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 + 25575801662460173