n^37+ 1092151855321023167343*n^36-242128929845596712243145*n^35+ 38681469626210667415247382*n^34-4742470048299336417165919524*n^33+ 464024031414643879406432478284*n^32-37199062705793320069882297083700*n^31+ 2489063477955116148759158199647274*n^30-140894277460150568206647021601249458*n^ 29+6813643018104448590716632466128375218*n^28-\ 283523087236443523731576666812892386430*n^27+ 10201429825469155226339897971871401739160*n^26-\ 318348317373741804728489645609683654249020*n^25+ 8626009042964429833435318928374919792557020*n^24-\ 202765796902705622969685021296353254225612900*n^23+ 4120488996961179825000304388830823458572364815*n^22-\ 71863224975107420992617818427849173230899968655*n^21+ 1060722368180308804439164242028409689516278172055*n^20-\ 12882644287981814711486483748419182497603688425025*n^19+ 120345645008931748003107095801463490711572081796266*n^18-\ 676042081463619417810256221437057632660842363005352*n^17-\ 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1241175948447884971834756572100177453953448951362355200000000*n-\ 215036852334848652178007197595136683410596833525760000000000)/(2*n-5)/(-1+2*n)/ (2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/( 2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67) /(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-\ 35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2* n-69), 6286707/34359738368*(2900778454383284*n^39-2187430260770557179*n^38+ 795098123043369355171*n^37-185505845950893209663571*n^36+ 31210110779969644847428377*n^35-4032809688274034287179414582*n^34+ 416206265718340133250477840148*n^33-35224759069389676219358298040668*n^32+ 2490663691040773794526466370133964*n^31-149137278977066413806702000765281874*n^ 30+7637905789086361407120835632483483546*n^29-\ 336985973723295024046662360077633270826*n^28+ 12872854464291385523810887537865217705870*n^27-\ 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)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-\ 35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2* n-69)/(2*n-77), 6286707/34359738368*(2851410407200445*n^39-2143147122447550596* n^38+776041612518368855620*n^37-180266309307653393769594*n^36+ 30175898065757978978085915*n^35-3876685922512025536039457058*n^34+ 397455520847193579550659828520*n^33-33384481007464580195818212308712*n^32+ 2340218726698997533433592765352850*n^31-138746100841753146463020062730416736*n^ 30+7024955159565570479427837545397026760*n^29-\ 305849668284065292585481031421260064204*n^28+ 11502011453938878024530936666730890085990*n^27-\ 374495571490468492761588190022013504702540*n^26+ 10556376939330933761945779385058732819949200*n^25-\ 256857135040040222500313241023112838205138760*n^24+ 5355316226760558271705718926375014112100666625*n^23-\ 94226704023919600490899447674404954520063121260*n^22+ 1353924199234046319318526308924248578368487508500*n^21-\ 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2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), 65231/ 2199023255552*(237225642344777390537*n^44-206243498095849351164214*n^43+ 85466430067278545755297553*n^42-22377656156097386896768505566*n^41+ 4128643892865921093208303397099*n^40-564450622886421740556099304451698*n^39+ 58017158385463321068231806417781391*n^38-4344766770980618078643813569998485362* n^37+198299942517222448055011720308954421486*n^36+ 1847448661945907050933767462975726577228*n^35-\ 1433936397844090873658676590104722003884246*n^34+ 170808649137754002793778206182956340030442132*n^33-\ 13847882551832121534132694334377801066151619882*n^32+ 890191469770225262404427925900284138246226120924*n^31-\ 47849862924222933213485809860645342987543354892578*n^30+ 2206973075569647815764235248690743960048543715185756*n^29-\ 88658392263401841304783507059363150413631551258233995*n^28+ 3131802779074720852420762495606743568318029746192891970*n^27-\ 97909914798135136238964023636714142420559188152196374315*n^26+ 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*n-133708772931609322778305480774614404700810044677886777139973324800000000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/( 2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13) /(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2* n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), 65231/1099511627776*(84415564309968866503*n^44-69540042809028403734386*n^43+ 26801089231263361273918747*n^42-6323520121783836785478146174*n^41+ 989301834793079337862243022881*n^40-99054872841835641644346085065182*n^39+ 3960707753438789520379760988732149*n^38+652445728327355671631113482285427862*n^ 37-168795682279113103638023021057471198506*n^36+ 22616833899683707709778221505068395695332*n^35-\ 2233088452968288753501789191489416221007234*n^34+ 177690957624632609353481148093682011812239988*n^33-\ 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6+962259824270535061773112614361769715421139757788775535728192974149386240000*n ^5-1338265716106059224124569988358105694783014491073513662938012579581132800000 *n^4+ 1364875635084174696380835528031790638937825219512697681776334396719104000000*n^ 3-944390647173731739523798359363893970346087229878659666172612108615680000000*n ^2+385554789586980033607420463123292270309793073637412772175528591360000000000* n-66854386465804661389152740387307202350405022338943388569986662400000000000)/( 2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2* n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/( 2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23) /(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-\ 57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), 1763/1099511627776*(3541368771599594112218*n^45-2672569123017979610581875*n^44+ 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13546702983848263680000000000*n+ 220151494631894749954479974095402617339883738562140578560966079283200000000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/( 2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13) /(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-\ 17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2* n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/( 2*n-85), 1763/1099511627776*(714536702786561827268*n^45+ 176877602345556932647725*n^44-530535280753514616563071110*n^43+ 309476561655272900037062302125*n^42-103914184766375669647231221438102*n^41+ 24362736288742094615144862410818035*n^40-4340246000034306322242073449244004770* n^39+616475893605349685258268887094639560475*n^38-\ 72017128324181634803426053518060235324182*n^37+ 7071440228422703274136453843222970123332770*n^36-\ 593011188710893151117733891296175187201312980*n^35+ 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819645337637159615171641879176558086370932129216818702623781920203278830796800* n^7+183199119344603512754897640665502184522207467849564564419280534084299430297\ 6000*n^6-3270255903992088062010683720729771459132728306436460820267373557292736\ 184320000*n^5+45051159470328626260608251875610709018568264297821455131142905814\ 41767014400000*n^4-455790597848992748128600739067609666949129711312443391300900\ 9660287516672000000*n^3+3133247698720365673989663411364141914347570415317856312\ 396356908209930240000000*n^2-12730218813638043630337981690279219552407011711675\ 71128242281308487680000000000*n+ 220151494631894749954479974095402617339883738562140578560966079283200000000000) /(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/( 2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13) /(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-\ 17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2* n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/( 2*n-85), -1763/4398046511104*(17369501606347269042281*n^46-\ 25279585472878699981200373*n^45+16516595810753207157966703755*n^44-\ 6654810597284368274022012913365*n^43+1887904797877618517284457631063141*n^42-\ 405549557407317408151560208372394583*n^41+ 69050883430762450924319877628963587725*n^40-\ 9613571732703306340743941363915420625355*n^39+ 1119276807629137560595934989098860270503956*n^38-\ 110823081400979213529438959818889349301135758*n^37+ 9453521324988687268821782237942862605265113670*n^36-\ 701878975272519649448665349116425692407733969970*n^35+ 45728109406357565670003431551901304338496274716506*n^34-\ 2631629768305643950879576175758044451899385690205038*n^33+ 134499696525013304206122373339008843057149726830080090*n^32-\ 6131651436255390951594950600083739060039819233672273110*n^31+ 250232674575807317950082659132579262353869402114388052961*n^30-\ 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000000*n^2-80495357734405002719490913252434358494810557726983753773742276836316\ 673616327475200000000000*n+1369874444756681266608361812289941847869848090037807\ 5711820477620456066780233728000000000000)/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/( 2*n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57) /(2*n-51)/(2*n-73)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-\ 23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2* n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/( 2*n-65)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-\ 1+2*n)/(2*n-5), -7/140737488355328/(2*n-99)/(2*n-95)*( 21326475829789728500134257719*n^50-26753248383393293080409477301625*n^49+ 16330308660218754763762847132569300*n^48-6463562221479453029440521584363733500* n^47+1865142663410579222661532036990815810510*n^46-\ 418295456431392146143061215644925450026450*n^45+ 75900077962069920337307335789128201865802300*n^44-\ 11453542229016546131408261193703511052904334500*n^43+ 1466337330791604719585301454757063146751595083305*n^42-\ 161678802210262642338492506047697619717340049001575*n^41+ 15533409788101057429600749282000034870275285720916800*n^40-\ 1312496891179940156792026125963835756633400732857758000*n^39+ 98263669317961082985582981398936800402872736217870006420*n^38-\ 6558435134995460687321776645559484301477749643058756262700*n^37+ 392193575352168144878117565178094719900491436370390415851800*n^36-\ 21100693052342706665950019493774301123884517303933314697289000*n^35+ 1024900583630429704811669663453600935584967027599509567177203065*n^34-\ 45070082089407488255727970006732850539116394609352370475454719575*n^33+ 1798570445110944425953728470212837099522526724899937061479785533300*n^32-\ 65255499455848177632417206163315647919762251338672703431940600157500*n^31+ 2155782000746714186295181480744248618294801895835073568873278243457262*n^30-\ 64920903096774067118764076452050224145221883475084702857958094873110450*n^29+ 1783649152514630671910771431707535850909769490951310792900959204899830700*n^28-\ 44730247950553573625252261033822163034759125541216698491181474611925736500*n^27 +1024134855378769400733563075436533582912525565396448954308258604843215321415*n ^26-\ 21406863758079168375989430051228822580615056593916829608434721472467688800425*n ^25+ 408362505232991113277444911823670768729291050366598379272440340515562892590200* n^24-71051405746383737097679142620227087209574670805909873855409500799017801462\ 23000*n^23+11265437212050299152469477901961793420887591278769865623710373228889\ 3249520101520*n^22-162576952022750428465835095104855541291076660630983601067546\ 0849055924278779602800*n^21+213234421111317203878423459106557059376592281128086\ 89230176735019171133996758803200*n^20-25371689214725332852539924173132630127371\ 3292148259857583327888841845601374702912000*n^19+273263891917076127744392099299\ 8441213010782611955361812057478052444126407212537985280*n^18-265721026091143544\ 86392151812959716680217000781539298560663423898283269404831791404800*n^17+23256\ 8328903688180751393781729871813314809221162526882629847231375387940491939333683\ 200*n^16-1825565374638475975068926909602502780606735894967876572566173636468890\ 298140830169856000*n^15+1279797161041549729411380294222039104914839835585408758\ 5955201252105640179936701360271360*n^14-797336719743384936737882325723940565900\ 62462255604612712558596936983002340715457472204800*n^13+43891560497077508749138\ 4954045766489197274114178088407702138990979718003543389766677299200*n^12-212020\ 7753469486217638574437095274809414814576565284008125272160937021482341769592217\ 600000*n^11+8914100059079946382072668839533681126550502630426142043678259440160\ 227599889575235256582144*n^10-3229896899092924585966312380820653308902004627921\ 4138852795467618402066581510500989455564800*n^9+9964734345221641303945382789728\ 9915877036119055844798161507471354629665154722051517317120000*n^8-2578564473916\ 4070506511806299607888267825313827963899651996550449611435271291027130941440000\ 0*n^7+5490457217453304307832139350856436907565029683918705707218760618534619952\ 23949256865873920000*n^6-938094178598354673745793733160765443858715588940273042\ 902569732125384978183091495370752000000*n^5+12427913650341962770968507545708816\ 29211040622060872073470776125954966599613092672307200000000*n^4-121500404922185\ 7463151205453870158238679021324228656186462630621512384363767189733376000000000 *n^3+81122417911634826444602251307232279340628027376787169834939917334959606719\ 4424682086400000000*n^2-3219676826995152669872911483749292250084687478881634594\ 18823161462468163687730380800000000000*n+54794977790267250664334472491597673914\ 793923601512302847281910481824267120934912000000000000)/(2*n-85)/(2*n-97)/(2*n-\ 83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2* n-51)/(2*n-73)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/( 2*n-17)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59) /(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-\ 65)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2* n)/(2*n-5), (119476667783090503800569670048*(n-50)!*(n-49)!*(2*n-105)!*(n+1)+ 177475050202066476519292810848*(n-51)!*(103/102*(n-49)!*(n+1)*(2*n-104)!+(n-50) !*((2*n-103)!*(n+1)-1/177475050202066476519292810848*binomial(2*n,n)*(n-53)!*(n -49)!)))/(n-53)!/(n-51)!/(n-50)!/(n-49)!/binomial(2*n,n), ( 119476667783090503800569670048*(n-50)!*(2*n-105)!*(n+1)+ 179215001674635755700854505072*(n-51)!*((2*n-104)!*(n+1)-1/ 179215001674635755700854505072*binomial(2*n,n)*(n-53)!*(n-50)!))/(n-51)!/(n-53) !/(n-50)!/binomial(2*n,n), ((119476667783090503800569670048*n+ 119476667783090503800569670048)*(2*n-105)!-(n-53)!*(n-51)!*binomial(2*n,n))/(n-\ 53)!/(n-51)!/binomial(2*n,n)] The limits, as n goes to infinity are 2251799813685221 562949953421133 4503599627351163 18014398508773527 [----------------, ---------------, ----------------, -----------------, 2251799813685248 562949953421312 4503599627370496 18014398509481984 9007199252264451 9007199240651157 4503599593405647 144115183509900033 ----------------, ----------------, ----------------, ------------------, 9007199254740992 9007199254740992 4503599627370496 144115188075855872 576460684074079857 288230261054276391 1152920084661338493 ------------------, ------------------, -------------------, 576460752303423488 288230376151711744 1152921504606846976 4611669846973619799 4611643170474946863 4611579655001916063 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 4611437243131200663 1152783773153041539 36884207595346747071 -------------------, -------------------, --------------------, 4611686018427387904 1152921504606846976 36893488147419103232 2304677405168360637 73715211010432895709 294617566804127745861 -------------------, --------------------, ---------------------, 2305843009213693952 73786976294838206464 295147905179352825856 73551172644505737063 73382321302737388461 2284861087154745531 --------------------, --------------------, -------------------, 73786976294838206464 73786976294838206464 2305843009213693952 2326649391159123645369 9229289526893247120351 4559086053655143051447 ----------------------, ----------------------, ----------------------, 2361183241434822606848 9444732965739290427392 4722366482869645213696 17925981766819887984615 70017460221293823392721 67786683969989949654971 -----------------------, -----------------------, -----------------------, 18889465931478580854784 75557863725914323419136 75557863725914323419136 64910935274675141600471 61295160581765989606613 227432242893590540306063 -----------------------, -----------------------, ------------------------, 75557863725914323419136 75557863725914323419136 302231454903657293676544 3298609618899110497458245 725044854005567078454455 -------------------------, -------------------------, 4835703278458516698824704 1208925819614629174706176 4888545174480446812413715 15474465875792173962119047 -------------------------, --------------------------, 9671406556917033397649408 38685626227668133590597632 5506511675503579130857193 3121716572165042209920167 --------------------------, --------------------------, 19342813113834066795298816 19342813113834066795298816 314932051753177125368371 -30622431331990235321541403 -------------------------, ---------------------------, 9671406556917033397649408 309485009821345068724781056 -284748083187153995908371655 -221241779995519266664282493 ----------------------------, ----------------------------, 1237940039285380274899124224 618970019642690137449562112 -1183682290339616648511488771 -5834649362287391410951539809 -----------------------------, -----------------------------, 2475880078570760549798248448 9903520314283042199192993792 -6816396318488415214470574109 -7662880360723964627282541461 -----------------------------, -----------------------------, 9903520314283042199192993792 9903520314283042199192993792 -8364194434363562329201635185 -17836315889317340176860562051 -----------------------------, ------------------------------, 9903520314283042199192993792 19807040628566084398385987584 -149285330808528099500939804033 -1201691050273520291949922261 -------------------------------, -----------------------------, 158456325028528675187087900672 1237940039285380274899124224 -313179004188835772130407999155 -1263916954360007823252935403187 -------------------------------, --------------------------------] 316912650057057350374175801344 1267650600228229401496703205376 and in Maple notation [2251799813685221/2251799813685248, 562949953421133/562949953421312, 4503599627351163/4503599627370496, 18014398508773527/18014398509481984, 9007199252264451/9007199254740992, 9007199240651157/9007199254740992, 4503599593405647/4503599627370496, 144115183509900033/144115188075855872, 576460684074079857/576460752303423488, 288230261054276391/288230376151711744, 1152920084661338493/1152921504606846976, 4611669846973619799/ 4611686018427387904, 4611643170474946863/4611686018427387904, 4611579655001916063/4611686018427387904, 4611437243131200663/ 4611686018427387904, 1152783773153041539/1152921504606846976, 36884207595346747071/36893488147419103232, 2304677405168360637/ 2305843009213693952, 73715211010432895709/73786976294838206464, 294617566804127745861/295147905179352825856, 73551172644505737063/ 73786976294838206464, 73382321302737388461/73786976294838206464, 2284861087154745531/2305843009213693952, 2326649391159123645369/ 2361183241434822606848, 9229289526893247120351/9444732965739290427392, 4559086053655143051447/4722366482869645213696, 17925981766819887984615/ 18889465931478580854784, 70017460221293823392721/75557863725914323419136, 67786683969989949654971/75557863725914323419136, 64910935274675141600471/ 75557863725914323419136, 61295160581765989606613/75557863725914323419136, 227432242893590540306063/302231454903657293676544, 3298609618899110497458245/ 4835703278458516698824704, 725044854005567078454455/1208925819614629174706176, 4888545174480446812413715/9671406556917033397649408, 15474465875792173962119047 /38685626227668133590597632, 5506511675503579130857193/ 19342813113834066795298816, 3121716572165042209920167/ 19342813113834066795298816, 314932051753177125368371/9671406556917033397649408, -30622431331990235321541403/309485009821345068724781056, -\ 284748083187153995908371655/1237940039285380274899124224, -\ 221241779995519266664282493/618970019642690137449562112, -\ 1183682290339616648511488771/2475880078570760549798248448, -\ 5834649362287391410951539809/9903520314283042199192993792, -\ 6816396318488415214470574109/9903520314283042199192993792, -\ 7662880360723964627282541461/9903520314283042199192993792, -\ 8364194434363562329201635185/9903520314283042199192993792, -\ 17836315889317340176860562051/19807040628566084398385987584, -\ 149285330808528099500939804033/158456325028528675187087900672, -\ 1201691050273520291949922261/1237940039285380274899124224, -\ 313179004188835772130407999155/316912650057057350374175801344, -\ 1263916954360007823252935403187/1267650600228229401496703205376] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999997, .9999999984, .9999999925, .9999999683, .9999998816, .9999996007, .9999987684, .9999964934, .\ 9999907088, .9999769361, .9999460555, .9998805370, .9997484501, .9994944998, .9\ 990273990, .9982031437, .9968042646, .9945159022, .9909005418, .9853743455, .97\ 71890386, .9654240242, .9489935730, .9266733702, .8971492923, .8590890752, .811\ 2346956, .7525101680, .6821364813, .5997430465, .5054637240, .4000055676, .2846\ 799813, .1613889641, .3256321094e-1, -.9894641214e-1, -.2300176698, -.357435373\ 2, -.4780854697, -.5891490275, -.6882801370, -.7737531825, -.8445678071, -.9005\ 038271, -.9421228896, -.9707182999, -.9882186910, -.9970546727] The cut off is at j=, 40 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 54], vs. those in the, 2, -th row from j=1 to j=, 53, are as follws 27 26 25 [53 (169947155749829 n - 61945738270811928 n + 10732290345967997313 n 24 23 - 1176065651795471821800 n + 91505712107916461191005 n 22 21 - 5379723079900399412053560 n + 248311114944564943348791285 n 20 19 - 9229787692404278737833037800 n + 281181597707619435711312171315 n 18 - 7108439826813465837692572485480 n 17 + 150434888093580970846090882346355 n 16 - 2680986459163129279145852819815800 n 15 + 40385443785857541046322660021580735 n 14 - 515135742117660462165869372631082920 n 13 + 5564350724708750435480508111552505095 n 12 - 50816640774942033165363513650191327800 n 11 + 391068524382427935138959617869195357420 n 10 - 2522897459692949996782114455002028903840 n 9 + 13544196159154993302455208217267039449040 n 8 - 59906603647895154485007153528487963036800 n 7 + 215388423334124138429688088162171861413696 n 6 - 617945285097724181310528400822887435382272 n 5 + 1375794781670613471082178338245744942630912 n 4 - 2252715459072692485557465773953374504960000 n 3 + 2270403243374705307465541839339855169536000 n 2 + 333145352961379419979539150202465566720000 n - 6675523254483515276541028670736893214720000 n + 10799984326629793341693071819217567744000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) 27 26 (2 n - 45)), 53 (84973577874901 n - 30972869135391762 n 25 24 + 5366145172976826147 n - 588032825895420995700 n 23 22 + 45752856053422024644345 n - 2689861539855301245260190 n 21 20 + 124155557458943004028635315 n - 4614893844673845230129946600 n 19 18 + 140590798708491796870623653235 n - 3554219901789721647486677829870 n 17 + 75217443258717274603759596512145 n 16 - 1340493183917736739180604129006100 n 15 + 20192719623128388057167046383876715 n 14 - 257567774037186237209603744907884130 n 13 + 2782171793534651947275234444776697705 n 12 - 25408207528029299967579476791526667600 n 11 + 195531202113481846780188656739038200980 n 10 - 1261377921254564685268156717281707242560 n 9 + 6770709510801575352459685132790253946160 n 8 - 29930452989250561429782351756932337081600 n 7 + 107383093301074934767412194121136935173824 n 6 - 305535674228288754431552329517291833471488 n 5 + 657971570024952884176090616342357488662528 n 4 - 930050932036359024007050718413524979302400 n 3 + 238529151205787234974532054775856920576000 n 2 + 2566802217685130265054882173163216814080000 n - 5110846907608874825391684339316095713280000 n + 99999854876201790200861776103866368000000)/(33554432 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) 28 27 (2 n - 45)), 1113 (16185443404709 n - 6344693814621618 n 26 25 + 1184361728400977193 n - 140114632023168099570 n 24 23 + 11795001668275492960605 n - 752012396203405906147410 n 22 21 + 37738428500636067725005485 n - 1529366029439815499743003050 n 20 19 + 50952452859601071229130366115 n - 1413422230943887930493956658430 n 18 + 32944472404430771567306369863755 n 17 - 649328051220867682263933661171350 n 16 + 10867857548005285889186260469347935 n 15 - 154831672197101437230264701935124070 n 14 + 1879093014426774705561323454467258895 n 13 - 19412654763357012205131868335922411950 n 12 + 170326931702550489642478083525573677820 n 11 - 1264294937365259829242281316345253138840 n 10 + 7893374603074179534521952926300084433040 n 9 - 41109034102827172520335814944762072452000 n 8 + 176447831025094635170205566561630981514816 n 7 - 612011949245904534282035084213416262869632 n 6 + 1650278585999835105292381436958015747781632 n 5 - 3128575561215264757940467560987901919662080 n 4 + 2655680394514237696391373049039577929728000 n 3 + 5292965263405918826269273522479664066560000 n 2 - 18890451505240869393189726752356314808320000 n + 13423326805745422125981801673149893836800000 n - 523808763637247472480704541496442880000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 28 27 (2 n - 51) (2 n - 45)), 7 (2573485501300111 n - 1008806316475001762 n 26 25 + 188313514791267959307 n - 22278226484006419550130 n 24 23 + 1875405263531295085040295 n - 119569970700921586481598690 n 22 21 + 6000410091489272692860213015 n - 243169194251897917775219469450 n 20 + 8101439599825562838501431934585 n 19 - 224734103694095625264332745345870 n 18 + 5238169100962478457086755541009745 n 17 - 103243049166032728061525954510304150 n 16 + 1727984118417752324414196959873596365 n 15 - 24618024771063405280005697586624652630 n 14 + 298768497865779177624295730736001744605 n 13 - 3086396953132452123977505385653113729550 n 12 + 27076579184988807118911781204159207803780 n 11 - 200908141155365594385286125911564160793560 n 10 + 1253003750123955924459345527729776588382960 n 9 - 6506244591998023880926255675005346521700000 n 8 + 27694999694145773671948314694399828133052864 n 7 - 93896987425103652874712044988275811213647488 n 6 + 237800378088232572275904857866264098333370368 n 5 - 371300607282219270434405170261761229276446720 n 4 + 17909984077235573046825821823846305206272000 n 3 + 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(n - 50)! (2 n - 107)! (n + 1) \ / + 690309636080078466403291426944 (n - 52)! | \ 105 --- (n - 50)! (n + 1) (2 n - 106)! + (n - 51)! ((2 n - 105)! (n + 1) 104 \\ - 1/690309636080078466403291426944 binomial(2 n, n) (n - 54)! (n - 50)!)|| // /((n - 54)! (n - 52)! (n - 51)! (n - 50)! binomial(2 n, n)), ( 464631485823129737002215383520 (n - 51)! (2 n - 107)! (n + 1) + 696947228734694605503323075280 ((2 n - 106)! (n + 1) - 1/696947228734694605503323075280 binomial(2 n, n) (n - 54)! (n - 51)!) (n - 52)!)/((n - 52)! (n - 54)! (n - 51)! binomial(2 n, n)), ( (464631485823129737002215383520 n + 464631485823129737002215383520) (2 n - 107)! - (n - 54)! (n - 52)! binomial(2 n, n))/((n - 54)! (n - 52)! binomial(2 n, n))] and in Maple notation [53/67108864*(169947155749829*n^27-61945738270811928*n^26+10732290345967997313* n^25-1176065651795471821800*n^24+91505712107916461191005*n^23-\ 5379723079900399412053560*n^22+248311114944564943348791285*n^21-\ 9229787692404278737833037800*n^20+281181597707619435711312171315*n^19-\ 7108439826813465837692572485480*n^18+150434888093580970846090882346355*n^17-\ 2680986459163129279145852819815800*n^16+40385443785857541046322660021580735*n^ 15-515135742117660462165869372631082920*n^14+ 5564350724708750435480508111552505095*n^13-\ 50816640774942033165363513650191327800*n^12+ 391068524382427935138959617869195357420*n^11-\ 2522897459692949996782114455002028903840*n^10+ 13544196159154993302455208217267039449040*n^9-\ 59906603647895154485007153528487963036800*n^8+ 215388423334124138429688088162171861413696*n^7-\ 617945285097724181310528400822887435382272*n^6+ 1375794781670613471082178338245744942630912*n^5-\ 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28-6344693814621618*n^27+1184361728400977193*n^26-140114632023168099570*n^25+ 11795001668275492960605*n^24-752012396203405906147410*n^23+ 37738428500636067725005485*n^22-1529366029439815499743003050*n^21+ 50952452859601071229130366115*n^20-1413422230943887930493956658430*n^19+ 32944472404430771567306369863755*n^18-649328051220867682263933661171350*n^17+ 10867857548005285889186260469347935*n^16-154831672197101437230264701935124070*n ^15+1879093014426774705561323454467258895*n^14-\ 19412654763357012205131868335922411950*n^13+ 170326931702550489642478083525573677820*n^12-\ 1264294937365259829242281316345253138840*n^11+ 7893374603074179534521952926300084433040*n^10-\ 41109034102827172520335814944762072452000*n^9+ 176447831025094635170205566561630981514816*n^8-\ 612011949245904534282035084213416262869632*n^7+ 1650278585999835105292381436958015747781632*n^6-\ 3128575561215264757940467560987901919662080*n^5+ 2655680394514237696391373049039577929728000*n^4+ 5292965263405918826269273522479664066560000*n^3-\ 18890451505240869393189726752356314808320000*n^2+ 13423326805745422125981801673149893836800000*n-\ 523808763637247472480704541496442880000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/( 2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/( 2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 7/67108864*( 2573485501300111*n^28-1008806316475001762*n^27+188313514791267959307*n^26-\ 22278226484006419550130*n^25+1875405263531295085040295*n^24-\ 119569970700921586481598690*n^23+6000410091489272692860213015*n^22-\ 243169194251897917775219469450*n^21+8101439599825562838501431934585*n^20-\ 224734103694095625264332745345870*n^19+5238169100962478457086755541009745*n^18-\ 103243049166032728061525954510304150*n^17+1727984118417752324414196959873596365 *n^16-24618024771063405280005697586624652630*n^15+ 298768497865779177624295730736001744605*n^14-\ 3086396953132452123977505385653113729550*n^13+ 27076579184988807118911781204159207803780*n^12-\ 200908141155365594385286125911564160793560*n^11+ 1253003750123955924459345527729776588382960*n^10-\ 6506244591998023880926255675005346521700000*n^9+ 27694999694145773671948314694399828133052864*n^8-\ 93896987425103652874712044988275811213647488*n^7+ 237800378088232572275904857866264098333370368*n^6-\ 371300607282219270434405170261761229276446720*n^5+ 17909984077235573046825821823846305206272000*n^4+ 1393703456012944280819570474145419912663040000*n^3-\ 2574282419936167312904966415166313746759680000*n^2+ 1452600956726513268568162877747740291891200000*n-\ 83285593418322348124432022097934417920000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 203/134217728*( 354963517374761*n^29-149262159023443813*n^28+29939930295497976999*n^27-\ 3813125738293115581617*n^26+346253062952516434225875*n^25-\ 23864692229520508063983285*n^24+1297676410705226500273464555*n^23-\ 57128393236772350451593628565*n^22+2073346041613752606137241589785*n^21-\ 62844815749527396136520084471355*n^20+1605941634689903073890721104421165*n^19-\ 34831735265878104493437515298840195*n^18+644189469897606758708342148383771265*n ^17-10188145402308167960759380835605261095*n^16+ 137976463166677038073550152738369624185*n^15-\ 1599962099625992003606410760987201046855*n^14+ 15861921224909004972120360126276473232330*n^13-\ 134031785184349644780451608901295993793540*n^12+ 960457682662877647546884348313091698934920*n^11-\ 5790717661863214212594827798430610037213360*n^10+ 28999020829503814218688839141400118262874464*n^9-\ 117915481267237011044055917878142057049534912*n^8+ 372221941387234334727618992912879539680898176*n^7-\ 820334788807822080612514853845599223490769408*n^6+ 841577542287590063354162418931214166225131520*n^5+ 1407739966616110679330975841595625134783488000*n^4-\ 6471565563478145901937742799992237760184320000*n^3+ 8864423125943238631653417625840690128814080000*n^2-\ 4433338235777723132302760061855543931699200000*n+ 327398539644439575385698293764293918720000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 609/ 33554432*(29580293093228*n^29-12438513230596899*n^28+2494994181055810602*n^27-\ 317760475065114315591*n^26+28854421229705523338250*n^25-\ 1988724238827095941226055*n^24+108139685932858966624426890*n^23-\ 4760697837881463974799583995*n^22+172778695678148700604425891930*n^21-\ 5237057556935638984550550214665*n^20+133827820044421693625092299857670*n^19-\ 2902610247220327088835361975852485*n^18+53680907642081474279582443029361470*n^ 17-848952604579239321900329525812154685*n^16+ 11496087863534236603502425775419719630*n^15-\ 133275778369115538260345906999665072665*n^14+ 1320542079012974241412545494564194203090*n^13-\ 11143794059536492185873959018678500843920*n^12+ 79615826954018891809515034900061069847160*n^11-\ 476817731728236923938553799752165140133280*n^10+ 2353636929104697656214198730000831267911072*n^9-\ 9284577067259679688970444912179315263787776*n^8+ 27499217675637722858430282905365631215698048*n^7-\ 52118873720341915618643579624275186183881984*n^6+ 22202298238656135877462135296107582629800960*n^5+ 188940876767309777984349206223947491597824000*n^4-\ 549677788334584340838670042867086356367360000*n^3+ 650282874911393665662996510519260174499840000*n^2-\ 307552382138199384505557102398261369241600000*n+ 27283211637036631282141524480357826560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 9541/ 33554432*(3776207616067*n^30-1699293418166865*n^29+365352800762905370*n^28-\ 49961207037241496535*n^27+4880215143999343587978*n^26-362544142450312720157715* n^25+21294507858541298034043650*n^24-1014997116443448068479104075*n^23+ 39985351355121438906420975480*n^22-1319231134760971388874567474825*n^21+ 36806521978896168845481915820950*n^20-874513596343645601963414308147725*n^19+ 17782972829172584843673632010880710*n^18-310497351366430206210587494375850625*n ^17+4663386209842759046601787059882788550*n^16-\ 60270228324340236307113039245475272025*n^15+ 669585548626755705600919229267537609205*n^14-\ 6377242923850476723356297512426212544050*n^13+ 51814798668926157950636990886488522366600*n^12-\ 356222395633849885307889550764089566783800*n^11+ 2044370355049533917428805587548889109381248*n^10-\ 9570832346971670006019758915817085729020960*n^9+ 35049665039902894774523390859053263104266880*n^8-\ 91945264354985117134100677155350272278675840*n^7+ 131084385807382486353114792504614705941869312*n^6+ 99187094381229733037347795157377770998615040*n^5-\ 960611476361555484881331031941308494152192000*n^4+ 2124820407835370905940613247990810288404480000*n^3-\ 2237797407211122486592395121147318620241920000*n^2+ 1012538123641638952192083188236874946969600000*n-\ 102747414037350717807213826234539048960000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 1363/33554432*(26433452942539*n^30-11895053564636655*n^29+2557469435401894190*n ^28-349728398553527138145*n^27+34161495195398899885326*n^26-\ 2537807243607817938580005*n^25+149061330272005392233867550*n^24-\ 7104956467143786733249656525*n^23+279895458390364131761755683660*n^22-\ 9234474691724122553901282869775*n^21+257637014328667280966466824302650*n^20-\ 6121153659778991766526735200498075*n^19+124461624842264194172909022389247570*n^ 18-2172771631424238567662025458774238375*n^17+ 32621351277227015717692767076634093850*n^16-\ 421293946186754139150402073008667520175*n^15+ 4673588455527694429768669107610709738985*n^14-\ 44384164734386992774738396145869909891350*n^13+ 358649184640948138410333201762699975039200*n^12-\ 2440876687079404742134782351822609361638600*n^11+ 13757058689503398861913712318072722087222816*n^10-\ 62394996081605187797581868960015583258753120*n^9+ 215989963968403727064609924404985669441826560*n^8-\ 505692510235414845652332287365974527283108480*n^7+ 469285171740013555762116035691325834083919104*n^6+ 1598855929124898756298261172247692662396769280*n^5-\ 7566205927139401408550829746179652601921024000*n^4+ 14542044917060469662744971986983905979074560000*n^3-\ 14344435435912122888299904077308625741168640000*n^2+ 6372340427009178778390649009062074954547200000*n-\ 719231898261455024650496783641773342720000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-\ 23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 126759/1073741824*(18190762385167*n^31-8740660677308006*n^30+ 2009646754602018740*n^29-294352534944963130120*n^28+30849493838649264486708*n^ 27-2463467494678716784946184*n^26+155845912532505010188495720*n^25-\ 8018055438168292884404216400*n^24+341738534202492577088553594330*n^23-\ 12229311940985037736646675320740*n^22+371099379835252229040875234584800*n^21-\ 9618773377494178405675078287667200*n^20+214074520485712549742216328126952260*n^ 19-4105522345391322149498576869868695080*n^18+ 67987033655001069044287307961237103800*n^17-\ 972780420917149807718117998205496994800*n^16+ 12015889891512383902786083734318015672655*n^15-\ 127790210029760405647632363217771257108390*n^14+ 1164338306090049281829839195793683446191500*n^13-\ 9013899198473117909295420808528328176396600*n^12+ 58512884313600860718085973675524550294116048*n^11-\ 311770138703000507055684549158075814148876064*n^10+ 1315038905913460455689007923972443354257975360*n^9-\ 4091791195525137172282701605182957706788490880*n^8+ 7729133391652596158441565270991891269477752832*n^7+ 287826348364667352504384378542216524994494464*n^6-\ 54528522960980542321480326006505124083775569920*n^5+ 180683703402636224860382424076425366832336896000*n^4-\ 306774721534184082380228899548821215652904960000*n^3+ 283749589136862538394335546514364369610997760000*n^2-\ 123452645826021416920428066966472968948940800000*n+ 15096136187165163528148061523319801774080000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 899/536870912*(1282448531899361*n^31-616216369712988598*n^30+ 141680000541244325920*n^29-20751825736099056400460*n^28+ 2174883472538485290153264*n^27-173673531001495030807970172*n^26+ 10987020612470022424085353260*n^25-565261111448182361171522483700*n^24+ 24091579606784612749660055176890*n^23-862097567286571481400588138862920*n^22+ 26158454633627319346776773511525900*n^21-677921848044167511617979611050595100*n ^20+15083722963283515987932891487326535080*n^19-\ 289132186304289873231288532422783187140*n^18+ 4783675027495950412537615722234315732900*n^17-\ 68336083029346259174235447510279529275900*n^16+ 841724618540579704390284007229714057230365*n^15-\ 8909160328188545080509246266838622383830370*n^14+ 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2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-45), 38657/1073741824*(119297477238503*n^32-61080266644644496*n^31+ 14985164925608577016*n^30-2345549688361214700320*n^29+263120716592807375283492* n^28-22528318635017242824932064*n^27+1530917609873995471518659064*n^26-\ 84773387405268993566776261920*n^25+3897120578762373946198890395370*n^24-\ 150767521853114566174904612970240*n^23+4958200822320361551113341321780440*n^22-\ 139647241953210947966582288327301600*n^21+3386762462081688072092575269768537540 *n^20-70989619408614595297606110226019208480*n^19+ 1288891183627048188064359039282971822280*n^18-\ 20284372032383862335646054864732198588000*n^17+ 276479066214066634960088623288844461318695*n^16-\ 3255067396253635355737949249536159779005040*n^15+ 32938886195283303029616532828938110010054640*n^14-\ 284214440263897977287240330807124232481302400*n^13+ 2065533042542562619958532169496532638602224032*n^12-\ 12405124655424301971490250021114812429890356224*n^11+ 59679325101319494312615473105925149155978984704*n^10-\ 216895034513206632789832349088452918688927713280*n^9+ 512078305247721865407556809025048149836137498368*n^8-\ 256697468593166638611861133586586268238056083456*n^7-\ 3668120272399529896214370396754626152745030918144*n^6+ 17309888336179472766395992610969049583320601067520*n^5-\ 42121684393097010770275136505353103009366122496000*n^4+ 61883892064224697830487331568268960321958707200000*n^3-\ 52994598727164948266484504602484671663725608960000*n^2+ 22671133891530322105809237188209549235768524800000*n-\ 3118580877920654595826493732829065096724480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 115971/1073741824*(39765769582573*n^32-20360035694624016*n^ 31+4995030911236760936*n^30-781842973893335735520*n^29+87705484221908356287372* n^28-7509217857821641312079904*n^27+510278576475352275657732264*n^26-\ 28255074941979127850473139040*n^25+1298816654778953270247450950670*n^24-\ 50240514922953880493535305394240*n^23+1651849012468720615066014608477640*n^22-\ 46505833494303968712663074915940000*n^21+1127123040058327942040017201502284140* n^20-23599533380001432206180453525400403680*n^19+ 427708283046952764514769114594460253080*n^18-\ 6712059583827218914118703300452094439200*n^17+ 91081732464614884499206312249841551459245*n^16-\ 1065131051182796722907669129014697374584240*n^15+ 10671014633315450870570507364874789536788240*n^14-\ 90742291519084800906613980989721875887219200*n^13+ 645740570933415890962144694650775446709286112*n^12-\ 3761031286234658106251017547694587249401019904*n^11+ 17260529561972433923987341478337373273774166784*n^10-\ 57664359111536468240178181688825011216335175680*n^9+ 108072882635048718872641999264767530767394757888*n^8+ 113159703303785656093903504740297372590349225984*n^7-\ 1662208242348008513838001370143323419138353618944*n^6+ 6361626538633028277852741343247823767213768048640*n^5-\ 14306381571672061755462474402829142750235844608000*n^4+ 20153671310584126194027870032227872286135418880000*n^3-\ 16894249105477207340380092083643807487882690560000*n^2+ 7225936043471618115219031579991702093011353600000*n-\ 1039526959306884865275497910943021698908160000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 1584937/1073741824*(5819359666378*n^33-3168626963972811*n^ 32+827805144217716016*n^31-138170189395625675640*n^30+16552906465065218908792*n ^29-1515956522811320514993684*n^28+110378321836288867031828784*n^27-\ 6560691078727278593588493720*n^26+324359758987827439967343387420*n^25-\ 13522925389788059025997712500290*n^24+480290146006668902570233404273840*n^23-\ 14642378616577069994704674805999800*n^22+385287354117417456837259074548234040*n ^21-8783406740340485995598816760416505780*n^20+ 173865062745627044158948598293447926480*n^19-\ 2990514643188418364943168840415871125800*n^18+ 44657540869868541731420433632184964830570*n^17-\ 577462816118185008856871950426620103131915*n^16+ 6435467992239515461111991348141473296849440*n^15-\ 61352443511492228655223953169741089345402000*n^14+ 494779286286684370506108929870817990119724032*n^13-\ 3318290508180164655719919381696170181537972384*n^12+ 18002812163642652380593461808119329639503426304*n^11-\ 75035304237249586068440456601251305469605896960*n^10+ 211017969825881979864641226713427340107511469568*n^9-\ 185354921991341317435576598123126070628370363136*n^8-\ 1699553608567499524350036918521476137173428260864*n^7+ 10723184188468665106662555330836878708311411793920*n^6-\ 34689117417401073491174188287188519961386454220800*n^5+ 71818093441335459272026049711544970824089149440000*n^4-\ 96239239280122742500934141290888075842152693760000*n^3+ 78367439506872982283418116689592405707181260800000*n^2-\ 33266713006488809245380209720113305964511232000000*n+ 4944091635727867042163953478875347104563200000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2* n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/( 2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51) /(2*n-57)/(2*n-63)/(2*n-45), 36859/1073741824*(250230254212714*n^33-\ 136248891752990973*n^32+35594697694528918528*n^31-5941055928788707849320*n^30+ 711721839013653638272696*n^29-65177949907691518272710412*n^28+ 4745273653361587306894348272*n^27-282011862175227869965005571560*n^26+ 13939528985248209077640357626460*n^25-580947761195660793711880102183470*n^24+ 20621759726516292833849013029784720*n^23-628135241041198141031863211941211400*n ^22+16506049522365731854626040745221838520*n^21-\ 375528629077897348261227492955043442540*n^20+ 7411287639614739839853272274748072797840*n^19-\ 126923994762206693832026827526290255277400*n^18+ 1883707262207371474289769570629902481302410*n^17-\ 24149098356199571400692016023145960211342845*n^16+ 265956322602291671530443170183253616513537520*n^15-\ 2494848735648771497102926355272893766316994000*n^14+ 19679774562071199902818076540031772801119799616*n^13-\ 127946226088056086640524406222745406426562834912*n^12+ 662382682124775548477670902895408552444663660032*n^11-\ 2540921282063771843175452961785459789774808180480*n^10+ 5729975207356534295474104777340928914753103081984*n^9+ 4671507338730475773658588006280807175326701585152*n^8-\ 108751349390445453634862852185754963098217052966912*n^7+ 531408446850349432758528040703274564568602446684160*n^6-\ 1574641110914648494617640739495692670691735660134400*n^5+ 3112647358921610912612270702851497555327050833920000*n^4-\ 4059608542629384797290894622021416914096392110080000*n^3+ 3261676781065087060805130299084613900160165478400000*n^2-\ 1387010657230217083044692785854615277116850176000000*n+ 212595940336298282813049999591639925496217600000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 110577/1073741824*(166816809688676*n^34-\ 96417888461627328*n^33+26771249621764295549*n^32-4755236987444031052272*n^31+ 607079295199271284958744*n^30-59333690348828530684438752*n^29+ 4617497067050910990013751916*n^28-293818893692812740561836770848*n^27+ 15577436208741545614092318324480*n^26-697652470414187152479947217805920*n^25+ 26665863955059463449038363188079310*n^24-876490237846607734502082441530023680*n ^23+24912134374378089529994943685136756280*n^22-\ 614591442362835955011903464806168580640*n^21+ 13189707696578428362915920741688175012620*n^20-\ 246417004927797775620327994929523846011360*n^19+ 4004473170442197194480931426878023959756540*n^18-\ 56466352079610414470030925370070566710426720*n^17+ 687843169700371751577533898533486668486993885*n^16-\ 7188872889693140685811063805423089805697131280*n^15+ 63801119503730249462643655316658208304042094544*n^14-\ 473325634226926753470102590026641943220269141632*n^13+ 2860559027516842682050856439859293827768790924256*n^12-\ 13408418549092715392035488131682725801535221920768*n^11+ 42987763771303688952202191929565312148875235320576*n^10-\ 44687042129456693722917584654203666943834705860608*n^9-\ 461500362274276093846460880344554561477946343457536*n^8+ 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n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-69), 99789/2147483648*( 739177886454611*n^35-452664363228871990*n^34+133311930770004792995*n^33-\ 25144451470231460292310*n^32+3412600372292955998921228*n^31-\ 354992483642788987542214360*n^30+29438128051853675509506243180*n^29-\ 1998331874878456257677006510040*n^28+113147544547075770538700717542026*n^27-\ 5417389576795771429397485129086900*n^26+221562253865667508061591975023731450*n^ 25-7798176353130070865600972920851978900*n^24+ 237460741715507213969711834499114716940*n^23-\ 6278119948037645537263119460889422354200*n^22+ 144396411112871407953588926385719738627100*n^21-\ 2890479026909651684757614759049183207247800*n^20+ 50304092029436877502451993204162900322751035*n^19-\ 759027213245811474040444114830047809967857350*n^18+ 9881979855278963170876122948899251944590383675*n^17-\ 110172558576103561694590256873165538054319838150*n^16+ 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n^3+ 339521625507560737857702548957136469529709455444450588677806550501294080000000* n^2-\ 137863025500360625169178658085141446308318199689786204238482306498560000000000* n+23800161581826459454538375577881364036744187952663846330915251814400000000000 )/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/ (2*n-65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13 )/(2*n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-89)/(2*n-15)/(2*n-37)/(2*n-\ 17)/(2*n-23)/(2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2* n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/( 2*n-85), 1763/1099511627776*(5795854594381345786388*n^45-\ 4945090832381985298145235*n^44+1966988357462129241967959210*n^43-\ 475431167925842876864679814515*n^42+74811324552616713797423763388698*n^41-\ 7063746849760309690093169872638285*n^40+101061869500641866131713540854933870*n^ 39+98054731033816374222396878130665187195*n^38-\ 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n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-91)/(2*n-83)/( 2*n-85), 1763/1099511627776*(574043952660539815261*n^46+ 1221569497865165707663087*n^45-1568604006705964514042623470*n^44+ 833848402529133702782179166685*n^43-276868711943601125332138373346204*n^42+ 65918519274169684752682998381955827*n^41-12073479208576381583742993183698853650 *n^40+1775460507083268731759469517907694005995*n^39-\ 215721199735468660746923214081895446839439*n^38+ 22103913027927469785709401960873385243304652*n^37-\ 1939404137607444733685505750289808440001491980*n^36+ 147429399377794723454809127003005112840442513930*n^35-\ 9799801844027434764429644293146480926978821295264*n^34+ 573797599230500894777158198913834736642421127851222*n^33-\ 29769995001298415891452499787959374567961504182847460*n^32+ 1375177175417370795123453006953553537130276792073197590*n^31-\ 56778514605624160111720710608831946042774371559044795409*n^30+ 2101903335709689386828895420949596346770120026504108310327*n^29-\ 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8730*n^26-208602212638775363059893213919503487939683814628442543613588735078095\ 81108710525*n^25+38832772266357853412118973291931378993227227022591701900505209\ 6498626607108575400*n^24-660269844944421450369926540133934348841068682756918681\ 8427250591947763370642066280*n^23+102438119790427425855264612186927149494263158\ 322386208887206841324585909353917111840*n^22-1448335094935990290873012224997072\ 172888588934319578174351243025422286204253321300400*n^21+1863214019493076655229\ 8368315423581427265186907636976472557401362593680607338513782400*n^20-217680426\ 705947472654107174994771663995124733478850790520924547309804067409127490161920* n^19+23044079091125801576026707643961662415675738523891483479759625517024657690\ 50892414748160*n^18-22045941349290100662679483631364149137572667511130907304446\ 530294077237795676955920454400*n^17+1900096908792053621404560380015808083802156\ 72916691145324912575831501544497710164797798400*n^16-14700168699425026212357263\ 86149291661016707536585692760421984636950918985192708997330135040*n^15+10165425\ 0456752246053676946971564936689590373329973462041163748928869625765492799547804\ 46720*n^14-62521530242962995523184186465399345661640906504560127144935143218468\ 422514027722696882278400*n^13+3400182325707009998223362204437457014822509443634\ 16918307991963214304256461801502399501926400*n^12-16238670376483468819700969124\ 10761146248063039079954527076871319942891059329919287466634838016*n^11+67547168\ 9310966850131785227127688736640594254806978447252961641988134168521890568029651\ 1275008*n^10-242311213224266999539427966338502961351459590815438041831596389755\ 14085488665253516143991193600*n^9+740620991737619519439853981363262687131036195\ 48140403026377375182459066490510572738644541440000*n^8-189993415411996925306082\ 860137145161267292008260092148124016057912290635768425291624931655680000*n^7+40\ 1311016822278678942326501552651851299753010648620958082099452056438177850581394\ 602265149440000*n^6-68063255602275491978207338144287192806236282017800709470796\ 3452879854531718547844538302464000000*n^5+8956618503925976852350047099617980993\ 55249492225783912331689224838568139683996593710694400000000*n^4-870361756902707\ 3598831906889338225973484161116925119963024886346962234605773648000188416000000\ 00*n^3+578040647840342731883184606752758434142973116386870768856463337915455403\ 636353741409484800000000*n^2-22839759074506777368996079034133897429902984536042\ 3049426804156767317466645742066073600000000000*n+387400492977189462196844720515\ 59555457759303986269198113028310710649756854500982784000000000000)/(2*n-91)/(2* n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/( 2*n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17 )/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-\ 13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2* n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n -5), (464631485823129737002215383520*(n-51)!*(n-50)!*(2*n-107)!*(n+1)+ 690309636080078466403291426944*(n-52)!*(105/104*(n-50)!*(n+1)*(2*n-106)!+(n-51) !*((2*n-105)!*(n+1)-1/690309636080078466403291426944*binomial(2*n,n)*(n-54)!*(n -50)!)))/(n-54)!/(n-52)!/(n-51)!/(n-50)!/binomial(2*n,n), ( 464631485823129737002215383520*(n-51)!*(2*n-107)!*(n+1)+ 696947228734694605503323075280*((2*n-106)!*(n+1)-1/ 696947228734694605503323075280*binomial(2*n,n)*(n-54)!*(n-51)!)*(n-52)!)/(n-52) !/(n-54)!/(n-51)!/binomial(2*n,n), ((464631485823129737002215383520*n+ 464631485823129737002215383520)*(2*n-107)!-(n-54)!*(n-52)!*binomial(2*n,n))/(n-\ 54)!/(n-52)!/binomial(2*n,n)] The limits, as n goes to infinity are 9007199254740937 4503599627369753 18014398509441117 18014398509100777 [----------------, ----------------, -----------------, -----------------, 9007199254740992 4503599627370496 18014398509481984 18014398509481984 72057594027076483 4503599623443963 36028796864895247 36028796360680657 -----------------, ----------------, -----------------, -----------------, 72057594037927936 4503599627370496 36028797018963968 36028797018963968 2305842849181383753 1152921230177525539 4611682577608810471 -------------------, -------------------, -------------------, 2305843009213693952 1152921504606846976 4611686018427387904 4611676064260573383 4611659225775074093 4611618470013212663 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 4611525591236181513 4611325335531558253 147549283975571809403 -------------------, -------------------, ---------------------, 4611686018427387904 4611686018427387904 147573952589676412928 73761822111419177079 294951476738093796611 294779751629196970511 --------------------, ---------------------, ---------------------, 73786976294838206464 295147905179352825856 295147905179352825856 1177935057293782542099 4593625983788481337 293203237249656126493 ----------------------, -------------------, ---------------------, 1180591620717411303424 4611686018427387904 295147905179352825856 291978946485906102843 37137629178174101611279 18396847154018182564337 ---------------------, -----------------------, -----------------------, 295147905179352825856 37778931862957161709568 18889465931478580854784 72612024599733405770781 71266694922023992685861 138919532316935709916567 -----------------------, -----------------------, ------------------------, 75557863725914323419136 75557863725914323419136 151115727451828646838272 67094374098895644012221 128147900784114564685267 120618276782034932755405 -----------------------, ------------------------, ------------------------, 75557863725914323419136 151115727451828646838272 151115727451828646838272 14265773021109728750288345 6434895821811359364544465 --------------------------, -------------------------, 19342813113834066795298816 9671406556917033397649408 22492246927850521098703285 18798677953082969974185265 --------------------------, --------------------------, 38685626227668133590597632 38685626227668133590597632 58758329874616277392635871 2554522912473578155350511 ---------------------------, -------------------------, 154742504910672534362390528 9671406556917033397649408 10917922141726920865379993 1012039488540531694305143 --------------------------, --------------------------, 77371252455336267181195264 77371252455336267181195264 -580102433455839906601443583 -610390549249389168184941581 ----------------------------, ----------------------------, 4951760157141521099596496896 2475880078570760549798248448 -3683694302693049546780618569 -4856942517017236018146575609 -----------------------------, -----------------------------, 9903520314283042199192993792 9903520314283042199192993792 -46364391467393629500974803 -6894774470547262382323534049 ---------------------------, -----------------------------, 77371252455336267181195264 9903520314283042199192993792 -3860661600096965551221232969 -2101307192114874059323161497 -----------------------------, -----------------------------, 4951760157141521099596496896 2475880078570760549798248448 -572471872378818798496663435757 -299074119797776476542840746441 -------------------------------, -------------------------------, 633825300114114700748351602688 316912650057057350374175801344 -1231558690168754145140281117549 -1253130866296256597215383974641 --------------------------------, --------------------------------, 1267650600228229401496703205376 1267650600228229401496703205376 -5056082666980944801705493590769 --------------------------------] 5070602400912917605986812821504 and in Maple notation [9007199254740937/9007199254740992, 4503599627369753/4503599627370496, 18014398509441117/18014398509481984, 18014398509100777/18014398509481984, 72057594027076483/72057594037927936, 4503599623443963/4503599627370496, 36028796864895247/36028797018963968, 36028796360680657/36028797018963968, 2305842849181383753/2305843009213693952, 1152921230177525539/ 1152921504606846976, 4611682577608810471/4611686018427387904, 4611676064260573383/4611686018427387904, 4611659225775074093/ 4611686018427387904, 4611618470013212663/4611686018427387904, 4611525591236181513/4611686018427387904, 4611325335531558253/ 4611686018427387904, 147549283975571809403/147573952589676412928, 73761822111419177079/73786976294838206464, 294951476738093796611/ 295147905179352825856, 294779751629196970511/295147905179352825856, 1177935057293782542099/1180591620717411303424, 4593625983788481337/ 4611686018427387904, 293203237249656126493/295147905179352825856, 291978946485906102843/295147905179352825856, 37137629178174101611279/ 37778931862957161709568, 18396847154018182564337/18889465931478580854784, 72612024599733405770781/75557863725914323419136, 71266694922023992685861/ 75557863725914323419136, 138919532316935709916567/151115727451828646838272, 67094374098895644012221/75557863725914323419136, 128147900784114564685267/ 151115727451828646838272, 120618276782034932755405/151115727451828646838272, 14265773021109728750288345/19342813113834066795298816, 6434895821811359364544465/9671406556917033397649408, 22492246927850521098703285 /38685626227668133590597632, 18798677953082969974185265/ 38685626227668133590597632, 58758329874616277392635871/ 154742504910672534362390528, 2554522912473578155350511/ 9671406556917033397649408, 10917922141726920865379993/ 77371252455336267181195264, 1012039488540531694305143/ 77371252455336267181195264, -580102433455839906601443583/ 4951760157141521099596496896, -610390549249389168184941581/ 2475880078570760549798248448, -3683694302693049546780618569/ 9903520314283042199192993792, -4856942517017236018146575609/ 9903520314283042199192993792, -46364391467393629500974803/ 77371252455336267181195264, -6894774470547262382323534049/ 9903520314283042199192993792, -3860661600096965551221232969/ 4951760157141521099596496896, -2101307192114874059323161497/ 2475880078570760549798248448, -572471872378818798496663435757/ 633825300114114700748351602688, -299074119797776476542840746441/ 316912650057057350374175801344, -1231558690168754145140281117549/ 1267650600228229401496703205376, -1253130866296256597215383974641/ 1267650600228229401496703205376, -5056082666980944801705493590769/ 5070602400912917605986812821504] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999998, .9999999991, .9999999957, .9999999817, .9999999306, .9999997620, .9999992539, .9999978415, .\ 9999941903, .9999853528, .9999652129, .9999217894, .9998328390, .9996590972, .9\ 993344746, .9987526473, .9977498033, .9960838542, .9934112088, .9892631503, .98\ 30248593, .9739209791, .9610121438, .9432068538, .9192923507, .8879866475, .848\ 0116725, .7981848006, .7375231791, .6653526334, .5814109560, .4859344358, .3797\ 168070, .2641314784, .1411108363, .1308030381e-1, -.1171507535, -.2465347795, -\ .3719580700, -.4904258650, -.5992457146, -.6961943079, -.7796544012, -.84871121\ 60, -.9032013589, -.9437115235, -.9715285032, -.9885459495, -.9971364874] The cut off is at j=, 41 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 55], vs. those in the, 2, -th row from j=1 to j=, 54, are as follws 27 26 25 [(2251799813685241 n - 820781032088265147 n + 142202847084079550652 n 24 23 - 15582869886291159096450 n + 1212450685430161213756395 n 22 21 - 71281330808727741440092965 n + 3290122273022155233194364690 n 20 19 - 122294686925120840345681037000 n + 3725656169698616483667402486135 n 18 - 94186827711086927985106389639045 n 17 + 1993262267633984469120347113419720 n 16 - 35523070606743376898878711003010250 n 15 + 535107131297512610096772387099401565 n 14 - 6825548631569823120634434658065677355 n 13 + 73727648886800804905349242283570469930 n 12 - 673320546697702797748617695881819591500 n 11 + 5181659478106036200985860512864618224680 n 10 - 33428426745227542613924466783886536580560 n 9 + 179461293393191621906915468366709905588960 n 8 - 793773922751959304832705396756121332456000 n 7 + 2854052168360138401417083093128751661497984 n 6 - 8189493511705132070476357246350334460924928 n 5 + 18244243767540805417521362258146377981186048 n 4 - 29946633231463169043022262589163793327308800 n 3 + 30531179209955628033297548803700116328448000 n 2 + 3214061156136057037196337441151676743680000 n - 87564140481723018820608044885290010542080000 n + 145749788482064109217756038671385231360000000)/(16777216 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 35) (2 n - 53) (2 n - 29) (2 n - 51) 28 27 (2 n - 45)), 371 (48556330214233 n - 19034081443976366 n 26 25 + 3553085185259249061 n - 420343896087680669910 n 24 23 + 35385005009036688572385 n - 2256037189355346373350270 n 22 21 + 113215285606647726912097545 n - 4588098100319385663034687950 n 20 19 + 152857359719818000680979086255 n - 4240266784046715255490647143010 n 18 + 98833423401126414468163519522335 n 17 - 1947984512208214709131516670594850 n 16 + 32603590466152194067217600997143595 n 15 - 464495778390879374696207423524755690 n 14 + 5637307065124077203298198420139022115 n 13 - 58238850445687478185828322308383058650 n 12 + 511004822384659232161868482630743599340 n 11 - 3793440790669297501877025732979949356680 n 10 + 23691026645537179912410259626262192003280 n 9 - 123506507973509938016448174521325795896800 n 8 + 531786348393405109320081595666714687248192 n 7 - 1863022413810363502084139201883219138777984 n 6 + 5185808869547468889557873688261794514545664 n 5 - 10927098575127175398991647858348535459491840 n 4 + 15007580581397967488312883567768473819136000 n 3 - 2967350977757850842444174646973623152640000 n 2 - 42749351573580799154765919485331847741440000 n + 81884734839051203959311438956599404134400000 n - 1571426290911742417442113624489328640000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 28 27 (2 n - 51) (2 n - 45)), 371 (48556330214179 n - 19034081443918154 n 26 25 + 3553085185229121327 n - 420343896077715323610 n 24 23 + 35385005006670872780955 n - 2256037188926174886993930 n 22 21 + 113215285544808590421450315 n - 4588098093056379918142290450 n 20 19 + 152857359011798943619936869765 n - 4240266726014532834994390149990 n 18 + 98833419364460244351239919502845 n 17 - 1947984272355742333870345900750350 n 16 + 32603578239433064225811635370346185 n 15 - 464495242375556422107485432031104910 n 14 + 5637286841599124117282852393909041305 n 13 - 58238194401559071713404654932642690150 n 12 + 510986573669915341370474073118332805620 n 11 - 3793007543524347101057578284209375553720 n 10 + 23682308634639478305736602081094838150960 n 9 - 123359286084258519218555186974619717888800 n 8 + 529728739533940882323573245555373690991296 n 7 - 1839686021539847472446245930075176461959296 n 6 + 4977139412636661689637126974628122543413248 n 5 - 9520638883593396226846139765247465305456640 n 4 + 8399493802420283378670806305379104468992000 n 3 + 15288390999702886759617707215491234631680000 n 2 - 57130498226307774939582228533494976839680000 n + 40999079888238414880044809926704326246400000 n - 1571426290911742417442113624489328640000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 28 (2 n - 51) (2 n - 45)), 10759 (3348712428530 n - 1408133576171671 n 27 26 + 282452172834802980 n - 35972884397607255459 n 25 24 + 3266538346424259426780 n - 225138608804726040672495 n 23 22 + 12242230679969990923629900 n - 538947149246528179865991855 n 21 20 + 19559872284165482277606483600 n - 592875925550809473061908175185 n 19 + 15150412661859812355731689310100 n 18 - 328602378835095735209167683323865 n 17 + 6077311445208814080442816438457700 n 16 - 96116703917239775117298990226233765 n 15 + 1301738712962244792958217762977630500 n 14 - 15096185020031267263286447739483227685 n 13 + 149696701419274489083792125040843419550 n 12 - 1265654796007184513116874773799011592580 n 11 + 9082710797294293646661297094217131514200 n 10 - 54957139369842211584258860822540160078320 n 9 + 277606777558928658621194287625612605948320 n 8 - 1152044621644542189822071652497578678010304 n 7 + 3812811527377562344742285044964761374472320 n 6 - 9420253201060332144061941362387076770082816 n 5 + 14235823691887955599439802648268254957035520 n 4 + 231779767228250435283194244362800889856000 n 3 - 54580464413890284780213177592975688847360000 n 2 + 98396628333330968413896566255574993960960000 n - 54897736171145129994601688214753784627200000 n + 3088665468343769579110361261927301120000000)/(67108864 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 (2 n - 51) (2 n - 57) (2 n - 45)), 203 (177481758699493 n 28 27 - 74631079523846519 n + 14969965153559972487 n 26 25 - 1906562870921232126171 n + 173126531864011981363875 n 24 23 - 11932346179271722673209455 n + 648838213845244849338770715 n 22 21 - 28564197525892607434010107095 n + 1036673100924941588915643684705 n 20 - 31422413791629455159556505664865 n 19 + 802971186085210331258783825691645 n 18 - 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48556330214179*n^28-19034081443918154*n^27+3553085185229121327*n^26-\ 420343896077715323610*n^25+35385005006670872780955*n^24-\ 2256037188926174886993930*n^23+113215285544808590421450315*n^22-\ 4588098093056379918142290450*n^21+152857359011798943619936869765*n^20-\ 4240266726014532834994390149990*n^19+98833419364460244351239919502845*n^18-\ 1947984272355742333870345900750350*n^17+32603578239433064225811635370346185*n^ 16-464495242375556422107485432031104910*n^15+ 5637286841599124117282852393909041305*n^14-\ 58238194401559071713404654932642690150*n^13+ 510986573669915341370474073118332805620*n^12-\ 3793007543524347101057578284209375553720*n^11+ 23682308634639478305736602081094838150960*n^10-\ 123359286084258519218555186974619717888800*n^9+ 529728739533940882323573245555373690991296*n^8-\ 1839686021539847472446245930075176461959296*n^7+ 4977139412636661689637126974628122543413248*n^6-\ 9520638883593396226846139765247465305456640*n^5+ 8399493802420283378670806305379104468992000*n^4+ 15288390999702886759617707215491234631680000*n^3-\ 57130498226307774939582228533494976839680000*n^2+ 40999079888238414880044809926704326246400000*n-\ 1571426290911742417442113624489328640000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 10759/67108864*( 3348712428530*n^29-1408133576171671*n^28+282452172834802980*n^27-\ 35972884397607255459*n^26+3266538346424259426780*n^25-225138608804726040672495* n^24+12242230679969990923629900*n^23-538947149246528179865991855*n^22+ 19559872284165482277606483600*n^21-592875925550809473061908175185*n^20+ 15150412661859812355731689310100*n^19-328602378835095735209167683323865*n^18+ 6077311445208814080442816438457700*n^17-96116703917239775117298990226233765*n^ 16+1301738712962244792958217762977630500*n^15-\ 15096185020031267263286447739483227685*n^14+ 149696701419274489083792125040843419550*n^13-\ 1265654796007184513116874773799011592580*n^12+ 9082710797294293646661297094217131514200*n^11-\ 54957139369842211584258860822540160078320*n^10+ 277606777558928658621194287625612605948320*n^9-\ 1152044621644542189822071652497578678010304*n^8+ 3812811527377562344742285044964761374472320*n^7-\ 9420253201060332144061941362387076770082816*n^6+ 14235823691887955599439802648268254957035520*n^5+ 231779767228250435283194244362800889856000*n^4-\ 54580464413890284780213177592975688847360000*n^3+ 98396628333330968413896566255574993960960000*n^2-\ 54897736171145129994601688214753784627200000*n+ 3088665468343769579110361261927301120000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19)/ (2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/ (2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39 )/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 203/67108864* (177481758699493*n^29-74631079523846519*n^28+14969965153559972487*n^27-\ 1906562870921232126171*n^26+173126531864011981363875*n^25-\ 11932346179271722673209455*n^24+648838213845244849338770715*n^23-\ 28564197525892607434010107095*n^22+1036673100924941588915643684705*n^21-\ 31422413791629455159556505664865*n^20+802971186085210331258783825691645*n^19-\ 17415887138746977247459083949066785*n^18+322095613879994550444171996672036945*n ^17-5094106487346312274438806143979652485*n^16+ 68989339049959647181865589555733960905*n^15-\ 800011931741592368007839196566469010365*n^14+ 7931689953933555378299197246595391794790*n^13-\ 67030381534628902785446740568027266996020*n^12+ 480468596173690399132874363771694552527960*n^11-\ 2898618709352147855269530125812369996469680*n^10+ 14535247206674559761988242551637811724614432*n^9-\ 59265280874310679542062225660343295656574656*n^8+ 188108927830220752411814353135913879506236288*n^7-\ 419388473728910737664910846974811806971259904*n^6+ 447999019877199055864516697929683225884405760*n^5+ 663204790094537770574076096783268056274944000*n^4-\ 3229889495353748112755104068423713755484160000*n^3+ 4482408650149107485426495857892597355479040000*n^2-\ 2251806587421408742802622903364596439449600000*n+ 163699269822219787692849146882146959360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19 )/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 203/ 134217728*(709927034512117*n^30-319467165320349225*n^29+68686327845462950915*n^ 28-9392707322517172949925*n^27+917480534812119507035583*n^26-\ 68158313460416247399639945*n^25+4003369421895521295257711175*n^24-\ 190819667097147811821103877625*n^23+7517264664442556080739006816205*n^22-\ 248016839335166660492229469748775*n^21+6919713329029288650917800722911025*n^20-\ 164413218253389890587874506831953375*n^19+3343411536914390258309366285876718885 *n^18-58381790923177989873275213436736419075*n^17+ 876992728371069074938168290569978167725*n^16-\ 11338649565884767888958788289584551889875*n^15+ 126071599372533263590095673448372448433530*n^14-\ 1202789528879853842584474646402415965787300*n^13+ 9807336483240354891062052179208344508930200*n^12-\ 67907292484213889204954970035073196526310000*n^11+ 395180219181739952877298383526980337494290848*n^10-\ 1899419376578276922680764822205537557075128000*n^9+ 7306657783327592017346015913239198184613080960*n^8-\ 21110593331924090092412126992387791844211059200*n^7+ 38886330617632313983226150692709987560875512832*n^6-\ 14574625639442234454691560423207661618470727680*n^5-\ 143140527002570905610701533243364771570378752000*n^4+ 405382347964407960263878376600096549300183040000*n^3-\ 473131034887321281544599060942637496385208320000*n^2+ 221588783222736988877136982551061858536652800000*n-\ 19316513839021934947756199332093341204480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 203/33554432*(177481758284723*n^30-79866790976825175*n^29+17171581787256256510* n^28-2348176775925823128825*n^27+229370121401009671476702*n^26-\ 17039576257413319375115805*n^25+1000842069572375571349459950*n^24-\ 47704885273752188992319701125*n^23+1879313296484867830212265231020*n^22-\ 62003990957920767663337649193975*n^21+1729914230038241556585712275143850*n^20-\ 41102532377545885927073946716947875*n^19+835816815028116052869160953287010690*n ^18-14594007906565817344893835200955508175*n^17+ 219199065633372267605147034992395168650*n^16-\ 2833233184918495619797405042424125236375*n^15+ 31482557310150612344378667582225915318945*n^14-\ 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295771/33554432*(243626296292*n^31-117062433932305*n^30+26914918567448335*n^29-\ 3942223456388418440*n^28+413163295079025650313*n^27-32992938760036886512560*n^ 26+2087231167409037912648165*n^25-107385628900545018694600800*n^24+ 4576938414946425585797609805*n^23-163791449496274512317112815250*n^22+ 4970449401584781891603286871775*n^21-128841992468054106073655797283400*n^20+ 2867916308189314257653947376412435*n^19-55016669628414011446977546770989800*n^ 18+911572747776692513625406612332815175*n^17-\ 13056690850011975901054413789251919600*n^16+ 161590179614884752466663820690404541355*n^15-\ 1724569825691088644756047547296645253925*n^14+ 15810766136659034017256826334456976542150*n^13-\ 123708575423890032322504489663633676031200*n^12+ 817382841448352049214541212243344439381448*n^11-\ 4482750355485389391651833654806391236105120*n^10+ 19821435355454515590890842300500208893702240*n^9-\ 66982053206108414667924027034240799676130560*n^8+ 152999303437464585643695923247360762876788352*n^7-\ 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5661051070186936323055523071244925665280000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 42253/1073741824*(109144566447187*n^32-55882010094234224*n^31+ 13709861440549026584*n^30-2145937209780930578080*n^29+240729315566671762318068* n^28-20611312194930079426072416*n^27+1400664480940474280626168536*n^26-\ 77562514866009863242458216480*n^25+3565780764414941071234322687730*n^24-\ 137959687010697183237135598186560*n^23+4537631003436267158119324453477560*n^22-\ 127833807465139606362040469738450400*n^21+3101623063612781965279833060981360660 *n^20-65062506622466545861956063796033545120*n^19+ 1182820085024714654393267926575538047720*n^18-\ 18655765708416094552398130678551132252000*n^17+ 255193506994570166029647247996863004027155*n^16-\ 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n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 4495/1073741824*(1025958607534537*n^32-525290580353333072*n ^31+128872548305650691912*n^30-20171764699932619911520*n^29+ 2262845837766891578996508*n^28-193744737371942462294061984*n^27+ 13166038829722650615242300040*n^26-729065824328110242701918173152*n^25+ 33516444083322341381132224978230*n^24-1296683446688440533553917325761600*n^23+ 42645303884800668810222270860737320*n^22-1201199815444152427804075194057860640* n^21+29135865189334585062746689917480456060*n^20-\ 610854561597584057181729097313437588320*n^19+ 11094843581267756227489580789732821585080*n^18-\ 174712347384125186154477972770773555343520*n^17+ 2383540933317372627289207596054838019834505*n^16-\ 28100993244650705395973258898496695878897520*n^15+ 284940937411042916806931201650632480605911760*n^14-\ 2465821804541481038891583943049593888014561280*n^13+ 17994544559402205436373153211649752029232984928*n^12-\ 108703727470745586729743105462043278124094197248*n^11+ 527458121419071597661669276318599735543800940288*n^10-\ 1944320819735052052517294249400255293796638269440*n^9+ 4742020808347044318326324017830756092151898290432*n^8-\ 3280716071209375584470434714898103001663669088256*n^7-\ 29172526324275864945904244727474165525201212006400*n^6+ 145670055387648164070113519900539867323579903639552*n^5-\ 360811057201179778531364057880394900446134113075200*n^4+ 534762652390782575989317750135966677003239292928000*n^3-\ 459914881592938542333386350614164024868179804160000*n^2+ 196759737837168612078233296225424120749731348480000*n-\ 26819795550117629524107846102329959831830528000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2 *n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/ (2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57 )/(2*n-63)/(2*n-45), 193285/1073741824*(47718963324182*n^33-25982945959884507*n ^32+6788095926257690528*n^31-1133022853333917934872*n^30+ 135739513970323338642248*n^29-12431742588780410840390292*n^28+ 905214713749557720771686160*n^27-53809077285450836789229308376*n^26+ 2660705710705471602518691343140*n^25-110954919843389377016169916074210*n^24+ 3942334828318347466184507132092080*n^23-120266050427676384725328414194454840*n^ 22+3167856605910139776339333398319573960*n^21-\ 72335136526272970837963939124190530100*n^20+ 1435446193353486581768562252203441201520*n^19-\ 24783751560058972494843457701341622046440*n^18+ 372183046748257080615855162876532935341430*n^17-\ 4852093338481101017891254650061611073149915*n^16+ 54705212217763290142532933627060171114363440*n^15-\ 530082099660025618198438026915133011083314800*n^14+ 4372500433064656468252501901834798250320339008*n^13-\ 30265608413955085886024069912889210351190698528*n^12+ 171911666127952606814107775804382950109166646272*n^11-\ 771402087074842303018007768036373140573600740608*n^10+ 2526768918669637521528190668052020425323968763392*n^9-\ 4663997160167696322034325345301106698229654088448*n^8-\ 4687087172102223308175846465539009933227937280000*n^7+ 68924064829929431595613317752544052486229166759936*n^6-\ 260887168337267325357710976004943351830363062927360*n^5+ 580584697301959495636499965889430871887138594816000*n^4-\ 810175884140461474802487746482282986142246502400000*n^3+ 673246000516501273336299657113170719165359063040000*n^2-\ 285406798936470655764959711263180327382050406400000*n+ 40541551412968509745744418526777846257418240000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 38657/1073741824*(238594269025862*n^33-\ 129914194251465591*n^32+33940229196359580512*n^31-5665039743364317258360*n^30+ 678681725191200522032648*n^29-62156151025292112575399844*n^28+ 4525746184611204306144357648*n^27-269011463404738175505689979960*n^26+ 13300626999700358146523266237380*n^25-554567111842181959652784029953290*n^24+ 19699156262146445181683629640716080*n^23-600689005336587256939722420888139800*n ^22+15811272194466512898725609201651757960*n^21-\ 360629716169190219349438417370845136580*n^20+ 7143809950021870624457978473200682158960*n^19-\ 123005386800427677438795061074336145989000*n^18+ 1839602283620259041208183560414177432296230*n^17-\ 23837128835017785212986208367265663408468215*n^16+ 266399540660585541412243317114323797139242480*n^15-\ 2549322590507365490426584756383705571924369200*n^14+ 20663124405523713088501009702473977777860836928*n^13-\ 139533973652259167543073379258133320132773407904*n^12+ 764526155652151666705095996189226953159672982528*n^11-\ 3238497626487114567914737388594730842261790685440*n^10+ 9442088898079944463598855452111862894013580641792*n^9-\ 10587606288856920951295894385456859200499660408576*n^8-\ 61250478407875504804868851096447624176894644318208*n^7+ 423031463587205108805228475851498315378117836021760*n^6-\ 1402633490375930734075178840485536190042707610828800*n^5+ 2938758168689764373227668813658510539631549194240000*n^4-\ 3964405605092425364233582609205102282482992414720000*n^3+ 3238621393063535279183125208662055435598653030400000*n^2-\ 1374207124659394311580343707932524443302100992000000*n+ 202707757064842548728722092633889231287091200000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15) /(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 1584937/1073741824*(11638679124728*n^34-\ 6727110737148892*n^33+1867888985156371397*n^32-331797430905092897008*n^31+ 42361930030471814077832*n^30-4140744308143222228076848*n^29+ 322297689173610451206425388*n^28-20513739454210147898017205952*n^27+ 1088022240553993578770052487800*n^26-48758499100896436697988634257480*n^25+ 1865402391317490580757323251139230*n^24-61399741589414822694753235526817120*n^ 23+1748687923740380748865450772437064040*n^22-\ 43267149661629410536789986213992594160*n^21+ 932409527473174997681209437523124821260*n^20-\ 17520586511954589764676055550367393291840*n^19+ 286982616775998193135546251198947320248720*n^18-\ 4090039208425202969519618664985630638040380*n^17+ 50534588583758749788949758403682076808736005*n^16-\ 538157315109688422550484812069623337991507120*n^15+ 4896585756636118088571336345329678114837874832*n^14-\ 37574893308534395380899667501721004551288939648*n^13+ 238343160246644426292570604433583030815047871968*n^12-\ 1207676133597966334032037406214356569094393149952*n^11+ 4553423448922036502512482591166771203898757468928*n^10-\ 10205710164865205492772029280498299241666749832192*n^9-\ 7028855919606752352422493404663901531827563285248*n^8+ 180048059962288454654875636851021723270789925588992*n^7-\ 874993054625140711983716083051810360285152129146880*n^6+ 2569372080988418162002254417502472537924400404889600*n^5-\ 5032668322708713617174264784502607468090921902080000*n^4+ 6506953017552964422964275152556295478256188129280000*n^3-\ 5184648111337303238769037503530975132291707699200000*n^2+ 2185818206755847501307602548180120177159962624000000*n-\ 331254139593767091824984883084648256005734400000000)/(2*n-5)/(-1+2*n)/(2*n-3)/( 2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/( 2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-25) /(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-\ 29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 36859/1073741824*(500456659942748*n^34 -289259435176724704*n^33+80316301350172111367*n^32-14266428945670836345136*n^31 +1821382072738981400551112*n^30-178023076282678640724811936*n^29+ 13855143792858883930155223428*n^28-881715828611970967843140294624*n^27+ 46753200643489822418242065371040*n^26-2094365360911519514762651580718560*n^25+ 80077656021663823126387120470278730*n^24-2633338491879890358522700388962635840* n^23+74895881035263345247237814963915422440*n^22-\ 1849403988872062739173130010239165207520*n^21+ 39739547272173189151561159069899238703460*n^20-\ 743677339371529052418090233784588504187680*n^19+ 12111986757300627231889873410537595394496420*n^18-\ 171277749804489535585948112647706063976000960*n^17+ 2094093921237217848471812282619351499263692455*n^16-\ 21989427329936725544805762388152243984693898640*n^15+ 196351632565186659514690878971622015243380971312*n^14-\ 1468658163820679781337862138165230318584171128576*n^13+ 8981023683147976489258101191745428664946497804448*n^12-\ 42928857862160887347023117715024553721076611792384*n^11+ 143868402474426541770296484162998771532912556117248*n^10-\ 199787508671043100341850400118262800938452369772544*n^9-\ 1158487049609039350180315297683033578876219391813888*n^8+ 10037793046486701226115254386280837566936355677794304*n^7-\ 41927341558523695780357601657306736631811980006072320*n^6+ 115265863451318918567483887570320076589843390172364800*n^5-\ 217466603268223618098569830672772908167407993856000000*n^4+ 274922535903350221089763923670199894787997270671360000*n^3-\ 216665906122332855626091627967582562881633242316800000*n^2+ 91539828842513422767887965450577748560515694592000000*n-\ 14243928002531984948474349972639875008246579200000000)/(2*n-5)/(-1+2*n)/(2*n-3) /(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11) /(2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-\ 25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2* n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 36859/1073741824*(1000884657852584*n ^35-613022291097245260*n^34+180578329080800061950*n^33-34070745923901330287875* n^32+4626306134818535990833472*n^31-481584475411992955853130280*n^30+ 39976211826703946363798692200*n^29-2717585596497146396049583516500*n^28+ 154184317934456253822666015693504*n^27-7403035420657957615629346583691360*n^26+ 303948492128620320547759871900831700*n^25-\ 10754355532419294097319255517200171250*n^24+ 329797763474772691303091289596983914960*n^23-\ 8801113011600081237711897817340780867400*n^22+ 204905610550269317881391630790209382609000*n^21-\ 4166688980500008449528363921753603560072500*n^20+ 73984801572063280869386216704044975015345240*n^19-\ 1145156491560743751089108486681996895304202100*n^18+ 15399088801448621944691210041729920562350334750*n^17-\ 178934215410690002307432835990885411347017691875*n^16+ 1782186018890672004174537133410439987314407193136*n^15-\ 15032493898383116949035482437602767281630491821040*n^14+ 105370041040023554054138738272531397228092684340800*n^13-\ 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4656891833376566123729968372919678800110737676042240000000000)/(2*n-5)/(-1+2*n) /(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/ (2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67 )/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-\ 35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2* n-69)/(2*n-77), 232841/4294967296*(10013470798588409*n^39-7586894424755055657*n ^38+2773141707298805803051*n^37-651238411721953326439623*n^36+ 110400310620832895383840392*n^35-14391086492733410762162245476*n^34+ 1500329164949252317972815648268*n^33-128460749905430007469090221770664*n^32+ 9204862981974680563703072916285854*n^31-559633579197684977862014071600403022*n^ 30+29165448980029071002023299399913875546*n^29-\ 1312834531373821869433580725047481911858*n^28+ 51325190904488435847988468512892683532140*n^27-\ 1749393933338719096227116018316835513437780*n^26+ 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n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/( 2*n-69)/(2*n-77), 232841/68719476736*(318163056149114737*n^40-\ 253214999463977466020*n^39+97284068407313974501250*n^38-\ 24029784839188539910954000*n^37+4287763553778564595212839349*n^36-\ 588752629405302484994912908140*n^35+64707402939908350245897869503700*n^34-\ 5845759239010579282390293579396400*n^33+442380458529834886088045299386336946*n^ 32-28433528584285394497516484233860253160*n^31+ 1568283868388455448931916226092070031900*n^30-\ 74803378169644418861552968521499073973600*n^29+ 3102942008430262759981704766835407954684098*n^28-\ 112381243696262198632598448097528045822059480*n^27+ 3562296567446681920383736330612879232924870400*n^26-\ 98919581584881840632997290045359996610374836000*n^25+ 2404610188534545473410372074486830200488507672765*n^24-\ 51031890203915116552370066567585555867247321198900*n^23+ 940303358660467264228001932082094701994930608071250*n^22-\ 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250068375595682160980*n^39+95827318649864273581850*n^38-\ 23598427360179816120854800*n^37+4195963418567872548809811729*n^36-\ 573793807586965346475838003260*n^35+62765760504449185240728686783300*n^34-\ 5639550268322303784239639655890800*n^33+424114089274377161727187140146849866*n^ 32-27064551931128183320115898635070506440*n^31+ 1480532805000369253156091528424958113900*n^30-\ 69951993072924947359375940209590044930400*n^29+ 2870091718944282162926898806326653941391258*n^28-\ 102629100602356967211595516971665895017894520*n^27+ 3204491504204452676024636423503990793465203200*n^26-\ 87384814407709796906433797764922334308514948000*n^25+ 2077177578919882306208444888406080554286203675065*n^24-\ 42835848619412694639876992195676356569383786248100*n^23+ 759259056862554945907680711436205615575120703374250*n^22-\ 11360869774883956379462169664648690113674155710490000*n^21+ 138122666208698313238486897803293026459224691154536213*n^20-\ 1230339026227283819664844170041255556696602110196004620*n^19+ 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269797247554069393959311108928164366401112557976939397120000000000*n+ 46516215948139155708484788477366672611759782070625239040000000000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/( 2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45) /(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), 9546481/68719476736*(14834104717449988*n^ 41-12304109883461687153*n^40+4925052259849347219500*n^39-\ 1266930017044283234641570*n^38+235318375660790473989504996*n^37-\ 33614613910730863030659304341*n^36+3840790697636979770204942287100*n^35-\ 360432296455903746806782916075860*n^34+28305320236981563785339880370255864*n^33 -1885692158694817837925523799509026354*n^32+ 107641622961114308927247460634771597400*n^31-\ 5303377209596162316458578625417321235420*n^30+ 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46516215948139155708484788477366672611759782070625239040000000000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/( 2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45) /(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), 47027/68719476736*(5858691383729667367*n^ 42-5081289356383177043169*n^41+2127780060514189804163749*n^40-\ 572901600932370663814656390*n^39+111436226700588410267367730009*n^38-\ 16679607075785005006872078383613*n^37+1998117490046598835990747646400023*n^36-\ 196712435315393638604064258877563480*n^35+ 16216397780866305110807986021169073386*n^34-\ 1134760240938588466701783751112922678402*n^33+ 68079422532449049245436687880499996298242*n^32-\ 3527059339486770264162342619246330754755620*n^31+ 158564439820930666707436191188719190934858018*n^30-\ 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2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), 65231/ 2199023255552*(428443102384356833141*n^44-397603897115192015805502*n^43+ 177969381663849969223935509*n^42-51153955523410102113924586798*n^41+ 10604319821576209916162055111647*n^40-1687917574283005779091287212318554*n^39+ 214402610497151207427920864266932763*n^38-\ 22292709578570742203906989813649443346*n^37+ 1930249218164224008894926968689172062838*n^36-\ 140758720273565514224170690035889877115716*n^35+ 8698967467137328853848699232503342709515282*n^34-\ 456054456363738604035462515259995785615566284*n^33+ 20154779114825960266943051861094111758603346174*n^32-\ 736106073299392646609289624956010183565375331508*n^31+ 21045750163846388673820801208839587931230651716406*n^30-\ 387481564051293640889213314900751603965434847933092*n^29-\ 1564640591662817828364556761675891120627284175192015*n^28+ 519946244387113571499871131542120442569004634388315610*n^27-\ 27826818671517101889181030583378667497714037416280267095*n^26+ 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5470720000*n^8+6780982251935780554637339906727227217971853493374429998108016973\ 350274521306283047738408960000*n^7-14327853124336945184091948965277968012325855\ 661809232656066114132139839469300752798323834880000*n^6+24306574824920543085876\ 624893287428173061398787690121879945503053917277920980086437183488000000*n^5-31\ 9913203149522918457693674348065303717766751699928188606544684652689213399856152\ 83814400000000*n^4+310907688463048051432498904115953297338425216978750812238876\ 17242237436630097477186355200000000*n^3-206490306210311999352254082461446407573\ 97003183879777928605608589096685522552545188249600000000*n^2+815841567505622080\ 7576765289652684589946325484982180111760638328410643669126112870400000000000*n-\ 1383573189204248079274445430412841266348546570938185646893868239666062744803606\ 528000000000000)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77) /(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-101)/(2*n-\ 29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2* n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/( 2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-9)/(2*n-61)/ (2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -7/ 70368744177664/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/( 2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-101 )/(2*n-29)/(2*n-53)/(2*n-35)*(19301658564783812280850552109*n^51-\ 25337422645942906558902095989227*n^50+16188512761095315374480238440470850*n^49-\ 6708755868855920122268256240773037750*n^48+ 2027614443163029205257008164776148733810*n^47-\ 476453829349268776819425644423838403232430*n^46+ 90619390076630666368970237481354716811063200*n^45-\ 14340133807627034898875179308141016216660062000*n^44+ 1926158224769224848185415464339362337426552624555*n^43-\ 222937828620508350632739474397817046489928926351165*n^42+ 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n^18+15667784599439296947708784542777615343070729029741966381902836423786820378\ 46029705740800*n^17-13513781751605789322393679067076137132904962136722525559771\ 458404103608551846313363392000*n^16+1046203359899743851398993392618591686413993\ 83967623734920567842275404277180687246998261760*n^15-72390855404301431561406338\ 9262176156034184463191911681338603434133712259856146642665185280*n^14+445476829\ 2898259189088949404315728997834777560601870801124139663973500535077872060045516\ 800*n^13-2423869669233616967984884689302642878097639021378980891249346261618713\ 9356118413432566784000*n^12+115809557387026173628850609002869122050575470191437\ 076574003978181851232428862602401868611584*n^11-4819083044640032836010974450024\ 76239438160874340189272350359676835661430068516189237168177152*n^10+17293050007\ 0637618977303296665250394000110843616405009516017600739242216650719708388107223\ 0400*n^9-5287050162851664710304738263637377736388356456904110313047163879963476\ 338784585946673971200000*n^8+13566091020139435610101095562114640245988281250154\ 464643739775530537408691796643616988856320000*n^7-28660007910499652282879496667\ 229399436808890924947173636613636806111500123868236813812367360000*n^6+48614744\ 7871241924386684224750157561721223584893473623727825613079639777451228256807157\ 76000000*n^5-639795271784576018355099840563752632806061287881089819508667596188\ 07895503349774234419200000000*n^4+621756434193461838762361425526132450012279244\ 24812528401764571832434496092050700002918400000000*n^3-412937235249141367331626\ 20353325631050658240667255420834258023195470282721420893172531200000000*n^2+163\ 1558247682753771617804107343652457784694628576106334414948335215621158851481436\ 1600000000000*n-276714637840849615854889086082568253269709314187637129378773647\ 9332125489607213056000000000000)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/( 2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13) /(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-9 )/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5) , -7/281474976710656/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n -81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2 *n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17) /(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)*(164030235148538199783782089177*n^52-\ 223019083582758447307480481966954*n^51+147687076153162942852204552228855339*n^ 50-63481079720843305256362005850713454550*n^49+ 19914584914490799094619217246953373682630*n^48-\ 4860867980195043305618597357827379033943260*n^47+ 961064241075158376710177670145865297996237410*n^46-\ 158220373628379165707676535680801061180464534500*n^45+ 22127284841224132706809284554813305898089397886115*n^44-\ 2668740449792430541358868882967791720019649387937230*n^43+ 280864226773998550144619137035055390271594728941230305*n^42-\ 26034766282087742961102606485491420356056789315072785250*n^41+ 2141696200570888116434409060029009620761463446356575087660*n^40-\ 157324059456746450163207981364418340196609633865554030695320*n^39+ 10372550258807356220272842382128700766086013883831994761062620*n^38-\ 616417181761382843909297432580473547160857361464534630705995000*n^37+ 33136112148152495541448995849042798195907473860468592513980927695*n^36-\ 1616027214210288039041485497587040238484757812832228358704543452390*n^35+ 71677211586855988707801709650468221712278175456938674451532146946365*n^34-\ 2897181945647616037413651157084498520149454835299070639318378033288250*n^33+ 106891795242416339379715075538904532604056067420863040701085666349679046*n^32-\ 3604550074651479466064100610822339059977262057763238488082987982612420892*n^31+ 111205453046775306453530451758487904525226063579299929509199488084467983122*n^ 30-3141058887772501596140241489145409388882564392140640355842018897761010632900 *n^29+ 81262785363216701253828849442773456261695059215768436943255388749107946474645*n ^28-192598499721550802784883374277199396641875066089855022867076603583985722513\ 6290*n^27+418149995589291297027933919158909863783013505672202855194670243334829\ 52999021015*n^26-83137226251519968795028414535232393173424560222037985680033728\ 3505759776343916750*n^25+151287394351808780781215338694653219265908370565691714\ 33508188640386683667159904360*n^24-25177016343393591159653753351415671370696128\ 6367922327317589788230288310453241228720*n^23+382769564445512473435641406174607\ 0743200706549121389781874817730178965850235268005520*n^22-530908876762700871547\ 32850883565821158904181780396222088470264744500117893900342996000*n^21+67072087\ 5425653206888311276833902389485308561150138698352188499189115626526414055613440 *n^20-7702885077618070431454793301639692251650953095385512538978731536101073432\ 440278011066880*n^19+8023272647712856763985180665854718211780202842858562992895\ 6004820495368039874750642510080*n^18-755894000676671648213035534273110589868528\ 111104199927231358648031491949716274392297280000*n^17+6421119592696704967656103\ 927683264274585576513130021408621080729525259088254053593607342080*n^16-4900106\ 0946062738058463078762510801957358264415001088280720714640589254474082303598808\ 412160*n^15+3344927571930181958662265403336172828867284134748510457403301441527\ 60430576472649778256834560*n^14-20322855377350762885451554908590415246604390089\ 45486619266337120626916907539287083726220288000*n^13+10925840527552608018330735\ 138351468687221345346412649053148446390367886707830888296230160433152*n^12-5161\ 7069172177665435187450770953310080819727031226339471235391330814221585191252749\ 587802619904*n^11+2125316232333780456837879741835052827504650535740811724598444\ 43534716081529481967308269015793664*n^10-75515985085079365588366654099793930952\ 5133674626430325908118000736072362063244907446691482828800*n^9+2287597253055111\ 9157761932272219504625670604585583971694161435784843076519301879700283772108800\ 00*n^8-581975973928067942038270726181527062405866178867240390738006506734640122\ 7548872815547116421120000*n^7+1219812623644562620094551569676878039654617220802\ 3885266814568381313558657519308360165904875520000*n^6-2054156421507518391412763\ 6645482732603382545048201996928335337568120500375059344755955597312000000*n^5+ 2685601038598536436079353293315239292907836991814175996088246570377701260201385\ 9830982246400000000*n^4-2594494053237394993116681881265559753401780975945502880\ 3460141441011594408737532625184358400000000*n^3+1714224523028354543482058924510\ 6987522704667704463885733853723457183069028392006989538918400000000*n^2-6743787\ 0007308939302761719247805509300996253673443959950966940499913652577025851916288\ 00000000000*n+11400643079043004173221430346601812034712023744530649730405474294\ 84835701718171779072000000000000)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/ (2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-\ 103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2 *n)/(2*n-5), -5/281474976710656/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2 *n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/ (2*n-73)/(2*n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-\ 23)*(239643974503431889295751704045*n^52-325048826284665147335694265684546*n^51 +214762092664892458611567170862108191*n^50-\ 92110900667716998430749529762350996550*n^49+ 28835740044890110596656248672409724480350*n^48-\ 7024361798928937995765843845110766007158860*n^47+ 1386172762325439821280809104432748180129875130*n^46-\ 227791017550191283500875664064763468203378419940*n^45+ 31801500929784046990190433506281197045798467839775*n^44-\ 3829181380317383535772604399855234299899852933972790*n^43+ 402355309244901851883982239684663613572076738083521885*n^42-\ 37240296942063791910737516413786374168843302003909807890*n^41+ 3059107706271271671191911155534374114354237811769416465500*n^40-\ 224408711845402341719396985753606547933565835121030081814200*n^39+ 14776342392457150199646748208068997087201252046839437054027820*n^38-\ 877043439856595321647567739787314298462071021807401235099922040*n^37+ 47091231084483895067466398175958972954164079268864836153368414875*n^36-\ 2294063748777121903542049461070890322270040276908556870906729954830*n^35+ 101643942267155723575541626886226129641664921292621945360401408910825*n^34-\ 4104347503551924489453607430305725869019805079313716427164366713838090*n^33+ 151287893628111650636203449876023634078737625885106722368284852033574110*n^32-\ 5097121310726537130503697371103988194053266496652584234020166091676639628*n^31+ 157121610451421482787159829410505867298543168469015610761825338870072777258*n^ 30-4434481851405719495309640492238349250704458173775199148480791853731061948740 *n^29+ 114639882913693866548306411391995606122272519726851322477990094558203936208825* n^28-27151489976604081871981890694063453558751628409505574322222107334750765528\ 76090*n^27+58909913444229800252389138406655660091355034396171136741716036212110\ 195314139195*n^26-1170537727497245629398033611038969133870435501873857071654344\ 412778756252198873310*n^25+2128842907793016347891197419951521487657453215362505\ 5068088663598651210145492718600*n^24-354089333166095348329814138210980510603004\ 976328937890907016714490241490090359779760*n^23+5380588339882041771408854492397\ 828556983648006732277030078625029866415513881279081040*n^22-7459532118437259262\ 0868952311214082921134313955196321199888782695932906070008623683360*n^21+941992\ 6083139177257823123737189777367161872319926684608119289096275920568458370396160\ 00*n^20-10814025865682822291628642955039934617878434899323797037065950635306314\ 303478894458585600*n^19+1125971813114092131670594570187307680790127899777044623\ 39765550539825333867246596620596480*n^18-10604543042942323301664938952649961458\ 85656075760355940526419766178238652734072864108904960*n^17+90055342084299003145\ 72138009947626598845033138944732140028522568065370792984718508525824000*n^16-68\ 7042983336532908102888401356235701689122384168208213952323583341800509386644439\ 25809541120*n^15+46887424880668722467428049531399136757945573886241160045895107\ 5339043444257980310255963648000*n^14-284810940943507448905211010060056269267250\ 4618421345299684858184770253113972074880261938012160*n^13+153087469294392071195\ 60657909205258659046851398549413293667948402592816390435582582224389406720*n^12 -723104524909688393055437378788388055166008098429698731288703877633453351752816\ 48875736777752576*n^11+29768977243408556751961247337975710534972548130965984261\ 5910303969870600644986176183963458994176*n^10-105759918932061436636419970735656\ 9976161171550092283862877888228635941907056682446989759306792960*n^9+3203409743\ 3611739054348090174399572069083547040310091226174180628802538881082240282064398\ 05747200*n^8-814886576668285936129255723645642054573637812300614632046705680695\ 5299367180620154222139146240000*n^7+1707861107383813641513759690852635912778922\ 2208019560115764681022242164057840792093231226552320000*n^6-2875862349554717533\ 9577447415232089998509760216923041161164982534440559823479802903762095308800000 *n^5+37597490288139689116902662210648605936634993703707461299264681642227373891\ 216955828284162048000000*n^4-36321211395429052376960460766686496639565863026261\ 740361963001238604209402878341000934195200000000*n^3+23997900972010401114926544\ 319131356909120750204427993358472073384869028286304824321061683200000000*n^2-94\ 4094645630390470745137145530534475215397799665894824301390448637352647692853827\ 9673856000000000*n+159609003106602058425100024852425368485968332423429096225676\ 6401278769982405440490700800000000000)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n -25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2 *n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/( 2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), ( 1807838872111813885790438037696*(n-52)!*(n-51)!*(2*n-109)!*(n+1)+ 2686414772577368297576445308352*(n-53)!*(107/106*(n-51)!*(n+1)*(2*n-108)!+(n-52 )!*((2*n-107)!*(n+1)-1/2686414772577368297576445308352*binomial(2*n,n)*(n-55)!* (n-51)!)))/(n-55)!/(n-53)!/(n-52)!/(n-51)!/binomial(2*n,n), ( 1807838872111813885790438037696*(n-52)!*(2*n-109)!*(n+1)+ 2711758308167720828685657056544*((2*n-108)!*(n+1)-1/ 2711758308167720828685657056544*binomial(2*n,n)*(n-55)!*(n-52)!)*(n-53)!)/(n-53 )!/(n-55)!/(n-52)!/binomial(2*n,n), ((1807838872111813885790438037696*n+ 1807838872111813885790438037696)*(2*n-109)!-(n-55)!*(n-53)!*binomial(2*n,n))/(n -55)!/(n-53)!/binomial(2*n,n)] The limits, as n goes to infinity are 2251799813685241 18014398509480443 18014398509460409 18014398509277135 [----------------, -----------------, -----------------, -----------------, 2251799813685248 18014398509481984 18014398509481984 18014398509481984 36028797015997079 144115188005959751 36028796931798769 18014398320145283 -----------------, ------------------, -----------------, -----------------, 36028797018963968 144115188075855872 36028797018963968 18014398509481984 288230364454417349 4611685366092992311 4611683940867743815 ------------------, -------------------, -------------------, 288230376151711744 4611686018427387904 4611686018427387904 4611679913057258935 4611669328866373667 2305821646988627767 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 2305843009213693952 4611583007207437133 4611450950473549207 36889409342303551961 -------------------, -------------------, --------------------, 4611686018427387904 4611686018427387904 36893488147419103232 295080387035568986905 295014189855784429727 294893714656010318741 ---------------------, ---------------------, ---------------------, 295147905179352825856 295147905179352825856 295147905179352825856 589365766112811249031 2354618511123245153399 293748244105777104763 ---------------------, ----------------------, ---------------------, 590295810358705651712 2361183241434822606848 295147905179352825856 146417975010265559699 2331546554214123739969 74081404156816024477817 ---------------------, ----------------------, -----------------------, 147573952589676412928 2361183241434822606848 75557863725914323419136 73321380247980187215557 72257346775610015048393 35403374709286669723057 -----------------------, -----------------------, -----------------------, 75557863725914323419136 75557863725914323419136 37778931862957161709568 275516679702655067267909 132760911536376789780637 63218311300911869623331 ------------------------, ------------------------, -----------------------, 302231454903657293676544 151115727451828646838272 75557863725914323419136 3795783053007440976769133 27947772011633980582620571 -------------------------, --------------------------, 4835703278458516698824704 38685626227668133590597632 25090116015366454186282945 21787111032667624974931663 --------------------------, --------------------------, 38685626227668133590597632 38685626227668133590597632 36103139366373612447997331 111337872436037254581025811 --------------------------, ---------------------------, 77371252455336267181195264 309485009821345068724781056 18878649655742164526498327 4693506502208148542197849 --------------------------, --------------------------, 77371252455336267181195264 38685626227668133590597632 -3628647013901817991685633 -1334935411923390294048825233 ---------------------------, -----------------------------, 618970019642690137449562112 9903520314283042199192993792 -2599651737722437583981329337 -3822350909840453402487504401 -----------------------------, -----------------------------, 9903520314283042199192993792 9903520314283042199192993792 -4974509745105506769926015519 -12061310688196944713489301421 -----------------------------, ------------------------------, 9903520314283042199192993792 19807040628566084398385987584 -1742478816037876248310859237 -7777279110324750068021117033 -----------------------------, -----------------------------, 2475880078570760549798248448 9903520314283042199192993792 -135111609953486685965953864763 -1148211646039767398486474624239 -------------------------------, --------------------------------, 158456325028528675187087900672 1267650600228229401496703205376 -1198219872517159446478758520225 -1232539243629095165689149429235 --------------------------------, --------------------------------, 1267650600228229401496703205376 1267650600228229401496703205376 -2507053718079711711027930816413 -10112957319449088120008150048669 --------------------------------, ---------------------------------] 2535301200456458802993406410752 10141204801825835211973625643008 and in Maple notation [2251799813685241/2251799813685248, 18014398509480443/18014398509481984, 18014398509460409/18014398509481984, 18014398509277135/18014398509481984, 36028797015997079/36028797018963968, 144115188005959751/144115188075855872, 36028796931798769/36028797018963968, 18014398320145283/18014398509481984, 288230364454417349/288230376151711744, 4611685366092992311/4611686018427387904, 4611683940867743815/4611686018427387904, 4611679913057258935/ 4611686018427387904, 4611669328866373667/4611686018427387904, 2305821646988627767/2305843009213693952, 4611583007207437133/ 4611686018427387904, 4611450950473549207/4611686018427387904, 36889409342303551961/36893488147419103232, 295080387035568986905/ 295147905179352825856, 295014189855784429727/295147905179352825856, 294893714656010318741/295147905179352825856, 589365766112811249031/ 590295810358705651712, 2354618511123245153399/2361183241434822606848, 293748244105777104763/295147905179352825856, 146417975010265559699/ 147573952589676412928, 2331546554214123739969/2361183241434822606848, 74081404156816024477817/75557863725914323419136, 73321380247980187215557/ 75557863725914323419136, 72257346775610015048393/75557863725914323419136, 35403374709286669723057/37778931862957161709568, 275516679702655067267909/ 302231454903657293676544, 132760911536376789780637/151115727451828646838272, 63218311300911869623331/75557863725914323419136, 3795783053007440976769133/ 4835703278458516698824704, 27947772011633980582620571/ 38685626227668133590597632, 25090116015366454186282945/ 38685626227668133590597632, 21787111032667624974931663/ 38685626227668133590597632, 36103139366373612447997331/ 77371252455336267181195264, 111337872436037254581025811/ 309485009821345068724781056, 18878649655742164526498327/ 77371252455336267181195264, 4693506502208148542197849/ 38685626227668133590597632, -3628647013901817991685633/ 618970019642690137449562112, -1334935411923390294048825233/ 9903520314283042199192993792, -2599651737722437583981329337/ 9903520314283042199192993792, -3822350909840453402487504401/ 9903520314283042199192993792, -4974509745105506769926015519/ 9903520314283042199192993792, -12061310688196944713489301421/ 19807040628566084398385987584, -1742478816037876248310859237/ 2475880078570760549798248448, -7777279110324750068021117033/ 9903520314283042199192993792, -135111609953486685965953864763/ 158456325028528675187087900672, -1148211646039767398486474624239/ 1267650600228229401496703205376, -1198219872517159446478758520225/ 1267650600228229401496703205376, -1232539243629095165689149429235/ 1267650600228229401496703205376, -2507053718079711711027930816413/ 2535301200456458802993406410752, -10112957319449088120008150048669/ 10141204801825835211973625643008] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999995, .9999999976, .9999999895, .9999999594, .9999998585, .9999995495, .9999986761, .\ 9999963810, .9999907356, .9999776630, .9999490278, .9998894438, .9997712396, .9\ 995469549, .9991387690, .9984244438, .9972197286, .9952577638, .9921667912, .98\ 74483747, .9804592203, .9704003876, .9563180219, .9371195257, .9116082235, .878\ 5380170, .8366873835, .7849495377, .7224329741, .6485642980, .5631836203, .4666\ 221396, .3597520684, .2440008279, .1213242995, -.5862395429e-2, -.1347940298, -\ .2624977438, -.3859588094, -.5022971214, -.6089405739, -.7037815891, -.78530450\ 42, -.8526741355, -.9057792785, -.9452288133, -.9723020234, -.9888583327, -.997\ 2145832] The cut off is at j=, 41 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 56], vs. those in the, 2, -th row from j=1 to j=, 55, are as follws 28 27 26 [(36028797018963911 n - 14123288431433809222 n + 2636389207464141820887 n 25 24 - 311895170897664556918170 n + 26255673716853383246648295 n 23 22 - 1673979594529419385762234590 n + 84005741924269913605416931515 n 21 - 3404368790940842814546874066650 n 20 + 113420160963159294175694607429585 n 19 - 3146277958124080607630938941976170 n 18 + 73334400480788410935742643659049445 n 17 - 1445404527826261120547223712181941950 n 16 + 24191865187020664626669426795934090365 n 15 - 344655916770652166934733488123110096730 n 14 + 4182883815963932639837284976055337983705 n 13 - 43213295502293663553969455934918266357550 n 12 + 379167629492746340965185168102757155579780 n 11 - 2814785959986846200328122260875422051953560 n 10 + 17579909697213455019260172142408747327803760 n 9 - 91663750430011427118369084039350791615901600 n 8 + 394931654967526488501537095916050655733616064 n 7 - 1386897298769788441069743961901976591349849728 n 6 + 3896163098472145980683235219854752871259890688 n 5 - 8513996838433902142885643611383553184477614080 n 4 + 13711858094718006511742822925883994320668672000 n 3 - 13589340530333891534962487528023609467310080000 n 2 - 2279938976570590442397224691052028804628480000 n + 40491732227246990337639461343500750015692800000 n - 65295905239964720929554705324780583649280000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 28 27 (2 n - 51) (2 n - 45)), (36028797018962371 n - 14123288431432064402 n 26 25 + 2636389207463191315947 n - 311895170897333104096470 n 24 23 + 26255673716770282854973995 n - 1673979594513469224900699690 n 22 21 + 84005741921833189622269177215 n - 3404368790636716819956279390150 n 20 + 113420160931575480574465142954685 n 19 - 3146277955358547736137707247421470 n 18 + 73334400274644119213592730486108545 n 17 - 1445404514654438304495267565752102450 n 16 + 24191864462119704068514123994617476265 n 15 - 344655882306453877220775763021444118430 n 14 + 4182882398469001494712685243464478881605 n 13 - 43213245071913639991876266005653089652050 n 12 + 379166080049072607260869760373603047537580 n 11 - 2814744980973104882415287898712313202237960 n 10 + 17578981516288274289191938225178648697372560 n 9 - 91645877518298788025146602652358157700645600 n 8 + 394642054751541857357904523612764175335223104 n 7 - 1383004381834728909361473352772069774585778048 n 6 + 3853608591269269114049762811315177508999024128 n 5 - 8146584784261543073285804183454892740178913280 n 4 + 11317347177819178189343316731999915230927872000 n 3 - 2703807329876495609150279339272211968327680000 n 2 - 31324537334646961138753863759291744973946880000 n + 61882828761723345037488920004118872509644800000 n - 1165998307856512873742048309371081850880000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 28 (2 n - 51) (2 n - 45)), 1537 (93763947999847 n - 39427740133893341 n 27 26 + 7908660839937395433 n - 1007240763319400233449 n 25 24 + 91463073744183182573805 n - 6303881054579700680733645 n 23 22 + 342782460200348377951314885 n - 15090520315497581546483439405 n 21 + 547676437295319378691058509095 n 20 - 16600527010818644651292693793635 n 19 + 424211630890616246056446486082755 n 18 - 9200871155463467904296343845161515 n 17 + 170164952944320931885432311229008255 n 16 - 2691277933381535499712869000784384815 n 15 + 36449071089925409219020773175368043095 n 14 - 422705791700263132915503619960792259535 n 13 + 4191860178483841311839972157187471679910 n 12 - 35446753849881021027496896468056870682180 n 11 + 254486556339961528986989591395403629598840 n 10 - 1541707892715770151162926095762912368921520 n 9 + 7814105782093644206485331665029253036640928 n 8 - 32730800501193294619292866145354783702568384 n 7 + 111085202213341976989183063950491219420164992 n 6 - 293886257268276308860969887598354544377944576 n 5 + 548059513295178377613205186243712482884188160 n 4 - 457174528697858814917364761875030588498944000 n 3 - 919635469318575614946221968029201087897600000 n 2 + 3249365590669793300543020007408027098152960000 n - 2299578363448887317678874963218741959065600000 n + 86482633113625548215090115333964431360000000)/(268435456 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 (2 n - 51) (2 n - 57) (2 n - 45)), 1537 (46881973999667 n 28 27 - 19713870066663238 n + 3954330419818312023 n 26 25 - 503620381608696721782 n + 45731536859674411744635 n 24 23 - 3151940524979473539658710 n + 171391229758661533119320835 n 22 21 - 7545260116592049147513836190 n + 273838214529999672902128024245 n 20 - 8300263158938066876057393425530 n 19 + 212105790697279720501714154414805 n 18 - 4600434067211204934470049591379170 n 17 + 85082397345003251946185651208333705 n 16 - 1345635400336795985108421948226306770 n 15 + 18224397133178080568962498718590365345 n 14 - 211348274090635090645742474894454247530 n 13 + 2095797647738234857822928679602578647060 n 12 - 17720134254958463113239141390898325970840 n 11 + 127175893830412408389340009950840104291440 n 10 - 769676611516016243722686590687763387913760 n 9 + 3889983356490661203930740707004059941916608 n 8 - 16163783214255944706891504545201951968182912 n 7 + 53653284581732736588343495614746237151255552 n 6 - 133449916303614980722168591725021099523181568 n 5 + 205311328466683586056976704574354280905134080 n 4 - 9518229580989601700232668850137250465792000 n 3 - 757516210001962751248400061674287050792960000 n 2 + 1391917750617955789961415107061439666913280000 n - 781465824624449136530526901124540320972800000 n + 43241316556812774107545057666982215680000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 30 (2 n - 51) (2 n - 57) (2 n - 45)), 1537 (375055791982763 n 29 28 - 168775106381095005 n + 36287116685702678695 n 27 26 - 4962185029031858702445 n + 484706704225772467204227 n 25 24 - 36008166766044359891895345 n + 2114987782298942460751324275 n 23 22 - 100810408778661904550573591025 n + 3971386865135004369737303094045 n 21 - 131027903747996058179368412995575 n 20 + 3655706985663570391392745076222325 n 19 - 86860350510194221956032731988869575 n 18 + 1766356951387198183922406799484305665 n 17 - 30844317363185097574803972880584637275 n 16 + 463356862461515954354051921336257033425 n 15 - 5991465604065373648625413941031323735675 n 14 + 66636524058970234252307146471315356460020 n 13 - 636170037932344140946826195903526352111200 n 12 + 5195161942650430647448935420806539116517600 n 11 - 36096376917541661863039439962548650318253600 n 10 + 211659964602890145531954801310068393601139072 n 9 - 1033951898373068067274822152614064760664286720 n 8 + 4113628277560910991331787167241160670345831680 n 7 - 12748477794920323261480996675333330990555687680 n 6 + 27699217384292000747960168646833919467958494208 n 5 - 28274803169857907890851545689394118145798778880 n 4 - 45620244530150403705703296637874310475204608000 n 3 + 210512032148945353680495265737927330498723840000 n 2 - 287369086595315894461907070350935087636807680000 n + 142919418492610306027259610573858467531980800000 n - 10204950707407814689380633609407802900480000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 30 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 29 (19877956970627101 n 29 28 - 8945080633381950225 n + 1923217181847554513495 n 27 26 - 262995805713716020281525 n + 25689455128389067944602499 n 25 24 - 1908432803216282159732245785 n + 112094347383926043379635511275 n 23 - 5342951072155389252324158291625 n 22 + 210483446445431814510960797414865 n 21 - 6944474235099625457973210205309575 n 20 + 193752149346658395102432452858189325 n 19 - 4603579755537989815492640153631771375 n 18 + 93615973530555879991139940460211137905 n 17 - 1634708128959758251870809660164046287475 n 16 + 24556409848944010438905237912323895154425 n 15 - 317500038342941871660125311339854632395875 n 14 + 3530446203455123514696656274458075993590590 n 13 - 33687329458574994555976799870648611646343900 n 12 + 274766870580035864745471633278066436260660600 n 11 - 1903745243956600637622878513847948484688970000 n 10 + 11092729767459452639255200799254112965557360544 n 9 - 53445114033578896770915032221257441401677531200 n 8 + 206499544357381386928174493585746192918603370880 n 7 - 601504895541691304651457313952136229889204889600 n 6 + 1128300213104976667453667019001043572569310826496 n 5 - 501757758464839789938180049397517515451373731840 n 4 - 3900353543294349086523525261284925810729062400000 n 3 + 11365487605251444835499664681308760972204441600000 n 2 - 13396674117379567809496531954072656600724930560000 n + 6294044312581191932408523344696710257337958400000 n - 540862387492614178537173581298613553725440000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 6293 (366413952973837 n 30 29 - 176061904098778466 n + 40480039241990024540 n 28 27 - 5929104620810592277720 n + 621397718536101122927388 n 26 25 - 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63241303117710249564662273998130806381232\ 73548852850982526810269217767854814009491456000000000000)/(562949953421312 (2 n - 99) (2 n - 95) (2 n - 85) (2 n - 97) (2 n - 83) (2 n - 91) (2 n - 81) (2 n - 77) (2 n - 69) (2 n - 71) (2 n - 45) (2 n - 63) (2 n - 57) (2 n - 51) (2 n - 73) (2 n - 101) (2 n - 29) (2 n - 53) (2 n - 35) (2 n - 55) (2 n - 79) (2 n - 39) (2 n - 23) (2 n - 17) (2 n - 105) (2 n - 37) (2 n - 15) (2 n - 89) (2 n - 25) (2 n - 31) (2 n - 67) (2 n - 87) (2 n - 59) (2 n - 13) (2 n - 43) (2 n - 75) (2 n - 27) (2 n - 21) (2 n - 41) (2 n - 11) (2 n - 33) (2 n - 65) (2 n - 103) (2 n - 9) (2 n - 61) (2 n - 7) (2 n - 49) (2 n - 93) (2 n - 47) / (2 n - 19) (2 n - 3) (-1 + 2 n) (2 n - 5)), | \ 7037658466435275483969919503888 (n - 53)! (n - 52)! (2 n - 111)! (n + 1) + / 10459639188646923196358962932384 (n - 54)! | \ 109 --- (n - 52)! (n + 1) (2 n - 110)! + (n - 53)! ((2 n - 109)! (n + 1) 108 - 1/10459639188646923196358962932384 binomial(2 n, n) (n - 56)! (n - 52)!) \\ ||/((n - 54)! (n - 53)! (n - 56)! (n - 52)! binomial(2 n, n)), ( // 7037658466435275483969919503888 (n - 53)! (2 n - 111)! (n + 1) + 10556487699652913225954879255832 (n - 54)! ((2 n - 110)! (n + 1) - 1/10556487699652913225954879255832 binomial(2 n, n) (n - 56)! (n - 53)!) )/((n - 54)! (n - 56)! (n - 53)! binomial(2 n, n)), ( (7037658466435275483969919503888 n + 7037658466435275483969919503888) (2 n - 111)! - (n - 56)! (n - 54)! binomial(2 n, n))/((n - 56)! (n - 54)! binomial(2 n, n))] and in Maple notation [1/134217728*(36028797018963911*n^28-14123288431433809222*n^27+ 2636389207464141820887*n^26-311895170897664556918170*n^25+ 26255673716853383246648295*n^24-1673979594529419385762234590*n^23+ 84005741924269913605416931515*n^22-3404368790940842814546874066650*n^21+ 113420160963159294175694607429585*n^20-3146277958124080607630938941976170*n^19+ 73334400480788410935742643659049445*n^18-1445404527826261120547223712181941950* n^17+24191865187020664626669426795934090365*n^16-\ 344655916770652166934733488123110096730*n^15+ 4182883815963932639837284976055337983705*n^14-\ 43213295502293663553969455934918266357550*n^13+ 379167629492746340965185168102757155579780*n^12-\ 2814785959986846200328122260875422051953560*n^11+ 17579909697213455019260172142408747327803760*n^10-\ 91663750430011427118369084039350791615901600*n^9+ 394931654967526488501537095916050655733616064*n^8-\ 1386897298769788441069743961901976591349849728*n^7+ 3896163098472145980683235219854752871259890688*n^6-\ 8513996838433902142885643611383553184477614080*n^5+ 13711858094718006511742822925883994320668672000*n^4-\ 13589340530333891534962487528023609467310080000*n^3-\ 2279938976570590442397224691052028804628480000*n^2+ 40491732227246990337639461343500750015692800000*n-\ 65295905239964720929554705324780583649280000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1/134217728*( 36028797018962371*n^28-14123288431432064402*n^27+2636389207463191315947*n^26-\ 311895170897333104096470*n^25+26255673716770282854973995*n^24-\ 1673979594513469224900699690*n^23+84005741921833189622269177215*n^22-\ 3404368790636716819956279390150*n^21+113420160931575480574465142954685*n^20-\ 3146277955358547736137707247421470*n^19+73334400274644119213592730486108545*n^ 18-1445404514654438304495267565752102450*n^17+ 24191864462119704068514123994617476265*n^16-\ 344655882306453877220775763021444118430*n^15+ 4182882398469001494712685243464478881605*n^14-\ 43213245071913639991876266005653089652050*n^13+ 379166080049072607260869760373603047537580*n^12-\ 2814744980973104882415287898712313202237960*n^11+ 17578981516288274289191938225178648697372560*n^10-\ 91645877518298788025146602652358157700645600*n^9+ 394642054751541857357904523612764175335223104*n^8-\ 1383004381834728909361473352772069774585778048*n^7+ 3853608591269269114049762811315177508999024128*n^6-\ 8146584784261543073285804183454892740178913280*n^5+ 11317347177819178189343316731999915230927872000*n^4-\ 2703807329876495609150279339272211968327680000*n^3-\ 31324537334646961138753863759291744973946880000*n^2+ 61882828761723345037488920004118872509644800000*n-\ 1165998307856512873742048309371081850880000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-45), 1537/268435456*( 93763947999847*n^29-39427740133893341*n^28+7908660839937395433*n^27-\ 1007240763319400233449*n^26+91463073744183182573805*n^25-\ 6303881054579700680733645*n^24+342782460200348377951314885*n^23-\ 15090520315497581546483439405*n^22+547676437295319378691058509095*n^21-\ 16600527010818644651292693793635*n^20+424211630890616246056446486082755*n^19-\ 9200871155463467904296343845161515*n^18+170164952944320931885432311229008255*n^ 17-2691277933381535499712869000784384815*n^16+ 36449071089925409219020773175368043095*n^15-\ 422705791700263132915503619960792259535*n^14+ 4191860178483841311839972157187471679910*n^13-\ 35446753849881021027496896468056870682180*n^12+ 254486556339961528986989591395403629598840*n^11-\ 1541707892715770151162926095762912368921520*n^10+ 7814105782093644206485331665029253036640928*n^9-\ 32730800501193294619292866145354783702568384*n^8+ 111085202213341976989183063950491219420164992*n^7-\ 293886257268276308860969887598354544377944576*n^6+ 548059513295178377613205186243712482884188160*n^5-\ 457174528697858814917364761875030588498944000*n^4-\ 919635469318575614946221968029201087897600000*n^3+ 3249365590669793300543020007408027098152960000*n^2-\ 2299578363448887317678874963218741959065600000*n+ 86482633113625548215090115333964431360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 1537/ 134217728*(46881973999667*n^29-19713870066663238*n^28+3954330419818312023*n^27-\ 503620381608696721782*n^26+45731536859674411744635*n^25-\ 3151940524979473539658710*n^24+171391229758661533119320835*n^23-\ 7545260116592049147513836190*n^22+273838214529999672902128024245*n^21-\ 8300263158938066876057393425530*n^20+212105790697279720501714154414805*n^19-\ 4600434067211204934470049591379170*n^18+85082397345003251946185651208333705*n^ 17-1345635400336795985108421948226306770*n^16+ 18224397133178080568962498718590365345*n^15-\ 211348274090635090645742474894454247530*n^14+ 2095797647738234857822928679602578647060*n^13-\ 17720134254958463113239141390898325970840*n^12+ 127175893830412408389340009950840104291440*n^11-\ 769676611516016243722686590687763387913760*n^10+ 3889983356490661203930740707004059941916608*n^9-\ 16163783214255944706891504545201951968182912*n^8+ 53653284581732736588343495614746237151255552*n^7-\ 133449916303614980722168591725021099523181568*n^6+ 205311328466683586056976704574354280905134080*n^5-\ 9518229580989601700232668850137250465792000*n^4-\ 757516210001962751248400061674287050792960000*n^3+ 1391917750617955789961415107061439666913280000*n^2-\ 781465824624449136530526901124540320972800000*n+ 43241316556812774107545057666982215680000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-19) /(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27) /(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 1537/ 536870912*(375055791982763*n^30-168775106381095005*n^29+36287116685702678695*n^ 28-4962185029031858702445*n^27+484706704225772467204227*n^26-\ 36008166766044359891895345*n^25+2114987782298942460751324275*n^24-\ 100810408778661904550573591025*n^23+3971386865135004369737303094045*n^22-\ 131027903747996058179368412995575*n^21+3655706985663570391392745076222325*n^20-\ 86860350510194221956032731988869575*n^19+1766356951387198183922406799484305665* n^18-30844317363185097574803972880584637275*n^17+ 463356862461515954354051921336257033425*n^16-\ 5991465604065373648625413941031323735675*n^15+ 66636524058970234252307146471315356460020*n^14-\ 636170037932344140946826195903526352111200*n^13+ 5195161942650430647448935420806539116517600*n^12-\ 36096376917541661863039439962548650318253600*n^11+ 211659964602890145531954801310068393601139072*n^10-\ 1033951898373068067274822152614064760664286720*n^9+ 4113628277560910991331787167241160670345831680*n^8-\ 12748477794920323261480996675333330990555687680*n^7+ 27699217384292000747960168646833919467958494208*n^6-\ 28274803169857907890851545689394118145798778880*n^5-\ 45620244530150403705703296637874310475204608000*n^4+ 210512032148945353680495265737927330498723840000*n^3-\ 287369086595315894461907070350935087636807680000*n^2+ 142919418492610306027259610573858467531980800000*n-\ 10204950707407814689380633609407802900480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 29/536870912*(19877956970627101*n^30-8945080633381950225*n^29+ 1923217181847554513495*n^28-262995805713716020281525*n^27+ 25689455128389067944602499*n^26-1908432803216282159732245785*n^25+ 112094347383926043379635511275*n^24-5342951072155389252324158291625*n^23+ 210483446445431814510960797414865*n^22-6944474235099625457973210205309575*n^21+ 193752149346658395102432452858189325*n^20-4603579755537989815492640153631771375 *n^19+93615973530555879991139940460211137905*n^18-\ 1634708128959758251870809660164046287475*n^17+ 24556409848944010438905237912323895154425*n^16-\ 317500038342941871660125311339854632395875*n^15+ 3530446203455123514696656274458075993590590*n^14-\ 33687329458574994555976799870648611646343900*n^13+ 274766870580035864745471633278066436260660600*n^12-\ 1903745243956600637622878513847948484688970000*n^11+ 11092729767459452639255200799254112965557360544*n^10-\ 53445114033578896770915032221257441401677531200*n^9+ 206499544357381386928174493585746192918603370880*n^8-\ 601504895541691304651457313952136229889204889600*n^7+ 1128300213104976667453667019001043572569310826496*n^6-\ 501757758464839789938180049397517515451373731840*n^5-\ 3900353543294349086523525261284925810729062400000*n^4+ 11365487605251444835499664681308760972204441600000*n^3-\ 13396674117379567809496531954072656600724930560000*n^2+ 6294044312581191932408523344696710257337958400000*n-\ 540862387492614178537173581298613553725440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n -27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2 *n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 6293/1073741824*(366413952973837*n^31-176061904098778466*n^30+ 40480039241990024540*n^29-5929104620810592277720*n^28+621397718536101122927388* n^27-49621401066495709886652504*n^26+3139198569037187637450882360*n^25-\ 161508307263632436315423212400*n^24+6883744920853047376563339015630*n^23-\ 246344616308634459802459464073740*n^22+7475704218784238476968547663917600*n^21-\ 193786581393667995500764805619283200*n^20+4313735845580545629714240071145046860 *n^19-82760883982590088992241133735169610680*n^18+ 1371555052722200341538645558535079420200*n^17-\ 19653608977590804210137086388886580798800*n^16+ 243446318542469436387743884992506492384205*n^15-\ 2602684661728150884971891646357086267123490*n^14+ 23941589318349585293359091491761678432592100*n^13-\ 188517898127550891337475440073804683721914600*n^12+ 1260123775019465600997859918746373182909628528*n^11-\ 7054467315739905186979300609891601795644932704*n^10+ 32327656236263979837960769712382016602658925760*n^9-\ 116349381890882489295068629925097665183085713280*n^8+ 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4398985493520210698850162189915*n^22+133493017141525281839607142329975*n^21-\ 3460378145081056486334744705538075*n^20+77026317453826678192340298913974435*n^ 19-1477678648985902648678615716501737655*n^18+ 24485145797905529259940708710030348075*n^17-\ 350747128011585503394404307883542932175*n^16+ 4341802139926846164340770046700549547180*n^15-\ 46356399930911289759316845723420521757540*n^14+ 425298519487520706557727925016655300716600*n^13-\ 3331826516678661754639153496347624384621600*n^12+ 22060490499687843722057646724479119222157088*n^11-\ 121395993001660609169046484805299196442886784*n^10+ 539683673087645354531266489096943564682053760*n^9-\ 1840076243259964663038101538689807955197326080*n^8+ 4278747365570207642759881640456970681412987392*n^7-\ 4125011532931091627341177624233888726207783936*n^6-\ 12227204691706318302770735045782282465497845760*n^5+ 59417257169375072486151148122179196597811200000*n^4-\ 114233933846743331138655873642580686399774720000*n^3+ 112242723755150539264938796282684739517480960000*n^2-\ 49468905229786105544336200314180682933862400000*n+ 5429987761199714432318562945887989923840000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 2070397/1073741824*(2227440393935*n^32-1140449452154128*n^31+ 279793220304728152*n^30-43794677143361379200*n^29+4912852096524082770660*n^28-\ 420640532268574764710592*n^27+28585191160635650634718488*n^26-\ 1582930392613046064155368320*n^25+72773006348430692180463637050*n^24-\ 2815652100117537354615524001120*n^23+92614145226377126618434341284280*n^22-\ 2609363318990666721796122035990400*n^21+63321991587629206469951368809106500*n^ 20-1328737748260763267449952940970127040*n^19+ 24170724100203277028836681800901329960*n^18-\ 381643597918648462645595032216588291200*n^17+ 5230623144502830494699395404209610022575*n^16-\ 62143134616048402982426718004125077794320*n^15+ 638084120653148779843530686095660586847280*n^14-\ 5633192421452462250002664997718717560812800*n^13+ 42403200580999403505958056180375622906190240*n^12-\ 268590144383216311885579383367368367036268032*n^11+ 1401786144465169824561505927503869227077497088*n^10-\ 5817648271438221799501242376425829867704422400*n^9+ 17928889580895404711502348186786898963458359040*n^8-\ 34132462470469345772381456694942379662510624768*n^7+ 3210964554212147247958601898170982027386314752*n^6+ 215332390313807042655956382921128826632609464320*n^5-\ 723895221228998021801644536195126019617300480000*n^4+ 1229110023593955623314890466133465935126855680000*n^3-\ 1132407885712726560704747500001396121568542720000*n^2+ 488652609920380863188215429190119400983756800000*n-\ 58227953864779916465713951589947806842880000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-63)/(2*n-45), 42253/1073741824*(109144572738239*n^32-55882016334957808*n^31 +13709864401571053528*n^30-2145938104066806124160*n^29+240729508593652377135396 *n^28-20611343886604688241767232*n^27+1400668593859478400123067032*n^26-\ 77562947711493642444386570880*n^25+3565818365648633624576989518810*n^24-\ 137962417391889901270372846146720*n^23+4537798216680319588200661338436920*n^22-\ 127842497439904683187419541593283200*n^21+3102007817752178104384669189071526020 *n^20-65077050286228479868318323763677944640*n^19+ 1183289633000201986214838119434213366440*n^18-\ 18668699245618199763470838131083018915200*n^17+ 255496642558688954390549560201539079781535*n^16-\ 3027765526777882154029404780659089956046320*n^15+ 30953520899206866206618555949665085599855920*n^14-\ 271278472707711509931624594400762407327584000*n^13+ 2017938796237611179796048995731357355501837216*n^12-\ 12543921805677533825278032065160990521931223552*n^11+ 63563193158856481960784329432681047891015704832*n^10-\ 251516803494468422641670069973804861508031754240*n^9+ 710141635717473542968857860257906914419116502784*n^8-\ 1049196239737294075337527520843867948967006793728*n^7-\ 1444964092492399188066625223799913596938149404672*n^6+ 13089631263878884497347846840086717304824521031680*n^5-\ 37092138922039713634575096145902846288455639040000*n^4+ 58626999275767512764111375012960997784999034880000*n^3-\ 52090853549351446080743530184513052119680942080000*n^2+ 22347561484305946253120898524224781610595123200000*n-\ 2853169739374215906819983627907442535301120000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 211265/1073741824*(43657820916928*n^33-23771677416829353*n^ 32+6210407677442501632*n^31-1036602104562433009128*n^30+ 124188683913814529660992*n^29-11373966165753250779624828*n^28+ 828207295320212421524053440*n^27-49233014031932968672819396584*n^26+ 2434565836452980269324477008960*n^25-101534367726428646620440901432790*n^24+ 3608218113294149053312910355683520*n^23-110105123078661679297768033351116360*n^ 22+2901634080242714149339431112203531840*n^21-\ 66310505061412369150061307847016473500*n^20+ 1317667222593939219951130618298646146880*n^19-\ 22799858648639824726544442327391243379160*n^18+ 343572923073207231220469131062586082910720*n^17-\ 4503062225333933578016931175734114392098185*n^16+ 51181005687580477476552374605398321973559360*n^15-\ 501876408899000026421598356855741606975571600*n^14+ 4211951070970582510826876156248291023110536832*n^13-\ 29885913412700243385244538898771054352716047712*n^12+ 175962816021731900984611296497894558601403309568*n^11-\ 834060595110903929706688386102927160466309736192*n^10+ 3009647863006176333205924003153133766584557037568*n^9-\ 7196871721327126451107249784001629558231947237632*n^8+ 4853197308228891730645864494914686544786625945600*n^7+ 43152342958469891974008135752178096267501246849024*n^6-\ 212320727960747301508346937212396463511519908003840*n^5+ 519814718361365104662867958619401428934141599744000*n^4-\ 762814464921028516747077733821799772819747635200000*n^3+ 650273161224021936722452100253652733580198543360000*n^2-\ 275748617053859584423719821122044149877021081600000*n+ 37091206611864806788659787162796752958914560000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2 *n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/ (2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51 )/(2*n-57)/(2*n-63)/(2*n-45), 4495/1073741824*(2051916479015990*n^33-\ 1117267709117522193*n^32+291888607810729231808*n^31-48720126416875782287496*n^ 30+5836829658283277606032520*n^29-534569872160591124987989244*n^28+ 38924864074922965995916788816*n^27-2313855746246381141072026486728*n^26+ 114415941414144492935724809391780*n^25-4771461707488504139306575666996710*n^24+ 169544539242333433048037411146609200*n^23-5172676858650688543228181135523817320 *n^22+136271854950369797635464090436923467400*n^21-\ 3112428091663931684653184704679685856860*n^20+ 61788641132209221707361782935536981944880*n^19-\ 1067462818296565388418011548348692821894520*n^18+ 16044974556837033878661735636344427333902550*n^17-\ 209456450851077946388230503655659603491631825*n^16+ 2366068129046838229779728751504043113466164880*n^15-\ 22988485536124286597909359815292355623317502800*n^14+ 190330604301393683122326188548960636463131946560*n^13-\ 1324166865176707198166911306130696313231618771552*n^12+ 7575477980978069730068559919337221892510331983872*n^11-\ 34363135875643334570755272539339233704405385522944*n^10+ 114806592343668882346364816081951339666099878663680*n^9-\ 225107181231746808832450616505684317012983159439616*n^8-\ 128615737773335710599581418964154604858059598483456*n^7+ 2812513304485960939158989023973893726224408166391808*n^6-\ 11028767058886849212043082901609555226273176563220480*n^5+ 24896032239780036443353815492266483429340665438208000*n^4-\ 35004333948144615863937624193493199939985348362240000*n^3+ 29195680340567570422207747618466523090264714117120000*n^2-\ 12374372746455373349007573363341838665101030195200000*n+ 1743286710757645919067009996651447389068984320000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15 )/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-\ 51)/(2*n-57)/(2*n-63)/(2*n-45), 193285/1073741824*(95437838642017*n^34-\ 55162970495856593*n^33+15317003216653037911*n^32-2720833460119787534456*n^31+ 347388210194433021644932*n^30-33957450263613192050888780*n^29+ 2643275979266134155702427548*n^28-168258563252175533593631318616*n^27+ 8925788000125075129125978112782*n^26-400112532601713829113718673349750*n^25+ 15314403289679660346447511479020370*n^24-504430686093366636003352070870096280*n ^23+14382131995166742724651526456413192740*n^22-\ 356446947225037166310395727514569782220*n^21+ 7700712481692237828105469035099727463820*n^20-\ 145235733573080497346701046694158241042600*n^19+ 2391617104020301928507660680538728125680985*n^18-\ 34342919565533491167464462121951297814259265*n^17+ 428796902901994754390650620171402643484524175*n^16-\ 4632429793118082716781501837747283172788401520*n^15+ 42979244253400458437659038369304013782339426448*n^14-\ 338678542184013822709276354459479140808230571232*n^13+ 2229349224014988678274358431483043576868591641504*n^12-\ 11938189076195481993903740596887922589009392724224*n^11+ 49564459367490608498064943646320032804096993965568*n^10-\ 142206867230940666197399092992537111788211390593280*n^9+ 159935774161117625322841441443897490203182619516672*n^8+ 871671736458542604850261868363287572400658482077696*n^7-\ 5999788393120052841092196983549633591756690084585472*n^6+ 19703747071405203875047506199391786360140223130501120*n^5-\ 40891491648477050221968815928459844439152192069632000*n^4+ 54676630018829107221222571185940360254081197015040000*n^3-\ 44295734322297344545299156109672358028842387374080000*n^2+ 18634506441857966362022248892129915345938730188800000*n-\ 2716283944668890152964876041294115699247022080000000)/(2*n-5)/(-1+2*n)/(2*n-3)/ (2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/ (2*n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-67)/(2*n-31)/(2*n-25 )/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-\ 29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45), 38657/1073741824*(477187479718987*n^34 -275813121853273985*n^33+76584179644299191953*n^32-13603909812920848333880*n^31 +1736884360892171028390028*n^30-169777748875742090121773900*n^29+ 13215119466519619867393707972*n^28-841157059602382411761605334360*n^27+ 44616850787796712096789699900890*n^26-1999660988243510720430248927644950*n^25+ 76515159745264181772594698128885470*n^24-2519102004282244484822804841272127000* n^23+71770629794748867441977623732639208460*n^22-\ 1776723997771306310485599284014325445900*n^21+ 38317306228693508842112298624346899824340*n^20-\ 720771017150767757217463316285365476805800*n^19+ 11823353728832749686792050146911124974247155*n^18-\ 168840091584881869776153237751079869524589425*n^17+ 2091641478900210839460196926735709600036165145*n^16-\ 22352284224442194480786148427564946744887018800*n^15+ 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(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53 )/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 33263/34359738368*(283629560706987017*n^38-204685060539188726299*n^37+ 71252443358867623244139*n^36-15935837470920680668507185*n^35+ 2573202348188004028244561736*n^34-319592423081793269561414702952*n^33+ 31761208483902907392438123211132*n^32-2594080889426041486035119770572500*n^31+ 177469436630975430017167747158864702*n^30-\ 10313325515538655873512943266471805434*n^29+ 514474432035771168274374770064333205914*n^28-\ 22204820032028986100803039397441135265390*n^27+ 834073449051318932865988905424914991179180*n^26-\ 27382915885404949136684447857767957576767460*n^25+ 787972566249933366473452703887448215927220460*n^24-\ 19906222787835646548543894103652964781348709700*n^23+ 441651082172179205311860175716271637094422699745*n^22-\ 8598261049269238271038065075204354954923625081315*n^21+ 146558636329833880848024565861122592129819128266515*n^20-\ 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/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-\ 67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2* n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/( 2*n-69), 33263/68719476736*(1131910332398965243*n^39-859772233690119856314*n^38 +315241645048907462074202*n^37-74314596546683214878034996*n^36+ 12657103037295328977754438959*n^35-1659275614618943128828514805402*n^34+ 174170383605137125017849175931936*n^33-15034928993004939705965944554874128*n^32 +1087824294552942098629132666716492358*n^31-\ 66899066934566765897142896015500937844*n^30+ 3533736860810164107624291742062887164092*n^29-\ 161594684580922081403818935617770570711416*n^28+ 6435105597322822551041503703891942656377030*n^27-\ 224115337046009685548042381020927020903381060*n^26+ 6845787675978827876138444873276474344474144680*n^25-\ 183702023621661589975475170422245782963950518640*n^24+ 4332280862552866442053059252185377480573700042655*n^23-\ 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53575031941300149579*n^38+19626267139072075668017*n^37-\ 4621503015942952756460181*n^36+786026180342642874617064264*n^35-\ 102865309437422118528312124572*n^34+10774541944695374295292525209956*n^33-\ 927673634459111848736759586920808*n^32+66908662375998373464356140728056218*n^31 -4099162363974943143656048288611370634*n^30+ 215545585294090663597560431593964663982*n^29-\ 9803727176986730342117800571737209514326*n^28+ 387928357515109150415472763611039716055780*n^27-\ 13409213406113945333630337627762677085837660*n^26+ 405984351492150538695219406420734869087775780*n^25-\ 10780981496559320222480435667703427660341650540*n^24+ 251109929097732710957707040654576249524399672755*n^23-\ 5122773392246885823827844975234530553653759368515*n^22+ 91238848472482542999116591732630442544868580842345*n^21-\ 1410582999942995126929720416221788861541132066357085*n^20+ 18747652399492968593196387792606505661846829371400412*n^19-\ 210586159642295487201197714408405078471380495587735216*n^18+ 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82408357883431714125525520327186636046759613915243479040000000000)/(2*n-5)/(-1+ 2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-\ 33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/( 2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45) /(2*n-71)/(2*n-69)/(2*n-77), 33263/34359738368*(1119240720036187457*n^40-\ 891718400791692716980*n^39+343024714219789555023850*n^38-\ 84853544626989698685795800*n^37+15166635604129755801781231389*n^36-\ 2086618708059894330340760638260*n^35+229848677106446654727380390833300*n^34-\ 20818356539930885357998413438114800*n^33+1580061548000755993193614746875943506* n^32-101895356903715932915844056871581006440*n^31+ 5641423188262174833751609175631926589900*n^30-\ 270239014085039720392061865729123218840400*n^29+ 11264774040983722162578582210152496235174578*n^28-\ 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06193128291629216036120871339916054304991757476529055457626272093642752*n^13-13\ 2135634297636322753402601338670488260042441847217238429714146449728727658660548\ 911051666096128*n^12+6176171278307317961015956533190792329765667524057462684783\ 51434082622665257834714159771918401536*n^11-25176266270746441013028144547925917\ 68756025418702746515498102646775025683697597343046181706530816*n^10+88617957873\ 7405381612517647875868220678398290834003736454705773561550071115729092542804362\ 5267200*n^9-2660999449753028819428006780544490222979104463646201201490084020833\ 9707143709753711335735033856000*n^8+6714531074774260040702375196958970166899944\ 8321095001274169797509996039552523599160830402232320000*n^7-1396722408685630162\ 4432680324849728501069266735969619555060084380350990709773941052055494852608000\ 0*n^6+2335707429503646226319484617038485863789758909373722507703507734446584745\ 84292010104266424320000000*n^5-303431472924612163070403665812871415020964687288\ 980865862315799557738376585249138246984663040000000*n^4+29146279297302650786790\ 2489218783680231553402190895065759773593762642321809204564599190323200000000*n^ 3-19160575865560861364849928743961867500570781084970914138599144926781903959590\ 6269558970777600000000*n^2+7505843058642505828193010299417581287741921816824831\ 7923274211423875428773320223862292480000000000*n-126482606235420499129324547996\ 26161276246547097705701965053620538435535709628018982912000000000000)/(2*n-101) /(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-\ 105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2 *n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/ (2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/ (-1+2*n)/(2*n-5), -265/562949953421312/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n -83)/(2*n-91)*(18114089211400737509553005353*n^53-\ 25519927712604702731213878031090*n^52+17522210225490167426688060449573635*n^51-\ 7813949968408363139611456253471526038*n^50+ 2544812631234189353263733828246419090870*n^49-\ 645272095849134710060664943477862699591660*n^48+ 132623621076202113883817595548578316937751570*n^47-\ 22712999732297768676572170669767015843203801540*n^46+ 3306725509146879238680752375673324900869587535635*n^45-\ 415488952691875370842875949105517295345082681978310*n^44+ 45590007139609361773825256648042426399115457690404745*n^43-\ 4409574656911600760631645945058904992467508601456425730*n^42+ 378821069469764756453177180850189579827984751166103505340*n^41-\ 29085933410356942691888287213327318506043831525900196483960*n^40+ 2006224411121747742646023032974582433014767969042689382525820*n^39-\ 124849498301117639866218459457907194189814177319703314736831160*n^38+ 7034977707905016758086907159747701374853340115887026671269142655*n^37-\ 360006728005526278442462975408806642694009702248266637430220044510*n^36+ 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6763315026580467582904271765195440*n^24+180957992536959937725599614066508977168\ 9614676863091242316951506126738088172226196880*n^23-269558697267604052357126802\ 71090307772176301772832023798924908271050138033707127013920*n^22+36674184638546\ 9858455008219813209500664453209827432907985523165196945520221864799586560*n^21-\ 4549476235258832456830716419069511349306016143787773927228175266943207973705225\ 074199040*n^20+5135488362918077213775672283865632969166604012900113798354808745\ 8127467379725703886823680*n^19-526254368439737497130113943985053385797986204010\ 783073410172168437562691397158688182858240*n^18+4882109272145942092034740209984\ 246624165893484682933425598003079094566378941920362249656320*n^17-4087218180122\ 3069463728122712009734526745780063516890795837103871052301152444914188989706240 *n^16+3076410042413110622816558426622971062649280324194316975051705417001421881\ 50015795702377902080*n^15-20729228242584667321543285824707453502046506880455866\ 95012159510764468964771183382954339328000*n^14+12441167468970866355266384190984\ 482399977710110111498674252093170150769501201325053114602160128*n^13-6611824826\ 1643850752524466438603729759358529122033435963096727132767934381964143078826575\ 790080*n^12+3089933584563191139284696441956160740674747844292991648522393648321\ 45321685433351082252608471040*n^11-12593850017660110988469594689400941226099912\ 42425831545980384401713821852221949118480633370247168*n^10+44323585091016244955\ 35573030145718867775244880676930253909816574328485947007168758857351141785600*n ^9-1330797826086786557161651081128719980528802536080369951513419361603880585437\ 6104479985277861888000*n^8+3357724104272052724918146055854552155390026633812565\ 6083013481835289854325581361931572551352320000*n^7-6984076749640684644843490469\ 8536660119245447248778005741567413909490865360587458242372530339840000*n^6+1167\ 8692449534449907205922205260172449487080051383408594096680410401622335509338420\ 2656153600000000*n^5-1517121550288669519227938514700588851229580065611060193261\ 65570706926767856645258814496440320000000*n^4+145724928743659933061489828450799\ 948842640772361549044113613055414891977325117606862834892800000000*n^3-95798208\ 0997259909664386657085056967785858885659239562111692216618995100215518765617512\ 44800000000*n^2+375278866944075352281590711972139961276511798249752402954643059\ 61881156032068764450160640000000000*n-63241303117710249564662273998130806381232\ 73548852850982526810269217767854814009491456000000000000)/(2*n-81)/(2*n-77)/(2* n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-101)/(2*n-29)/ (2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-105)/(2*n-\ 37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2* n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/ (2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/( 2*n-5), (7037658466435275483969919503888*(n-53)!*(n-52)!*(2*n-111)!*(n+1)+ 10459639188646923196358962932384*(n-54)!*(109/108*(n-52)!*(n+1)*(2*n-110)!+(n-\ 53)!*((2*n-109)!*(n+1)-1/10459639188646923196358962932384*binomial(2*n,n)*(n-56 )!*(n-52)!)))/(n-54)!/(n-53)!/(n-56)!/(n-52)!/binomial(2*n,n), ( 7037658466435275483969919503888*(n-53)!*(2*n-111)!*(n+1)+ 10556487699652913225954879255832*(n-54)!*((2*n-110)!*(n+1)-1/ 10556487699652913225954879255832*binomial(2*n,n)*(n-56)!*(n-53)!))/(n-54)!/(n-\ 56)!/(n-53)!/binomial(2*n,n), ((7037658466435275483969919503888*n+ 7037658466435275483969919503888)*(2*n-111)!-(n-56)!*(n-54)!*binomial(2*n,n))/(n -56)!/(n-54)!/binomial(2*n,n)] The limits, as n goes to infinity are 36028797018963911 36028797018962371 144115188075764839 72057594037488179 [-----------------, -----------------, ------------------, -----------------, 36028797018963968 36028797018963968 144115188075855872 72057594037927936 576460752277506731 576460752148185929 2305843006064356241 ------------------, ------------------, -------------------, 576460752303423488 576460752303423488 2305843009213693952 288230374413289427 4611685909281842195 4611685631908812467 ------------------, -------------------, -------------------, 288230376151711744 4611686018427387904 4611686018427387904 144115149000231155 4611682286588437525 18446702641922255845 ------------------, -------------------, --------------------, 144115188075855872 4611686018427387904 18446744073709551616 18446636403496880459 73785922370621489795 18446133943728916643 --------------------, --------------------, --------------------, 18446744073709551616 73786976294838206464 18446744073709551616 590252846129060225711 590205629689100096531 2360458689855152057387 ---------------------, ---------------------, ----------------------, 590295810358705651712 590295810358705651712 2361183241434822606848 1179893257366301425427 9434370077796509146471 9426194974954694472421 ----------------------, ----------------------, ----------------------, 1180591620717411303424 9444732965739290427392 9444732965739290427392 37650733386586780877909 2347770381496333730789 37430479528863107268359 -----------------------, ----------------------, -----------------------, 37778931862957161709568 2361183241434822606848 37778931862957161709568 37229304070563703382191 295484161585338706656623 146072742564333496484419 -----------------------, ------------------------, ------------------------, 37778931862957161709568 302231454903657293676544 151115727451828646838272 1150107930787751156136275 1125197672725535140621277 -------------------------, -------------------------, 1208925819614629174706176 1208925819614629174706176 4369715285384289623703809 1050342184269701510110571 -------------------------, -------------------------, 4835703278458516698824704 1208925819614629174706176 63842019092666187783133619 59696032099936678243363631 --------------------------, --------------------------, 77371252455336267181195264 77371252455336267181195264 218887455527775194152695611 97765245860773879507641487 ---------------------------, ---------------------------, 309485009821345068724781056 154742504910672534362390528 674774304948479086692215231 554025454840936945445840501 ----------------------------, ----------------------------, 1237940039285380274899124224 1237940039285380274899124224 1684171358626838454292753325 138825664408518343786426093 ----------------------------, ---------------------------, 4951760157141521099596496896 618970019642690137449562112 1010348983445722125468867817 -240469360751137831607235143 ----------------------------, ----------------------------, 9903520314283042199192993792 9903520314283042199192993792 -3008634125400096826472428351 -688122283559297390323650305 -----------------------------, ----------------------------, 19807040628566084398385987584 2475880078570760549798248448 -31648669054326891061092938389 -40701345617123738948109811459 ------------------------------, ------------------------------, 79228162514264337593543950336 79228162514264337593543950336 -195932930759889217987682150911 -56336010574764357856502981359 -------------------------------, ------------------------------, 316912650057057350374175801344 79228162514264337593543950336 -2004708637628081183684626521893 -2171402725886054676992239508513 --------------------------------, --------------------------------, 2535301200456458802993406410752 2535301200456458802993406410752 -9210697281556835211887938941905 -4800233641021195440031546418545 --------------------------------, --------------------------------, 10141204801825835211973625643008 5070602400912917605986812821504 -39471238103860703430373947419765 -40124965553151136130146382603039 ---------------------------------, ---------------------------------, 40564819207303340847894502572032 40564819207303340847894502572032 -161819423175061158673829890319135 ----------------------------------] 162259276829213363391578010288128 and in Maple notation [36028797018963911/36028797018963968, 36028797018962371/36028797018963968, 144115188075764839/144115188075855872, 72057594037488179/72057594037927936, 576460752277506731/576460752303423488, 576460752148185929/576460752303423488, 2305843006064356241/2305843009213693952, 288230374413289427/288230376151711744, 4611685909281842195/4611686018427387904, 4611685631908812467/ 4611686018427387904, 144115149000231155/144115188075855872, 4611682286588437525 /4611686018427387904, 18446702641922255845/18446744073709551616, 18446636403496880459/18446744073709551616, 73785922370621489795/ 73786976294838206464, 18446133943728916643/18446744073709551616, 590252846129060225711/590295810358705651712, 590205629689100096531/ 590295810358705651712, 2360458689855152057387/2361183241434822606848, 1179893257366301425427/1180591620717411303424, 9434370077796509146471/ 9444732965739290427392, 9426194974954694472421/9444732965739290427392, 37650733386586780877909/37778931862957161709568, 2347770381496333730789/ 2361183241434822606848, 37430479528863107268359/37778931862957161709568, 37229304070563703382191/37778931862957161709568, 295484161585338706656623/ 302231454903657293676544, 146072742564333496484419/151115727451828646838272, 1150107930787751156136275/1208925819614629174706176, 1125197672725535140621277/ 1208925819614629174706176, 4369715285384289623703809/4835703278458516698824704, 1050342184269701510110571/1208925819614629174706176, 63842019092666187783133619 /77371252455336267181195264, 59696032099936678243363631/ 77371252455336267181195264, 218887455527775194152695611/ 309485009821345068724781056, 97765245860773879507641487/ 154742504910672534362390528, 674774304948479086692215231/ 1237940039285380274899124224, 554025454840936945445840501/ 1237940039285380274899124224, 1684171358626838454292753325/ 4951760157141521099596496896, 138825664408518343786426093/ 618970019642690137449562112, 1010348983445722125468867817/ 9903520314283042199192993792, -240469360751137831607235143/ 9903520314283042199192993792, -3008634125400096826472428351/ 19807040628566084398385987584, -688122283559297390323650305/ 2475880078570760549798248448, -31648669054326891061092938389/ 79228162514264337593543950336, -40701345617123738948109811459/ 79228162514264337593543950336, -195932930759889217987682150911/ 316912650057057350374175801344, -56336010574764357856502981359/ 79228162514264337593543950336, -2004708637628081183684626521893/ 2535301200456458802993406410752, -2171402725886054676992239508513/ 2535301200456458802993406410752, -9210697281556835211887938941905/ 10141204801825835211973625643008, -4800233641021195440031546418545/ 5070602400912917605986812821504, -39471238103860703430373947419765/ 40564819207303340847894502572032, -40124965553151136130146382603039/ 40564819207303340847894502572032, -161819423175061158673829890319135/ 162259276829213363391578010288128] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999997, .9999999986, .9999999940, .9999999763, .9999999162, .9999997289, .9999991908, .\ 9999977540, .9999941632, .9999857167, .9999669248, .9999272158, .9998472280, .9\ 996931405, .9994084632, .9989027866, .9980372139, .9966066146, .9943194329, .99\ 07765435, .9854514735, .9776750791, .9666283254, .9513469827, .9307417002, .903\ 6359416, .8688226914, .8251387572, .7715531312, .7072635138, .6317930934, .5450\ 783427, .4475381983, .3401156973, .2242849573, .1020191761, -.2428120033e-1, -.\ 1518972057, -.2779303770, -.3994623635, -.5137232056, -.6182553165, -.711060420\ 8, -.7907181353, -.8564673600, -.9082448744, -.9466791638, -.9730411444, -.9891\ 567703, -.9972891926] The cut off is at j=, 42 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 57], vs. those in the, 2, -th row from j=1 to j=, 56, are as follws 28 27 26 [3 (12009599006321313 n - 4707762810477946982 n + 878796402488053034265 n 25 24 - 103965056965890194444370 n + 8751891238951631387620185 n 23 22 - 557993198176569796228996590 n + 28001913974771405892612599925 n 21 - 1134789596982124126028294959650 n 20 + 37806720321244515110693532564255 n 19 - 1048759319391454341158756809110570 n 18 + 24444800161512162989018153420731275 n 17 - 481801509355249602703934911008706950 n 16 + 8063955066733560697121053191379948995 n 15 - 114885305799091318006813936799228613930 n 14 + 1394294613912189250522274384337178473975 n 13 - 14404432139736554660790383432422847432550 n 12 + 126389219221483166586603028202307864635940 n 11 - 938262235019941105066421567941280737619160 n 10 + 5859975524410092162359864859089158797937200 n 9 - 30554691797347522003657709718492643471453600 n 8 + 131645640141999645446837411198097409368165312 n 7 - 462322693025899538124510233416597692975792768 n 6 + 1298978938928308520025523693942733232009423360 n 5 - 2840225685866890769262486291236509609639802880 n 4 + 4585131552281146584656089497681841010463744000 n 3 - 4595753105265584426356054680182120716369920000 n 2 - 583951790020036991730246781452253745971200000 n + 13367601066449140084062551001405594717388800000 n - 22153967849273744601098917878050555166720000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 28 (2 n - 51) (2 n - 45)), 87 (828248207332486 n - 348278371183295733 n 27 26 + 69859837419758170204 n - 8897293409429880069237 n 25 24 + 807923818100823598104840 n - 55684282654009134866625885 n 23 22 + 3027911732567481023529034380 n - 133299596219793694746810305265 n 21 + 4837808539781926821697329604860 n 20 - 146637989500947479730036314227755 n 19 + 3747202807021491392059594106300940 n 18 - 81274366185464531266865251903933695 n 17 + 1503123988326935081050733356763464440 n 16 - 23772966361125622713342693319974142095 n 15 + 321967258688515001564760789790139443860 n 14 - 3733917669964832054534123917448812094955 n 13 + 37028605501619499169689679204581451876830 n 12 - 313126408089161950535781454033139183492340 n 11 + 2248268449758261006966937018324014351180920 n 10 - 13624270969847607466982254776171171552491760 n 9 + 69119410669679301876928712307048618620754464 n 8 - 290396537947613830113008909891972463908544192 n 7 + 995184154647367389980884817691700086570669696 n 6 - 2716278980747629709390895724595443263650585088 n 5 + 5625109631901668274464103196721986157951662080 n 4 - 7602091229837971769387147363118287236764672000 n 3 + 1385201626341870093567986596415773659955200000 n 2 + 21699691828320081842345961099163782598164480000 n - 41307852218000596562530634412286521822412800000 n + 763929925837025675899962685450019143680000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 (2 n - 51) (2 n - 57) (2 n - 45)), 29 (2484744621996688 n 28 27 - 1044835113548993999 n + 209579512258776305982 n 26 25 - 26691880228111747592811 n + 2423771454256798805301870 n 24 23 - 167052847953050000883596655 n + 9083735196297821282662397790 n 22 - 399898788479823847266812912295 n 21 + 14513425600245505688175527736630 n 20 - 439913966789637191139645461369265 n 19 + 11241608290240360106756779243006770 n 18 - 243823089992341325567658202855521585 n 17 + 4509371482076801893374965602428685770 n 16 - 71318875558266924798240538540213761285 n 15 + 965900784548715677053004901399892673130 n 14 - 11201716858834462820664030846315037168365 n 13 + 111084678113121937863513902004652928233890 n 12 - 939348363698110878651194697265199494939020 n 11 + 6744088802939309970035672235982269556892360 n 10 - 40858667231510160887513999703277867450635280 n 9 + 207123215036038879338290546065542635034258912 n 8 - 867949841544713814945551535431435936200723776 n 7 + 2949226932804699435205121826257883919409923968 n 6 - 7827045116460836635023986198413163294258609664 n 5 + 14722498868109059014709380797043254892722186240 n 4 - 12749272216400452769874099566783352300911616000 n 3 - 23508581192529247379883255875337894204211200000 n 2 + 86781981703245930217919252265541221444157440000 n - 62005025823000761219910717153015908558438400000 n + 2291789777511077027699888056350057431040000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 30 (2 n - 51) (2 n - 57) (2 n - 45)), 1537 (187527895999199 n 29 28 - 84387553199201475 n + 18143558347452008455 n 27 26 - 2481092516079556330875 n + 242353352494440772204851 n 25 24 - 18004083454193771531440935 n + 1057493901698780892894113475 n 23 22 - 50405205664505915368626180375 n + 1985693560572542472839585568885 n 21 - 65513962684597069379174350737825 n 20 + 1827854268211083892403226681821925 n 19 - 43430222800631846530455706832036625 n 18 + 883180979372117577767342428372071345 n 17 - 15422272198650160139266869726636511725 n 16 + 231682867075087984891717148464437597825 n 15 - 2995882099003900456025569363761782644125 n 14 + 33322580246501846661999624583346070761160 n 13 - 318191920702993836581518355961105759731400 n 12 + 2599832340602973354723271024510211787888400 n 11 - 18088178202534810817462975455987293115456000 n 10 + 106421985095384281407236443839838255611678656 n 9 - 524161225421929816318809139078560488356387200 n 8 + 2126685539548064343506701894635326329523473920 n 7 - 6900642995944662523601906255156426001211872000 n 6 + 16766461273401202597858160535500890868978555904 n 5 - 25002839416924386812899774371727355409251389440 n 4 - 383603007421789626743720002793860961531904000 n 3 + 94203872117716195084097621661533344739819520000 n 2 - 169098380795820042860966175222043786784931840000 n + 93910920506571875650276265036950003148390400000 n - 5102475353703907344690316804703901450240000000)/(268435456 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 30 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 1537 (375055791990433 n 29 28 - 168775106389378605 n + 36287116689993506795 n 27 26 - 4962185030450800035345 n + 484706704562161997333607 n 25 24 - 36008166826904471544007845 n + 2114987791032891389269013775 n 23 22 - 100810409798809293076298121525 n + 3971386963873778736928765540845 n 21 - 131027911769227853200925543514075 n 20 + 3655707537595993569310500556410825 n 19 - 86860382882919202947049406639331075 n 18 + 1766358576590097433590015364865208765 n 17 - 30844387351642359218412161567387078775 n 16 + 463359449082631081877030101450632209925 n 15 - 5991547541926610201286269853543889139175 n 14 + 66638742104498784133070171047375965161070 n 13 - 636221091513078996320415787427844007796700 n 12 + 5196153880009283347086572063435847917540600 n 11 - 36112480688909235721171899206042165939475600 n 10 + 211875390052655754534256837206489982688347552 n 9 - 1036281338655478990554234849324226423742253120 n 8 + 4133447256496836164383748340749993354573086080 n 7 - 12876039589879191215053469172479065962365537280 n 6 + 28283596999466578271503721999570682727732297728 n 5 - 29989297166310658713481321871266100853531770880 n 4 - 43070398325822268703774761138256857163296768000 n 3 + 210153673185482434708301568460323937901936640000 n 2 - 290523347813362290397552998755694122800250880000 n + 145122178842674210889723036813483146792140800000 n - 10204950707407814689380633609407802900480000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 47647 (24197147867332 n 30 29 - 11626729547866889 n + 2673210155217150005 n 28 27 - 391544650282213740835 n + 41035699706724187438953 n 26 25 - 3276885214831810831780671 n + 207305600334617752712368845 n 24 23 - 10665646981864287513576817575 n + 454587323351521975193576626005 n 22 - 16268073000561214916977472549685 n 21 + 493680821917218098533968091569075 n 20 - 12797359791345733238007270078042225 n 19 + 284876082726360915740084771480235435 n 18 - 5465634957338106530996296340139648645 n 17 + 90585392055607753722056572590099065775 n 16 - 1298234021175508881642265842218771760525 n 15 + 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(2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 899 (1282448836173217 n 30 29 - 616216665154869806 n + 141680137757175686840 n 28 27 - 20751866305472492682820 n + 2174892045588493801799208 n 26 25 - 173674909153820966968063764 n + 10987195749738912410163496260 n 24 - 565279161601056136311636237900 n 23 + 24093115329753842688841158717330 n 22 - 862206796222734957501672768585840 n 21 + 26165007381550097453525688160250100 n 20 - 678255453024553514476948523242637700 n 19 + 15098192700553653863632317002780351760 n 18 - 289667961146156158613003365741204930380 n 17 + 4800615766019052299350875765481921622700 n 16 - 68792896862410295586555001505996164431300 n 15 + 852198740635790997623925517484659228650405 n 14 - 9112405434868291707760897626970173772509090 n 13 + 83851426505271528456865966819694629437016100 n 12 - 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)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27 )/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-\ 39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 29/ 134217728*(2484744621996688*n^29-1044835113548993999*n^28+209579512258776305982 *n^27-26691880228111747592811*n^26+2423771454256798805301870*n^25-\ 167052847953050000883596655*n^24+9083735196297821282662397790*n^23-\ 399898788479823847266812912295*n^22+14513425600245505688175527736630*n^21-\ 439913966789637191139645461369265*n^20+11241608290240360106756779243006770*n^19 -243823089992341325567658202855521585*n^18+ 4509371482076801893374965602428685770*n^17-\ 71318875558266924798240538540213761285*n^16+ 965900784548715677053004901399892673130*n^15-\ 11201716858834462820664030846315037168365*n^14+ 111084678113121937863513902004652928233890*n^13-\ 939348363698110878651194697265199494939020*n^12+ 6744088802939309970035672235982269556892360*n^11-\ 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18004083454193771531440935*n^25+1057493901698780892894113475*n^24-\ 50405205664505915368626180375*n^23+1985693560572542472839585568885*n^22-\ 65513962684597069379174350737825*n^21+1827854268211083892403226681821925*n^20-\ 43430222800631846530455706832036625*n^19+883180979372117577767342428372071345*n ^18-15422272198650160139266869726636511725*n^17+ 231682867075087984891717148464437597825*n^16-\ 2995882099003900456025569363761782644125*n^15+ 33322580246501846661999624583346070761160*n^14-\ 318191920702993836581518355961105759731400*n^13+ 2599832340602973354723271024510211787888400*n^12-\ 18088178202534810817462975455987293115456000*n^11+ 106421985095384281407236443839838255611678656*n^10-\ 524161225421929816318809139078560488356387200*n^9+ 2126685539548064343506701894635326329523473920*n^8-\ 6900642995944662523601906255156426001211872000*n^7+ 16766461273401202597858160535500890868978555904*n^6-\ 25002839416924386812899774371727355409251389440*n^5-\ 383603007421789626743720002793860961531904000*n^4+ 94203872117716195084097621661533344739819520000*n^3-\ 169098380795820042860966175222043786784931840000*n^2+ 93910920506571875650276265036950003148390400000*n-\ 5102475353703907344690316804703901450240000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 1537/536870912*(375055791990433*n^30-168775106389378605*n^29+ 36287116689993506795*n^28-4962185030450800035345*n^27+484706704562161997333607* n^26-36008166826904471544007845*n^25+2114987791032891389269013775*n^24-\ 100810409798809293076298121525*n^23+3971386963873778736928765540845*n^22-\ 131027911769227853200925543514075*n^21+3655707537595993569310500556410825*n^20-\ 86860382882919202947049406639331075*n^19+1766358576590097433590015364865208765* n^18-30844387351642359218412161567387078775*n^17+ 463359449082631081877030101450632209925*n^16-\ 5991547541926610201286269853543889139175*n^15+ 66638742104498784133070171047375965161070*n^14-\ 636221091513078996320415787427844007796700*n^13+ 5196153880009283347086572063435847917540600*n^12-\ 36112480688909235721171899206042165939475600*n^11+ 211875390052655754534256837206489982688347552*n^10-\ 1036281338655478990554234849324226423742253120*n^9+ 4133447256496836164383748340749993354573086080*n^8-\ 12876039589879191215053469172479065962365537280*n^7+ 28283596999466578271503721999570682727732297728*n^6-\ 29989297166310658713481321871266100853531770880*n^5-\ 43070398325822268703774761138256857163296768000*n^4+ 210153673185482434708301568460323937901936640000*n^3-\ 290523347813362290397552998755694122800250880000*n^2+ 145122178842674210889723036813483146792140800000*n-\ 10204950707407814689380633609407802900480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 47647/536870912*(24197147867332*n^31-11626729547866889*n^30+2673210155217150005 *n^29-391544650282213740835*n^28+41035699706724187438953*n^27-\ 3276885214831810831780671*n^26+207305600334617752712368845*n^25-\ 10665646981864287513576817575*n^24+454587323351521975193576626005*n^23-\ 16268073000561214916977472549685*n^22+493680821917218098533968091569075*n^21-\ 12797359791345733238007270078042225*n^20+284876082726360915740084771480235435*n ^19-5465634957338106530996296340139648645*n^18+ 90585392055607753722056572590099065775*n^17-\ 1298234021175508881642265842218771760525*n^16+ 16086414327879860297464423996916292193155*n^15-\ 172105394484210106795581309734978665370910*n^14+ 1585639125812268688480739715821896470124700*n^13-\ 12526328724004264198483075395700206174551800*n^12+ 84289735969138967688897372028723833174254608*n^11-\ 478108504251538118494082288514359349598118816*n^10+ 2246670678290853576027327154228631986934982720*n^9-\ 8477283274791713586930049965299847420437063040*n^8+ 24120192237436625451397564598935798214517184512*n^7-\ 44035904240518957027988734377669322580223504384*n^6+ 17443738981418807582923222300659578110012538880*n^5+ 154646082270882345326619423434829730811006976000*n^4-\ 438928052158672687399656623455216969379020800000*n^3+ 510654955930979703015316997144177866895523840000*n^2-\ 237675385150138151342479151804184086760652800000*n+ 20080709456512151485555440328189547642880000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-\ 21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2* n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/( 2*n-45), 899/536870912*(1282448836173217*n^31-616216665154869806*n^30+ 141680137757175686840*n^29-20751866305472492682820*n^28+ 2174892045588493801799208*n^27-173674909153820966968063764*n^26+ 10987195749738912410163496260*n^25-565279161601056136311636237900*n^24+ 24093115329753842688841158717330*n^23-862206796222734957501672768585840*n^22+ 26165007381550097453525688160250100*n^21-678255453024553514476948523242637700*n ^20+15098192700553653863632317002780351760*n^19-\ 289667961146156158613003365741204930380*n^18+ 4800615766019052299350875765481921622700*n^17-\ 68792896862410295586555001505996164431300*n^16+ 852198740635790997623925517484659228650405*n^15-\ 9112405434868291707760897626970173772509090*n^14+ 83851426505271528456865966819694629437016100*n^13-\ 660678575664593360794864260067046424845458600*n^12+ 4421499455189921392796844950796101472025019248*n^11-\ 24805292421668549424156916788134090377187149664*n^10+ 114088673122635752583693414716763479390040096960*n^9-\ 413175123685497027238105837471893489581973751680*n^8+ 1083457022582019971760075987981010440088844408832*n^7-\ 1599300655024337682828300038169552798168252051456*n^6-\ 809249374851694565787828473780014828671177768960*n^5+ 10177391646125550892263671774657007449791795200000*n^4-\ 22980038999029521062903955831373212438432645120000*n^3+ 24312302534627146482891533899527211969810268160000*n^2-\ 10946231554261695141500090676750990482237030400000*n+ 1064277601195144028734438337394046025072640000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-45), 44051/1073741824*(104689700631157*n^32-53601126502893584*n^31+ 13150282502832799784*n^30-2058350199558454422880*n^29+230904135720989561811948* n^28-19770120526980600281778336*n^27+1343506172284179980232539496*n^26-\ 74397979499745608544311516640*n^25+3420355161043916534156409540030*n^24-\ 132337551841884560316103786426560*n^23+4352993357349302068699193386586760*n^22-\ 122647474669406977685842502644461600*n^21+2976498164347504264064782105470157260 *n^20-62466088493404727751781217948699418720*n^19+ 1136583766467346763142452259283647727320*n^18-\ 17954692446726432733627255648420685949600*n^17+ 246304757950387287177857897172365711201205*n^16-\ 2931322044444529255433361770918519744397360*n^15+ 30194490796026643082065739793170775759055760*n^14-\ 268083601228964243247917247585215989252672000*n^13+ 2037947883906033342474539199410695252493399008*n^12-\ 13124724607743230749288401022945647731104349696*n^11+ 70393748840539900024008012865483301160473623296*n^10-\ 305537159895984194566609096444347138868410040320*n^9+ 1018256129210926432194359953818318379070527579392*n^8-\ 2312878201900338092650209762479190472132386975744*n^7+ 2136653462015641333926494199405826445822979907584*n^6+ 6673503790233648534600734159891137896544789463040*n^5-\ 31445234291338371762022500034007074363845304320000*n^4+ 59603940497602031911934699352891037395719946240000*n^3-\ 57966903184309628005302187506855903434692362240000*n^2+ 25319286311005061054416988865122450427976089600000*n-\ 2736713831644656073888555724727546921615360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 44051/1073741824*(104689699551457*n^32-53601125328179984*n^ 31+13150281890116006184*n^30-2058349995626074526080*n^29+ 230904087084425734214748*n^28-19770111678921246872300736*n^27+ 1343504896078033761038843496*n^26-74397829746478262073074912640*n^25+ 3420340605548527315471116525030*n^24-132336364839293112894159285406560*n^23+ 4352911388704802785187880574754760*n^22-122642650496346281110826752244745600*n^ 21+2976255147256728442926074526013483260*n^20-\ 62455584596447808344368215121994646720*n^19+ 1136193903432147481649770248138670687320*n^18-\ 17942276005186305361105595742736987689600*n^17+ 245966218031940886911353817882809325480705*n^16-\ 2923450706334562770507135388542779528313360*n^15+ 30039311725350217495653486615289039659031760*n^14-\ 265509721062496116326731463917114686382240000*n^13+ 2002395379764555058253064812652451355575386208*n^12-\ 12721157091658545415008866447714743692206583296*n^11+ 66693053198274061666400693489788465920397360896*n^10-\ 278726515358544494924014834819023604099394949120*n^9+ 869243171419936856371361501762350219933649678592*n^8-\ 1702640644740854598803375389484656368532106809344*n^7+ 403910284939885826425095876078819057857695555584*n^6+ 9719831164985403730007952511458622514917909463040*n^5-\ 33767084911159936935262369958169606396664504320000*n^4+ 58020706668969016313429050512627451535628042240000*n^3-\ 53759560025956909731089832076205937372223242240000*n^2+ 23218738413909382050335860567634201694280089600000*n-\ 2736713831644656073888555724727546921615360000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45), 6211191/1073741824*(1484960224234*n^33-808560800922853*n^32 +211238529229653728*n^31-35258630286817970120*n^30+4224117565769637850776*n^29-\ 386871810718374125745612*n^28+28170622507074615817217232*n^27-\ 1674626790852270590365722120*n^26+82811574173698826482766342460*n^25-\ 3453800941332602192115364422270*n^24+122745110403092373247796497087920*n^23-\ 3746007910972133913326294259642600*n^22+98740166844544021213144805987964120*n^ 21-2257327179547583955566920900567957740*n^20+ 44885021036338380605870695967463931440*n^19-\ 777532957821340251429205073244759171000*n^18+ 11739223794352357437118854717825648375210*n^17-\ 154352485837878843511792681135767587194245*n^16+ 1763439637324965183972054120765315270999920*n^15-\ 17434026791170790790237660308482181976208400*n^14+ 148166711223152636978013491709468201475554496*n^13-\ 1071450100866380178992292783160078867825304032*n^12+ 6489480029224588896964067709397242587341765632*n^11-\ 32106434453716447227088978478758853297690868480*n^10+ 124264846706222208518512930271566098691650722304*n^9-\ 343567457333445568967832188186598008919714133248*n^8+ 495382366497658275331159825940366399662005424128*n^7+ 697890064784163584358930867192430199898730782720*n^6-\ 6122480721738197307370697980837483346351643033600*n^5+ 17110723367140555472887810898343782587829575680000*n^4-\ 26757833516458619452149324771441350539839406080000*n^3+ 23560629019927441574123702125237759729061068800000*n^2-\ 10019466902801575823320595425349280369475584000000*n+ 1261605667070231523423802284448869148262400000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2*n-33)/(2*n-11)/(2* n-41)/(2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/( 2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51) /(2*n-57)/(2*n-63)/(2*n-45), 126759/1073741824*(72763042752954*n^33-\ 39619470434291957*n^32+10350683415178537024*n^31-1727671407196741344360*n^30+ 206981414932592804905016*n^29-18956657003953876886248108*n^28+ 1380351773458571837025372336*n^27-82055708926385456073968218920*n^26+ 4057671574004220771777170395260*n^25-169228617641637130183123148719230*n^24+ 6013994992532098517727081347370960*n^23-183524719872175257712937677906029000*n^ 22+4836806907637412108040299995454923320*n^21-\ 110547282385295434171087666688127502860*n^20+ 2197124969187926729765045453739779482320*n^19-\ 38029287429399428547506071618703178072600*n^18+ 573357393623403062999484580854516963520410*n^17-\ 7520738673890400180102434271215259855775605*n^16+ 85583567322682138437399221746129404856648560*n^15-\ 840747076887167595097052097375685328682403600*n^14+ 7074461698126058988968709707289581502360118976*n^13-\ 50385022495775419709868514956838681006837591008*n^12+ 298237258731786193504754942621053663932366048256*n^11-\ 1424678531188907882441654066300513279927685098240*n^10+ 5206538140353916668700873814695577343915858430464*n^9-\ 12803495550085112946819744673639047960577028351232*n^8+ 10638289291402298330374236505479379197187158380544*n^7+ 66314408478660582672608313921052671675094446366720*n^6-\ 346365202032012892733106789194867311925600659046400*n^5+ 863036132132649026441888939603680237980686868480000*n^4-\ 1277418133500337306115126516900819818382471331840000*n^3+ 1093560363317863764999553053397417784861117644800000*n^2-\ 463768120564643360854334750943066825604202496000000*n+ 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99789/17179869184*(47304386960646247*n^38-34147199820579570759*n^37+ 11891353627953096498049*n^36-2660889444787881625333485*n^35+ 429954286981232156517254076*n^34-53449059164008131548361170832*n^33+ 5318179368716960983908304071812*n^32-435044284143016253022801201831300*n^31+ 29823632095357228106698389580828082*n^30-1737699307556813243471976913318964994* n^29+86973607641149039172999591960652816974*n^28-\ 3769579788316357360230015137685484148790*n^27+ 142337896299784583478197036047352929090380*n^26-\ 4703264816127782524544904024435157225957860*n^25+ 136415488487965412914392784182415282755061860*n^24-\ 3479495690765353450160345256535990772306027700*n^23+ 78101723397770110855450899836053482904734971295*n^22-\ 1542060557262183576062159383445385892903392110415*n^21+ 26736954565563383131560231458976160166597811439865*n^20-\ 405838384291509839218621591886179635612247173128325*n^19+ 5366797629610344660082829880291251195647105799021888*n^18-\ 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67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2* n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/( 2*n-69), 33263/34359738368*(567099585421900684*n^39-431039691421257335157*n^38+ 158178859648012337001101*n^37-37329809892332238721709373*n^36+ 6366840831812532443394826167*n^35-836136898237290517263433226826*n^34+ 87962642755962655119304264295468*n^33-7614234025020504063001294577094564*n^32+ 552796656166477975407127057457890804*n^31-\ 34138162406617383886568709294565033422*n^30+ 1812407536554812455451672562051324311046*n^29-\ 83387607492018822635943098807365407371958*n^28+ 3345069035998293548310822652290852768007890*n^27-\ 117516692889475547998477338542321860476464280*n^26+ 3626855111421169457171694760470640051904432340*n^25-\ 98518955873639499517743495277638518617055396820*n^24+ 2357217860326726496422706904467208059472416381640*n^23-\ 49660142273653147989234476717531861573880338224245*n^22+ 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33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79)/( 2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45) /(2*n-71)/(2*n-69)/(2*n-77), 1363783/34359738368*(54806731756850522*n^41-\ 45923247769317671689*n^40+18599267552078124427960*n^39-\ 4849727569385059248898010*n^38+914892406747902362119311114*n^37-\ 133034539799608878090344536293*n^36+15512003930957545375924137576700*n^35-\ 1489721774428366828240954553153780*n^34+120106546135780034992446413961783236*n^ 33-8244525176109387563158224779132499682*n^32+ 486967615123727344087573733452578018720*n^31-\ 24949145663642709753367041465702689146860*n^30+ 1115477360594367834236268293106596370996708*n^29-\ 43716926752999775960276836666433521336284546*n^28+ 1506481936050942454574319180304169808765483720*n^27-\ 45730564954999367449616192527004029697964818400*n^26+ 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4782331136999065403190616367744264254912634051346088839274070047457280000000000\ 000)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-101)/( 2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-105 )/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-\ 13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2* n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1 +2*n)/(2*n-5), -53/562949953421312/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83) /(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-\ 51)/(2*n-73)/(2*n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)*( 82287349720965066595482709628*n^53-116558271147084031926677061411829*n^52+ 80444010908633723929647552239590418*n^51-36050809264875521037454792764907797655 *n^50+11796203200984018775269889845785927360770*n^49-\ 3004538197213105405081925813213030345830510*n^48+ 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135097990054633147620447236142877752572173530572382450953952766098487294*n^33-\ 4812476946941658106803590513232856725709249613957835729155550743974852142*n^32+ 157034714767911391434369008075895263186296910401481979106272067077707851564*n^ 31-4697576352859756536759544199251311973801687248449011122742322387175159341690 *n^30+ 128894603833222312930428297578534982435763686551330931981897722267970202355080* n^29-32449144990841388842278899040139994529843985530597636720122823701313274560\ 29665*n^28+74955158142262246977084678116921298661356645866073234504761365066803\ 437121537130*n^27-1588367006984084315172824971226049552233335269143997829541559\ 333446372074074710675*n^26+3086541570463695570705938513460266194845101887780385\ 8667624723221016930058144109090*n^25-549647772721218773963082756265871638619919\ 358398180410818002972189257346739807847720*n^24+8961945307613527366400947569210\ 580397904050566369440298159892383536477392382271079440*n^23-1336381662273924228\ 06584251517207743622008875881232711414219662589262164849460365928400*n^22+18199\ 4329949925851145892896641490688892675209822141727331799740713323090912483153733\ 2960*n^21-225968509468115761278096049984144563033435395042558410634995507886099\ 58708514688787130880*n^20+25528633018762231117719661373696994492942860028528295\ 9377782877482421776818813917973404160*n^19-261801541307154143835440162413556665\ 6739133723851812751034948869663533908075521274199481600*n^18+243045296879304173\ 58330030599405129557677023427583438824940425407920189148590421002582801920*n^17 -203603152916222101989403617316809699080737854057151949108245969476088200481618\ 301572445276160*n^16+1533390142829230412232029657248499032428214644337340446894\ 258577916707215676473481085108817920*n^15-1033758839894239494536719044917750813\ 2561640716586534672283140248322553444295858908140998963200*n^14+620728683761223\ 2554907631283881967361730359799105079098851639720368991673505138340476191382732\ 8*n^13-330023227721656226055982465107573600129519805901592928019521217964294193\ 053839247742219250171904*n^12+1542885422188754085546465644519113953392797008438\ 328953145637843779754310632723454542501834784768*n^11-6290485994043768319821735\ 255733479929805566418553680167717433652683313039012156347723477449441280*n^10+ 2214534140347002333456108908630728688740168581009627883111931607188256570526194\ 9827198730633216000*n^9-6650631569434025440871591940207514535570351358767342333\ 1232676946512145668746695473403978055680000*n^8+1678345561013008158367867186462\ 72684929937330322116424339871789219478082493465423176406285680640000*n^7-349151\ 4968875698953041691730770182601028372202569952758246193700919024779762215470977\ 39388518400000*n^6+583917130565422064844391472529062817245305003639609189760657\ 003694764536908585913738969743360000000*n^5-75860111339464916682740652194948600\ 4084800987323649281444951980976511627351624841763278028800000000*n^4+7286974909\ 2727917408148825056656104592115590123165451509346425990687279641557332607557959\ 6800000000*n^3-4790436532780383544938538177224747531031060537670488073248393523\ 10031344431436712208498688000000000*n^2+187654397690157081263978502521491917825\ 966628675290338033162537520569121627846367562956800000000000*n-3162065155885512\ 4782331136999065403190616367744264254912634051346088839274070047457280000000000\ 000)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/( 2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21) /(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49 )/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -265/1125899906842624/(2 *n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)*( 69695324733590076387451249033*n^54-102137618530744764492897707069745*n^53+ 72977069235920046994477889548461531*n^52-33879788665852631614159694615806955247 *n^51+11491833490161066216905485711039029695502*n^50-\ 3036273193216683846916486190173320631543890*n^49+ 650571286634106209877106747031089704546855810*n^48-\ 116211021359118672962340062503144861149840556890*n^47+ 17656487628533360318793601048854967348542085099395*n^46-\ 2316572892421896905926166959447077536543851337451315*n^45+ 265580763625754322957567739179047910530410817496010065*n^44-\ 26855809040573349374208017602282237439030989261442038125*n^43+ 2413677404252766672292545726558113589081351023378198478660*n^42-\ 194015513993023257411986988928023835655689689017804288161440*n^41+ 14020483840592486569946173723517597896882731974946052164239100*n^40-\ 914829323709595952119757957494039267461359194208225048454023820*n^39+ 54093476821967891315507996859177391560131677219942815918645786495*n^38-\ 2907378652736667285841709649313582696600289063363929687584370152615*n^37+ 142403311365163792737390764172583217410731454302306442901909735769805*n^36-\ 6369803153216478816339694118858496247143849559675693945593540729164505*n^35+ 260665794908711072177418122381381721350935194301778660391261522218102334*n^34-\ 9772734616884240007873625840622815916182909327778429841427082936045627450*n^33+ 336059010567013119106987004543048252690410396150243293040325417123574563058*n^ 32-\ 10608602315705074702379936636927598479748470800539786488089152045076868089866*n ^31+ 307617216421697529685480205006593384622854529939402776720614855687021902193621* n^30-81966551368724932617620439380290695618261782991234001579503921046981175341\ 04085*n^29+20072493213876589444793618839495194125831109610172210347650635322129\ 3355016319015*n^28-451724513604390273736821633575700355050348361757023800418079\ 3146619777267274070635*n^27+933958878352810968310806674446298395284237913863636\ 55084592933565287514207192443280*n^26-17731368134526140062279571564821546908849\ 52935951470609809235373331199299032559827060*n^25+30888610537277232161665990040\ 487956060263960636226223584923040359726336607036734485360*n^24-4932606168134248\ 61220323626338694702808258156646620219967644543355039443756012159391600*n^23+72\ 1191515615241583779085658241172040712382621867507884303540224507553730078803240\ 6771840*n^22-964010511172301711376447634731415014850963052247316073834405762570\ 25534635887947537034560*n^21+11760025823846429485365382532678451722887420615886\ 30119303680577920516503198358837874579200*n^20-13065742294723227385041888423877\ 572947643184965586629441115382735477842933882934637928193280*n^19+1318905925583\ 0206093176598643940470533220185199506052504422046015742780807754871100532830208\ 0*n^18-120624058153417125207654005619498633177494657729221734292018139132484459\ 5507828844342126197760*n^17+996297669480298539248507517494429322160131645300986\ 0325603631853691538202855301830569327595520*n^16-740376096576023730321526879041\ 72714287085572152729963101369346021988311578774669519469650923520*n^15+49287540\ 2571785698769599109654372878108484224768342666589068196676848269882852496767238\ 364561408*n^14-2924477344065286356680630658328233496700490590824024669831279563\ 592422404983692280117675006279680*n^13+1537511787867672956027815575106658951466\ 9884736786725468540430443954837164284221128709750104129536*n^12-711252828769068\ 8098301127809631711249738849034457874042297994922262291587896394884281867510035\ 2512*n^11+287125023787215387180801658302839403838407762644481588339556800066055\ 261785308000989927447411556352*n^10-1001471256420584427462676859071190561510233\ 324947823307008595310996559521046429023888439700776550400*n^9+29816395021258535\ 1036350838356147697575510693850257246447173463562644208037510089550728065659699\ 2000*n^8-7464014387724635097987227325743417858854442090549839905850155202621955\ 185267166263597033838346240000*n^7+15412193768334270710447915774191519553531825\ 647202396611090939796512807073518533529684854712565760000*n^6-25598972376347621\ 5695845840511043050214000370055516041978733161231458618009913479826181508300800\ 00000*n^5+330499974777270674002058109744865146200864097486328404906644194712506\ 63668359193372520643297280000000*n^4-315695592412849242518124499195745489576420\ 61700346155849337518207308207615602356014050338406400000000*n^3+206518099478485\ 0533713257994825332022769349863121491607370723839540663745462416631378183454720\ 0000000*n^2-8056513677485620057371625254726746446998357735913681769756818845438\ 827486740347385502760960000000000*n+1353363886718999340683772663559999256558380\ 539454510110260737397612602320930198031171584000000000000)/(2*n-81)/(2*n-77)/(2 *n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-101)/(2*n-29) /(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-105)/(2*n-\ 37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2* n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-107)/(2*n-11)/(2*n-33)/(2*n-65)/ (2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/ (-1+2*n)/(2*n-5), -265/2251799813685248/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2* n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/( 2*n-51)/(2*n-73)/(2*n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)*( 145125100776745351895713333007*n^54-212217452125237329768118420532775*n^53+ 151313378897769132566092840296898491*n^52-\ 70107567888868929982186400750990985189*n^51+ 23734676899528616604362873183921024477244*n^50-\ 6259503869110527797538578015645884879992930*n^49+ 1338853715968290844774645749661575409887240930*n^48-\ 238757868193822747620297049559895498320891596830*n^47+ 36217519095309168454126767852277304256659018818785*n^46-\ 4744551592158701712896724408643207719918877828292205*n^45+ 543138294727141449031641390087340913347676071535974785*n^44-\ 54846061724113062384963487923950212388943391159777913775*n^43+ 4922748670350634466434976228535633273685933984842593671050*n^42-\ 395196445247468707846025656545861344530554387564727965699180*n^41+ 28524215138666213224440940521266082875054174137531692980111820*n^40-\ 1859045569102430996752375873914657892391034087657663865463056740*n^39+ 109804057409853067589166471982989110981089325844284926461015464625*n^38-\ 5895519462565503519510992558695419383519940371882212632094512258705*n^37+ 288476349398311494447595916522482030874290638559727021080813409974925*n^36-\ 12891625016005106410663891086894833660828911212281767554531101590286835*n^35+ 527082209233460004396867859620435936926611255798616316182810787084372936*n^34-\ 19744378903550706385472320501393174801822488767811557046235853859891981330*n^33 +678415352327144148229926071773124546845996319999669350276739216352000303858*n^ 32-\ 21399816397437951190431306741352718405013753559327394897008361106224026815342*n ^31+ 620086969623327294799985995562961399997126750614390234595792874203855858527327* n^30-16511495248444197885214897204703822615056165797365632111095219975720294978\ 934395*n^29+4040879373479279131705021704153445459107115247751809288438948527295\ 22148583063895*n^28-90884430301182876283715167260711674619737547984768273482513\ 74960531003174260000345*n^27+18780220628187047282067268320146279685774940147075\ 3547062344103998643642500408233490*n^26-356358848583111739191406900386175741849\ 2437324106307005801542531300745046539930961920*n^25+620484211053906133179050521\ 49646172370857064616357198526677359068140222539148465752640*n^24-99039858862335\ 8984352532382412193900336692539475523226259526171186397283690264233674800*n^23+ 1447434777728380926658335870677780820652378948782950824375426514529651906979153\ 3141020000*n^22-193400718221231235844242309553464832259711173667850655461632034\ 551750407421525645315368320*n^21+2358441449771258322451354160931696688891714500\ 547564975838646353829642446723958233585844480*n^20-2619410254990195024577891690\ 5766403208866401939254243726343220438926884204604400784008840960*n^19+264330630\ 6359447519243890104336687404021552790617709269689496473400392589186475075807253\ 37600*n^18-24168127761519211900195255144184888901685729355353313549504874082628\ 20601773338723056894545920*n^17+19956496542300126848423547546011992440491069079\ 843448615369022035528900529839226792445149798400*n^16-1482668182188301020647107\ 31370165906339727808166558471733020386422702502701337394163150676643840*n^15+98\ 6813248574649638885204167870616647618875228647892290174521341363058292469900097\ 613820867551232*n^14-5854122182980644359011848634205571706946025950789782290265\ 346346001590320103048610041119662571520*n^13+3077207067425445631048911180936108\ 4607308912386452351458264097780636662391767845470999023449931776*n^12-142329751\ 0552395797561524309327713318097972206610692864222309976622166152117378316184306\ 98370105344*n^11+57449261104544171030184801480906385396431925417929537543600932\ 6901333622954805443995244735657672704*n^10-200355637329956100944343760895307633\ 1918493146359726922799517431493234139476762659935739577158860800*n^9+5964520135\ 9069868968213029357296662467550274965276384841055918409461943797041043757325518\ 67695104000*n^8-149299191807151389350970229866745636403310426116033535584480391\ 26638539152613590102322681300910080000*n^7+308262785741829030177647542580006556\ 26603123026311212321267367202923219852250364793428070912491520000*n^6-511985446\ 6689118397632325866682274832733868120030166342209213078753499302305120645580645\ 7282560000000*n^5+6609849634649564412795314442080898296710901236207544499173790\ 3528667446802098766499619494952960000000*n^4-6313646693092033029745904623443599\ 8530225563395930456249694811811380811838187092502951808204800000000*n^3+4130172\ 0843555431574212155745396134840734677759356546366986875271480855319459398713096\ 955494400000000*n^2-16112490177910543520577701659463987456489150821299676586227\ 037414066663228493423913646161920000000000*n+2706727773437998681367545327119998\ 513116761078909020220521474795225204641860396062343168000000000000)/(2*n-39)/(2 *n-23)/(2*n-17)/(2*n-105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67) /(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-\ 107)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2* n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), ( 27409827711379493990198633857248*(n-54)!*(n-53)!*(2*n-113)!*(n+1)+ 40744338489888437012457428706720*(n-55)!*(111/110*(n-53)!*(n+1)*(2*n-112)!+(n-\ 54)!*((2*n-111)!*(n+1)-1/40744338489888437012457428706720*binomial(2*n,n)*(n-57 )!*(n-53)!)))/binomial(2*n,n)/(n-57)!/(n-55)!/(n-54)!/(n-53)!, ( 27409827711379493990198633857248*(n-54)!*(2*n-113)!*(n+1)+ 41114741567069240985297950785872*(n-55)!*((2*n-112)!*(n+1)-1/ 41114741567069240985297950785872*binomial(2*n,n)*(n-57)!*(n-54)!))/binomial(2*n ,n)/(n-57)!/(n-55)!/(n-54)!, ((27409827711379493990198633857248*n+ 27409827711379493990198633857248)*(2*n-113)!-(n-57)!*(n-55)!*binomial(2*n,n))/( n-57)!/(n-55)!/binomial(2*n,n)] The limits, as n goes to infinity are 36028797018963939 36028797018963141 4503599627368997 288230376150768863 [-----------------, -----------------, ----------------, ------------------, 36028797018963968 36028797018963968 4503599627370496 288230376151711744 576460752289295521 288230376108691951 1152921503719722083 ------------------, ------------------, -------------------, 576460752303423488 288230376151711744 1152921504606846976 4611686002503097007 4611685954941232307 4611685790060101347 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 4611686018427387904 4611685268160848043 9223367489932494035 18446718451583147155 -------------------, -------------------, --------------------, 4611686018427387904 9223372036854775808 18446744073709551616 4611669123793849285 36893152531958210875 147570798712903498913 -------------------, --------------------, ---------------------, 4611686018427387904 36893488147419103232 147573952589676412928 590267650383964503731 590235864777846495041 295086857436600560163 ---------------------, ---------------------, ---------------------, 590295810358705651712 590295810358705651712 295147905179352825856 4720457470415928341883 9437554065219110651101 4715858377472170612973 ----------------------, ----------------------, ----------------------, 4722366482869645213696 9444732965739290427392 4722366482869645213696 18843856289925389416667 75248409669794782080643 37524406949970638794159 -----------------------, -----------------------, -----------------------, 18889465931478580854784 75557863725914323419136 37778931862957161709568 37372244527776437722363 37146838422171741262861 589152551764417358500591 -----------------------, -----------------------, ------------------------, 37778931862957161709568 37778931862957161709568 604462909807314587353088 1163720359007584599312437 285942777688595103836843 -------------------------, ------------------------, 1208925819614629174706176 302231454903657293676544 2234298365826071043146605 8659687468939665971125843 -------------------------, -------------------------, 2417851639229258349412352 9671406556917033397649408 66451049808914734976352719 62932811418821997094587569 --------------------------, --------------------------, 77371252455336267181195264 77371252455336267181195264 14662209697721386418462501 428350171274090195179768579 --------------------------, ---------------------------, 19342813113834066795298816 618970019642690137449562112 761405751424527326533797101 326265547859071188231309943 ----------------------------, ---------------------------, 1237940039285380274899124224 618970019642690137449562112 1061390342950566055146890767 3177154036134113007251725123 ----------------------------, ----------------------------, 2475880078570760549798248448 9903520314283042199192993792 2030021949416729599249035961 823823440890050026606641937 ----------------------------, ----------------------------, 9903520314283042199192993792 9903520314283042199192993792 -417851494358002474642881323 -6674200920030267941396639357 ----------------------------, -----------------------------, 9903520314283042199192993792 39614081257132168796771975168 -23202391878511650209570018539 -4085123977532483117868217463 ------------------------------, -----------------------------, 79228162514264337593543950336 9903520314283042199192993792 -83146246407478754853863986343 -397541690062424933305568293397 ------------------------------, -------------------------------, 158456325028528675187087900672 633825300114114700748351602688 -1820467803237689427923580589313 -2017868697227394880524701231363 --------------------------------, --------------------------------, 2535301200456458802993406410752 2535301200456458802993406410752 -1090307383802787132390145902571 -18469261054401370242674580993745 --------------------------------, ---------------------------------, 1267650600228229401496703205376 20282409603651670423947251286016 -38458151705837518252364033246855 -19749955180204264712989000797235 ---------------------------------, ---------------------------------, 40564819207303340847894502572032 20282409603651670423947251286016 -80273081298626072508595297836025 -323661996542446117595962313268217 ---------------------------------, ----------------------------------] 81129638414606681695789005144064 324518553658426726783156020576256 and in Maple notation [36028797018963939/36028797018963968, 36028797018963141/36028797018963968, 4503599627368997/4503599627370496, 288230376150768863/288230376151711744, 576460752289295521/576460752303423488, 288230376108691951/288230376151711744, 1152921503719722083/1152921504606846976, 4611686002503097007/ 4611686018427387904, 4611685954941232307/4611686018427387904, 4611685790060101347/4611686018427387904, 4611685268160848043/ 4611686018427387904, 9223367489932494035/9223372036854775808, 18446718451583147155/18446744073709551616, 4611669123793849285/ 4611686018427387904, 36893152531958210875/36893488147419103232, 147570798712903498913/147573952589676412928, 590267650383964503731/ 590295810358705651712, 590235864777846495041/590295810358705651712, 295086857436600560163/295147905179352825856, 4720457470415928341883/ 4722366482869645213696, 9437554065219110651101/9444732965739290427392, 4715858377472170612973/4722366482869645213696, 18843856289925389416667/ 18889465931478580854784, 75248409669794782080643/75557863725914323419136, 37524406949970638794159/37778931862957161709568, 37372244527776437722363/ 37778931862957161709568, 37146838422171741262861/37778931862957161709568, 589152551764417358500591/604462909807314587353088, 1163720359007584599312437/ 1208925819614629174706176, 285942777688595103836843/302231454903657293676544, 2234298365826071043146605/2417851639229258349412352, 8659687468939665971125843/ 9671406556917033397649408, 66451049808914734976352719/ 77371252455336267181195264, 62932811418821997094587569/ 77371252455336267181195264, 14662209697721386418462501/ 19342813113834066795298816, 428350171274090195179768579/ 618970019642690137449562112, 761405751424527326533797101/ 1237940039285380274899124224, 326265547859071188231309943/ 618970019642690137449562112, 1061390342950566055146890767/ 2475880078570760549798248448, 3177154036134113007251725123/ 9903520314283042199192993792, 2030021949416729599249035961/ 9903520314283042199192993792, 823823440890050026606641937/ 9903520314283042199192993792, -417851494358002474642881323/ 9903520314283042199192993792, -6674200920030267941396639357/ 39614081257132168796771975168, -23202391878511650209570018539/ 79228162514264337593543950336, -4085123977532483117868217463/ 9903520314283042199192993792, -83146246407478754853863986343/ 158456325028528675187087900672, -397541690062424933305568293397/ 633825300114114700748351602688, -1820467803237689427923580589313/ 2535301200456458802993406410752, -2017868697227394880524701231363/ 2535301200456458802993406410752, -1090307383802787132390145902571/ 1267650600228229401496703205376, -18469261054401370242674580993745/ 20282409603651670423947251286016, -38458151705837518252364033246855/ 40564819207303340847894502572032, -19749955180204264712989000797235/ 20282409603651670423947251286016, -80273081298626072508595297836025/ 81129638414606681695789005144064, -323661996542446117595962313268217/ 324518553658426726783156020576256] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999998, .9999999992, .9999999965, .9999999862, .9999999505, .9999998373, .9999995070, .\ 9999986110, .9999963366, .9999909031, .9999786285, .9999522951, .9998984482, .9\ 997931622, .9995957509, .9992399043, .9986218551, .9975854457, .9959044097, .99\ 32627816, .9892350759, .9832686259, .9746711373, .9626069194, .9461052880, .924\ 0841454, .8953906981, .8588596888, .8133875234, .7580184749, .6920370255, .6150\ 586678, .5271104214, .4286921455, .3208105739, .2049798339, .8318490948e-1, -.\ 4219221864e-1, -.1684805177, -.2928553578, -.4124921087, -.5247265857, -.627210\ 1950, -.7180479396, -.7959088636, -.8601008698, -.9106048746, -.9480666365, -.9\ 737479701, -.9894421184, -.9973605296] The cut off is at j=, 43 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 58], vs. those in the, 2, -th row from j=1 to j=, 57, are as follws 29 28 [3 (48038396025285271 n - 20200145528632440201 n 27 26 + 4051870570346711025529 n - 516043017747203569749189 n 25 24 + 46859581449919261985479245 n - 3229688393947019365619694345 n 23 22 + 175618880491255663929410868405 n - 7731376581057872534452373703705 n 21 + 280592895341540004651991802774535 n 20 - 8505003394243633841948946800973735 n 19 + 217337763061128517314397933345745715 n 18 - 4713913256138321459029623857633372415 n 17 + 87181192351476812903800475797802235615 n 16 - 1378832101730039496394148303843795219715 n 15 + 18674103358045368989939274913257788172135 n 14 - 216567316146488927739124414014345782803635 n 13 + 2147662194775220771709275010359716634729430 n 12 - 18161421507438799916139558400272972141766980 n 11 + 130401836654269628130992396237356742975187320 n 10 - 790256838770729511692203373092138368400764720 n 9 + 4009833347012477100585443967680760477382918304 n 8 - 16857141865208164083203273471615836377966489024 n 7 + 57903917435507182726428492491676363943459840896 n 6 - 159478436083043523845946926528844089701845966336 n 5 + 342409645393443944286612613223507571368999577600 n 4 - 542876700774872167744285633499236888385596416000 n 3 + 529459303135060191123575068268931994439516160000 n 2 + 99709815154607849367381061972245018681999360000 n - 1598534303049587347047539872989579692605440000000 n + 2569860270515754373727474473853864399339520000000)/(268435456 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 29 (2 n - 51) (2 n - 57) (2 n - 45)), 3 (24019198012642360 n 28 27 - 10100072764315884817 n + 2025935285173159042970 n 26 25 - 258021508873527975820293 n + 23429790724939661941244730 n 24 23 - 1614844196969366196052320465 n + 87809440244942182663625077050 n 22 - 3865688290436046011339294162985 n 21 + 140296447660274154570339154186050 n 20 - 4252501696119311452406667769786095 n 19 + 108668881448817035230074011963491350 n 18 - 2356956622336962190310470732564428255 n 17 + 43590595828302500325848857435038149550 n 16 - 689416032599499220932282893315149477755 n 15 + 9337050844490930631290504211031134553950 n 14 - 108283624917528141099959202282644579539395 n 13 + 1073829952809565378344010652870548233892350 n 12 - 9080676495509662465286150232994675285918260 n 11 + 65200032577178187165067312396138337937635000 n 10 - 395108744814355535974398525379499925875533040 n 9 + 2004544097102281236241067497381800143560644640 n 8 - 8422618794547426997504095272151658683438204608 n 7 + 28872889304621120708718625529596159064471959680 n 6 - 78883254890587788797241890895669493981088236032 n 5 + 163871884243772433966094913560743623454080440320 n 4 - 223934882537831286552726537310955033116689408000 n 3 + 49727565988633183541674467616550510475018240000 n 2 + 621800604509510955080273853504968829904158720000 n - 1219317752245454019238587353148909668873011200000 n + 22153967849273744601098917878050555166720000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 30 (2 n - 51) (2 n - 57) (2 n - 45)), 3 (48038396025276873 n 29 28 - 21617278211368119075 n + 4647774863435359288410 n 27 26 - 635573199570170049698125 n + 62082850472848700782273602 n 25 24 - 4612046046592154513475422745 n + 270894688091547787353220681950 n 23 - 12912133552677910063192007147625 n 22 + 508668504157324325665818278188020 n 21 - 16782493775640676747325157672451275 n 20 + 468235360640287013365754781311633850 n 19 - 11125377086885302101168210650134671375 n 18 + 226241622478552719531103754879280608190 n 17 - 3950676698816433701754559423257068499075 n 16 + 59349623842585808697621421166837144380650 n 15 - 767452305293756271939434639138181766783875 n 14 + 8536361175013112591800173550503673938258195 n 13 - 81516344438464952285997290128140120220406550 n 12 + 666134749787213569445663708993511906976659300 n 11 - 4636459025625202886617878443386809811577547000 n 10 + 27309892134469858356810730550264883549017297712 n 9 - 134943806006584720504797750511206410265223306400 n 8 + 552428172479008683904200157922488079714036763840 n 7 - 1836886786937564395896666178499939716956246192000 n 6 + 4772993491096489285094405849961353115850077777408 n 5 - 8763281127052251260352023774142474279373577594880 n 4 + 7187760695220731262362718328172613766174190592000 n 3 + 14600266045551836740083748062480597568857047040000 n 2 - 51089071039354311131676752275377289731162439680000 n + 36016706549921491751556808515183970251099340800000 n - 1307084103107150931464836154804982754836480000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 30 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), (144115188075603469 n 29 28 - 64851834633834503025 n + 13943324590151907525830 n 27 26 - 1906719598654117637503575 n + 186248551403712520984020606 n 25 24 - 13836138136788697888681394235 n + 812684063795541552093404567850 n 23 - 38736400595253365746465461196875 n 22 + 1526005505624959811117857663603560 n 21 - 50347480697111920670724625385961825 n 20 + 1404706032587937762238018407179057550 n 19 - 33376127947040047536786415893600556125 n 18 + 678724675656711145867681067230607511570 n 17 - 11852020503418241996179932776782022739225 n 16 + 178048456186562600806710743970571976035950 n 15 - 2302341351126730245001683950085371056109625 n 14 + 25608579425120554162497261140582650229885835 n 13 - 244534966682305752026068931979704882924398650 n 12 + 1998067700786293265569732956939673589593730900 n 11 - 13902521530842412725848533059839376317242437000 n 10 + 81811931499600922489167148478928328972287815536 n 9 - 403149032321933531498297867674778335508572898400 n 8 + 1637655742264167992357215930861566415111820905920 n 7 - 5328588418438751829772199398878142498032924316800 n 6 + 13029381941160559436869842845996789855591202599424 n 5 - 19765132587763689425802227843657659406046395504640 n 4 + 868538877348268631730352509001319070169138176000 n 3 + 71797662676338519697276098305613801100938117120000 n 2 - 131263124520490342310072937829571470040384471040000 n + 73344783118988398798535120745557541055915622400000 n - 3921252309321452794394508464414948264509440000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 1643 (1403434576496003 n 30 29 - 674350313992968154 n + 155046189120840673060 n 28 27 - 22709589757632113700080 n + 2380070593331878574129772 n 26 25 - 190059344442271125336692016 n + 12023725121372894121081654960 n 24 - 618607562481978619740825078600 n 23 + 26366068630681366306884158834970 n 22 - 943548571025627503153724915839860 n 21 + 28633512568317866324746909569181800 n 20 - 742248441646369566724503997773445800 n 19 + 16522898300421899517824650622724527340 n 18 - 317010831467278103779373415185899912320 n 17 + 5254114537811984433831310453902541002000 n 16 - 75303213287295125770118347452538922914200 n 15 + 933181305071608490719471204920982555690395 n 14 - 9986471564187022242106786909588301552581410 n 13 + 92062839159927390651930039147787058961507700 n 12 - 728309002667454476183842062250347618208795400 n 11 + 4916589502191973401122262678967097814183272432 n 10 - 28088242687067660823511543837211984721244631776 n 9 + 134043939826637037581647285621686846357970338240 n 8 - 522456228257099394385106514331231927137943489920 n 7 + 1591156589383420694581099582527806917942599449088 n 6 - 3410572597170550699100849007768494823111642814464 n 5 + 3462885798217448465938090535329193621357964042240 n 4 + 5409448693787516006895774878964490114117607424000 n 3 - 25037192816748473130581767097456791437985382400000 n 2 + 34065635207201595819328142010189396174703165440000 n - 16852905845310857758935163064885514960528998400000 n + 1164681148477704786162215539034993763287040000000)/(1073741824 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 1643 (701717288199415 n 30 29 - 337175156940026564 n + 77523094528905189170 n 28 27 - 11354794867565659400770 n + 1190035293781347796462890 n 26 25 - 95029671655549867481422506 n + 6011862472527198857105287830 n 24 - 309303770029226167417723891650 n 23 + 13183033130598656221579510534200 n 22 - 471774180122569905127531014326010 n 21 + 14316748316779469667412993519792050 n 20 - 371123705473542166675001231727553950 n 19 + 8261420498119965066497292248554599750 n 18 - 158504042680026471018292613880884895870 n 17 + 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(n - 54)! (2 n - 115)! (n + 1) / + 158787967431439827253564499586816 (n - 56)! | \ 113 --- (n - 54)! (n + 1) (2 n - 114)! + (n - 55)! ((2 n - 113)! (n + 1) 112 - 1/158787967431439827253564499586816 binomial(2 n, n) (n - 58)! (n - 54)! \\ )||/((n - 58)! (n - 56)! (n - 55)! (n - 54)! binomial(2 n, n)), ( // 106803811427099407616980883650656 (n - 55)! (2 n - 115)! (n + 1) + 160205717140649111425471325475984 (n - 56)! ((2 n - 114)! (n + 1) - 1/160205717140649111425471325475984 binomial(2 n, n) (n - 58)! (n - 55)! ))/((n - 56)! (n - 58)! (n - 55)! binomial(2 n, n)), ( (106803811427099407616980883650656 n + 106803811427099407616980883650656) (2 n - 115)! - (n - 58)! (n - 56)! binomial(2 n, n))/((n - 58)! 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2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45) , 1/134217728*(144115188075603469*n^30-64851834633834503025*n^29+ 13943324590151907525830*n^28-1906719598654117637503575*n^27+ 186248551403712520984020606*n^26-13836138136788697888681394235*n^25+ 812684063795541552093404567850*n^24-38736400595253365746465461196875*n^23+ 1526005505624959811117857663603560*n^22-50347480697111920670724625385961825*n^ 21+1404706032587937762238018407179057550*n^20-\ 33376127947040047536786415893600556125*n^19+ 678724675656711145867681067230607511570*n^18-\ 11852020503418241996179932776782022739225*n^17+ 178048456186562600806710743970571976035950*n^16-\ 2302341351126730245001683950085371056109625*n^15+ 25608579425120554162497261140582650229885835*n^14-\ 244534966682305752026068931979704882924398650*n^13+ 1998067700786293265569732956939673589593730900*n^12-\ 13902521530842412725848533059839376317242437000*n^11+ 81811931499600922489167148478928328972287815536*n^10-\ 403149032321933531498297867674778335508572898400*n^9+ 1637655742264167992357215930861566415111820905920*n^8-\ 5328588418438751829772199398878142498032924316800*n^7+ 13029381941160559436869842845996789855591202599424*n^6-\ 19765132587763689425802227843657659406046395504640*n^5+ 868538877348268631730352509001319070169138176000*n^4+ 71797662676338519697276098305613801100938117120000*n^3-\ 131263124520490342310072937829571470040384471040000*n^2+ 73344783118988398798535120745557541055915622400000*n-\ 3921252309321452794394508464414948264509440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2* n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/( 2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45) , 1643/1073741824*(1403434576496003*n^31-674350313992968154*n^30+ 155046189120840673060*n^29-22709589757632113700080*n^28+ 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3410572597170550699100849007768494823111642814464*n^6+ 3462885798217448465938090535329193621357964042240*n^5+ 5409448693787516006895774878964490114117607424000*n^4-\ 25037192816748473130581767097456791437985382400000*n^3+ 34065635207201595819328142010189396174703165440000*n^2-\ 16852905845310857758935163064885514960528998400000*n+ 1164681148477704786162215539034993763287040000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2* n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/( 2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57) /(2*n-45), 1643/536870912*(701717288199415*n^31-337175156940026564*n^30+ 77523094528905189170*n^29-11354794867565659400770*n^28+ 1190035293781347796462890*n^27-95029671655549867481422506*n^26+ 6011862472527198857105287830*n^25-309303770029226167417723891650*n^24+ 13183033130598656221579510534200*n^23-471774180122569905127531014326010*n^22+ 14316748316779469667412993519792050*n^21-371123705473542166675001231727553950*n ^20+8261420498119965066497292248554599750*n^19-\ 158504042680026471018292613880884895870*n^18+ 2627000488020163601934824286394833428850*n^17-\ 37649581099965738458242479314438424518550*n^16+ 466528440140388983811273528519508018165425*n^15-\ 4991596467853554506036852793431069265882810*n^14+ 45994560952248657379624583412885931955945300*n^13-\ 363452759273043228916287935646192126262342600*n^12+ 2447099787291762056784771775111006555359882160*n^11-\ 13896585347039486074918353690629771441375630816*n^10+ 65446677483774723525131012440709975616262380480*n^9-\ 247965973197104623013276555504813129528038980480*n^8+ 710951055238728089688411428413996199119855196160*n^7-\ 1320253121194059579673738172472817490037281255424*n^6+ 607854756394491970944935755878832028043413176320*n^5+ 4368198192575071156239443142397648563118706688000*n^4-\ 12745569263506221544547562386775014873088983040000*n^3+ 14970900200216927727263032149481994315008573440000*n^2-\ 6989678161041984440676089776960795346416435200000*n+ 582340574238852393081107769517496881643520000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n -19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n -21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2 *n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/ (2*n-45), 1643/1073741824*(2806869151815133*n^32-1437117004927606064*n^31+ 352576447248191004968*n^30-55187089150847428877920*n^29+ 6190845269195781977833932*n^28-530063121967918277926866336*n^27+ 36021186690182572162041661992*n^26-1994709517797462323611672195680*n^25+ 91704382681053514280446266679470*n^24-3548156667888313008854223175599360*n^23+ 116710370319543922682618966203160520*n^22-3288394043820914549081724841099312800 *n^21+79806647787135168937277704644904660140*n^20-\ 1674925069652140751932732709361439117920*n^19+ 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73374912354095401528219578959204607087083520000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2 *n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2 *n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/ (2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57 )/(2*n-63)/(2*n-45), 31/1073741824*(148764064817132897*n^32-\ 76167201000253357264*n^31+18686551561519634387464*n^30-\ 2924915675174769233859680*n^29+328114786781385188164606908*n^28-\ 28093343073843541095940445856*n^27+1909122531161873580625952607816*n^26-\ 105719559418805048421598932361440*n^25+4860327653164464795648057731442630*n^24-\ 188051903315846615483260158720221760*n^23+6185620282396836093457861615953513960 *n^22-174283045636531697392885409339321997600*n^21+ 4229653467024321826748351718082254962460*n^20-\ 88766454544107329694393363281749647489120*n^19+ 1615165064209280403849931790715638736681720*n^18-\ 25516150920872446898478748897165269179421600*n^17+ 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(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/ (2*n-21)/(2*n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37 )/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-\ 57)/(2*n-63)/(2*n-45), 1519/1073741824*(6072002609074756*n^33-\ 3306205391112759267*n^32+863754491050890764512*n^31-144172587876376411359480*n^ 30+17272428419734783889547184*n^29-1581920980158322084174627668*n^28+ 115189987976925763518991084128*n^27-6847586011720919864978320393080*n^26+ 338620171316394579234549319175640*n^25-14122893091097397625988393647324530*n^24 +501925842290981860348997366693731680*n^23-\ 15318687129987733780193942066290517400*n^22+ 403813190281324512241444830038404780080*n^21-\ 9233061522212157378938985524805403879860*n^20+ 183643083548570348478519383402289898361760*n^19-\ 3182869744460965690000480575290342001645000*n^18+ 48101058203428644032452629617715786562843140*n^17-\ 633539298277633679338047834710162425947447555*n^16+ 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33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/(2* n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/( 2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71) /(2*n-69)/(2*n-77), 1147/34359738368*(16452732071295066551*n^39-\ 12507166247627908102428*n^38+4590636962322942252649564*n^37-\ 1083648513730483372490090142*n^36+184882867430409651784616702913*n^35-\ 24290125616690906624414274711654*n^34+2556696651743447961894840939082552*n^33-\ 221459948374192902482098933241555256*n^32+ 16091405105609985641111842843182566006*n^31-\ 994748658339229505213242444409203500288*n^30+ 52877848808203736832515893436739698610744*n^29-\ 2436579457509787787582933764853382128860932*n^28+ 97921873638019864456399623615621113079043410*n^27-\ 3447656847727020352871413448497990699662739620*n^26+ 106679575350021737996973276075106632290084555760*n^25-\ 2906703653026248410505404397017492899850454080280*n^24+ 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n^40-321003583051993727828603640*n^39+63522236996806504224431359964*n^38-\ 9693810572354909912915162797638*n^37+1186826699973108296006086775359663*n^36-\ 119738636424261877933157938769283660*n^35+ 10146900355921739357382165399474352456*n^34-\ 732498755039379858046596098832123523212*n^33+ 45526386470758543352472387074880017062222*n^32-\ 2455831734307811070390132352581622887152160*n^31+ 115677938201747053090031696749318843540718928*n^30-\ 4779226632063005960793823670805366069859572636*n^29+ 173722462961338420021519343147339426191694817846*n^28-\ 5565758381472354334004394122360794213721499780840*n^27+ 157245990310791911967900065601119433156171646055190*n^26-\ 3913906643503127914776082679899951213212879026922630*n^25+ 85588666861907692580550705434822961534592501527372955*n^24-\ 1635569050184830180253887164198646650253885333704093000*n^23+ 27051775711894962857919576523571127426040202927371401508*n^22-\ 380459083731938405130466091014885683168771475445707944606*n^21+ 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2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), 65231 /8796093022208*(4207791949314209087071*n^45-4179605311541712545274825*n^44+ 2008827974487647697585632805*n^43-622245208446691160451702662025*n^42+ 139590254482453950866451233342331*n^41-24161398630424444851026616897526835*n^40 +3356704379508097982898663305269607635*n^39-\ 384438409202740311320926216645010618775*n^38+ 36992311538597137315575433772145728356596*n^37-\ 3032576260055094657706125359773699585179270*n^36+ 214000158767776070628083662610136626538536490*n^35-\ 13098588990335086434960336548188128634475552250*n^34+ 699169296370275608873108415675182836223676511846*n^33-\ 32655463913041501484583972221789482719763492667430*n^32+ 1336367321057267903693184916149892815549862807661670*n^31-\ 47865554715444299391211387514861962170805883226787950*n^30+ 1493859322895652902346095906894129871567925321274914551*n^29-\ 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97137840848629132614108541691913601381854708830274846720000*n^6-114008629893694\ 8463515966922918943195444954793793770593885476811723559053096386560000000*n^5+ 1524713064484120087108992434612618189928160282506724574161595708880417470640291\ 840000000*n^4-15020747298460034098705474473147660631340251495273933061692829645\ 09464768335052800000000*n^3+100872254774496030288792664054507433174197152098588\ 8052416517158324579787499110400000000*n^2-4018510470760612458766468398691480559\ 94729998641437658361071618058032565452800000000000*n+68466677900063538232816128\ 243840775287253939195907994901116447105673592832000000000000)/(2*n-95)/(2*n-85) /(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-\ 63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2* n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/( 2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-11) /(2*n-33)/(2*n-65)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n-19) /(2*n-3)/(-1+2*n)/(2*n-5), 14147/35184372088832/(2*n-99)/(2*n-95)/(2*n-85)/(2*n -97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2 *n-57)/(2*n-51)/(2*n-73)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)*( 181481257739618026887944*n^50-99510152400958237073138500*n^49-\ 17322945610016885217154665575*n^48+37648229072372395015364119357000*n^47-\ 19705945547292443757875529798208740*n^46+6389894056680237983245884855236394600* n^45-1514188517073255951468765902229719501950*n^44+ 281524876500911784051584944257790633430000*n^43-\ 42757076593781395830955887466866309638464820*n^42+ 5445532382253212067416335249254425168263878100*n^41-\ 592434234704667473119933903376527296198230142825*n^40+ 55816807407591325778615347527679896450533524203000*n^39-\ 4602516311776914467937298346771340399160904867240080*n^38+ 334919878553868284827827867838618316411934003198719600*n^37-\ 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51038774691982048866436771143246075937570160640000*n^6-498648521348074769616566\ 4176289516852483186150485304839634115597084475704049777246208000000*n^5+6616526\ 4520629268824450832926937889361730853338844636265897777674395433540991005491200\ 00000*n^4-647439983244079911280718933125602585857634029091134299875702336502205\ 4167734753689600000000*n^3+4323677383762339829553662934439622348876143264734498\ 836508190509507794061054758092800000000*n^2-17151064759691860422484064137959634\ 16653204084116270094522363407195505954494873600000000000*n+29146264782057048225\ 7098257934030180397840019156980334294052715328852484685824000000000000)/(2*n-95 )/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-\ 45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2* n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/( 2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41) /(2*n-11)/(2*n-33)/(2*n-65)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47) /(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -329/140737488355328*( 177782671001340412672222139*n^51-271106192812430574275781164832*n^50+ 196745442065959002700423579653425*n^49-91048379609181516038378089411110900*n^48 +30321833197818132796799850530927369810*n^47-\ 7767158545145163386722466551772910409680*n^46+ 1596198008796093835818616129996201781447750*n^45-\ 270903567970060674607516937666997020174879400*n^44+ 38779152275267072021914054324054118993577491705*n^43-\ 4757292841050855078683290588479660614930025115440*n^42+ 506394328471664547064238021081160327392230790133875*n^41-\ 47236014710474400957049870617900474567955704085341900*n^40+ 3892077956702461994728993687200019545571476078771354020*n^39-\ 285137086748913481474867037198817982304499984093034208160*n^38+ 18673933516745171561561700365200823241096340189890581316500*n^37-\ 1098188170154214791132165224122213597879022512114938413868400*n^36+ 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33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-47)/(2*n -19)/(2*n-3)/(-1+2*n)/(2*n-5), -53/562949953421312/(2*n-99)*( 165241155468967723382528364313*n^54-242882984761326568625496102471105*n^53+ 174034209662911453511721058884985482*n^52-\ 81015555445633412231918235707391288165*n^51+ 27551275059061457619501247720223266948275*n^50-\ 7297353139530992902856558357981040041991300*n^49+ 1567260734073073513022778328620051225639360780*n^48-\ 280586861703343724573071878078928952565460806750*n^47+ 42722020980297836119309856352040338545125708879185*n^46-\ 5616640360957214351107839216425214688572784568482575*n^45+ 645160773887099857676642831268976614289755427783130790*n^44-\ 65359619644365458074730075083910729571884222292271660575*n^43+ 5884523830840464920587579582547595098692789517655084111315*n^42-\ 473796492382740564822955355755007101847070328283306161209750*n^41+ 34293011004398581119797814140604154666391784642965086146827760*n^40-\ 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4294110461848200529444951386046040174808570992509620906481767806098324263861522\ 944955489976320*n^13-7273147256004703673402250123876806194019853451524262356519\ 077868851136722949211802497611814154403840*n^12+3330976546701826029312384111345\ 6573101081504898316357084445819806072490985474411416658329286463193088*n^11-133\ 2048105243695528237734652017405272132018418235618013794599317026835368761389577\ 44883280187513372672*n^10+46051103428022095476801304010176563960574078905212501\ 5036592129701457728882574776654253687476479590400*n^9-1359740048935215405324242\ 588493690462118343124004940599922544219365337426406246233486699846929743872000* n^8+337765801686635501251389727120785405739215284578140619951050261525642612428\ 4045401026528374863953920000*n^7-6924558160660482639700625421038769392145572387\ 940829720080244747028218917738743325028387409740431360000*n^6+11425567358125544\ 8340247100222878325265390905377685997243075459730158002754250645833529573978931\ 20000000*n^5-146623153052711126766928890694319526625000837383451882852691889215\ 72792948892672207361304250286080000000*n^4+139294720871528917942457399310910861\ 70453409389705649875175474308208111195005104475564138299392000000000*n^3-906859\ 1152301714331259136445364287717676480791531543522324090968361207547033374206626\ 018571059200000000*n^2+35234527411637819431412810760747899742948450920583935688\ 63035605060961095840533469702190530560000000000*n-59006665460948371253812488131\ 2159675859453915202166408073681505359094611925566341590810624000000000000)/(2*n -81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2 *n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17) /(2*n-105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-\ 59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-107)/(2*n-11)/(2 *n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-109)/ (2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -5/2251799813685248/(2*n-99)*( 15404814309670959256261764368003*n^55-23365574090820319218698469745523740*n^54+ 17288501282267934586133915760780234720*n^53-\ 8316493680714678419055390824635865286990*n^52+ 2924641319613642230938139780088026323112847*n^51-\ 801622836659162841149333463295119581844315558*n^50+ 178295349486833032494427227437540347065797935800*n^49-\ 33081604917146722662757214467779284558078585975700*n^48+ 5224261072686834046067841490407255991896959389647615*n^47-\ 712926287462061104727545437881726450350159630745462120*n^46+ 85070802282979127385650234720647335069389467724799787800*n^45-\ 8960365331349719071173037049231486936603682229193211538650*n^44+ 839461135028641564507486116628237817297564512431420668297355*n^43-\ 70393897260267006158317907574072191557593044785990388894019210*n^42+ 5311241015626741194470990628793537674528706997883183607961268000*n^41-\ 362142856769072830196448915849923468605070409158006494572791749800*n^40+ 22396507290693836734724600550710179301121174865376475600310490726265*n^39-\ 1260198474017715156150762085561535495752395524941249886355346700381340*n^38+ 64682226930551785829475713684491642843885045688767423639676789299811600*n^37-\ 3035042645595820843502255546328495977498388591111410418609141876828153250*n^36+ 130426479142676774668506533099980677427116505436680939119436250584243707789*n^ 35-5140844477440049313981956746094811745692738695057662121660916905351659745850 *n^34+ 186076346604945552811447676689146811799608402705103054034770965838758213155960* n^33-61907200044296835951844584258778169262179833815222029694004787361526279267\ 93620*n^32+18944511824152320222896870543909250841732463632024128497859936334464\ 7642578900501*n^31-533478830557823637002250807384515183761088753783304114746006\ 8323468098287376640384*n^30+138275976842355350979892683182614112072731818754719\ 411158173051178045293979328612600*n^29-3299016084532277063638678474186314523032\ 459513402340744537280146215801750918311570550*n^28+7243566097108939091519180042\ 4620623873576362939525705860362420035158143612104176428185*n^27-146312985086713\ 5967231995787152117613399846087816300871053315789252665224901062652550230*n^26+ 2717169942394463046298043308711117249407538612336835909282641439890446039803299\ 9350193200*n^25-463555379052262902760910279463780426715223491227979428557397671\ 007846131521442927783919600*n^24+7257501211603081361710877116562902192570842482\ 332727251669241228457137720344932352262873520*n^23-1041405370777775065941523081\ 92239397412873401083497419427672501708491694927212660208012993440*n^22+13675259\ 2289940476342653265543802600814364877907670974715927312781410472159496297855193\ 2368000*n^21-164040087538209294558182982134374140029273177503714781286212268359\ 59970307865385734220750707200*n^20+17936858000516233484751126838329623826098651\ 5272962043761846263067439387917878179237916926498560*n^19-178344818236523211687\ 6181622843818200418878875381633370532343945029840746275930167959814096898560*n^ 18+1607909897724361192799873478184360844020788544086936928471798118722159243623\ 1228085987696714342400*n^17-131017440543753946391230171594759763957065944831934\ 274098338258297979303123396471914376434771456000*n^16+9612173281669878984323995\ 45545552820137502503986456294787917966913159834360034886567060409524932608*n^15 -632178514070474503004970455450542574678929667379382355089875458564023714412076\ 7106867688756618485760*n^14+370831721183890567611739345763212589083021200392098\ 86707913033783068536511532847527856966450982584320*n^13-19286589507348451872103\ 2689015776973831727077915808960110041746714206612469412191357876893207657840640 *n^12+8831665216911893779570304383476216199815483555688245454734999307354293569\ 72384184387335979360494026752*n^11-35313162524646933115605018780088095474509402\ 19882207869292270377234191129202671439805875763343741943808*n^10+12207024639342\ 9944740594024744412506126810699234326541806515681769806581790464984823628999725\ 52759705600*n^9-360400876952131334112654224847023427622060367319672681336375391\ 15383929894113399224142389249476395008000*n^8+895184609996249386004318239186638\ 39865738885395626729804791908117912362145933161073318815167131156480000*n^7-183\ 5111844684538799169798385388131496110957026904145123920556043793555366404317022\ 46247697946289111040000*n^6+302780657118703566404177988736288339533061400033406\ 068686736828233503783937256863001598635362222080000000*n^5-38854289367111404738\ 2167774723393375706277250225265663561719489344859325407107723874949791869829120\ 000000*n^4+36911632766504008950531613759885116852064212591393909420523495779656\ 6902295561357417074151915520000000000*n^3-2403072350488934611505155245979238350\ 94984586223423651900492673763821075480936592035052790795468800000000*n^2+933685\ 6295920257485674259842539185394907271463161373179764196037012992977612066098718\ 6344099840000000000*n-156367663471513183822603093547722314102755287528574098139\ 52559892016007216027508052156481536000000000000)/(2*n-95)/(2*n-85)/(2*n-97)/(2* n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/( 2*n-51)/(2*n-73)/(2*n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39 )/(2*n-23)/(2*n-17)/(2*n-105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n -67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2 *n-107)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/ (2*n-93)/(2*n-109)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), ( 106803811427099407616980883650656*(n-55)!*(n-54)!*(2*n-115)!*(n+1)+ 158787967431439827253564499586816*(n-56)!*(113/112*(n-54)!*(n+1)*(2*n-114)!+(n-\ 55)!*((2*n-113)!*(n+1)-1/158787967431439827253564499586816*binomial(2*n,n)*(n-\ 58)!*(n-54)!)))/(n-58)!/(n-56)!/(n-55)!/(n-54)!/binomial(2*n,n), ( 106803811427099407616980883650656*(n-55)!*(2*n-115)!*(n+1)+ 160205717140649111425471325475984*(n-56)!*((2*n-114)!*(n+1)-1/ 160205717140649111425471325475984*binomial(2*n,n)*(n-58)!*(n-55)!))/(n-56)!/(n-\ 58)!/(n-55)!/binomial(2*n,n), ((106803811427099407616980883650656*n+ 106803811427099407616980883650656)*(2*n-115)!-(n-58)!*(n-56)!*binomial(2*n,n))/ (n-58)!/(n-56)!/binomial(2*n,n)] The limits, as n goes to infinity are 144115188075855813 9007199254740885 144115188075830619 144115188075603469 [------------------, ----------------, ------------------, ------------------, 144115188075855872 9007199254740992 144115188075855872 144115188075855872 2305843009182932929 1152921504511638845 4611686016432263519 -------------------, -------------------, -------------------, 2305843009213693952 1152921504606846976 4611686018427387904 4611686009331119807 2305842990796138591 4611685883871779307 -------------------, -------------------, -------------------, 4611686018427387904 2305843009213693952 4611686018427387904 4611685569602658141 4611684637873426929 73786913132244554935 -------------------, -------------------, --------------------, 4611686018427387904 4611686018427387904 73786976294838206464 36893403627394630505 147573100905276754595 147571923039411367967 --------------------, ---------------------, ---------------------, 36893488147419103232 147573952589676412928 147573952589676412928 2361109730713044667001 295128070901219508913 2360855592004818381429 ----------------------, ---------------------, ----------------------, 2361183241434822606848 295147905179352825856 2361183241434822606848 2360534126980790581179 37759139965089595863549 18871283685775441333997 ----------------------, -----------------------, -----------------------, 2361183241434822606848 37778931862957161709568 18889465931478580854784 75428756207187769728143 75336006559385661795143 300752747232804209107697 -----------------------, -----------------------, ------------------------, 75557863725914323419136 75557863725914323419136 302231454903657293676544 37479807619327510893805 74616240754088134355699 74114474672263120397957 -----------------------, -----------------------, -----------------------, 37778931862957161709568 75557863725914323419136 75557863725914323419136 4697629825144347362889173 2317122141332477379332599 -------------------------, -------------------------, 4835703278458516698824704 2417851639229258349412352 9096925030352353535230837 8870209257767997020509729 -------------------------, -------------------------, 9671406556917033397649408 9671406556917033397649408 274478476645715172958728401 4103904945954578702764559 ---------------------------, -------------------------, 309485009821345068724781056 4835703278458516698824704 248038165823618633705300651 230370471813232440949431245 ---------------------------, ---------------------------, 309485009821345068724781056 309485009821345068724781056 3351222901370513959093596173 1481514219650081051316458017 ----------------------------, ----------------------------, 4951760157141521099596496896 2475880078570760549798248448 5043796737531342713309740843 4061353210480169183805522043 ----------------------------, ----------------------------, 9903520314283042199192993792 9903520314283042199192993792 5978523423171152139468086555 1842853215335003458666595509 -----------------------------, ----------------------------, 19807040628566084398385987584 9903520314283042199192993792 320926919155297028297967971 -36897604049526572860923863 ----------------------------, ---------------------------, 4951760157141521099596496896 618970019642690137449562112 -58490498759440995769161083731 -48692721673997007423747969773 ------------------------------, ------------------------------, 316912650057057350374175801344 158456325028528675187087900672 -269420190285198482460243248327 -339304644709140182921329636547 -------------------------------, -------------------------------, 633825300114114700748351602688 633825300114114700748351602688 -6448022609028726058465450679627 -918742560842538914504942046469 --------------------------------, -------------------------------, 10141204801825835211973625643008 1267650600228229401496703205376 -8121982190061428296522953724211 -8757781239855289339274003308589 --------------------------------, --------------------------------, 10141204801825835211973625643008 10141204801825835211973625643008 -148120895579322804850676099633255 -77024071548354796281308821840015 ----------------------------------, ---------------------------------, 162259276829213363391578010288128 81129638414606681695789005144064 -316218810569097375693451477350085 -321180934551329870295125367962173 ----------------------------------, ----------------------------------, 324518553658426726783156020576256 324518553658426726783156020576256 -1294736595526610050644593429690941 -----------------------------------] 1298074214633706907132624082305024 and in Maple notation [144115188075855813/144115188075855872, 9007199254740885/9007199254740992, 144115188075830619/144115188075855872, 144115188075603469/144115188075855872, 2305843009182932929/2305843009213693952, 1152921504511638845/ 1152921504606846976, 4611686016432263519/4611686018427387904, 4611686009331119807/4611686018427387904, 2305842990796138591/ 2305843009213693952, 4611685883871779307/4611686018427387904, 4611685569602658141/4611686018427387904, 4611684637873426929/ 4611686018427387904, 73786913132244554935/73786976294838206464, 36893403627394630505/36893488147419103232, 147573100905276754595/ 147573952589676412928, 147571923039411367967/147573952589676412928, 2361109730713044667001/2361183241434822606848, 295128070901219508913/ 295147905179352825856, 2360855592004818381429/2361183241434822606848, 2360534126980790581179/2361183241434822606848, 37759139965089595863549/ 37778931862957161709568, 18871283685775441333997/18889465931478580854784, 75428756207187769728143/75557863725914323419136, 75336006559385661795143/ 75557863725914323419136, 300752747232804209107697/302231454903657293676544, 37479807619327510893805/37778931862957161709568, 74616240754088134355699/ 75557863725914323419136, 74114474672263120397957/75557863725914323419136, 4697629825144347362889173/4835703278458516698824704, 2317122141332477379332599/ 2417851639229258349412352, 9096925030352353535230837/9671406556917033397649408, 8870209257767997020509729/9671406556917033397649408, 274478476645715172958728401/309485009821345068724781056, 4103904945954578702764559/4835703278458516698824704, 248038165823618633705300651/309485009821345068724781056, 230370471813232440949431245/309485009821345068724781056, 3351222901370513959093596173/4951760157141521099596496896, 1481514219650081051316458017/2475880078570760549798248448, 5043796737531342713309740843/9903520314283042199192993792, 4061353210480169183805522043/9903520314283042199192993792, 5978523423171152139468086555/19807040628566084398385987584, 1842853215335003458666595509/9903520314283042199192993792, 320926919155297028297967971/4951760157141521099596496896, -\ 36897604049526572860923863/618970019642690137449562112, -\ 58490498759440995769161083731/316912650057057350374175801344, -\ 48692721673997007423747969773/158456325028528675187087900672, -\ 269420190285198482460243248327/633825300114114700748351602688, -\ 339304644709140182921329636547/633825300114114700748351602688, -\ 6448022609028726058465450679627/10141204801825835211973625643008, -\ 918742560842538914504942046469/1267650600228229401496703205376, -\ 8121982190061428296522953724211/10141204801825835211973625643008, -\ 8757781239855289339274003308589/10141204801825835211973625643008, -\ 148120895579322804850676099633255/162259276829213363391578010288128, -\ 77024071548354796281308821840015/81129638414606681695789005144064, -\ 316218810569097375693451477350085/324518553658426726783156020576256, -\ 321180934551329870295125367962173/324518553658426726783156020576256, -\ 1294736595526610050644593429690941/1298074214633706907132624082305024] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999996, .9999999980, .9999999920, .9999999708, .9999999027, .9999997006, .\ 9999991440, .9999977091, .9999942288, .9999862472, .9999688670, .9999327989, .9\ 998612351, .9997250893, .9994761128, .9990374399, .9982912762, .9970637448, .99\ 51073667, .9920822472, .9875377237, .9808969049, .9714470791, .9583392561, .940\ 6000024, .9171581409, .8868877908, .8486676518, .8014545388, .7443671406, .6767\ 740753, .5983788280, .5092933197, .4100918746, .3018382976, .1860806215, .\ 6481067519e-1, -.5961129437e-1, -.1845634712, -.3072942760, -.4250701104, -.535\ 3283383, -.6358241190, -.7247600882, -.8008892778, -.8635839046, -.9128654982, -.9493949813, -.9744244420, -.9897151671, -.9974287918] The cut off is at j=, 44 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 59], vs. those in the, 2, -th row from j=1 to j=, 58, are as follws 29 28 27 [59 (1221315153185219 n - 513563021914384372 n + 103013658568136896251 n 26 25 - 13119737739335749818828 n + 1191345291099660067473375 n 24 23 - 82110721880012663164066740 n + 4464886792151179367787056295 n 22 21 - 196560421552401656991774793260 n + 7133717678184108813860323930665 n 20 - 216228899854546003536229071773820 n 19 + 5525536349084665447714064277487185 n 18 - 119845252279816556754280661140770180 n 17 + 2216470992296632536917363907851228085 n 16 - 35055053450108515766724369240102508380 n 15 + 474765340355775887668999525667658093165 n 14 - 5505948745165611742492996257032455951620 n 13 + 54601582244131879572903625912186649574720 n 12 - 461731085834268987602819842439715893688960 n 11 + 3315301722030901898870394450033191086835680 n 10 - 20091293112878501608474129611322491309319040 n 9 + 101945247962764990794591264554291038091475456 n 8 - 428576713020953908488305829977437641639241728 n 7 + 1472204028835938606920345813317590004724231424 n 6 - 4055300081639317040798589632864236491857667072 n 5 + 8711871393546924130549215762641370157222932480 n 4 - 13844325994588279730625831252602064879630336000 n 3 + 13652624643111607813596790822642709481062400000 n 2 + 2024785044579943511662602090713770483384320000 n - 40265992003740034830633424588357628382412800000 n + 66461903547821233803296753634151665500160000000)/(134217728 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) (2 n - 29) 30 (2 n - 51) (2 n - 57) (2 n - 45)), 177 (1628420204246949 n 29 28 - 732789091911118785 n + 157551690286140737625 n 27 26 - 21544854222791394719665 n + 2104503405879235373131581 n 25 24 - 156340543956417489267647685 n + 9182870783449580345607370125 n 23 - 437699442556493934092181713925 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n - 33) (2 n - 65) (2 n - 103) (2 n - 9) (2 n - 61) (2 n - 7) (2 n - 49) (2 n - 93) (2 n - 109) (2 n - 47) (2 n - 19) (2 n - 3) / (-1 + 2 n) (2 n - 5)), | \ 416353841156489216133993275248320 (n - 56)! (n - 55)! (2 n - 117)! (n + 1) / + 619100059458779617034024783195328 (n - 57)! | \ 115 --- (n - 55)! (n + 1) (2 n - 116)! + (n - 56)! ((2 n - 115)! (n + 1) 114 - 1/619100059458779617034024783195328 binomial(2 n, n) (n - 59)! (n - 55)! \\ )||/((n - 59)! (n - 57)! (n - 56)! (n - 55)! binomial(2 n, n)), ( // 416353841156489216133993275248320 (n - 56)! (2 n - 117)! (n + 1) + 624530761734733824200989912872480 ((2 n - 116)! (n + 1) - 1/624530761734733824200989912872480 binomial(2 n, n) (n - 59)! (n - 56)! ) (n - 57)!)/((n - 57)! (n - 59)! (n - 56)! binomial(2 n, n)), ( (416353841156489216133993275248320 n + 416353841156489216133993275248320) (2 n - 117)! - (n - 59)! (n - 57)! binomial(2 n, n))/((n - 59)! (n - 57)! binomial(2 n, n))] and in Maple notation [59/134217728*(1221315153185219*n^29-513563021914384372*n^28+ 103013658568136896251*n^27-13119737739335749818828*n^26+ 1191345291099660067473375*n^25-82110721880012663164066740*n^24+ 4464886792151179367787056295*n^23-196560421552401656991774793260*n^22+ 7133717678184108813860323930665*n^21-216228899854546003536229071773820*n^20+ 5525536349084665447714064277487185*n^19-119845252279816556754280661140770180*n^ 18+2216470992296632536917363907851228085*n^17-\ 35055053450108515766724369240102508380*n^16+ 474765340355775887668999525667658093165*n^15-\ 5505948745165611742492996257032455951620*n^14+ 54601582244131879572903625912186649574720*n^13-\ 461731085834268987602819842439715893688960*n^12+ 3315301722030901898870394450033191086835680*n^11-\ 20091293112878501608474129611322491309319040*n^10+ 101945247962764990794591264554291038091475456*n^9-\ 428576713020953908488305829977437641639241728*n^8+ 1472204028835938606920345813317590004724231424*n^7-\ 4055300081639317040798589632864236491857667072*n^6+ 8711871393546924130549215762641370157222932480*n^5-\ 13844325994588279730625831252602064879630336000*n^4+ 13652624643111607813596790822642709481062400000*n^3+ 2024785044579943511662602090713770483384320000*n^2-\ 40265992003740034830633424588357628382412800000*n+ 66461903547821233803296753634151665500160000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2* n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 177/ 268435456*(1628420204246949*n^30-732789091911118785*n^29+157551690286140737625* n^28-21544854222791394719665*n^27+2104503405879235373131581*n^26-\ 156340543956417489267647685*n^25+9182870783449580345607370125*n^24-\ 437699442556493934092181713925*n^23+17243000151415961747978081735235*n^22-\ 568898095091434016712920381940675*n^21+15872385188149853206076826340804875*n^20 -377131432402926372827106758211088275*n^19+ 7669207888878700881561588764584424895*n^18-\ 133921262282520897910319987589976200375*n^17+ 2011852489724883700381858724517079587375*n^16-\ 26015365519177542786971564669174725749975*n^15+ 289369319333265630138855630261913109968860*n^14-\ 2763300151355709143920425693304864649371200*n^13+ 22581724208114680904489707602084832247268000*n^12-\ 157187765635702824900321072654160374521911200*n^11+ 926131414056118522339772965376700881657917056*n^10-\ 4580314964616478033129786927236806282796592640*n^9+ 18805401928836369658463542129911261130815456000*n^8-\ 63122811589991717713037771539195421140198536960*n^7+ 169125042219836404920226196034474493356704615424*n^6-\ 344536081468062405912693873021845669867804528640*n^5+ 458529331772481667312350091630419100392523776000*n^4-\ 76687148516696371159146752707979280636641280000*n^3-\ 1312027815926487933644017445641927473882071040000*n^2+ 2482943440189455527679372542053920448079462400000*n-\ 44307935698547489202197835756101110333440000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2*n-\ 19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-\ 27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2* n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45), 3/134217728*(48038396025280863*n^30-21617278211372859195*n^29+ 4647774863438067365250*n^28-635573199571160611535005*n^27+ 62082850473109259838569322*n^26-4612046046644636067296519145*n^25+ 270894688099963443183136452750*n^24-12912133553780677635844488483225*n^23+ 508668504277595435205151726183320*n^22-16782493786703597007898747291862475*n^21 +468235361506843755072373276964972250*n^20-\ 11125377145090572963942613828944470175*n^19+ 226241625847243006041230387006310609990*n^18-\ 3950676867322680348662964197710191277875*n^17+ 59349631138255486676453701787239778912250*n^16-\ 767452578696224283815104080783101006205075*n^15+ 8536370029772857018898936506473756274263945*n^14-\ 81516591525702572925571899523846986855537150*n^13+ 666140661426287469056675639869422274152673500*n^12-\ 4636579446618698161883915678951771299311581400*n^11+ 27311960380854170590133413897197512311871397072*n^10-\ 134973357927933187657622846622132120906629941280*n^9+ 552772961441983386333555140648416023002088072000*n^8-\ 1840084968976182163971031217263402141982946165120*n^7+ 4795645915235579510009525220723009313397586175488*n^6-\ 8877890838985503854573739977005302173249767802880*n^5+ 7551273904118695433042346907865816159707671552000*n^4+ 14108487531295463384092864646537360441436733440000*n^3-\ 51475583535521225674617450944962480101494292480000*n^2+ 36626322630739438240924978738173085527755980800000*n-\ 1307084103107150931464836154804982754836480000000)/(2*n-5)/(-1+2*n)/(2*n-3)/(2* n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-9)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2* n-27)/(2*n-43)/(2*n-13)/(2*n-59)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/( 2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-51)/(2*n-57)/(2*n-45) , 93/134217728*(3099251356467116*n^31-1489190276780218163*n^30+ 342393667668039507445*n^29-50150344057700263969260*n^28+ 5255989229470744523972059*n^27-419714386154016970214797452*n^26+ 26552393058459293724410544495*n^25-1366091711268107136124059604200*n^24+ 58225069406121583490824004908215*n^23-2083669869921120701556823052164170*n^22+ 63232348817282232869533571052367725*n^21-1639132552794586776071453345476857600* n^20+36488100658222334398229222003836596105*n^19-\ 700067274587895409282604626896165090540*n^18+ 11602909810092821832186445688652629887125*n^17-\ 166297370761808928507066743402568720437400*n^16+ 2060865963889999462176501515787642260350065*n^15-\ 22056009291840619676585615768467840320346395*n^14+ 203367154979904150114333501361710813834950650*n^13-\ 1609623189904044378426738208186896496185961300*n^12+ 10879716005504488785393909989673495951700365704*n^11-\ 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45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), 47027/68719476736*(3202029004962119543 *n^41-2688041998766271148579*n^40+1091171278140794233698730*n^39-\ 285313382418537450280979390*n^38+54004875804067439893779349251*n^37-\ 7884535997942332374794816080023*n^36+923768510816400617308083280727200*n^35-\ 89220945285035464890679528553584220*n^34+7241526453781990757859302073154068014* n^33-500983874526494136417921309419390873302*n^32+ 29861347305289649312819175966742563194460*n^31-\ 1546125866024499835319492023627074623027940*n^30+ 69974926653542613081222310602383596396802862*n^29-\ 2781239829142285801952842304045516332031744406*n^28+ 97410053591041627368985606506752959465765109160*n^27-\ 3013099507280443713339406462451660619599267618400*n^26+ 82410466389149580758799230447234027699067119427435*n^25-\ 1993284426578640689541038325258422631387110348970855*n^24+ 42593580166823333312788786316775452761000071914005450*n^23-\ 802238086830329368667084597736385382157511157725415150*n^22+ 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12779108198476119707*n^42-11251071010844893165329*n^41+ 4793228635609622658014894*n^40-1316253090451526209065440490*n^39+ 261847128938870294765222115839*n^38-40208381017192489442195712263133*n^37+ 4958745345273184081915194619398988*n^36-504549291778307876137137399395312580*n^ 35+43179312084161578401460099918233200806*n^34-\ 3152708666474449342344755364758313531282*n^33+ 198527064230538628023629959411633322202052*n^32-\ 10871190766010352246820700847057534266548620*n^31+ 520968199229303687675184529552667427019426278*n^30-\ 21953755030643009716341936968755023567296324466*n^29+ 816394743498049085061423617766303207488904653376*n^28-\ 26855268804918452738328176841226902676378013132160*n^27+ 782513561892277296007083433468449601879257064921215*n^26-\ 20204093204005293533321349268462592111766711808267005*n^25+ 461903111196295359809984453671904037833697134998326430*n^24-\ 9331509003310644558408483031047513064488012651855829050*n^23+ 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79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2* n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83), 47027/34359738368*( 3183731763788785742*n^42-2800540621297142978814*n^41+1191793731757739008268279* n^40-326839376436678357161474040*n^39+64914871895037628549915713884*n^38-\ 9948879381756055178663933243718*n^37+1224133289064426896193689534268463*n^36-\ 124215049813190621645889222659734860*n^35+ 10596158023843913810383290110058571336*n^34-\ 770758643680506424745522882167242568332*n^33+ 48321913183376767210116358729469901919822*n^32-\ 2632590993918488981772866906114520801430560*n^31+ 125415054787520345530735182851772924209317968*n^30-\ 5249037125218090180124996397049372921888477596*n^29+ 193660442813203676709753615780004695887113281846*n^28-\ 6312418251573090344987517842549510383377332292840*n^27+ 181981599822928046701360815935947396179454734570390*n^26-\ 4640109603930902755682877791769990230069300883287430*n^25+ 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4544391528672699740023272443499680039418531241084996382883840000000000*n+ 774638330387985045758629268484742333500243584757872040345600000000000)/(2*n-5)/ (-1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/( 2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59) /(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-\ 79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2* n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), 2022161/ 549755813888*(4629678538062153311*n^44-4450005929728642112698*n^43+ 2071004739872767283338919*n^42-621634455582337760798800642*n^41+ 135247595942855962642254809477*n^40-22725586754532703592939553454366*n^39+ 3068358242700334443059986761734473*n^38-341965065621770734885411942265092814*n^ 37+32069759227686033516526478460580615738*n^36-\ 2567042852368198527167097898847383024844*n^35+ 177285433289420885931947261895127649927782*n^34-\ 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2029557149342605035887129*n^42-607176938277092840751796702*n^41+ 131606342549404202077003922747*n^40-22019940528578239456368911060386*n^39+ 2958861542737183170311231978570743*n^38-327985975583632953854222145544102034*n^ 37+30572466314098392436672315476491271718*n^36-\ 2430511212428891691309334215973644086324*n^35+ 166566817934474339822553629785740936772762*n^34-\ 9920135591089516563488693787774809249358956*n^33+ 516570931828902522466803603359690026533791094*n^32-\ 23623952364013064440713905745625344954489890052*n^31+ 951648397359959215155236021101032068952626271726*n^30-\ 33820383568570447019943306121904067535309486949988*n^29+ 1060411073288236425548102855722297018797035863423005*n^28-\ 29280156273668372728178144787261998568469362287068110*n^27+ 708857003874420974617469595607487977949168193058273405*n^26-\ 14917349921761902282111512965491996185102095891059050390*n^25+ 268403011330524714089772495239827849660461004311872995999*n^24-\ 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65)/(2*n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2* n-59)/(2*n-87)/(2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/( 2*n-39)/(2*n-79)/(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57) /(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81)/(2*n-83)/(2*n-85), 65231 /549755813888*(277205829966107769346*n^45-277249956913340887562055*n^44+ 134272713301033823788568340*n^43-41943736094979199885424658495*n^42+ 9497500433532985436056380628896*n^41-1660950995355107157796539911002845*n^40+ 233407050975871046355698738147918980*n^39-\ 27073662819664585955559395915998157265*n^38+ 2642356390791458893072195846239196125426*n^37-\ 220093022697591940243409381514821147074410*n^36+ 15813667879706700601374851580236853621036720*n^35-\ 988074023212911266330609023968708697838813910*n^34+ 54015508731144051719473986355072420336581489616*n^33-\ 2594956831453494653423632935301145546390252129850*n^32+ 109869608925414695129043490299031600176407059557960*n^31-\ 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6800433688666875040842925995081300220126842478296135538162923717490193543223946\ 4448*n^13-181677418768673676643342452227209770472291805753714685415349271750505\ 4646535935916485705901118324736*n^12+832201059098094073205258647855540254710051\ 2952809449306450777085210396609078457137348935248930144256*n^11-332847256565461\ 1510066618773918293005057564709842478775307502477348183029717893703458474908311\ 1456768*n^10+115086443965953587490394110225083396101600327610727204648515596528\ 340366889104934320891827325082009600*n^9-33985224140497423066799122809403817720\ 3332208762498886167363476679609921583874917257753092002152448000*n^8+8442896485\ 0698584570187292183316698652782152521325619627704070178814129553399151021796990\ 2813184000000*n^7-1731016230882650101019405754973251119164077434493889339054800\ 457804929056142338324594439189524643840000*n^6+28563547973303075676287839258776\ 77719671334489564578351503822053185991532467370280076854857564160000000*n^5-366\ 5679221961834219768741757723875604523029039478115345868819466897838600222891149\ 028247071621120000000*n^4+34825422226553090524518956506124974832482888904650978\ 79655573288232767308250458116005659305574400000000*n^3-226727153961731634619607\ 4176476001634443010169518709202763514535792431499653487231459482455244800000000 *n^2+88089800354766332723533628704264759965538036141381175567440192185489921437\ 6361048305974640640000000000*n-147516663652370928134531220328039918964863478800\ 541602018420376339773652981391585397702656000000000000)/(2*n-73)/(2*n-101)/(2*n -29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-105)/( 2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13) /(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-107)/(2*n-11)/(2*n-33)/(2*n-\ 65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-109)/(2*n-47)/(2* n-19)/(2*n-3)/(-1+2*n)/(2*n-5), -5/4503599627370496*( 59389004782989147411401256152717*n^56-93559088981043142305351746706042076*n^55+ 71926374572850069589428641114571102270*n^54-\ 35963495113300589907475275541568332522320*n^53+ 13151130671748271743413100394474958978169303*n^52-\ 3749871599388491670060820161059651782660574796*n^51+ 868037579283042775080742642996375024408376786976*n^50-\ 167704027808761999435092424452455691794745540313200*n^49+ 27590323722022116751192180853160712455064648984150485*n^48-\ 3924452588639484988815740518104797601696604950988808460*n^47+ 488378584867365877681260960793250018198273211354053533290*n^46-\ 53677805389794341198162167690377740685825746685678725967600*n^45+ 5250816920264349616363000611385753168781062380158540773150695*n^44-\ 460041079887617446587283260630902818773214313711406297094741500*n^43+ 36289891873019917846425208227697164157878256996728746767158102420*n^42-\ 2588844836384942091088750504352836731071237419422615908078003470800*n^41+ 167636237659532537809853307800157005449659355619795118444449594493135*n^40-\ 9883961152988277386939493411140537457353662789821120193127821642640340*n^39+ 532039235236900399797299415633527578674342237541933742635462874095339730*n^38-\ 26204319317684756788664025472909511565467572426571187705998415054978900400*n^37 +1183118494822077272954786008782123289879961216457025565550049277735037664221*n ^36-\ 49043373154527431182662613892181443632122416373707713688930552190183303588708*n ^35+186885425740617267988963005446085542865599938036990527611476145666584647887\ 5320*n^34-655310373526572898580573092639483565381450902653335404077637297777078\ 70423452560*n^33+21160447734449665734977894922999496819180367892393683343023635\ 63184846242722035199*n^32-62956654547775433699110901842601002565114647006717442\ 112736239064611207919670962948*n^31+1726387866316459967278496778907437189903970\ 381812850876746967638242114134691524996998*n^30-4363829224284078529048632711007\ 6945538493664939426938607591694978883723609411128355600*n^29+101670819681431528\ 0294656269707147203783592287078651744763108436667583541882199124644965*n^28-218\ 2757011818606498427750643125445501818481928807871074090833871556218102372210224\ 1459540*n^27+431609031333234703515716668887350026492610181382915936692569776094\ 523200154933011507736660*n^26-7855248215856615769776937237974346622410487764864\ 950845872648858624795540240391675031908400*n^25+1314714297187288374316691532348\ 60580759868049054969587297595060251783341331615344747988425680*n^24-20212871524\ 7923873611147454130598452823239902200725929628068907812389059210354792675089011\ 3600*n^23+285083562488041334609797096722732527969735733408609773084781225618674\ 10146814135003166098722880*n^22-36827973962955615065350199921560181136186835269\ 5790837511182391723262782314821268147658818835200*n^21+434951370714510614414170\ 1618954305583873052114182281615173899016176433089426693472229222095031040*n^20-\ 4686256273156326469206944830161444942315572589537330637369133569231803046837488\ 5862052658052828160*n^19+459465819338800725303652924447462300703180517565114946\ 078121951962256736738545637736123320411192320*n^18-4087677842739998488231517008\ 674550431124174620242812155548419927188876069159977408991202701484953600*n^17+ 3288985577572717499310238079421613824662733993721903620566857510629443197538075\ 1213325124522805030912*n^16-238427188764372132773420170965968912972081970098807\ 675641098803595785663596156734942038812244815036416*n^15+1550413814665671296287\ 6276020255156757379966209218342947787783889612654253483086882646956632799791513\ 60*n^14-89974817170707982361643619514972809911354336987436350746140407005393017\ 97111348003949943757221628805120*n^13+46322227092513134705777178492097392935892\ 975217837661685293539943572235474243338374383453604862182555648*n^12-2100934460\ 1975996940340007798864996388985348128057059364740001470585585247963793142452203\ 6812776872083456*n^11+832495475124144259785416197434613727560520742134159181045\ 227899771041947717029071266984737646091253579776*n^10-2853412233695354967945424\ 1302364009261872759152110648813539306455410294999516192914795740044810520625152\ 00*n^9+835756870602789967686953839702102290236646806426853101922991774230300879\ 9172060144184747001842547818496000*n^8-2060504492411574556409269405220237209154\ 3793221538818668861396636815231430801376881711158711819561861120000*n^7+4194856\ 0286578033335353712818563375718141355488887650407543019826981764887688873209083\ 744317179040890880000*n^6-68770898415984831664032718659593090501635351029487634\ 677500999597159444216431632557892261774192803840000000*n^5+87734698619801016252\ 7744523461947680143897773042729312912856252430188843089567627546168744294639206\ 40000000*n^4-829079657499086752387567079939105842565410774785620837504670347031\ 39321856854480388455657125393203200000000*n^3+537237323147662211052350112371820\ 09433091320957538748217617804298574924955920893482589080165915033600000000*n^2-\ 2079094224512522083701016224795390910891637041534470533799886119843265530796994\ 2982912064859668480000000000*n+347136212906759268086178867675943537308116738313\ 4344978697468296027553601958106787578738900992000000000000)/(2*n-99)/(2*n-95)/( 2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45) /(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-111)/(2*n-101)/(2*n-29)/(2*n-53)/(2*n -35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-105)/(2*n-37)/(2*n-15)/( 2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75) /(2*n-27)/(2*n-21)/(2*n-41)/(2*n-107)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n -9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93)/(2*n-109)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+ 2*n)/(2*n-5), -5/4503599627370496/(2*n-99)/(2*n-95)/(2*n-85)*( 61701859181487344047774679743097*n^56-97009867743602451686820894702889036*n^55+ 74437232436403696839841993498523534070*n^54-\ 37150900730620261041373505699796354049920*n^53+ 13561514082486320123739496297807775727253323*n^52-\ 3860382202486368359150332478119648108827378636*n^51+ 892177875527418595597740181818358925384451074976*n^50-\ 172101500998747755603849447224249897974022911957200*n^49+ 28271867528176793482473182225878625443292357125304385*n^48-\ 4015694935600210294104048399528246349794262712543167260*n^47+ 499055303722315399049488354555957760088602162919062854290*n^46-\ 54780111784275591348210690649021802148723513261101437209600*n^45+ 5351998417742865664690076748204807959958021549998885432671995*n^44-\ 468350179675200380978196536732140640780877465218453171231291100*n^43+ 36903536863814045708548483303884159766697382225096080424772656420*n^42-\ 2629778010221586083005550720331563325787135937258881155388270558800*n^41+ 170111433403642854503350560590831307312441659960778364405885653562035*n^40-\ 10020057872494834139076322185887626828342303617595127782877035856119140*n^39+ 538861111512980961584023575145347526120099126750142132823920412772672730*n^38-\ 26516717267285934454528320040212908246419336819949100870695472219003846400*n^37 +1196211219224916468091505464919191625748829093748163593614379234715954283161*n ^36-\ 49546290045321814604083721559359122019100856687764551871745760711420812657188*n ^35+188658003296603527252539509269908718133005516816648433044242455288047366022\ 1720*n^34-661048078619964654447028094815642990246476820383340120703506018283130\ 76591797360*n^33+21331121697728632587275586086790557679205555586416633636788114\ 77695965322591961859*n^32-63423380192512942602033391624975957983450093394293441\ 254760542386504034332467847668*n^31+1738123107758790346030604691239818394013460\ 389110713168015587963922740529856164387998*n^30-4390958167109523518412778922485\ 8792533418146359743015925908653465244589803037745961600*n^29+102247272072772329\ 4900917388142710417531996854791592796969075842370330271173307892364065*n^28-219\ 4009979941308713244300254037129824411872781795627997147466041303142565601831127\ 0066740*n^27+433625673963225378858170042418242324856711084800065545078039297753\ 765630999822362327730660*n^26-7888394283218100301916884650516272099543743017354\ 702122670810069427115832740855957283596400*n^25+1319704880743475334748408731170\ 38807775403113549516074944945154979837302445131210792767476880*n^24-20281601483\ 0271672862153221486011881241634582219163514176636192832508624147231543022792778\ 4000*n^23+285947831834848203304560495208130729694518107163347342664283369891647\ 24044641797331659787458880*n^22-36927000929546218189531479725130703703275177938\ 9835517807918202077408674439178392262501345427200*n^21+435982694703082308336572\ 3524919565840908628219359993787192577436911708170624131394583811386576640*n^20-\ 4695991154095044435681633349771308105915450369416537966842299172433431657337822\ 8070688504083983360*n^19+460295877651419378712877565558506225898029126227191598\ 551087578395206620689996012883572344857144320*n^18-4094046267176365226724525193\ 140495946223376195779562730818194907945886981681824724529748816378777600*n^17+ 3293361892153024088313926540918536451061523659083965630410832464737716342882573\ 5301730232898375790592*n^16-238695095213950535017340930340325028759141564744127\ 428395651970900394200806476295307607238546760318976*n^15+1551865405759909152301\ 8257370771312154450243093992126532148923132794358540740557949257780638684547481\ 60*n^14-90043887020956907458451641959895393897508786229980626952217391256056945\ 56691013470134139666541290782720*n^13+46350810708481160127088787237017219378992\ 768012427344437963602366507839653868984335419732094809432915968*n^12-2101950737\ 5788616371601222505994073519918207433259052523599085165158108112707738276120777\ 1910605972176896*n^11+832800953405836312672863466821590116301366311357382207658\ 300090940305721164437491699066647330311714635776*n^10-2854171329946627781529921\ 2063328346089085166951018548717720222477240595668323020573653064396129647460352\ 00*n^9+835907599360554055670165797226066210525140955164791440165621755751011316\ 6866515492614851427557070340096000*n^8-2060729715321043649739642505772185171131\ 9036379332140452729130931351256271388990808635468537974300344320000*n^7+4195075\ 6681160698670417411927823150870875226721013803645788037267179293994279039867940\ 875132264649850880000*n^6-68771521444413480490772045299949975759840040772662052\ 977551944844333615880373049729023387011530096640000000*n^5+87732862873124480897\ 7934313728305979611991856791796040003511662383857631588980652832413153541645926\ 40000000*n^4-829048393233716054453872725518348800443111320375149900082957630790\ 52630818329175178342840231211827200000000*n^3+537215288211900916455697248526503\ 58911691886530828849471564006161113460897237073044409951152871833600000000*n^2-\ 2079032550879130696985298963155253764258849471492486126576443390420389015258459\ 7051990063164948480000000000*n+347136212906759268086178867675943537308116738313\ 4344978697468296027553601958106787578738900992000000000000)/(2*n-97)/(2*n-83)/( 2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51) /(2*n-73)/(2*n-111)/(2*n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n -39)/(2*n-23)/(2*n-17)/(2*n-105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/( 2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41) /(2*n-107)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-\ 49)/(2*n-93)/(2*n-109)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), ( 416353841156489216133993275248320*(n-56)!*(n-55)!*(2*n-117)!*(n+1)+ 619100059458779617034024783195328*(n-57)!*(115/114*(n-55)!*(n+1)*(2*n-116)!+(n-\ 56)!*((2*n-115)!*(n+1)-1/619100059458779617034024783195328*binomial(2*n,n)*(n-\ 59)!*(n-55)!)))/(n-59)!/(n-57)!/(n-56)!/(n-55)!/binomial(2*n,n), ( 416353841156489216133993275248320*(n-56)!*(2*n-117)!*(n+1)+ 624530761734733824200989912872480*((2*n-116)!*(n+1)-1/ 624530761734733824200989912872480*binomial(2*n,n)*(n-59)!*(n-56)!)*(n-57)!)/(n-\ 57)!/(n-59)!/(n-56)!/binomial(2*n,n), ((416353841156489216133993275248320*n+ 416353841156489216133993275248320)*(2*n-117)!-(n-59)!*(n-57)!*binomial(2*n,n))/ (n-59)!/(n-57)!/binomial(2*n,n)] The limits, as n goes to infinity are 72057594037927921 288230376151709973 144115188075842589 72057594037860447 [-----------------, ------------------, ------------------, -----------------, 72057594037927936 288230376151711744 144115188075855872 72057594037927936 576460752299243401 4611686018217034703 4611686017307811647 ------------------, -------------------, -------------------, 576460752303423488 4611686018427387904 4611686018427387904 4611686013242766767 4611685997107223057 9223371878714293489 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 9223372036854775808 4611685750706897457 4611685182698146875 9223367187252094065 -------------------, -------------------, -------------------, 4611686018427387904 4611686018427387904 9223372036854775808 147573741921431502845 147573414290952409295 147572651719141649525 ---------------------, ---------------------, ---------------------, 147573952589676412928 147573952589676412928 147573952589676412928 1180567733178122987377 4722157393110133238833 2360964423492744351179 ----------------------, ----------------------, ----------------------, 1180591620717411303424 4722366482869645213696 2361183241434822606848 1180371948265184022027 9441339383634767965791 75507325164958686394443 ----------------------, ----------------------, -----------------------, 1180591620717411303424 9444732965739290427392 75557863725914323419136 75466978943400648106343 75399673690504057918643 150581818016353595748661 -----------------------, -----------------------, ------------------------, 75557863725914323419136 75557863725914323419136 151115727451828646838272 600963121249736481461089 74860676827847613544517 37237958170687179632911 ------------------------, -----------------------, -----------------------, 604462909807314587353088 75557863725914323419136 37778931862957161709568 295683380826671255446403 9361955382206302001525071 ------------------------, -------------------------, 302231454903657293676544 9671406556917033397649408 9224870717454259691372881 9041206747259587951104463 -------------------------, -------------------------, 9671406556917033397649408 9671406556917033397649408 140811864442592611279289893 543543883413043944705448177 ---------------------------, ---------------------------, 154742504910672534362390528 618970019642690137449562112 259430378247412236256063901 122147505156471796433596363 ---------------------------, ---------------------------, 309485009821345068724781056 154742504910672534362390528 904463006864393911453794281 6551115758205128041939808563 ----------------------------, ----------------------------, 1237940039285380274899124224 9903520314283042199192993792 5761567699756954774562154043 4868955092886642339414922243 ----------------------------, ----------------------------, 9903520314283042199192993792 9903520314283042199192993792 1939821393469356361896703499 11218682052147020224350359293 ----------------------------, -----------------------------, 4951760157141521099596496896 39614081257132168796771975168 103728120033416863816412287 464331265829650609305239317 ---------------------------, ----------------------------, 618970019642690137449562112 9903520314283042199192993792 -3032614887747385247240958857 -126869542753259877744678486737 -----------------------------, -------------------------------, 39614081257132168796771975168 633825300114114700748351602688 -203627583731996049514597973747 -277119325094615788443244291097 -------------------------------, -------------------------------, 633825300114114700748351602688 633825300114114700748351602688 -2766258257351891792102374880251 -13064199398647747673427764793379 --------------------------------, ---------------------------------, 5070602400912917605986812821504 20282409603651670423947251286016 -7415367749733626314385588689757 -4085237668963530882942660956425 --------------------------------, --------------------------------, 10141204801825835211973625643008 5070602400912917605986812821504 -35166657090635363720185708941065 -296945023914945737057006280763585 ---------------------------------, ----------------------------------, 40564819207303340847894502572032 324518553658426726783156020576256 -308509295907436720238873398715485 -316429070059869939024030879494665 ----------------------------------, ----------------------------------, 324518553658426726783156020576256 324518553658426726783156020576256 -642531578548783309564218396226757 -2589642900499343670263154519684293 ----------------------------------, -----------------------------------] 649037107316853453566312041152512 2596148429267413814265248164610048 and in Maple notation [72057594037927921/72057594037927936, 288230376151709973/288230376151711744, 144115188075842589/144115188075855872, 72057594037860447/72057594037927936, 576460752299243401/576460752303423488, 4611686018217034703/4611686018427387904, 4611686017307811647/4611686018427387904, 4611686013242766767/ 4611686018427387904, 4611685997107223057/4611686018427387904, 9223371878714293489/9223372036854775808, 4611685750706897457/ 4611686018427387904, 4611685182698146875/4611686018427387904, 9223367187252094065/9223372036854775808, 147573741921431502845/ 147573952589676412928, 147573414290952409295/147573952589676412928, 147572651719141649525/147573952589676412928, 1180567733178122987377/ 1180591620717411303424, 4722157393110133238833/4722366482869645213696, 2360964423492744351179/2361183241434822606848, 1180371948265184022027/ 1180591620717411303424, 9441339383634767965791/9444732965739290427392, 75507325164958686394443/75557863725914323419136, 75466978943400648106343/ 75557863725914323419136, 75399673690504057918643/75557863725914323419136, 150581818016353595748661/151115727451828646838272, 600963121249736481461089/ 604462909807314587353088, 74860676827847613544517/75557863725914323419136, 37237958170687179632911/37778931862957161709568, 295683380826671255446403/ 302231454903657293676544, 9361955382206302001525071/9671406556917033397649408, 9224870717454259691372881/9671406556917033397649408, 9041206747259587951104463/ 9671406556917033397649408, 140811864442592611279289893/ 154742504910672534362390528, 543543883413043944705448177/ 618970019642690137449562112, 259430378247412236256063901/ 309485009821345068724781056, 122147505156471796433596363/ 154742504910672534362390528, 904463006864393911453794281/ 1237940039285380274899124224, 6551115758205128041939808563/ 9903520314283042199192993792, 5761567699756954774562154043/ 9903520314283042199192993792, 4868955092886642339414922243/ 9903520314283042199192993792, 1939821393469356361896703499/ 4951760157141521099596496896, 11218682052147020224350359293/ 39614081257132168796771975168, 103728120033416863816412287/ 618970019642690137449562112, 464331265829650609305239317/ 9903520314283042199192993792, -3032614887747385247240958857/ 39614081257132168796771975168, -126869542753259877744678486737/ 633825300114114700748351602688, -203627583731996049514597973747/ 633825300114114700748351602688, -277119325094615788443244291097/ 633825300114114700748351602688, -2766258257351891792102374880251/ 5070602400912917605986812821504, -13064199398647747673427764793379/ 20282409603651670423947251286016, -7415367749733626314385588689757/ 10141204801825835211973625643008, -4085237668963530882942660956425/ 5070602400912917605986812821504, -35166657090635363720185708941065/ 40564819207303340847894502572032, -296945023914945737057006280763585/ 324518553658426726783156020576256, -308509295907436720238873398715485/ 324518553658426726783156020576256, -316429070059869939024030879494665/ 324518553658426726783156020576256, -642531578548783309564218396226757/ 649037107316853453566312041152512, -2589642900499343670263154519684293/ 2596148429267413814265248164610048] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999998, .9999999989, .9999999954, .9999999829, .9999999419, .9999998188, .\ 9999994742, .9999985725, .9999963523, .9999911850, .9999797665, .9999557235, .9\ 999073270, .9998139302, .9996406905, .9993311277, .9987971499, .9979063723, .99\ 64668837, .9942100855, .9907728082, .9856805456, .9783342403, .9680034985, .953\ 8292763, .9348388669, .9099753460, .8781425047, .8382647625, .7893597511, .7306\ 193985, .6614936457, .5817696654, .4916388252, .3917438107, .2831993497, .16758\ 18161, .4688547618e-1, -.7655396242e-1, -.2001648447, -.3212676801, -.437217203\ 3, -.5455482482, -.6441147602, -.7312117145, -.8056710714, -.8669250296, -.9150\ 325014, -.9506676658, -.9750723541, -.9899766459, -.9974941615] The cut off is at j=, 45 --------------------------------------------- The rational functions describing the sorting probabilities of the cell, [1, 60], vs. those in the, 2, -th row from j=1 to j=, 59, are as follws 30 29 28 [59 (9770521225481753 n - 4396734551466787935 n + 945310141716890665525 n 27 26 - 129269125336766615337375 n + 12627020435280605518724217 n 25 24 - 938043263739640325126429115 n + 55097224700895784507212269625 n 23 - 2626196655367373103666061792875 n 22 + 103458000911897314855304769417495 n 21 - 3413388570893666218845064655106525 n 20 + 95234311158857957333155706488065375 n 19 - 2262788596660529969741870043257044125 n 18 + 46015247478877656696245347545557880915 n 17 - 803527581922317336180506162219121148425 n 16 + 12071115343946342936437012777021025329875 n 15 - 156092210581326326648931280012081761101625 n 14 + 1736216572946838470345687198825871320858820 n 13 - 16579822461602293076864112046086269355230400 n 12 + 135490960460452741279681921050582091434964000 n 11 - 943141809190046790511538535566920799033916000 n 10 + 5557112652720838291679348616989991960003228032 n 9 - 27487792265374674283303282721377209371424944640 n 8 + 112923276588643224591496419788142700648788729600 n 7 - 379902304594920974133406944519152936605100448000 n 6 + 1026952610020361728209955770892439804598192288768 n 5 - 2168472459730770089502526024001992046088806952960 n 4 + 3387263068785534411265132053098896476839294976000 n 3 - 3253160760319726131587277357093315714867855360000 n 2 - 678042901567887970596292124878478895928442880000 n + 9950621215405364382255510972953235879729561600000 n - 15950856851477096112791220872196399720038400000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 30 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 59 (9770521225481723 n 29 28 - 4396734551466748815 n + 945310141716866070625 n 27 26 - 129269125336756687845675 n + 12627020435277715282079397 n 25 24 - 938043263738993917122370935 n + 55097224700780283315835666125 n 23 - 2626196655350443558376773596375 n 22 + 103458000909823317409694467148295 n 21 - 3413388570678384496664416354629225 n 20 + 95234311139731237934754944712118875 n 19 - 2262788595195072307379882903921144625 n 18 + 46015247381515564376842069300130619015 n 17 - 803527576291698623529930279478061459325 n 16 + 12071115059813362430254739228407839411375 n 15 - 156092198056254823236921016625276799436125 n 14 + 1736216090670179598869606094457462382408370 n 13 - 16579806262094190630681014920731970494830100 n 12 + 135490487022113317831329619187190258586357000 n 11 - 943129819355511933069825074271249683967150000 n 10 + 5556851055100229209960672098094697666768815712 n 9 - 27482913882633976369443324143991740605101447360 n 8 + 112846351190272179484754020163214697875117072000 n 7 - 378891521981554330899725457608104897238755507200 n 6 + 1016108448619015583452144176643997754811281727488 n 5 - 2076240513821162766141782390953103107542619914240 n 4 + 2793108210867658521436710941327999144824418304000 n 3 - 575005406275883534556899420468386204663480320000 n 2 - 7782547506326016502691340208598753477524193280000 n + 15154646668188308603448982699708909128410726400000 n - 265847614191284935213187014536606662000640000000)/(536870912 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 5487 (420237472063711 n 30 29 - 201924105326596330 n + 46426260023064057620 n 28 27 - 6800046651998914203280 n + 712676505720696230372764 n 26 25 - 56910425247545595780615120 n + 3600324483574921192481342640 n 24 23 - 185232774558050429506960770600 n + 7894924682470437287289499177890 n 22 - 282531509473306851176216798843700 n 21 + 8573877946815580356341007040053000 n 20 - 222255271542089750182234335710281800 n 19 + 4947539703212290701507892662439511580 n 18 - 94924410254762113893894218855523602400 n 17 + 1573277503144794959969057929319684509200 n 16 - 22548861193124012812658319584139043366200 n 15 + 279441764379535720885410816686628372654615 n 14 - 2990716482571443362095548846397990715034450 n 13 + 27577101021324693823626110005735373022313700 n 12 - 218297330038101796565109068855062403565955400 n 11 + 1476062544538474449702688767626653031756398384 n 10 - 8468600239610975434828312563767278883641495520 n 9 + 40857686710951482529213528261683834212566374080 n 8 - 163690520986070646860854558553653906900542990720 n 7 + 533868337909601932269062281654999881146291821056 n 6 - 1363780455659512979335736948590652194704731412480 n 5 + 2467080043811058171163029360976702469767240949760 n 4 - 1990723971048451132063961077357413229772562432000 n 3 - 4085037476561475848430411830185922094292992000000 n 2 + 14160789426275571956829904864540248315435417600000 n - 9946640565775470862979683594436855300987289600000 n + 348746332595018947268911997564150674882560000000)/(1073741824 (2 n - 5) (-1 + 2 n) (2 n - 3) (2 n - 19) (2 n - 47) (2 n - 49) (2 n - 7) (2 n - 61) (2 n - 9) (2 n - 33) (2 n - 11) (2 n - 41) (2 n - 21) (2 n - 27) (2 n - 43) (2 n - 13) (2 n - 59) (2 n - 31) (2 n - 25) (2 n - 15) (2 n - 37) (2 n - 17) (2 n - 23) (2 n - 39) (2 n - 55) (2 n - 35) (2 n - 53) 31 (2 n - 29) (2 n - 51) (2 n - 57) (2 n - 45)), 31 (18595508138810809 n 30 29 - 8935141660691182499 n + 2054362006014016624175 n 28 27 - 300902064348368062663345 n + 31535935377408597003175671 n 26 25 - 2518286317044728133805374501 n + 159314358370592386941529762095 n 24 - 8196550270275090179101668300525 n 23 + 349350416735143505327197598853885 n 22 - 12502019247703676078592355807362135 n 21 + 379394095170250814900520474652295025 n 20 - 9834795473184254742888529845899394075 n 19 + 218928613255520394348723651373357748445 n 18 - 4200404126373860786071424585510788297095 n 17 + 69617480203167977809441860626229861272325 n 16 - 997785048684970166418576852259809541292175 n 15 + 12365223316386914812937271427701155807517510 n 14 - 132336849404523362630879580990841630736428410 n 13 + 1220222580139994285082513358661258192827904300 n 12 - 9658154336976851621373942199211132470290628600 n 11 + 65285716063020875748941238125544989347719590896 n 10 - 374227492453758429250358564204229323843660845856 n 9 + 1800790192188224059003438715113181333964654388800 n 8 - 7160683391692081097184977687449607338798515857280 n 7 + 22864489945366852011698322091173819141038680022784 n 6 - 55011230693820345755271562136572454465379894549504 n 5 + 82334756619099382743500980234218174448073059553280 n 4 - 3382309505921143323092959535159273815123236864000 n 3 - 294880428029796672408811310959155065746556190720000 n 2 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n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-55)/(2*n-35)/(2*n-53)/( 2*n-29)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-45)/(2*n-71)/(2*n-69), 10199/1073741824 *(7234720157181719*n^36-4688095232889599958*n^35+1462694621376356706045*n^34-\ 292655205185237898216390*n^33+42194201086482099272950497*n^32-\ 4670219777298121216607189544*n^31+412829104759750769472296811820*n^30-\ 29935083381883105805763662919960*n^29+1814994294357958629111421727684094*n^28-\ 93325381476459734090207074414918068*n^27+4113401880495875359004756581129594110* n^26-156687450524780157866209820563747653300*n^25+ 5190627308086511867902733604678681754410*n^24-\ 150249412767788501040326856679125504201720*n^23+ 3813375617286759856358724079122285742688700*n^22-\ 85057613688348421184596086478768996127374200*n^21+ 1669408570662264607353394712764524197430276315*n^20-\ 28835758445704993806068411478219503891169573030*n^19+ 437919323679754338184926276467932218710218406125*n^18-\ 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194497577114356054153251334604488357493600058193173634690213478400000*n^3+ 139330955064274247308477817899108180352892388418002102278684672000000*n^2-\ 56947690366015373105933601176319938463568654187050784155238400000000*n+ 9442791837472248608822412060905434540187235760336923525120000000000)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2* n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/( 2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-\ 45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), 47027/68719476736*(3208610611491982388 *n^41-2695294929162180003769*n^40+1095013949113254655751980*n^39-\ 286616946596573186128515890*n^38+54323034042171528910685356116*n^37-\ 7944054103570181580704086736253*n^36+932646360677488070121691948620700*n^35-\ 90305508336958395654668281833487220*n^34+7352142070845085523811994134989470424* n^33-510537581047506125427500975700506717122*n^32+ 30567700293747431452755023550779478783960*n^31-\ 1591207490741018575264007549060702894954940*n^30+ 72474913410383782125052995509275325484846792*n^29-\ 2902312713408750645875098457072833509165541266*n^28+ 102550962358802813467552261455789683049406139160*n^27-\ 3205063221155871483270597025471881010621521974400*n^26+ 88727873999994712243207763038621720367767249163460*n^25-\ 2176790034037763085834044258378544097648398357810405*n^24+ 47302560066208164025230678820378962487034260364856700*n^23-\ 909018915497338334944244053254633631623743191026507650*n^22+ 15403788284120292095278365486657213384423276048817151972*n^21-\ 229115824940622333250791492820263215236194570707000733361*n^20+ 2970174183518348264078088357945915137585056172284474867020*n^19-\ 33186369372182264389267869133676419453079662419748906284060*n^18+ 313608239647974138991795479892766305064551977781374509693744*n^17-\ 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193092061357098162460214885857197180264402076645834819767789158400000*n^3+ 138016579834040339750205622894202934631021783620096136820293632000000*n^2-\ 56510385842843898088015145598005279670304383650850727683686400000000*n+ 9442791837472248608822412060905434540187235760336923525120000000000)/(2*n-5)/(-\ 1+2*n)/(2*n-3)/(2*n-19)/(2*n-47)/(2*n-49)/(2*n-7)/(2*n-61)/(2*n-9)/(2*n-65)/(2* n-33)/(2*n-11)/(2*n-41)/(2*n-21)/(2*n-27)/(2*n-75)/(2*n-43)/(2*n-13)/(2*n-59)/( 2*n-67)/(2*n-31)/(2*n-25)/(2*n-15)/(2*n-37)/(2*n-17)/(2*n-23)/(2*n-39)/(2*n-79) /(2*n-55)/(2*n-35)/(2*n-53)/(2*n-29)/(2*n-73)/(2*n-51)/(2*n-57)/(2*n-63)/(2*n-\ 45)/(2*n-71)/(2*n-69)/(2*n-77)/(2*n-81), 47027/274877906944*( 25641827176201978519*n^42-22596204148345742823783*n^41+ 9637392623495069379839368*n^40-2650189045509274553882107230*n^39+ 528116928006267495193406298463*n^38-81265823537005550583697029090891*n^37+ 10047662573422334782366908101556086*n^36-1025470527421937388100004471235756660* n^35+88080745448666175525841705459973727302*n^34-\ 6459105034419930789833459366933005730814*n^33+ 408818879045717361798910648772021642148344*n^32-\ 22521471013639570090045947131678095361116740*n^31+ 1086867672346945794663561944409399910128895926*n^30-\ 46176043203068146249428841853022589450610089182*n^29+ 1733473771859810317734367288819721281629564740772*n^28-\ 57651285133101339632878231608708853740607038478320*n^27+ 1701370551535781620760066460659584287096984837766155*n^26-\ 44585240185804521296605659598912560904681464746884635*n^25+ 1037245327679035664757861238274202226903136038158770960*n^24-\ 21395424179177993002873254328898257158442740122639074350*n^23+ 390330325380607311377991667993399748960419529675264155211*n^22-\ 6272643649618397587028506034538522353395675539419216490327*n^21+ 88230108236190384903281278575932035839530739587682424193942*n^20-\ 1075378803158052025129036709612250997819372258612721346796620*n^19+ 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n^40-528136586096040711945728790*n^39+105141024027353168474706811361*n^38-\ 16159252563655404775274523521883*n^37+1994935126392926887795935867742600*n^36-\ 203233267054403211968745517933539084*n^35+ 17417817518471740466752426820824394074*n^34-\ 1273892239012952568343967618800558213470*n^33+ 80374030634093980271407482206939365288036*n^32-\ 4411133352708407696900725234290791578407732*n^31+ 211936827544669414070587318146819276760449402*n^30-\ 8957546849043323196883476960493394075517640254*n^29+ 334233696337232418707881029071797813457341838856*n^28-\ 11037277658684500920810882045306167866512922910304*n^27+ 323040640507721877872225808629691076229866198744665*n^26-\ 8383804121468042884972328126023055912233038769843515*n^25+ 192828538525346355447349283538007958622606331701583270*n^24-\ 3923686165016521531835130534397834463281089816287205510*n^23+ 70406828551025107327051387972522954408709741107723149157*n^22-\ 1108261263013478215255398212474178562192103743363916605287*n^21+ 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10110805/549755813888*( 475022215132062071*n^43-438281778547058679316*n^42+195817214190506290391639*n^ 41-56436273379256508564292396*n^40+11792717650206266262486102917*n^39-\ 1903710891303335343789048120268*n^38+247040771820673336161449541583225*n^37-\ 26474826906534900074683988882164964*n^36+2388846507999303792854346614519040658* n^35-184104423388584502948943030742267646616*n^34+ 12251597544596825456861756347291851619558*n^33-\ 709927048877998918691026130693437509265784*n^32+ 36052563662060675731561152277204650709889994*n^31-\ 1612536728827581674180489903617552028772632280*n^30+ 63758538425238559925674600590024307879606303218*n^29-\ 2234367287767063314622555269766408799421869171976*n^28+ 69512031051232553138943083469704556085071021875595*n^27-\ 1921049805830917626632298785868325079143815575723460*n^26+ 47145267291331611877052220585208877628246947731240435*n^25-\ 1025909218538799050419218585263352563259932305204511900*n^24+ 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15603179225289874583019974310840182127984640000000*n^4-367320388790602813004764\ 70744156626643487179704686487716392977674864012526878720000000*n^3+246761254885\ 98898249313516567012613983742364999369212737310375146654844479078400000000*n^2-\ 9822764166306862194456963881538792874223675285402995264913177006511960883200000\ 000000*n+1669918973172281420312588493752214031396437541363609631734547490382282\ 752000000000000)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77) /(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-29)/(2*n-\ 53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/(2*n-23)/(2*n-17)/(2*n-37)/(2*n-15)/(2* n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/( 2*n-27)/(2*n-21)/(2*n-41)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-9)/(2*n-61)/(2*n-7)/( 2*n-49)/(2*n-93)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 580027/ 70368744177664*(64766991044841080248763*n^50-75048725754745848811663975*n^49+ 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29797051515622749570264021938555015135232000*n^8+153287959568372833225256426700\ 8283136820605636231422447956787359842872362691597434880000*n^7-3288358139339278\ 870516566441170227502884737459245113942425360967141806102291425525760000*n^6+56\ 5066419138893004685832986652921747665521957901078051128941457036578509890886041\ 6000000*n^5-7516071593915748007225212235189915604006266136383744603628602411538\ 816331499438080000000*n^4+73649578573706622516784751416314903685296833753546913\ 12701462924323934153421619200000000*n^3-492010180739129785841113715902252899407\ 3780499680640906193457072017460807545651200000000*n^2+1950103322041437082460297\ 633511779224293978929475035986268475389288288511590400000000000*n-3306439566881\ 11721221892521762938378216494633189994707083440403095691984896000000000000)/(2* n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/(2*n-77)/(2*n-69)/( 2*n-71)/(2*n-45)/(2*n-63)/(2*n-57)/(2*n-51)/(2*n-73)/(2*n-29)/(2*n-53)/(2*n-35) 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94180982592813917451113557021266066996397495812096000000000000)/(2*n-39)/(2*n-\ 23)/(2*n-17)/(2*n-105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67)/(2 *n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-107) /(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2*n-93 )/(2*n-109)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), 1/4503599627370496*(-\ 617810569230116762356262545508888*n^57+1006228975692784667904583869504564801*n^ 56-800172477003473714775322924747222342876*n^55+ 414066956628754811408657625020998340885670*n^54-\ 156790469318046338464379600397767431600086792*n^53+ 46319294393961490579770718668104118818208416739*n^52-\ 11115296143845130287719926374794537043796324069644*n^51+ 2227503526352449242205951179078908736340227624063840*n^50-\ 380353556871476220104375025762285087893248864125699240*n^49+ 56187210721512917373510373376136454214138757862122895305*n^48-\ 7266474983468035545601634612255888148219099413332844914380*n^47+ 830542045673050580383118056218946085045005450755195645689250*n^46-\ 84546362082190704901848511472364454021784660377531449293644680*n^45+ 7713982739783095044469261047788941717193934914950342809316946835*n^44-\ 634171534138382675605188406124285120227784983528137010399289112060*n^43+ 47185033163222242275517798566487390916926291482325847172829185120900*n^42-\ 3189301801261999863945247617591133749651830540565642015019859320026840*n^41+ 196451688054715835592864621509046652584137971455026898560537291974106955*n^40-\ 11057308940137306343671990086571963537225080725332952013680833615100016980*n^39 +569982543243135356148297504099147971458717249186118177065391410370548756650*n^ 38-\ 26960065083564123375439804775505104800174309671444913210371036428808987152344*n ^37+117197704913772364047120796170087223701599856656976249906120799692286251577\ 7313*n^36-468840469014481515629664298319029574808969179614655901604625803071624\ 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38320*n^25+94865529930841879058704005796419167564497346500515855526586571244588\ 927581426657351765559265040*n^24-1432139103879864514839301680210291845739620967\ 086952618888010720881170030058296982003746784229440*n^23+1985224879315601060834\ 1512452426128437259586208686844147865543068931521716837719528380917278369600*n^ 22-2522755122013117170095408706538855540021710896017872036462969934676130820076\ 18461895583504865775360*n^21+29333110457467857805783024570656293522397030581470\ 42139750289485009450527245836656092505746383128320*n^20-31139144049327407168266\ 790894877950515980442178970402137127459870396908592231583623901554989184977920* n^19+30104007923923390938194684936050011622995458755285574994019897472599559544\ 3589292309076604628967705600*n^18-264271941881756446544161734032874285168814866\ 6004889572558128990872472131633120959941308479134652551168*n^17+209961317234157\ 0441297791329611867430771655896701854830597898820082861082491896297357277991891\ 7809319936*n^16-150391636274219516780470579333351315507981134875670140741080174\ 695472178973333065849643201221310293262336*n^15+9669030493891911410460466641818\ 57050291163978407121395280421121586203605398738448617656876949554096619520*n^14 -555124886029629572522764890828794212160770471085482918285901183302590488677384\ 8766911614964280899631644672*n^13+282912334050476792682141044102945160217904017\ 33257055338084580294940248101878763477315801127769426997018624*n^12-12709202662\ 0782402849542651671334373387726835435155207851017383058812659771268867014629346\ 200269753657851904*n^11+4990856174645050425072198659210319414331979130153435273\ 28550730090267960605186287148480131833005416686551040*n^10-16962256656677888583\ 3747379258237342311997999060539614144464467892301263699744676197031322873294706\ 7043840000*n^9+4929006442308051037893744112224016328445196029774949097023764774\ 344596573457713689865406132234525563944960000*n^8-12062713012068485225116225090\ 9541866753919955375292007632205994771094505512754193037128958737290255086387200\ 00*n^7+243899726908298736380245755448514734296805921144353081160560319822391224\ 54785866619726343484904605692723200000*n^6-397332601874539242683891365345786594\ 29106727226469941847405245265966550752678388191514991684390412615680000000*n^5+ 5039804813974038106631105287911310232705554358952778247979594436047520608360955\ 2935257550715277541376000000000*n^4-4737833521888851734530954654721149641695604\ 6631346418063910731638932571733905424027935058033662912102400000000*n^3+3056048\ 6832915249578187851338183217347050100874053116319834893233654430665805238021664\ 851423515377664000000000*n^2-11781225148097741272954924866172791372024598172189\ 257057982595080864951808498952118733252664177459200000000000*n+1961319602923189\ 8646869106023690809857908595714709049129640695872555677851063303349819874790604\ 80000000000000)/(2*n-99)/(2*n-95)/(2*n-85)/(2*n-97)/(2*n-83)/(2*n-91)/(2*n-81)/ (2*n-77)/(2*n-69)/(2*n-71)/(2*n-45)/(2*n-63)/(2*n-113)/(2*n-57)/(2*n-51)/(2*n-\ 73)/(2*n-111)/(2*n-101)/(2*n-29)/(2*n-53)/(2*n-35)/(2*n-55)/(2*n-79)/(2*n-39)/( 2*n-23)/(2*n-17)/(2*n-105)/(2*n-37)/(2*n-15)/(2*n-89)/(2*n-25)/(2*n-31)/(2*n-67 )/(2*n-87)/(2*n-59)/(2*n-13)/(2*n-43)/(2*n-75)/(2*n-27)/(2*n-21)/(2*n-41)/(2*n-\ 107)/(2*n-11)/(2*n-33)/(2*n-65)/(2*n-103)/(2*n-9)/(2*n-61)/(2*n-7)/(2*n-49)/(2* n-93)/(2*n-109)/(2*n-47)/(2*n-19)/(2*n-3)/(-1+2*n)/(2*n-5), ( 1623779980510307942922573773468448*(n-57)!*(n-56)!*(2*n-119)!*(n+1)+ 2414852278707637453577160996440256*(n-58)!*(117/116*(n-56)!*(n+1)*(2*n-118)!+(n -57)!*((2*n-117)!*(n+1)-1/2414852278707637453577160996440256*binomial(2*n,n)*(n -60)!*(n-56)!)))/(n-60)!/(n-58)!/(n-57)!/(n-56)!/binomial(2*n,n), ( 1623779980510307942922573773468448*(n-57)!*(2*n-119)!*(n+1)+ 2435669970765461914383860660202672*(n-58)!*((2*n-118)!*(n+1)-1/ 2435669970765461914383860660202672*binomial(2*n,n)*(n-60)!*(n-57)!))/(n-58)!/(n -60)!/(n-57)!/binomial(2*n,n), ((1623779980510307942922573773468448*n+ 1623779980510307942922573773468448)*(2*n-119)!-(n-60)!*(n-58)!*binomial(2*n,n)) /(n-60)!/(n-58)!/binomial(2*n,n)] The limits, as n goes to infinity are 576460752303423427 576460752303421657 2305843009213582257 [------------------, ------------------, -------------------, 576460752303423488 576460752303423488 2305843009213693952 576460752303135079 4611686018409235943 4611686018311388039 ------------------, -------------------, -------------------, 576460752303423488 4611686018427387904 4611686018427387904 4611686017800307469 4611686015478541451 18446744024464939847 -------------------, -------------------, --------------------, 4611686018427387904 4611686018427387904 18446744073709551616 18446743888328759953 73786973746892652757 9223371028073169429 --------------------, --------------------, -------------------, 18446744073709551616 73786976294838206464 9223372036854775808 73786952541232631029 73786910883096352081 590294454397975431485 --------------------, --------------------, ---------------------, 73786976294838206464 73786976294838206464 590295810358705651712 295146243943516569433 9444609268300034943131 9444458603049920844521 ---------------------, ----------------------, ----------------------, 295147905179352825856 9444732965739290427392 9444732965739290427392 37776603979323751572769 9443548956223650574741 75539332402174785901233 -----------------------, ----------------------, -----------------------, 37778931862957161709568 9444732965739290427392 75557863725914323419136 75522918434371657949033 150988425285904647901621 37722832806658363940119 -----------------------, ------------------------, -----------------------, 75557863725914323419136 151115727451828646838272 37778931862957161709568 1205858206615250443813013 1203836053641422025069155 -------------------------, -------------------------, 1208925819614629174706176 1208925819614629174706176 4802856987868328847777155 299006645426720765995547 -------------------------, ------------------------, 4835703278458516698824704 302231454903657293676544 2378344696938561896949157 2358806192996891498720569 -------------------------, -------------------------, 2417851639229258349412352 2417851639229258349412352 37306167375248097618943021 18357909449811183512611553 --------------------------, --------------------------, 38685626227668133590597632 19342813113834066795298816 1149836072856343052260500335 1117299914173988523375520493 ----------------------------, ----------------------------, 1237940039285380274899124224 1237940039285380274899124224 4303921438851556507695608741 1024604194113109454087233919 ----------------------------, ----------------------------, 4951760157141521099596496896 1237940039285380274899124224 7696245272701894205058539527 7098786395573146641692245111 ----------------------------, ----------------------------, 9903520314283042199192993792 9903520314283042199192993792 1599946428417307041423106195 5597922055164064161451828213 ----------------------------, ----------------------------, 2475880078570760549798248448 9903520314283042199192993792 37566603514766037253449256601 29603845930307091566739499751 -----------------------------, -----------------------------, 79228162514264337593543950336 79228162514264337593543950336 83948018748498929937342908639 5921408489315667757195972003 ------------------------------, -----------------------------, 316912650057057350374175801344 39614081257132168796771975168 9316668562007366788710819869 -29484099405265862897182547191 ------------------------------, ------------------------------, 316912650057057350374175801344 316912650057057350374175801344 -545857191502407596863673690453 -424403240448603371445880088209 -------------------------------, -------------------------------, 2535301200456458802993406410752 1267650600228229401496703205376 -18211699975104380546983199108153 -22529899332663170437822652096483 ---------------------------------, ---------------------------------, 40564819207303340847894502572032 40564819207303340847894502572032 -105809054996449618354673954243531 -29913173528332317074708932500059 ----------------------------------, ---------------------------------, 162259276829213363391578010288128 40564819207303340847894502572032 -262946064257022246502316953984999 -282374041204407098247853712144191 ----------------------------------, ----------------------------------, 324518553658426726783156020576256 324518553658426726783156020576256 -595239212895682088818563725288225 -77226321153764595294532818188611 ----------------------------------, ---------------------------------, 649037107316853453566312041152512 81129638414606681695789005144064 -5066089600434266834889879617660449 -5141553734143880505314165898799207 -----------------------------------, -----------------------------------, 5192296858534827628530496329220096 5192296858534827628530496329220096 -20718444309748363390905654886459495 ------------------------------------] 20769187434139310514121985316880384 and in Maple notation [576460752303423427/576460752303423488, 576460752303421657/576460752303423488, 2305843009213582257/2305843009213693952, 576460752303135079/576460752303423488, 4611686018409235943/4611686018427387904, 4611686018311388039/ 4611686018427387904, 4611686017800307469/4611686018427387904, 4611686015478541451/4611686018427387904, 18446744024464939847/ 18446744073709551616, 18446743888328759953/18446744073709551616, 73786973746892652757/73786976294838206464, 9223371028073169429/ 9223372036854775808, 73786952541232631029/73786976294838206464, 73786910883096352081/73786976294838206464, 590294454397975431485/ 590295810358705651712, 295146243943516569433/295147905179352825856, 9444609268300034943131/9444732965739290427392, 9444458603049920844521/ 9444732965739290427392, 37776603979323751572769/37778931862957161709568, 9443548956223650574741/9444732965739290427392, 75539332402174785901233/ 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324518553658426726783156020576256, -595239212895682088818563725288225/ 649037107316853453566312041152512, -77226321153764595294532818188611/ 81129638414606681695789005144064, -5066089600434266834889879617660449/ 5192296858534827628530496329220096, -5141553734143880505314165898799207/ 5192296858534827628530496329220096, -20718444309748363390905654886459495/ 20769187434139310514121985316880384] and in floating point [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, .9999999999, .9999999994, .9999999973, .9999999900, .9999999655, .9999998906, .\ 9999996781, .9999991135, .9999977029, .9999943715, .9999869030, .9999709507, .9\ 999383814, .9998746381, .9997547400, .9995375029, .9991575849, .9985150703, .99\ 74625300, .9957898443, .9932075463, .9893300005, .9836603116, .9755793758, .964\ 3418244, .9490816740, .9288301827, .9025476830, .8691700127, .8276686767, .7771\ 221776, .7167942479, .6462132162, .5652456781, .4741571977, .3736530672, .26489\ 32402, .1494773652, .2939822238e-1, -.9303541339e-1, -.2153026991, -.3347951244 , -.4489530665, -.5554048994, -.6520986477, -.7374166609, -.8102651183, -.87013\ 21944, -.9171112194, -.9518879002, -.9756933662, -.9902272297, -.9975568074] The cut off is at j=, 46 ------------------------- This ends this article that took, 733.999, seconds to produce ----------------------- This took, 733.999, seconds.