Theorem: For any integers p and q define the Abel-sum type sequence by n ----- \ A[n](r, s) = ) / ----- k = 0 (k - 1 + p) (n - k + q) binomial(n, k) binomial(n + k, k) (r + k) (s - k) Then we have the following functional recurrence relating , A[n](r, s), A[n + 1](r, s), A[n + 2](r, s), A[n + 3](r, s), A[n + 4](r, s), A[n + 5](r, s) 2 2 (r + s) (n + 2) (n + 1) (n + s + 3) (2 + n + s) %3 A[n](r - 1, s + 1) - ----------------------------------------------------------------------- (n + 4) (n + 5) %2 2 2 (r + s) (n + 2) (2 n + 3) (n + s + 3) (2 + n - r) %3 A[n + 1](r, s) - ---------------------------------------------------------------------- (n + 4) (n + 5) %2 10 9 9 8 2 8 + (r + s) (n + 2) (12 n + 94 n r + 74 n s + 180 n r + 320 n r s 8 2 7 2 7 2 7 3 6 2 2 + 84 n s + 190 n r s + 72 n r s - 110 n s - 40 n r s 6 3 6 4 5 2 3 5 4 5 5 9 - 264 n r s - 180 n s - 50 n r s - 110 n r s - 48 n s + 260 n 8 8 7 2 7 7 2 6 3 + 1954 n r + 1384 n s + 3687 n r + 5883 n r s + 1086 n s - 54 n r 6 2 6 2 6 3 5 3 5 2 2 + 3162 n r s + 348 n r s - 2552 n s - 57 n r s - 976 n r s 5 3 5 4 4 3 2 4 2 3 4 4 - 4757 n r s - 3078 n s + 12 n r s - 742 n r s - 1616 n r s 4 5 3 3 3 3 2 4 8 7 7 - 704 n s + 15 n r s + 9 n r s + 2384 n + 17280 n r + 10418 n s 6 2 6 6 2 5 3 5 2 + 32473 n r + 44997 n r s + 3286 n s - 702 n r + 21755 n r s 5 2 5 3 4 3 4 2 2 4 3 - 5807 n r s - 24332 n s - 552 n r s - 8733 n r s - 35175 n r s 4 4 3 3 2 3 2 3 3 4 3 5 - 21670 n s + 184 n r s - 4511 n r s - 9623 n r s - 4158 n s 2 3 3 2 5 2 6 3 4 2 5 + 134 n r s - 116 n r s - 36 n s + 10 n r s + 16 n r s 6 7 6 6 5 2 + 6 n r s + 11780 n + 83980 n r + 37806 n s + 160011 n r 5 5 2 4 3 4 2 4 2 + 182811 n r s - 18882 n s - 3696 n r + 78218 n r s - 66278 n r s 4 3 3 3 3 2 2 3 3 - 124582 n s - 1909 n r s - 39460 n r s - 137137 n r s 3 4 2 3 2 2 2 3 2 4 - 80352 n s + 952 n r s - 14066 n r s - 28878 n r s 2 5 3 3 2 4 5 6 - 12200 n s + 391 n r s - 259 n r s - 716 n r s - 222 n s 3 4 2 5 6 6 5 5 + 26 r s + 44 r s + 18 r s + 31904 n + 239858 n r + 47568 n s 4 2 4 4 2 3 3 + 480195 n r + 411675 n r s - 177934 n s - 10014 n r 3 2 3 2 3 3 2 3 + 151251 n r s - 287119 n r s - 370722 n s - 2652 n r s 2 2 2 2 3 2 4 3 2 - 97793 n r s - 297603 n r s - 165020 n s + 2024 n r s 2 3 4 5 3 3 2 4 - 22599 n r s - 43689 n r s - 17658 n s + 392 r s - 466 r s 5 6 5 4 4 3 2 - 1056 r s - 342 s + 33968 n + 389050 n r - 126358 n s + 893070 n r 3 3 2 2 3 2 2 + 460330 n r s - 597704 n s - 14562 n r + 137908 n r s 2 2 2 3 3 2 2 - 630536 n r s - 640342 n s - 802 n r s - 127698 n r s 3 4 3 2 2 3 4 - 340704 n r s - 177084 n s + 1556 r s - 14992 r s - 26928 r s 5 4 3 3 2 2 - 10188 s - 59068 n + 275928 n r - 614860 n s + 996080 n r 2 2 2 3 2 2 + 90096 n r s - 1033552 n s - 10572 n r + 20892 n r s - 697312 n r s 3 3 2 2 3 4 - 592368 n s + 756 r s - 69252 r s - 161100 r s - 77220 s 3 2 2 2 - 264744 n - 103776 n r - 1062432 n s + 600336 n r - 282432 n r s 2 3 2 2 3 2 - 914160 n s - 2880 r - 33408 r s - 307920 r s - 225072 s - 406512 n 2 2 - 298224 n r - 882144 n s + 145728 r - 188064 r s - 325728 s - 304992 n - 138240 r - 292032 s - 93312) A[n + 1](r - 1, s + 1)/((n + 4) (n + 5) %2) 2 (2 + s) (r + s) (n + 2) (n + 1) (2 + n + s) %3 A[n + 1](r - 2, 2 + s) + ---------------------------------------------------------------------- (n + 4) (n + 5) %2 9 8 8 7 2 7 + 2 (r + s) (2 n + 5) (12 n + 58 n r + 26 n s + 42 n r + 46 n r s 7 2 6 3 6 2 6 2 6 3 5 3 - 20 n s - 60 n r - 62 n r s - 82 n r s - 48 n s + 30 n r s 5 2 2 5 3 8 7 7 6 2 + 58 n r s + 16 n r s + 224 n + 1096 n r + 398 n s + 879 n r 6 6 2 5 3 5 2 5 2 + 761 n r s - 458 n s - 911 n r - 967 n r s - 1432 n r s 5 3 4 4 4 3 4 2 2 4 3 - 816 n s + 18 n r + 469 n r s + 855 n r s + 240 n r s 3 4 3 3 2 7 6 6 5 2 - 9 n r s - 3 n r s + 1712 n + 8652 n r + 2172 n s + 7596 n r 5 5 2 4 3 4 2 4 2 + 4990 n r s - 4344 n s - 5474 n r - 5993 n r s - 10195 n r s 4 3 3 4 3 3 3 2 2 3 3 - 5686 n s + 132 n r + 2770 n r s + 4790 n r s + 1274 n r s 3 4 2 4 2 3 2 2 2 3 2 4 - 36 n s - 63 n r s + 69 n r s + 96 n r s + 18 n r s 4 2 3 3 2 4 6 5 5 - 6 n r s - 8 n r s - 2 n r s + 6644 n + 36724 n r + 3346 n s 4 2 4 4 2 3 3 3 2 + 34779 n r + 15591 n r s - 22278 n s - 16295 n r - 18784 n r s 3 2 3 3 2 4 2 3 2 2 2 - 37957 n r s - 20738 n s + 318 n r + 7971 n r s + 12739 n r s 2 3 2 4 4 3 2 2 3 + 2802 n r s - 306 n s - 152 n r s + 494 n r s + 624 n r s 4 4 2 3 3 2 4 5 4 + 134 n r s - 20 r s - 32 r s - 12 r s + 11972 n + 89462 n r 4 3 2 3 3 2 2 3 - 14666 n s + 90032 n r + 19308 n r s - 66740 n s - 24404 n r 2 2 2 2 2 3 4 3 - 31674 n r s - 78374 n r s - 41616 n s + 272 n r + 11596 n r s 2 2 3 4 4 3 2 + 16144 n r s + 1912 n r s - 888 n s - 140 r s + 760 r s 2 3 4 4 3 3 + 1020 r s + 264 r s - 1948 n + 122008 n r - 79068 n s 2 2 2 2 2 3 2 + 129552 n r - 10864 n r s - 116256 n s - 15988 n r - 28620 n r s 2 3 4 3 2 2 3 - 86100 n r s - 43428 n s + 40 r + 7132 r s + 8028 r s - 588 r s 4 3 2 2 2 - 900 s - 53224 n + 79632 n r - 154704 n s + 93160 n r - 49128 n r s 2 3 2 2 3 2 - 108000 n s - 2448 r - 12192 r s - 40176 r s - 18432 s - 105072 n 2 2 + 8880 n r - 141696 n s + 24096 r - 32832 r s - 40608 s - 91296 n 11 - 9216 r - 50112 s - 31104) A[n + 2](r, s)/((n + 4) (n + 5) %2) - (6 n 10 10 9 2 9 9 2 8 3 + 104 n r + 82 n s + 418 n r + 558 n r s + 144 n s + 460 n r 8 2 8 2 8 3 7 3 7 2 2 + 858 n r s + 246 n r s - 124 n s - 10 n r s - 504 n r s 7 3 7 4 6 3 2 6 2 3 6 4 - 764 n r s - 280 n s - 110 n r s - 296 n r s - 212 n r s 6 5 10 9 9 8 2 8 - 32 n s + 142 n + 2247 n r + 1686 n s + 8754 n r + 10937 n r s 8 2 7 3 7 2 7 2 7 3 + 2366 n s + 9209 n r + 15933 n r s + 2974 n r s - 3220 n s 6 4 6 3 6 2 2 6 3 6 4 - 138 n r - 963 n r s - 10843 n r s - 15214 n r s - 5396 n s 5 4 5 3 2 5 2 3 5 4 5 5 + 3 n r s - 1874 n r s - 5038 n r s - 3640 n r s - 576 n s 4 4 2 4 3 3 4 2 4 9 8 + 33 n r s + 36 n r s + 6 n r s + 1442 n + 20680 n r 8 7 2 7 7 2 6 3 + 14410 n s + 78838 n r + 89852 n r s + 13748 n s + 79100 n r 6 2 6 2 6 3 5 4 5 3 + 123947 n r s + 5259 n r s - 35298 n s - 1746 n r - 13194 n r s 5 2 2 5 3 5 4 4 4 - 97314 n r s - 128234 n r s - 44068 n s + 163 n r s 4 3 2 4 2 3 4 4 4 5 - 13461 n r s - 36244 n r s - 26404 n r s - 4412 n s 3 4 2 3 3 3 3 2 4 3 5 3 6 + 356 n r s + 232 n r s - 236 n r s - 172 n r s - 24 n s 2 4 3 2 3 4 2 2 5 2 6 8 + 22 n r s + 46 n r s + 28 n r s + 4 n r s + 8084 n 7 7 6 2 6 6 2 + 103990 n r + 63618 n s + 397274 n r + 393970 n r s + 17572 n s 5 3 5 2 5 2 5 3 + 378480 n r + 516253 n r s - 96274 n r s - 214722 n s 4 4 4 3 4 2 2 4 3 - 8872 n r - 81380 n r s - 476787 n r s - 594010 n r s 4 4 3 4 3 3 2 3 2 3 - 197666 n s + 1485 n r s - 52384 n r s - 140964 n r s 3 4 3 5 2 4 2 2 3 3 - 103226 n r s - 18188 n s + 1475 n r s + 348 n r s 2 2 4 2 5 2 6 4 3 3 4 - 2296 n r s - 1604 n r s - 276 n s + 106 n r s + 238 n r s 2 5 6 7 6 6 + 168 n r s + 36 n r s + 26246 n + 297904 n r + 132536 n s 5 2 5 5 2 4 3 + 1217248 n r + 945582 n r s - 176840 n s + 1095086 n r 4 2 4 2 4 3 3 4 + 1206741 n r s - 735507 n r s - 794518 n s - 22910 n r 3 3 3 2 2 3 3 3 4 - 276900 n r s - 1385266 n r s - 1635464 n r s - 526058 n s 2 4 2 3 2 2 2 3 2 4 + 5427 n r s - 116915 n r s - 312988 n r s - 229738 n r s 2 5 4 2 3 3 2 4 5 - 42604 n s + 2768 n r s - 220 n r s - 6498 n r s - 4812 n r s 6 4 3 3 4 2 5 6 6 - 996 n s + 128 r s + 312 r s + 252 r s + 68 r s + 42938 n 5 5 4 2 4 4 2 + 414243 n r - 33960 n s + 2291864 n r + 980953 n r s - 1038754 n s 3 3 3 2 3 2 3 3 + 1941115 n r + 1445822 n r s - 2414946 n r s - 1831898 n s 2 4 2 3 2 2 2 2 3 - 31074 n r - 542651 n r s - 2396290 n r s - 2681230 n r s 2 4 4 3 2 2 3 - 831912 n s + 8946 n r s - 141888 n r s - 376762 n r s 4 5 4 2 3 3 2 4 - 277528 n r s - 54420 n s + 1976 r s - 488 r s - 5788 r s 5 6 5 4 4 - 4680 r s - 1140 s - 6886 n - 111752 n r - 919892 n s 3 2 3 3 2 2 3 + 2502232 n r - 652192 n r s - 2654964 n s + 2023286 n r 2 2 2 2 2 3 4 + 437890 n r s - 4240576 n r s - 2569780 n s - 20124 n r 3 2 2 3 4 - 578126 n r s - 2293660 n r s - 2431652 n r s - 726924 n s 4 3 2 2 3 4 5 + 5576 r s - 72800 r s - 192184 r s - 143148 r s - 30060 s 4 3 3 2 2 2 - 191276 n - 1456600 n r - 2402256 n s + 1261508 n r - 3008380 n r s 2 2 3 2 2 - 3701528 n s + 1097024 n r - 744036 n r s - 3888344 n r s 3 4 3 2 2 3 - 2005224 n s - 4320 r - 260840 r s - 941296 r s - 946856 r s 4 3 2 2 2 - 273240 s - 420552 n - 2436368 n r - 3108800 n s - 62776 n r 2 3 2 2 - 3160960 n r s - 2733840 n s + 218400 r - 573888 r s - 1465968 r s 3 2 2 - 666912 s - 460080 n - 1840080 n r - 2090784 n s - 232896 r 2 - 1169760 r s - 836928 s - 262656 n - 548928 r - 581760 s - 62208) 8 A[n + 2](r - 1, s + 1)/((n + 4) (n + 5) %2) - (2 + s) (n + 2) (12 n r 8 7 2 7 7 2 6 2 6 2 + 18 n s + 40 n r + 98 n r s + 36 n s + 80 n r s + 38 n r s 6 3 5 2 2 5 3 5 4 8 7 - 32 n s - 50 n r s - 120 n r s - 64 n s + 12 n + 276 n r 7 6 2 6 6 2 5 3 5 2 + 332 n s + 822 n r + 1645 n r s + 454 n s - 12 n r + 1121 n r s 5 2 5 3 4 3 4 2 2 4 3 + 254 n r s - 708 n s - 24 n r s - 775 n r s - 1764 n r s 4 4 3 3 2 3 2 3 7 6 6 - 928 n s + 15 n r s + 12 n r s + 200 n + 2506 n r + 2386 n s 5 2 5 5 2 4 3 4 2 + 6960 n r + 11131 n r s + 1586 n s - 156 n r + 6095 n r s 4 2 4 3 3 3 3 2 2 3 3 - 1093 n r s - 6134 n s - 164 n r s - 4571 n r s - 10128 n r s 3 4 2 3 2 2 2 3 2 4 2 5 - 5260 n s + 110 n r s - 22 n r s - 150 n r s - 48 n s 3 3 2 4 5 6 5 5 + 10 n r s + 18 n r s + 8 n r s + 1316 n + 11550 n r + 7774 n s 4 2 4 4 2 3 3 3 2 + 31368 n r + 38113 n r s - 3030 n s - 736 n r + 15605 n r s 3 2 3 3 2 3 2 2 2 - 15182 n r s - 26996 n s - 332 n r s - 13221 n r s 2 3 2 4 3 2 2 3 4 - 28606 n r s - 14546 n s + 283 n r s - 396 n r s - 856 n r s 5 3 3 2 4 5 5 4 - 276 n s + 26 r s + 46 r s + 20 r s + 4060 n + 27938 n r 4 3 2 3 3 2 2 3 + 6252 n s + 80648 n r + 65995 n r s - 36058 n s - 1572 n r 2 2 2 2 2 3 3 2 2 + 16853 n r s - 54699 n r s - 63718 n s - 112 n r s - 19295 n r s 3 4 3 2 2 3 4 5 - 40078 n r s - 19566 n s + 260 r s - 662 r s - 1198 r s - 384 s 4 3 3 2 2 2 + 4048 n + 29438 n r - 32658 n s + 116810 n r + 42922 n r s 2 2 3 2 2 3 - 100620 n s - 1492 n r + 606 n r s - 84014 n r s - 75900 n s 3 2 2 3 4 3 2 + 176 r s - 11660 r s - 22480 r s - 10176 s - 9828 n - 6672 n r 2 2 2 3 - 104976 n s + 86456 n r - 18064 n r s - 121488 n s - 480 r 2 2 3 2 - 8160 r s - 47160 r s - 35016 s - 34128 n - 39864 n r - 121656 n s 2 2 + 24288 r - 25824 r s - 53856 s - 38448 n - 23040 r - 50688 s - 15552) 9 8 A[n + 2](r - 2, 2 + s)/((n + 4) (n + 5) %2) - 2 (2 n + 7) (6 n + 50 n r 8 7 2 7 7 2 6 3 6 2 + 22 n s + 112 n r + 76 n r s - 4 n s + 40 n r - 22 n r s 6 2 6 3 5 3 5 2 2 5 3 8 - 106 n r s - 48 n s - 20 n r s - 42 n r s - 16 n r s + 100 n 7 7 6 2 6 6 2 5 3 + 811 n r + 328 n s + 1817 n r + 1149 n r s - 142 n s + 512 n r 5 2 5 2 5 3 4 4 4 3 - 536 n r s - 1800 n r s - 816 n s - 12 n r - 267 n r s 4 2 2 4 3 3 4 3 3 2 7 6 - 545 n r s - 208 n r s + 6 n r s + 3 n r s + 670 n + 5265 n r 6 5 2 5 5 2 4 3 + 1742 n s + 12225 n r + 6835 n r s - 1750 n s + 2638 n r 4 2 4 2 4 3 3 4 3 3 - 4593 n r s - 12457 n r s - 5658 n s - 60 n r - 1330 n r s 3 2 2 3 3 3 4 2 4 2 3 2 - 2852 n r s - 1188 n r s - 36 n s + 28 n r s - 49 n r s 2 2 3 2 4 4 2 3 3 2 4 6 - 56 n r s - 6 n r s + 4 n r s + 6 n r s + 2 n r s + 2194 n 5 5 4 2 4 4 2 + 16751 n r + 2632 n s + 43701 n r + 19175 n r s - 10754 n s 3 3 3 2 3 2 3 3 2 4 + 6784 n r - 19508 n r s - 45157 n r s - 20440 n s - 100 n r 2 3 2 2 2 2 3 2 4 4 - 3315 n r s - 7683 n r s - 3740 n r s - 342 n s + 50 n r s 3 2 2 3 4 4 2 3 3 5 - 246 n r s - 220 n r s - 2 n r s + 4 r s + 4 r s + 2848 n 4 4 3 2 3 3 2 + 22877 n r - 10812 n s + 88031 n r + 20553 n r s - 36690 n s 2 3 2 2 2 2 2 3 4 + 8810 n r - 44607 n r s - 90772 n r s - 40606 n s - 60 n r 3 2 2 3 4 4 - 4292 n r s - 10770 n r s - 6398 n r s - 1104 n s + 28 r s 3 2 2 3 4 4 3 3 - 284 r s - 204 r s + 36 r s - 3326 n - 8526 n r - 56696 n s 2 2 2 2 2 3 2 + 96450 n r - 14456 n r s - 70292 n s + 4976 n r - 52994 n r s 2 3 4 3 2 2 3 - 96728 n r s - 42324 n s - 8 r - 2324 r s - 6264 r s - 4608 r s 4 3 2 2 2 - 1188 s - 17780 n - 68116 n r - 107040 n s + 48592 n r - 50748 n r s 2 3 2 2 3 2 - 70200 n s + 600 r - 25992 r s - 43344 r s - 18360 s - 26904 n 2 2 - 79416 n r - 94320 n s + 5664 r - 31104 r s - 28080 s - 18864 n 10 - 30528 r - 31968 s - 5184) A[n + 3](r, s)/((n + 4) (n + 5) %2) + (30 n 9 9 8 2 8 8 2 7 3 + 232 n r + 92 n s + 530 n r + 326 n r s - 44 n s + 300 n r 7 2 7 3 6 3 6 2 2 6 3 - 480 n r s - 200 n s - 150 n r s - 320 n r s - 108 n r s 6 4 9 8 8 7 2 7 + 32 n s + 608 n + 4535 n r + 1610 n s + 10056 n r + 5680 n r s 7 2 6 3 6 2 6 2 6 3 - 1208 n s + 5113 n r - 919 n r s - 9522 n r s - 3876 n s 5 4 5 3 5 2 2 5 3 5 4 - 90 n r - 2642 n r s - 5480 n r s - 1904 n r s + 512 n s 4 4 4 3 2 4 2 3 8 7 + 45 n r s + 24 n r s - 6 n r s + 5126 n + 37201 n r 7 6 2 6 6 2 5 3 + 10610 n s + 80592 n r + 39463 n r s - 14130 n s + 36251 n r 5 2 5 2 5 3 4 4 4 3 - 15182 n r s - 79930 n r s - 31712 n s - 894 n r - 18749 n r s 4 2 2 4 3 4 4 3 4 3 3 2 - 38792 n r s - 14204 n r s + 3188 n s + 432 n r s - 130 n r s 3 2 3 3 4 3 5 2 4 2 2 3 3 - 408 n r s - 8 n r s + 24 n s + 30 n r s + 46 n r s 2 2 4 2 5 7 6 6 + 12 n r s - 4 n r s + 22366 n + 163819 n r + 25850 n s 5 2 5 5 2 4 3 + 352960 n r + 131558 n r s - 92016 n s + 136737 n r 4 2 4 2 4 3 3 4 - 104797 n r s - 368514 n r s - 141958 n s - 3294 n r 3 3 3 2 2 3 3 3 4 - 69464 n r s - 145378 n r s - 57082 n r s + 9456 n s 2 4 2 3 2 2 2 3 2 4 2 5 + 1573 n r s - 2288 n r s - 3440 n r s - 272 n r s + 156 n s 4 2 3 3 2 4 5 6 + 178 n r s + 294 n r s + 104 n r s - 12 n r s + 46268 n 5 5 4 2 4 4 2 + 402843 n r - 47042 n s + 910398 n r + 164423 n r s - 364554 n s 3 3 3 2 3 2 3 3 + 290637 n r - 384744 n r s - 1009762 n r s - 376398 n s 2 4 2 3 2 2 2 2 3 - 5446 n r - 143905 n r s - 305634 n r s - 130594 n r s 2 4 4 3 2 2 3 4 + 12136 n s + 2654 n r s - 7434 n r s - 10086 n r s - 1408 n r s 5 4 2 3 3 2 4 5 5 + 276 n s + 256 r s + 464 r s + 212 r s + 4 r s - 10718 n 4 4 3 2 3 3 2 + 473990 n r - 480864 n s + 1378156 n r - 229426 n r s - 899248 n s 2 3 2 2 2 2 2 3 + 336546 n r - 788802 n r s - 1649784 n r s - 594968 n s 4 3 2 2 3 4 - 3684 n r - 161050 n r s - 345120 n r s - 162804 n r s + 1112 n s 4 3 2 2 3 4 5 4 + 1792 r s - 7160 r s - 9752 r s - 1940 r s + 84 s - 313288 n 3 3 2 2 2 - 62700 n r - 1378992 n s + 1122228 n r - 1036080 n r s 2 2 3 2 2 - 1349104 n s + 181480 n r - 854572 n r s - 1496536 n r s 3 4 3 2 2 3 4 - 525536 n s - 512 r - 77432 r s - 165680 r s - 87416 r s - 7656 s 3 2 2 2 - 821464 n - 899632 n r - 2031600 n s + 358280 n r - 1288384 n r s 2 3 2 2 3 - 1126848 n s + 25280 r - 382528 r s - 586512 r s - 203904 s 2 2 - 1071360 n - 1009392 n r - 1556160 n s - 25152 r - 570912 r s 2 - 402048 s - 730368 n - 379584 r - 490752 s - 207360) 9 9 8 2 A[n + 3](r - 1, s + 1)/((n + 4) (n + 5) %1) + (6 n r + 18 n s + 20 n r 8 8 2 7 2 7 2 7 3 + 106 n r s + 48 n s + 130 n r s + 102 n r s - 24 n s 6 2 2 6 3 6 4 9 8 8 - 70 n r s - 172 n r s - 96 n s + 24 n + 264 n r + 424 n s 7 2 7 7 2 6 3 6 2 + 631 n r + 2151 n r s + 718 n s - 6 n r + 2131 n r s 6 2 6 3 5 3 5 2 2 5 3 + 1248 n r s - 824 n s - 39 n r s - 1263 n r s - 2956 n r s 5 4 4 3 2 4 2 3 8 7 7 - 1632 n s + 21 n r s + 18 n r s + 472 n + 3658 n r + 3930 n s 6 2 6 6 2 5 3 5 2 + 7463 n r + 18113 n r s + 3254 n s - 144 n r + 14010 n r s 5 2 5 3 4 3 4 2 2 4 3 + 3823 n r s - 9874 n s - 352 n r s - 9059 n r s - 20702 n r s 4 4 3 3 2 3 4 3 5 2 3 3 - 11312 n s + 202 n r s - 238 n r s - 72 n s + 14 n r s 2 2 4 2 5 7 6 6 + 26 n r s + 12 n r s + 3784 n + 24858 n r + 17438 n s 5 2 5 5 2 4 3 4 2 + 45629 n r + 80843 n r s - 3658 n s - 1070 n r + 46227 n r s 4 2 4 3 3 3 3 2 2 - 13564 n r s - 58960 n s - 1089 n r s - 33915 n r s 3 3 3 4 2 3 2 2 2 3 2 4 - 76284 n r s - 41010 n s + 755 n r s - 796 n r s - 2040 n r s 2 5 3 3 2 4 5 6 5 - 648 n s + 78 n r s + 150 n r s + 72 n r s + 15168 n + 94736 n r 5 4 2 4 4 2 3 3 + 29088 n s + 160441 n r + 199261 n r s - 88078 n s - 3568 n r 3 2 3 2 3 3 2 3 + 75312 n r s - 122569 n r s - 196042 n s - 1152 n r s 2 2 2 2 3 2 4 3 2 - 71719 n r s - 157918 n r s - 82490 n s + 1342 n r s 2 3 4 5 3 3 2 4 - 2758 n r s - 5726 n r s - 1932 n s + 104 r s + 212 r s 5 5 4 4 3 2 + 108 r s + 26248 n + 207486 n r - 58254 n s + 332540 n r 3 3 2 2 3 2 2 + 240934 n r s - 353396 n s - 5788 n r + 37898 n r s 2 2 2 3 3 2 2 - 335884 n r s - 368756 n s + 368 n r s - 83134 n r s 3 4 3 2 2 3 4 - 176924 n r s - 88344 n s + 960 r s - 2592 r s - 5244 r s 5 4 3 3 2 2 - 1908 s - 19704 n + 240992 n r - 375164 n s + 392044 n r 2 2 2 3 2 2 + 40272 n r s - 681624 n s - 4224 n r - 43628 n r s - 419924 n r s 3 3 2 2 3 4 - 367056 n s + 1032 r s - 41784 r s - 85176 r s - 40176 s 3 2 2 2 - 182216 n + 92336 n r - 745720 n s + 232336 n r - 198224 n r s 2 3 2 2 3 2 - 658800 n s - 944 r - 47552 r s - 204864 r s - 150768 s - 353056 n 2 2 - 71344 n r - 691632 n s + 47904 r - 142464 r s - 254880 s - 310944 n - 59520 r - 251136 s - 107136) A[n + 3](r - 2, 2 + s)/((n + 4) (n + 5) %1) 2 (2 n + 9) (n + r + 4) A[n + 4](r, s) 8 7 7 + -------------------------------------- - (18 n + 72 n r - 6 n s n + 5 6 2 6 6 2 5 2 5 2 5 3 + 40 n r - 82 n r s - 60 n s - 20 n r s - 2 n r s + 48 n s 7 6 6 5 2 5 5 2 + 270 n + 1177 n r - 98 n s + 564 n r - 1303 n r s - 886 n s 4 3 4 2 4 2 4 3 3 3 3 2 2 - 12 n r - 269 n r s - 24 n r s + 640 n s + 6 n r s - 9 n r s 6 5 5 4 2 4 4 2 + 1438 n + 7790 n r - 818 n s + 3074 n r - 8484 n r s - 5256 n s 3 3 3 2 3 2 3 3 2 3 - 144 n r - 1457 n r s - 139 n r s + 3258 n s + 70 n r s 2 2 2 2 3 2 4 3 2 2 3 4 - 48 n r s + 62 n r s + 36 n s + 4 n r s - 2 n r s - 6 n r s 5 4 4 3 2 3 + 2260 n + 26836 n r - 4398 n s + 8410 n r - 28780 n r s 3 2 2 3 2 2 2 2 2 3 - 16092 n s - 436 n r - 3823 n r s - 528 n r s + 7822 n s 3 2 2 3 4 3 2 2 3 + 204 n r s - 199 n r s + 242 n r s + 210 n s + 32 r s + 14 r s 4 4 3 3 2 2 2 - 18 r s - 8166 n + 50580 n r - 15312 n s + 12018 n r - 54120 n r s 2 2 3 2 2 3 - 27292 n s - 424 n r - 5045 n r s - 1235 n r s + 8730 n s 3 2 2 3 4 3 2 + 224 r s - 280 r s + 216 r s + 306 s - 44362 n + 48387 n r 2 2 2 3 2 - 32300 n s + 8066 n r - 54497 n r s - 25326 n s - 64 r - 2866 r s 2 3 2 2 - 1212 r s + 3546 s - 85258 n + 15878 n r - 36804 n s + 1644 r 2 - 23550 r s - 10728 s - 77232 n - 3360 r - 17208 s - 27432) 8 7 7 6 2 A[n + 4](r - 1, s + 1)/((n + 5) %1) - (6 n + 38 n r + 28 n s + 60 n r 6 5 2 5 2 5 3 7 6 + 82 n r s - 30 n r s - 88 n r s - 64 n s + 112 n + 615 n r 6 5 2 5 5 2 4 3 4 2 + 346 n s + 827 n r + 1014 n r s - 188 n s - 18 n r - 445 n r s 4 2 4 3 3 3 3 2 2 6 5 - 1212 n r s - 864 n s + 9 n r s + 12 n r s + 786 n + 4075 n r 5 4 2 4 4 2 3 3 + 1290 n s + 4505 n r + 4549 n r s - 2522 n s - 114 n r 3 2 3 2 3 3 2 3 2 2 2 - 2423 n r s - 6440 n r s - 4500 n s + 54 n r s + 2 n r s 2 3 2 4 3 2 2 3 4 5 - 130 n r s - 48 n s + 6 n r s + 14 n r s + 8 n r s + 2314 n 4 4 3 2 3 3 2 + 14075 n r - 666 n s + 12135 n r + 8000 n r s - 13104 n s 2 3 2 2 2 2 2 3 3 - 246 n r - 6327 n r s - 16790 n r s - 11406 n s + 119 n r s 2 2 3 4 3 2 2 3 4 - 204 n r s - 716 n r s - 300 n s + 14 r s + 42 r s + 28 r s 4 3 3 2 2 2 + 484 n + 26551 n r - 17186 n s + 16491 n r + 179 n r s 2 2 3 2 2 3 - 32990 n s - 194 n r - 8261 n r s - 22146 n r s - 14422 n s 3 2 2 3 4 3 2 + 98 r s - 298 r s - 942 r s - 456 s - 14650 n + 25314 n r 2 2 2 3 2 - 47996 n s + 9870 n r - 15276 n r s - 40420 n s - 28 r - 4450 r s 2 3 2 2 - 12240 r s - 7596 s - 38420 n + 8124 n r - 56248 n s + 1400 r 2 - 12964 r s - 19488 s - 40632 n - 1896 r - 24624 s - 15984) / 6 5 5 4 A[n + 4](r - 2, 2 + s) / ((n + 5) (6 n + 20 n r + 4 n s - 10 n r s / 4 2 5 4 4 3 2 3 3 2 - 16 n s + 46 n + 191 n r + 10 n s - 6 n r - 98 n r s - 144 n s 2 2 4 3 3 2 2 2 + 3 n r s + 76 n + 696 n r - 108 n s - 24 n r - 343 n r s 2 2 2 2 3 2 2 3 3 - 474 n s + 11 n r s - 16 n r s - 12 n s + 2 r s + 2 r s - 254 n 2 2 2 2 2 + 1167 n r - 574 n s - 26 n r - 547 n r s - 698 n s + 14 r s 2 3 2 2 2 - 34 r s - 30 s - 1098 n + 830 n r - 988 n s - 4 r - 358 r s - 408 s - 1440 n + 144 r - 576 s - 648)) - A[n + 5](r - 1, s + 1) + A[n + 5](r - 2, 2 + s) = 0 6 5 5 4 4 2 5 4 %1 := 6 n + 20 n r + 4 n s - 10 n r s - 16 n s + 46 n + 191 n r 4 3 2 3 3 2 2 2 4 3 + 10 n s - 6 n r - 98 n r s - 144 n s + 3 n r s + 76 n + 696 n r 3 2 2 2 2 2 2 2 - 108 n s - 24 n r - 343 n r s - 474 n s + 11 n r s - 16 n r s 3 2 2 3 3 2 2 2 - 12 n s + 2 r s + 2 r s - 254 n + 1167 n r - 574 n s - 26 n r 2 2 2 3 2 - 547 n r s - 698 n s + 14 r s - 34 r s - 30 s - 1098 n + 830 n r 2 2 - 988 n s - 4 r - 358 r s - 408 s - 1440 n + 144 r - 576 s - 648 6 5 5 4 4 2 5 4 %2 := 6 n + 20 n r + 4 n s - 10 n r s - 16 n s + 46 n + 191 n r 4 3 2 3 3 2 2 2 4 3 + 10 n s - 6 n r - 98 n r s - 144 n s + 3 n r s + 76 n + 696 n r 3 2 2 2 2 2 2 2 - 108 n s - 24 n r - 343 n r s - 474 n s + 11 n r s - 16 n r s 3 2 2 3 3 2 2 2 - 12 n s + 2 r s + 2 r s - 254 n + 1167 n r - 574 n s - 26 n r 2 2 2 3 2 - 547 n r s - 698 n s + 14 r s - 34 r s - 30 s - 1098 n + 830 n r 2 2 - 988 n s - 4 r - 358 r s - 408 s - 1440 n + 144 r - 576 s - 648 6 5 5 4 4 2 5 4 %3 := 6 n + 20 n r + 4 n s - 10 n r s - 16 n s + 82 n + 291 n r 4 3 2 3 3 2 2 2 4 + 30 n s - 6 n r - 138 n r s - 208 n s + 3 n r s + 396 n 3 3 2 2 2 2 2 2 + 1660 n r - 28 n s - 42 n r - 697 n r s - 1002 n s + 17 n r s 2 3 2 2 3 3 2 2 - 16 n r s - 12 n s + 2 r s + 2 r s + 630 n + 4601 n r - 798 n s 2 2 2 2 3 2 - 92 n r - 1567 n r s - 2142 n s + 28 r s - 50 r s - 42 s - 854 n 2 2 + 6116 n r - 2400 n s - 60 r - 1356 r s - 1740 s - 3828 n + 3048 r - 2232 s - 3312 and in Maple notation -(r+s)^2*(n+2)*(n+1)*(n+s+3)*(2+n+s)^2*(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^ 4*s^2+82*n^5+291*n^4*r+30*n^4*s-6*n^3*r^2-138*n^3*r*s-208*n^3*s^2+3*n^2*r^2*s+ 396*n^4+1660*n^3*r-28*n^3*s-42*n^2*r^2-697*n^2*r*s-1002*n^2*s^2+17*n*r^2*s-16*n *r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3+630*n^3+4601*n^2*r-798*n^2*s-92*n*r^2-1567*n* r*s-2142*n*s^2+28*r^2*s-50*r*s^2-42*s^3-854*n^2+6116*n*r-2400*n*s-60*r^2-1356*r *s-1740*s^2-3828*n+3048*r-2232*s-3312)/(n+4)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s-10*n ^4*r*s-16*n^4*s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s^2+3* n^2*r^2*s+76*n^4+696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11*n*r^ 2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26*n*r^2 -547*n*r*s-698*n*s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-988*n*s-4*r^2-\ 358*r*s-408*s^2-1440*n+144*r-576*s-648)*A[n](r-1,s+1)-2*(r+s)^2*(n+2)*(2*n+3)*( n+s+3)*(2+n-r)*(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+82*n^5+291*n^4*r+ 30*n^4*s-6*n^3*r^2-138*n^3*r*s-208*n^3*s^2+3*n^2*r^2*s+396*n^4+1660*n^3*r-28*n^ 3*s-42*n^2*r^2-697*n^2*r*s-1002*n^2*s^2+17*n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^ 2+2*r*s^3+630*n^3+4601*n^2*r-798*n^2*s-92*n*r^2-1567*n*r*s-2142*n*s^2+28*r^2*s-\ 50*r*s^2-42*s^3-854*n^2+6116*n*r-2400*n*s-60*r^2-1356*r*s-1740*s^2-3828*n+3048* r-2232*s-3312)/(n+4)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+46*n^5 +191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s^2+3*n^2*r^2*s+76*n^4+696*n^3 *r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11*n*r^2*s-16*n*r*s^2-12*n*s^3+ 2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26*n*r^2-547*n*r*s-698*n*s^2+14* r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-988*n*s-4*r^2-358*r*s-408*s^2-1440*n+144 *r-576*s-648)*A[n+1](r,s)+(r+s)*(n+2)*(12*n^10+94*n^9*r+74*n^9*s+180*n^8*r^2+ 320*n^8*r*s+84*n^8*s^2+190*n^7*r^2*s+72*n^7*r*s^2-110*n^7*s^3-40*n^6*r^2*s^2-\ 264*n^6*r*s^3-180*n^6*s^4-50*n^5*r^2*s^3-110*n^5*r*s^4-48*n^5*s^5+260*n^9+1954* n^8*r+1384*n^8*s+3687*n^7*r^2+5883*n^7*r*s+1086*n^7*s^2-54*n^6*r^3+3162*n^6*r^2 *s+348*n^6*r*s^2-2552*n^6*s^3-57*n^5*r^3*s-976*n^5*r^2*s^2-4757*n^5*r*s^3-3078* n^5*s^4+12*n^4*r^3*s^2-742*n^4*r^2*s^3-1616*n^4*r*s^4-704*n^4*s^5+15*n^3*r^3*s^ 3+9*n^3*r^2*s^4+2384*n^8+17280*n^7*r+10418*n^7*s+32473*n^6*r^2+44997*n^6*r*s+ 3286*n^6*s^2-702*n^5*r^3+21755*n^5*r^2*s-5807*n^5*r*s^2-24332*n^5*s^3-552*n^4*r ^3*s-8733*n^4*r^2*s^2-35175*n^4*r*s^3-21670*n^4*s^4+184*n^3*r^3*s^2-4511*n^3*r^ 2*s^3-9623*n^3*r*s^4-4158*n^3*s^5+134*n^2*r^3*s^3-116*n^2*r*s^5-36*n^2*s^6+10*n *r^3*s^4+16*n*r^2*s^5+6*n*r*s^6+11780*n^7+83980*n^6*r+37806*n^6*s+160011*n^5*r^ 2+182811*n^5*r*s-18882*n^5*s^2-3696*n^4*r^3+78218*n^4*r^2*s-66278*n^4*r*s^2-\ 124582*n^4*s^3-1909*n^3*r^3*s-39460*n^3*r^2*s^2-137137*n^3*r*s^3-80352*n^3*s^4+ 952*n^2*r^3*s^2-14066*n^2*r^2*s^3-28878*n^2*r*s^4-12200*n^2*s^5+391*n*r^3*s^3-\ 259*n*r^2*s^4-716*n*r*s^5-222*n*s^6+26*r^3*s^4+44*r^2*s^5+18*r*s^6+31904*n^6+ 239858*n^5*r+47568*n^5*s+480195*n^4*r^2+411675*n^4*r*s-177934*n^4*s^2-10014*n^3 *r^3+151251*n^3*r^2*s-287119*n^3*r*s^2-370722*n^3*s^3-2652*n^2*r^3*s-97793*n^2* r^2*s^2-297603*n^2*r*s^3-165020*n^2*s^4+2024*n*r^3*s^2-22599*n*r^2*s^3-43689*n* r*s^4-17658*n*s^5+392*r^3*s^3-466*r^2*s^4-1056*r*s^5-342*s^6+33968*n^5+389050*n ^4*r-126358*n^4*s+893070*n^3*r^2+460330*n^3*r*s-597704*n^3*s^2-14562*n^2*r^3+ 137908*n^2*r^2*s-630536*n^2*r*s^2-640342*n^2*s^3-802*n*r^3*s-127698*n*r^2*s^2-\ 340704*n*r*s^3-177084*n*s^4+1556*r^3*s^2-14992*r^2*s^3-26928*r*s^4-10188*s^5-\ 59068*n^4+275928*n^3*r-614860*n^3*s+996080*n^2*r^2+90096*n^2*r*s-1033552*n^2*s^ 2-10572*n*r^3+20892*n*r^2*s-697312*n*r*s^2-592368*n*s^3+756*r^3*s-69252*r^2*s^2 -161100*r*s^3-77220*s^4-264744*n^3-103776*n^2*r-1062432*n^2*s+600336*n*r^2-\ 282432*n*r*s-914160*n*s^2-2880*r^3-33408*r^2*s-307920*r*s^2-225072*s^3-406512*n ^2-298224*n*r-882144*n*s+145728*r^2-188064*r*s-325728*s^2-304992*n-138240*r-\ 292032*s-93312)/(n+4)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+46*n^ 5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s^2+3*n^2*r^2*s+76*n^4+696*n^ 3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11*n*r^2*s-16*n*r*s^2-12*n*s^3 +2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26*n*r^2-547*n*r*s-698*n*s^2+14 *r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-988*n*s-4*r^2-358*r*s-408*s^2-1440*n+ 144*r-576*s-648)*A[n+1](r-1,s+1)+(2+s)^2*(r+s)*(n+2)*(n+1)*(2+n+s)*(6*n^6+20*n^ 5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+82*n^5+291*n^4*r+30*n^4*s-6*n^3*r^2-138*n^3*r *s-208*n^3*s^2+3*n^2*r^2*s+396*n^4+1660*n^3*r-28*n^3*s-42*n^2*r^2-697*n^2*r*s-\ 1002*n^2*s^2+17*n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3+630*n^3+4601*n^2* r-798*n^2*s-92*n*r^2-1567*n*r*s-2142*n*s^2+28*r^2*s-50*r*s^2-42*s^3-854*n^2+ 6116*n*r-2400*n*s-60*r^2-1356*r*s-1740*s^2-3828*n+3048*r-2232*s-3312)/(n+4)/(n+ 5)/(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^ 3*r^2-98*n^3*r*s-144*n^3*s^2+3*n^2*r^2*s+76*n^4+696*n^3*r-108*n^3*s-24*n^2*r^2-\ 343*n^2*r*s-474*n^2*s^2+11*n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^ 3+1167*n^2*r-574*n^2*s-26*n*r^2-547*n*r*s-698*n*s^2+14*r^2*s-34*r*s^2-30*s^3-\ 1098*n^2+830*n*r-988*n*s-4*r^2-358*r*s-408*s^2-1440*n+144*r-576*s-648)*A[n+1](r -2,2+s)+2*(r+s)*(2*n+5)*(12*n^9+58*n^8*r+26*n^8*s+42*n^7*r^2+46*n^7*r*s-20*n^7* s^2-60*n^6*r^3-62*n^6*r^2*s-82*n^6*r*s^2-48*n^6*s^3+30*n^5*r^3*s+58*n^5*r^2*s^2 +16*n^5*r*s^3+224*n^8+1096*n^7*r+398*n^7*s+879*n^6*r^2+761*n^6*r*s-458*n^6*s^2-\ 911*n^5*r^3-967*n^5*r^2*s-1432*n^5*r*s^2-816*n^5*s^3+18*n^4*r^4+469*n^4*r^3*s+ 855*n^4*r^2*s^2+240*n^4*r*s^3-9*n^3*r^4*s-3*n^3*r^3*s^2+1712*n^7+8652*n^6*r+ 2172*n^6*s+7596*n^5*r^2+4990*n^5*r*s-4344*n^5*s^2-5474*n^4*r^3-5993*n^4*r^2*s-\ 10195*n^4*r*s^2-5686*n^4*s^3+132*n^3*r^4+2770*n^3*r^3*s+4790*n^3*r^2*s^2+1274*n ^3*r*s^3-36*n^3*s^4-63*n^2*r^4*s+69*n^2*r^3*s^2+96*n^2*r^2*s^3+18*n^2*r*s^4-6*n *r^4*s^2-8*n*r^3*s^3-2*n*r^2*s^4+6644*n^6+36724*n^5*r+3346*n^5*s+34779*n^4*r^2+ 15591*n^4*r*s-22278*n^4*s^2-16295*n^3*r^3-18784*n^3*r^2*s-37957*n^3*r*s^2-20738 *n^3*s^3+318*n^2*r^4+7971*n^2*r^3*s+12739*n^2*r^2*s^2+2802*n^2*r*s^3-306*n^2*s^ 4-152*n*r^4*s+494*n*r^3*s^2+624*n*r^2*s^3+134*n*r*s^4-20*r^4*s^2-32*r^3*s^3-12* r^2*s^4+11972*n^5+89462*n^4*r-14666*n^4*s+90032*n^3*r^2+19308*n^3*r*s-66740*n^3 *s^2-24404*n^2*r^3-31674*n^2*r^2*s-78374*n^2*r*s^2-41616*n^2*s^3+272*n*r^4+ 11596*n*r^3*s+16144*n*r^2*s^2+1912*n*r*s^3-888*n*s^4-140*r^4*s+760*r^3*s^2+1020 *r^2*s^3+264*r*s^4-1948*n^4+122008*n^3*r-79068*n^3*s+129552*n^2*r^2-10864*n^2*r *s-116256*n^2*s^2-15988*n*r^3-28620*n*r^2*s-86100*n*r*s^2-43428*n*s^3+40*r^4+ 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*s^2+7822*n^2*s^3+204*n*r^3*s-199*n*r^2*s^2+242*n*r*s^3+210*n*s^4+32*r^3*s^2+14 *r^2*s^3-18*r*s^4-8166*n^4+50580*n^3*r-15312*n^3*s+12018*n^2*r^2-54120*n^2*r*s-\ 27292*n^2*s^2-424*n*r^3-5045*n*r^2*s-1235*n*r*s^2+8730*n*s^3+224*r^3*s-280*r^2* s^2+216*r*s^3+306*s^4-44362*n^3+48387*n^2*r-32300*n^2*s+8066*n*r^2-54497*n*r*s-\ 25326*n*s^2-64*r^3-2866*r^2*s-1212*r*s^2+3546*s^3-85258*n^2+15878*n*r-36804*n*s +1644*r^2-23550*r*s-10728*s^2-77232*n-3360*r-17208*s-27432)/(n+5)/(6*n^6+20*n^5 *r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s -144*n^3*s^2+3*n^2*r^2*s+76*n^4+696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474* n^2*s^2+11*n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574 *n^2*s-26*n*r^2-547*n*r*s-698*n*s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-\ 988*n*s-4*r^2-358*r*s-408*s^2-1440*n+144*r-576*s-648)*A[n+4](r-1,s+1)-(6*n^8+38 *n^7*r+28*n^7*s+60*n^6*r^2+82*n^6*r*s-30*n^5*r^2*s-88*n^5*r*s^2-64*n^5*s^3+112* n^7+615*n^6*r+346*n^6*s+827*n^5*r^2+1014*n^5*r*s-188*n^5*s^2-18*n^4*r^3-445*n^4 *r^2*s-1212*n^4*r*s^2-864*n^4*s^3+9*n^3*r^3*s+12*n^3*r^2*s^2+786*n^6+4075*n^5*r +1290*n^5*s+4505*n^4*r^2+4549*n^4*r*s-2522*n^4*s^2-114*n^3*r^3-2423*n^3*r^2*s-\ 6440*n^3*r*s^2-4500*n^3*s^3+54*n^2*r^3*s+2*n^2*r^2*s^2-130*n^2*r*s^3-48*n^2*s^4 +6*n*r^3*s^2+14*n*r^2*s^3+8*n*r*s^4+2314*n^5+14075*n^4*r-666*n^4*s+12135*n^3*r^ 2+8000*n^3*r*s-13104*n^3*s^2-246*n^2*r^3-6327*n^2*r^2*s-16790*n^2*r*s^2-11406*n ^2*s^3+119*n*r^3*s-204*n*r^2*s^2-716*n*r*s^3-300*n*s^4+14*r^3*s^2+42*r^2*s^3+28 *r*s^4+484*n^4+26551*n^3*r-17186*n^3*s+16491*n^2*r^2+179*n^2*r*s-32990*n^2*s^2-\ 194*n*r^3-8261*n*r^2*s-22146*n*r*s^2-14422*n*s^3+98*r^3*s-298*r^2*s^2-942*r*s^3 -456*s^4-14650*n^3+25314*n^2*r-47996*n^2*s+9870*n*r^2-15276*n*r*s-40420*n*s^2-\ 28*r^3-4450*r^2*s-12240*r*s^2-7596*s^3-38420*n^2+8124*n*r-56248*n*s+1400*r^2-\ 12964*r*s-19488*s^2-40632*n-1896*r-24624*s-15984)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s -10*n^4*r*s-16*n^4*s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s ^2+3*n^2*r^2*s+76*n^4+696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11 *n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26* n*r^2-547*n*r*s-698*n*s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-988*n*s-4*r ^2-358*r*s-408*s^2-1440*n+144*r-576*s-648)*A[n+4](r-2,2+s)-A[n+5](r-1,s+1)+A[n+ 5](r-2,2+s) = 0 Proof: We claim that: Let a(n,k,r,s) be the summand on the sum defining A[n](r,s) in other words a(n, k, r, s) = (k - 1 + p) (n - k + q) binomial(n, k) binomial(n + k, k) (r + k) (s - k) and in Maple notation a(n,k,r,s) = binomial(n,k)*binomial(n+k,k)*(r+k)^(k-1+p)*(s-k)^(n-k+q) Then the following identity is true 2 2 - (r + s) (n + 2) (n + 1) (n + s + 3) (2 + n + s) %3 a(n, k + 1, r - 1, s + 1)/((n + 4) (n + 5) %2) - 2 2 (r + s) (n + 2) (2 n + 3) (n + s + 3) (2 + n - r) %3 a(n + 1, k, r, s) ------------------------------------------------------------------------- (n + 4) (n + 5) %2 10 9 9 8 2 8 + (r + s) (n + 2) (12 n + 94 n r + 74 n s + 180 n r + 320 n r s 8 2 7 2 7 2 7 3 6 2 2 + 84 n s + 190 n r s + 72 n r s - 110 n s - 40 n r s 6 3 6 4 5 2 3 5 4 5 5 9 - 264 n r s - 180 n s - 50 n r s - 110 n r s - 48 n s + 260 n 8 8 7 2 7 7 2 6 3 + 1954 n r + 1384 n s + 3687 n r + 5883 n r s + 1086 n s - 54 n r 6 2 6 2 6 3 5 3 5 2 2 + 3162 n r s + 348 n r s - 2552 n s - 57 n r s - 976 n r s 5 3 5 4 4 3 2 4 2 3 4 4 - 4757 n r s - 3078 n s + 12 n r s - 742 n r s - 1616 n r s 4 5 3 3 3 3 2 4 8 7 7 - 704 n s + 15 n r s + 9 n r s + 2384 n + 17280 n r + 10418 n s 6 2 6 6 2 5 3 5 2 + 32473 n r + 44997 n r s + 3286 n s - 702 n r + 21755 n r s 5 2 5 3 4 3 4 2 2 4 3 - 5807 n r s - 24332 n s - 552 n r s - 8733 n r s - 35175 n r s 4 4 3 3 2 3 2 3 3 4 3 5 - 21670 n s + 184 n r s - 4511 n r s - 9623 n r s - 4158 n s 2 3 3 2 5 2 6 3 4 2 5 + 134 n r s - 116 n r s - 36 n s + 10 n r s + 16 n r s 6 7 6 6 5 2 + 6 n r s + 11780 n + 83980 n r + 37806 n s + 160011 n r 5 5 2 4 3 4 2 4 2 + 182811 n r s - 18882 n s - 3696 n r + 78218 n r s - 66278 n r s 4 3 3 3 3 2 2 3 3 - 124582 n s - 1909 n r s - 39460 n r s - 137137 n r s 3 4 2 3 2 2 2 3 2 4 - 80352 n s + 952 n r s - 14066 n r s - 28878 n r s 2 5 3 3 2 4 5 6 - 12200 n s + 391 n r s - 259 n r s - 716 n r s - 222 n s 3 4 2 5 6 6 5 5 + 26 r s + 44 r s + 18 r s + 31904 n + 239858 n r + 47568 n s 4 2 4 4 2 3 3 + 480195 n r + 411675 n r s - 177934 n s - 10014 n r 3 2 3 2 3 3 2 3 + 151251 n r s - 287119 n r s - 370722 n s - 2652 n r s 2 2 2 2 3 2 4 3 2 - 97793 n r s - 297603 n r s - 165020 n s + 2024 n r s 2 3 4 5 3 3 2 4 - 22599 n r s - 43689 n r s - 17658 n s + 392 r s - 466 r s 5 6 5 4 4 3 2 - 1056 r s - 342 s + 33968 n + 389050 n r - 126358 n s + 893070 n r 3 3 2 2 3 2 2 + 460330 n r s - 597704 n s - 14562 n r + 137908 n r s 2 2 2 3 3 2 2 - 630536 n r s - 640342 n s - 802 n r s - 127698 n r s 3 4 3 2 2 3 4 - 340704 n r s - 177084 n s + 1556 r s - 14992 r s - 26928 r s 5 4 3 3 2 2 - 10188 s - 59068 n + 275928 n r - 614860 n s + 996080 n r 2 2 2 3 2 2 + 90096 n r s - 1033552 n s - 10572 n r + 20892 n r s - 697312 n r s 3 3 2 2 3 4 - 592368 n s + 756 r s - 69252 r s - 161100 r s - 77220 s 3 2 2 2 - 264744 n - 103776 n r - 1062432 n s + 600336 n r - 282432 n r s 2 3 2 2 3 2 - 914160 n s - 2880 r - 33408 r s - 307920 r s - 225072 s - 406512 n 2 2 - 298224 n r - 882144 n s + 145728 r - 188064 r s - 325728 s - 304992 n - 138240 r - 292032 s - 93312) a(n + 1, k + 1, r - 1, s + 1)/((n + 4) 2 (n + 5) %2) + (2 + s) (r + s) (n + 2) (n + 1) (2 + n + s) %3 a(n + 1, k + 2, r - 2, 2 + s)/((n + 4) (n + 5) %2) + 2 (r + s) (2 n + 5) ( 9 8 8 7 2 7 7 2 6 3 12 n + 58 n r + 26 n s + 42 n r + 46 n r s - 20 n s - 60 n r 6 2 6 2 6 3 5 3 5 2 2 - 62 n r s - 82 n r s - 48 n s + 30 n r s + 58 n r s 5 3 8 7 7 6 2 6 + 16 n r s + 224 n + 1096 n r + 398 n s + 879 n r + 761 n r s 6 2 5 3 5 2 5 2 5 3 - 458 n s - 911 n r - 967 n r s - 1432 n r s - 816 n s 4 4 4 3 4 2 2 4 3 3 4 + 18 n r + 469 n r s + 855 n r s + 240 n r s - 9 n r s 3 3 2 7 6 6 5 2 5 - 3 n r s + 1712 n + 8652 n r + 2172 n s + 7596 n r + 4990 n r s 5 2 4 3 4 2 4 2 4 3 - 4344 n s - 5474 n r - 5993 n r s - 10195 n r s - 5686 n s 3 4 3 3 3 2 2 3 3 3 4 + 132 n r + 2770 n r s + 4790 n r s + 1274 n r s - 36 n s 2 4 2 3 2 2 2 3 2 4 4 2 - 63 n r s + 69 n r s + 96 n r s + 18 n r s - 6 n r s 3 3 2 4 6 5 5 4 2 - 8 n r s - 2 n r s + 6644 n + 36724 n r + 3346 n s + 34779 n r 4 4 2 3 3 3 2 3 2 + 15591 n r s - 22278 n s - 16295 n r - 18784 n r s - 37957 n r s 3 3 2 4 2 3 2 2 2 2 3 - 20738 n s + 318 n r + 7971 n r s + 12739 n r s + 2802 n r s 2 4 4 3 2 2 3 4 - 306 n s - 152 n r s + 494 n r s + 624 n r s + 134 n r s 4 2 3 3 2 4 5 4 4 - 20 r s - 32 r s - 12 r s + 11972 n + 89462 n r - 14666 n s 3 2 3 3 2 2 3 2 2 + 90032 n r + 19308 n r s - 66740 n s - 24404 n r - 31674 n r s 2 2 2 3 4 3 2 2 - 78374 n r s - 41616 n s + 272 n r + 11596 n r s + 16144 n r s 3 4 4 3 2 2 3 4 + 1912 n r s - 888 n s - 140 r s + 760 r s + 1020 r s + 264 r s 4 3 3 2 2 2 - 1948 n + 122008 n r - 79068 n s + 129552 n r - 10864 n r s 2 2 3 2 2 3 - 116256 n s - 15988 n r - 28620 n r s - 86100 n r s - 43428 n s 4 3 2 2 3 4 3 + 40 r + 7132 r s + 8028 r s - 588 r s - 900 s - 53224 n 2 2 2 2 + 79632 n r - 154704 n s + 93160 n r - 49128 n r s - 108000 n s 3 2 2 3 2 - 2448 r - 12192 r s - 40176 r s - 18432 s - 105072 n + 8880 n r 2 2 - 141696 n s + 24096 r - 32832 r s - 40608 s - 91296 n - 9216 r 11 - 50112 s - 31104) a(n + 2, k, r, s)/((n + 4) (n + 5) %2) - (6 n 10 10 9 2 9 9 2 8 3 + 104 n r + 82 n s + 418 n r + 558 n r s + 144 n s + 460 n r 8 2 8 2 8 3 7 3 7 2 2 + 858 n r s + 246 n r s - 124 n s - 10 n r s - 504 n r s 7 3 7 4 6 3 2 6 2 3 6 4 - 764 n r s - 280 n s - 110 n r s - 296 n r s - 212 n r s 6 5 10 9 9 8 2 8 - 32 n s + 142 n + 2247 n r + 1686 n s + 8754 n r + 10937 n r s 8 2 7 3 7 2 7 2 7 3 + 2366 n s + 9209 n r + 15933 n r s + 2974 n r s - 3220 n s 6 4 6 3 6 2 2 6 3 6 4 - 138 n r - 963 n r s - 10843 n r s - 15214 n r s - 5396 n s 5 4 5 3 2 5 2 3 5 4 5 5 + 3 n r s - 1874 n r s - 5038 n r s - 3640 n r s - 576 n s 4 4 2 4 3 3 4 2 4 9 8 + 33 n r s + 36 n r s + 6 n r s + 1442 n + 20680 n r 8 7 2 7 7 2 6 3 + 14410 n s + 78838 n r + 89852 n r s + 13748 n s + 79100 n r 6 2 6 2 6 3 5 4 5 3 + 123947 n r s + 5259 n r s - 35298 n s - 1746 n r - 13194 n r s 5 2 2 5 3 5 4 4 4 - 97314 n r s - 128234 n r s - 44068 n s + 163 n r s 4 3 2 4 2 3 4 4 4 5 - 13461 n r s - 36244 n r s - 26404 n r s - 4412 n s 3 4 2 3 3 3 3 2 4 3 5 3 6 + 356 n r s + 232 n r s - 236 n r s - 172 n r s - 24 n s 2 4 3 2 3 4 2 2 5 2 6 8 + 22 n r s + 46 n r s + 28 n r s + 4 n r s + 8084 n 7 7 6 2 6 6 2 + 103990 n r + 63618 n s + 397274 n r + 393970 n r s + 17572 n s 5 3 5 2 5 2 5 3 + 378480 n r + 516253 n r s - 96274 n r s - 214722 n s 4 4 4 3 4 2 2 4 3 - 8872 n r - 81380 n r s - 476787 n r s - 594010 n r s 4 4 3 4 3 3 2 3 2 3 - 197666 n s + 1485 n r s - 52384 n r s - 140964 n r s 3 4 3 5 2 4 2 2 3 3 - 103226 n r s - 18188 n s + 1475 n r s + 348 n r s 2 2 4 2 5 2 6 4 3 3 4 - 2296 n r s - 1604 n r s - 276 n s + 106 n r s + 238 n r s 2 5 6 7 6 6 + 168 n r s + 36 n r s + 26246 n + 297904 n r + 132536 n s 5 2 5 5 2 4 3 + 1217248 n r + 945582 n r s - 176840 n s + 1095086 n r 4 2 4 2 4 3 3 4 + 1206741 n r s - 735507 n r s - 794518 n s - 22910 n r 3 3 3 2 2 3 3 3 4 - 276900 n r s - 1385266 n r s - 1635464 n r s - 526058 n s 2 4 2 3 2 2 2 3 2 4 + 5427 n r s - 116915 n r s - 312988 n r s - 229738 n r s 2 5 4 2 3 3 2 4 5 - 42604 n s + 2768 n r s - 220 n r s - 6498 n r s - 4812 n r s 6 4 3 3 4 2 5 6 6 - 996 n s + 128 r s + 312 r s + 252 r s + 68 r s + 42938 n 5 5 4 2 4 4 2 + 414243 n r - 33960 n s + 2291864 n r + 980953 n r s - 1038754 n s 3 3 3 2 3 2 3 3 + 1941115 n r + 1445822 n r s - 2414946 n r s - 1831898 n s 2 4 2 3 2 2 2 2 3 - 31074 n r - 542651 n r s - 2396290 n r s - 2681230 n r s 2 4 4 3 2 2 3 - 831912 n s + 8946 n r s - 141888 n r s - 376762 n r s 4 5 4 2 3 3 2 4 - 277528 n r s - 54420 n s + 1976 r s - 488 r s - 5788 r s 5 6 5 4 4 - 4680 r s - 1140 s - 6886 n - 111752 n r - 919892 n s 3 2 3 3 2 2 3 + 2502232 n r - 652192 n r s - 2654964 n s + 2023286 n r 2 2 2 2 2 3 4 + 437890 n r s - 4240576 n r s - 2569780 n s - 20124 n r 3 2 2 3 4 - 578126 n r s - 2293660 n r s - 2431652 n r s - 726924 n s 4 3 2 2 3 4 5 + 5576 r s - 72800 r s - 192184 r s - 143148 r s - 30060 s 4 3 3 2 2 2 - 191276 n - 1456600 n r - 2402256 n s + 1261508 n r - 3008380 n r s 2 2 3 2 2 - 3701528 n s + 1097024 n r - 744036 n r s - 3888344 n r s 3 4 3 2 2 3 - 2005224 n s - 4320 r - 260840 r s - 941296 r s - 946856 r s 4 3 2 2 2 - 273240 s - 420552 n - 2436368 n r - 3108800 n s - 62776 n r 2 3 2 2 - 3160960 n r s - 2733840 n s + 218400 r - 573888 r s - 1465968 r s 3 2 2 - 666912 s - 460080 n - 1840080 n r - 2090784 n s - 232896 r 2 - 1169760 r s - 836928 s - 262656 n - 548928 r - 581760 s - 62208) a(n + 2, k + 1, r - 1, s + 1)/((n + 4) (n + 5) %2) - (2 + s) (n + 2) ( 8 8 7 2 7 7 2 6 2 12 n r + 18 n s + 40 n r + 98 n r s + 36 n s + 80 n r s 6 2 6 3 5 2 2 5 3 5 4 8 + 38 n r s - 32 n s - 50 n r s - 120 n r s - 64 n s + 12 n 7 7 6 2 6 6 2 5 3 + 276 n r + 332 n s + 822 n r + 1645 n r s + 454 n s - 12 n r 5 2 5 2 5 3 4 3 4 2 2 + 1121 n r s + 254 n r s - 708 n s - 24 n r s - 775 n r s 4 3 4 4 3 3 2 3 2 3 7 - 1764 n r s - 928 n s + 15 n r s + 12 n r s + 200 n 6 6 5 2 5 5 2 + 2506 n r + 2386 n s + 6960 n r + 11131 n r s + 1586 n s 4 3 4 2 4 2 4 3 3 3 - 156 n r + 6095 n r s - 1093 n r s - 6134 n s - 164 n r s 3 2 2 3 3 3 4 2 3 2 2 2 3 - 4571 n r s - 10128 n r s - 5260 n s + 110 n r s - 22 n r s 2 4 2 5 3 3 2 4 5 6 - 150 n r s - 48 n s + 10 n r s + 18 n r s + 8 n r s + 1316 n 5 5 4 2 4 4 2 + 11550 n r + 7774 n s + 31368 n r + 38113 n r s - 3030 n s 3 3 3 2 3 2 3 3 2 3 - 736 n r + 15605 n r s - 15182 n r s - 26996 n s - 332 n r s 2 2 2 2 3 2 4 3 2 2 3 - 13221 n r s - 28606 n r s - 14546 n s + 283 n r s - 396 n r s 4 5 3 3 2 4 5 5 - 856 n r s - 276 n s + 26 r s + 46 r s + 20 r s + 4060 n 4 4 3 2 3 3 2 + 27938 n r + 6252 n s + 80648 n r + 65995 n r s - 36058 n s 2 3 2 2 2 2 2 3 3 - 1572 n r + 16853 n r s - 54699 n r s - 63718 n s - 112 n r s 2 2 3 4 3 2 2 3 - 19295 n r s - 40078 n r s - 19566 n s + 260 r s - 662 r s 4 5 4 3 3 2 2 - 1198 r s - 384 s + 4048 n + 29438 n r - 32658 n s + 116810 n r 2 2 2 3 2 2 + 42922 n r s - 100620 n s - 1492 n r + 606 n r s - 84014 n r s 3 3 2 2 3 4 3 - 75900 n s + 176 r s - 11660 r s - 22480 r s - 10176 s - 9828 n 2 2 2 2 - 6672 n r - 104976 n s + 86456 n r - 18064 n r s - 121488 n s 3 2 2 3 2 - 480 r - 8160 r s - 47160 r s - 35016 s - 34128 n - 39864 n r 2 2 - 121656 n s + 24288 r - 25824 r s - 53856 s - 38448 n - 23040 r - 50688 s - 15552) a(n + 2, k + 2, r - 2, 2 + s)/((n + 4) (n + 5) %2) - 2 9 8 8 7 2 7 7 2 (2 n + 7) (6 n + 50 n r + 22 n s + 112 n r + 76 n r s - 4 n s 6 3 6 2 6 2 6 3 5 3 + 40 n r - 22 n r s - 106 n r s - 48 n s - 20 n r s 5 2 2 5 3 8 7 7 6 2 - 42 n r s - 16 n r s + 100 n + 811 n r + 328 n s + 1817 n r 6 6 2 5 3 5 2 5 2 + 1149 n r s - 142 n s + 512 n r - 536 n r s - 1800 n r s 5 3 4 4 4 3 4 2 2 4 3 - 816 n s - 12 n r - 267 n r s - 545 n r s - 208 n r s 3 4 3 3 2 7 6 6 5 2 + 6 n r s + 3 n r s + 670 n + 5265 n r + 1742 n s + 12225 n r 5 5 2 4 3 4 2 4 2 + 6835 n r s - 1750 n s + 2638 n r - 4593 n r s - 12457 n r s 4 3 3 4 3 3 3 2 2 3 3 - 5658 n s - 60 n r - 1330 n r s - 2852 n r s - 1188 n r s 3 4 2 4 2 3 2 2 2 3 2 4 - 36 n s + 28 n r s - 49 n r s - 56 n r s - 6 n r s 4 2 3 3 2 4 6 5 5 + 4 n r s + 6 n r s + 2 n r s + 2194 n + 16751 n r + 2632 n s 4 2 4 4 2 3 3 3 2 + 43701 n r + 19175 n r s - 10754 n s + 6784 n r - 19508 n r s 3 2 3 3 2 4 2 3 2 2 2 - 45157 n r s - 20440 n s - 100 n r - 3315 n r s - 7683 n r s 2 3 2 4 4 3 2 2 3 - 3740 n r s - 342 n s + 50 n r s - 246 n r s - 220 n r s 4 4 2 3 3 5 4 4 - 2 n r s + 4 r s + 4 r s + 2848 n + 22877 n r - 10812 n s 3 2 3 3 2 2 3 2 2 + 88031 n r + 20553 n r s - 36690 n s + 8810 n r - 44607 n r s 2 2 2 3 4 3 2 2 - 90772 n r s - 40606 n s - 60 n r - 4292 n r s - 10770 n r s 3 4 4 3 2 2 3 4 - 6398 n r s - 1104 n s + 28 r s - 284 r s - 204 r s + 36 r s 4 3 3 2 2 2 - 3326 n - 8526 n r - 56696 n s + 96450 n r - 14456 n r s 2 2 3 2 2 3 - 70292 n s + 4976 n r - 52994 n r s - 96728 n r s - 42324 n s 4 3 2 2 3 4 3 - 8 r - 2324 r s - 6264 r s - 4608 r s - 1188 s - 17780 n 2 2 2 2 - 68116 n r - 107040 n s + 48592 n r - 50748 n r s - 70200 n s 3 2 2 3 2 + 600 r - 25992 r s - 43344 r s - 18360 s - 26904 n - 79416 n r 2 2 - 94320 n s + 5664 r - 31104 r s - 28080 s - 18864 n - 30528 r - 31968 s 10 9 - 5184) a(n + 3, k, r, s)/((n + 4) (n + 5) %2) + (30 n + 232 n r 9 8 2 8 8 2 7 3 7 2 + 92 n s + 530 n r + 326 n r s - 44 n s + 300 n r - 480 n r s 7 3 6 3 6 2 2 6 3 6 4 9 - 200 n s - 150 n r s - 320 n r s - 108 n r s + 32 n s + 608 n 8 8 7 2 7 7 2 + 4535 n r + 1610 n s + 10056 n r + 5680 n r s - 1208 n s 6 3 6 2 6 2 6 3 5 4 + 5113 n r - 919 n r s - 9522 n r s - 3876 n s - 90 n r 5 3 5 2 2 5 3 5 4 4 4 - 2642 n r s - 5480 n r s - 1904 n r s + 512 n s + 45 n r s 4 3 2 4 2 3 8 7 7 + 24 n r s - 6 n r s + 5126 n + 37201 n r + 10610 n s 6 2 6 6 2 5 3 5 2 + 80592 n r + 39463 n r s - 14130 n s + 36251 n r - 15182 n r s 5 2 5 3 4 4 4 3 4 2 2 - 79930 n r s - 31712 n s - 894 n r - 18749 n r s - 38792 n r s 4 3 4 4 3 4 3 3 2 3 2 3 - 14204 n r s + 3188 n s + 432 n r s - 130 n r s - 408 n r s 3 4 3 5 2 4 2 2 3 3 2 2 4 - 8 n r s + 24 n s + 30 n r s + 46 n r s + 12 n r s 2 5 7 6 6 5 2 - 4 n r s + 22366 n + 163819 n r + 25850 n s + 352960 n r 5 5 2 4 3 4 2 + 131558 n r s - 92016 n s + 136737 n r - 104797 n r s 4 2 4 3 3 4 3 3 - 368514 n r s - 141958 n s - 3294 n r - 69464 n r s 3 2 2 3 3 3 4 2 4 - 145378 n r s - 57082 n r s + 9456 n s + 1573 n r s 2 3 2 2 2 3 2 4 2 5 4 2 - 2288 n r s - 3440 n r s - 272 n r s + 156 n s + 178 n r s 3 3 2 4 5 6 5 + 294 n r s + 104 n r s - 12 n r s + 46268 n + 402843 n r 5 4 2 4 4 2 3 3 - 47042 n s + 910398 n r + 164423 n r s - 364554 n s + 290637 n r 3 2 3 2 3 3 2 4 - 384744 n r s - 1009762 n r s - 376398 n s - 5446 n r 2 3 2 2 2 2 3 2 4 - 143905 n r s - 305634 n r s - 130594 n r s + 12136 n s 4 3 2 2 3 4 5 + 2654 n r s - 7434 n r s - 10086 n r s - 1408 n r s + 276 n s 4 2 3 3 2 4 5 5 4 + 256 r s + 464 r s + 212 r s + 4 r s - 10718 n + 473990 n r 4 3 2 3 3 2 - 480864 n s + 1378156 n r - 229426 n r s - 899248 n s 2 3 2 2 2 2 2 3 + 336546 n r - 788802 n r s - 1649784 n r s - 594968 n s 4 3 2 2 3 4 - 3684 n r - 161050 n r s - 345120 n r s - 162804 n r s + 1112 n s 4 3 2 2 3 4 5 4 + 1792 r s - 7160 r s - 9752 r s - 1940 r s + 84 s - 313288 n 3 3 2 2 2 - 62700 n r - 1378992 n s + 1122228 n r - 1036080 n r s 2 2 3 2 2 - 1349104 n s + 181480 n r - 854572 n r s - 1496536 n r s 3 4 3 2 2 3 4 - 525536 n s - 512 r - 77432 r s - 165680 r s - 87416 r s - 7656 s 3 2 2 2 - 821464 n - 899632 n r - 2031600 n s + 358280 n r - 1288384 n r s 2 3 2 2 3 - 1126848 n s + 25280 r - 382528 r s - 586512 r s - 203904 s 2 2 - 1071360 n - 1009392 n r - 1556160 n s - 25152 r - 570912 r s 2 - 402048 s - 730368 n - 379584 r - 490752 s - 207360) 9 9 a(n + 3, k + 1, r - 1, s + 1)/((n + 4) (n + 5) %1) + (6 n r + 18 n s 8 2 8 8 2 7 2 7 2 7 3 + 20 n r + 106 n r s + 48 n s + 130 n r s + 102 n r s - 24 n s 6 2 2 6 3 6 4 9 8 8 - 70 n r s - 172 n r s - 96 n s + 24 n + 264 n r + 424 n s 7 2 7 7 2 6 3 6 2 + 631 n r + 2151 n r s + 718 n s - 6 n r + 2131 n r s 6 2 6 3 5 3 5 2 2 5 3 + 1248 n r s - 824 n s - 39 n r s - 1263 n r s - 2956 n r s 5 4 4 3 2 4 2 3 8 7 7 - 1632 n s + 21 n r s + 18 n r s + 472 n + 3658 n r + 3930 n s 6 2 6 6 2 5 3 5 2 + 7463 n r + 18113 n r s + 3254 n s - 144 n r + 14010 n r s 5 2 5 3 4 3 4 2 2 4 3 + 3823 n r s - 9874 n s - 352 n r s - 9059 n r s - 20702 n r s 4 4 3 3 2 3 4 3 5 2 3 3 - 11312 n s + 202 n r s - 238 n r s - 72 n s + 14 n r s 2 2 4 2 5 7 6 6 + 26 n r s + 12 n r s + 3784 n + 24858 n r + 17438 n s 5 2 5 5 2 4 3 4 2 + 45629 n r + 80843 n r s - 3658 n s - 1070 n r + 46227 n r s 4 2 4 3 3 3 3 2 2 - 13564 n r s - 58960 n s - 1089 n r s - 33915 n r s 3 3 3 4 2 3 2 2 2 3 2 4 - 76284 n r s - 41010 n s + 755 n r s - 796 n r s - 2040 n r s 2 5 3 3 2 4 5 6 5 - 648 n s + 78 n r s + 150 n r s + 72 n r s + 15168 n + 94736 n r 5 4 2 4 4 2 3 3 + 29088 n s + 160441 n r + 199261 n r s - 88078 n s - 3568 n r 3 2 3 2 3 3 2 3 + 75312 n r s - 122569 n r s - 196042 n s - 1152 n r s 2 2 2 2 3 2 4 3 2 - 71719 n r s - 157918 n r s - 82490 n s + 1342 n r s 2 3 4 5 3 3 2 4 - 2758 n r s - 5726 n r s - 1932 n s + 104 r s + 212 r s 5 5 4 4 3 2 + 108 r s + 26248 n + 207486 n r - 58254 n s + 332540 n r 3 3 2 2 3 2 2 + 240934 n r s - 353396 n s - 5788 n r + 37898 n r s 2 2 2 3 3 2 2 - 335884 n r s - 368756 n s + 368 n r s - 83134 n r s 3 4 3 2 2 3 4 - 176924 n r s - 88344 n s + 960 r s - 2592 r s - 5244 r s 5 4 3 3 2 2 - 1908 s - 19704 n + 240992 n r - 375164 n s + 392044 n r 2 2 2 3 2 2 + 40272 n r s - 681624 n s - 4224 n r - 43628 n r s - 419924 n r s 3 3 2 2 3 4 - 367056 n s + 1032 r s - 41784 r s - 85176 r s - 40176 s 3 2 2 2 - 182216 n + 92336 n r - 745720 n s + 232336 n r - 198224 n r s 2 3 2 2 3 2 - 658800 n s - 944 r - 47552 r s - 204864 r s - 150768 s - 353056 n 2 2 - 71344 n r - 691632 n s + 47904 r - 142464 r s - 254880 s - 310944 n - 59520 r - 251136 s - 107136) a(n + 3, k + 2, r - 2, 2 + s)/((n + 4) 2 (2 n + 9) (n + r + 4) a(n + 4, k, r, s) 8 7 (n + 5) %1) + ----------------------------------------- - (18 n + 72 n r n + 5 7 6 2 6 6 2 5 2 5 2 - 6 n s + 40 n r - 82 n r s - 60 n s - 20 n r s - 2 n r s 5 3 7 6 6 5 2 5 + 48 n s + 270 n + 1177 n r - 98 n s + 564 n r - 1303 n r s 5 2 4 3 4 2 4 2 4 3 3 3 - 886 n s - 12 n r - 269 n r s - 24 n r s + 640 n s + 6 n r s 3 2 2 6 5 5 4 2 4 - 9 n r s + 1438 n + 7790 n r - 818 n s + 3074 n r - 8484 n r s 4 2 3 3 3 2 3 2 3 3 - 5256 n s - 144 n r - 1457 n r s - 139 n r s + 3258 n s 2 3 2 2 2 2 3 2 4 3 2 2 3 + 70 n r s - 48 n r s + 62 n r s + 36 n s + 4 n r s - 2 n r s 4 5 4 4 3 2 3 - 6 n r s + 2260 n + 26836 n r - 4398 n s + 8410 n r - 28780 n r s 3 2 2 3 2 2 2 2 2 3 - 16092 n s - 436 n r - 3823 n r s - 528 n r s + 7822 n s 3 2 2 3 4 3 2 2 3 + 204 n r s - 199 n r s + 242 n r s + 210 n s + 32 r s + 14 r s 4 4 3 3 2 2 2 - 18 r s - 8166 n + 50580 n r - 15312 n s + 12018 n r - 54120 n r s 2 2 3 2 2 3 - 27292 n s - 424 n r - 5045 n r s - 1235 n r s + 8730 n s 3 2 2 3 4 3 2 + 224 r s - 280 r s + 216 r s + 306 s - 44362 n + 48387 n r 2 2 2 3 2 - 32300 n s + 8066 n r - 54497 n r s - 25326 n s - 64 r - 2866 r s 2 3 2 2 - 1212 r s + 3546 s - 85258 n + 15878 n r - 36804 n s + 1644 r 2 - 23550 r s - 10728 s - 77232 n - 3360 r - 17208 s - 27432) 8 7 7 a(n + 4, k + 1, r - 1, s + 1)/((n + 5) %1) - (6 n + 38 n r + 28 n s 6 2 6 5 2 5 2 5 3 7 + 60 n r + 82 n r s - 30 n r s - 88 n r s - 64 n s + 112 n 6 6 5 2 5 5 2 4 3 + 615 n r + 346 n s + 827 n r + 1014 n r s - 188 n s - 18 n r 4 2 4 2 4 3 3 3 3 2 2 - 445 n r s - 1212 n r s - 864 n s + 9 n r s + 12 n r s 6 5 5 4 2 4 4 2 + 786 n + 4075 n r + 1290 n s + 4505 n r + 4549 n r s - 2522 n s 3 3 3 2 3 2 3 3 2 3 - 114 n r - 2423 n r s - 6440 n r s - 4500 n s + 54 n r s 2 2 2 2 3 2 4 3 2 2 3 4 + 2 n r s - 130 n r s - 48 n s + 6 n r s + 14 n r s + 8 n r s 5 4 4 3 2 3 + 2314 n + 14075 n r - 666 n s + 12135 n r + 8000 n r s 3 2 2 3 2 2 2 2 2 3 - 13104 n s - 246 n r - 6327 n r s - 16790 n r s - 11406 n s 3 2 2 3 4 3 2 2 3 + 119 n r s - 204 n r s - 716 n r s - 300 n s + 14 r s + 42 r s 4 4 3 3 2 2 2 + 28 r s + 484 n + 26551 n r - 17186 n s + 16491 n r + 179 n r s 2 2 3 2 2 3 - 32990 n s - 194 n r - 8261 n r s - 22146 n r s - 14422 n s 3 2 2 3 4 3 2 + 98 r s - 298 r s - 942 r s - 456 s - 14650 n + 25314 n r 2 2 2 3 2 - 47996 n s + 9870 n r - 15276 n r s - 40420 n s - 28 r - 4450 r s 2 3 2 2 - 12240 r s - 7596 s - 38420 n + 8124 n r - 56248 n s + 1400 r 2 - 12964 r s - 19488 s - 40632 n - 1896 r - 24624 s - 15984) / 6 5 5 a(n + 4, k + 2, r - 2, 2 + s) / ((n + 5) (6 n + 20 n r + 4 n s / 4 4 2 5 4 4 3 2 3 - 10 n r s - 16 n s + 46 n + 191 n r + 10 n s - 6 n r - 98 n r s 3 2 2 2 4 3 3 2 2 - 144 n s + 3 n r s + 76 n + 696 n r - 108 n s - 24 n r 2 2 2 2 2 3 2 2 - 343 n r s - 474 n s + 11 n r s - 16 n r s - 12 n s + 2 r s 3 3 2 2 2 2 + 2 r s - 254 n + 1167 n r - 574 n s - 26 n r - 547 n r s - 698 n s 2 2 3 2 2 + 14 r s - 34 r s - 30 s - 1098 n + 830 n r - 988 n s - 4 r - 358 r s 2 - 408 s - 1440 n + 144 r - 576 s - 648)) - a(n + 5, k + 1, r - 1, s + 1) + a(n + 5, k + 2, r - 2, 2 + s) = 0 6 5 5 4 4 2 5 4 %1 := 6 n + 20 n r + 4 n s - 10 n r s - 16 n s + 46 n + 191 n r 4 3 2 3 3 2 2 2 4 3 + 10 n s - 6 n r - 98 n r s - 144 n s + 3 n r s + 76 n + 696 n r 3 2 2 2 2 2 2 2 - 108 n s - 24 n r - 343 n r s - 474 n s + 11 n r s - 16 n r s 3 2 2 3 3 2 2 2 - 12 n s + 2 r s + 2 r s - 254 n + 1167 n r - 574 n s - 26 n r 2 2 2 3 2 - 547 n r s - 698 n s + 14 r s - 34 r s - 30 s - 1098 n + 830 n r 2 2 - 988 n s - 4 r - 358 r s - 408 s - 1440 n + 144 r - 576 s - 648 6 5 5 4 4 2 5 4 %2 := 6 n + 20 n r + 4 n s - 10 n r s - 16 n s + 46 n + 191 n r 4 3 2 3 3 2 2 2 4 3 + 10 n s - 6 n r - 98 n r s - 144 n s + 3 n r s + 76 n + 696 n r 3 2 2 2 2 2 2 2 - 108 n s - 24 n r - 343 n r s - 474 n s + 11 n r s - 16 n r s 3 2 2 3 3 2 2 2 - 12 n s + 2 r s + 2 r s - 254 n + 1167 n r - 574 n s - 26 n r 2 2 2 3 2 - 547 n r s - 698 n s + 14 r s - 34 r s - 30 s - 1098 n + 830 n r 2 2 - 988 n s - 4 r - 358 r s - 408 s - 1440 n + 144 r - 576 s - 648 6 5 5 4 4 2 5 4 %3 := 6 n + 20 n r + 4 n s - 10 n r s - 16 n s + 82 n + 291 n r 4 3 2 3 3 2 2 2 4 + 30 n s - 6 n r - 138 n r s - 208 n s + 3 n r s + 396 n 3 3 2 2 2 2 2 2 + 1660 n r - 28 n s - 42 n r - 697 n r s - 1002 n s + 17 n r s 2 3 2 2 3 3 2 2 - 16 n r s - 12 n s + 2 r s + 2 r s + 630 n + 4601 n r - 798 n s 2 2 2 2 3 2 - 92 n r - 1567 n r s - 2142 n s + 28 r s - 50 r s - 42 s - 854 n 2 2 + 6116 n r - 2400 n s - 60 r - 1356 r s - 1740 s - 3828 n + 3048 r - 2232 s - 3312 and in Maple notation: -(r+s)^2*(n+2)*(n+1)*(n+s+3)*(2+n+s)^2*(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^ 4*s^2+82*n^5+291*n^4*r+30*n^4*s-6*n^3*r^2-138*n^3*r*s-208*n^3*s^2+3*n^2*r^2*s+ 396*n^4+1660*n^3*r-28*n^3*s-42*n^2*r^2-697*n^2*r*s-1002*n^2*s^2+17*n*r^2*s-16*n *r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3+630*n^3+4601*n^2*r-798*n^2*s-92*n*r^2-1567*n* r*s-2142*n*s^2+28*r^2*s-50*r*s^2-42*s^3-854*n^2+6116*n*r-2400*n*s-60*r^2-1356*r *s-1740*s^2-3828*n+3048*r-2232*s-3312)/(n+4)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s-10*n ^4*r*s-16*n^4*s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s^2+3* n^2*r^2*s+76*n^4+696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11*n*r^ 2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26*n*r^2 -547*n*r*s-698*n*s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-988*n*s-4*r^2-\ 358*r*s-408*s^2-1440*n+144*r-576*s-648)*a(n,k+1,r-1,s+1)-2*(r+s)^2*(n+2)*(2*n+3 )*(n+s+3)*(2+n-r)*(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+82*n^5+291*n^4* r+30*n^4*s-6*n^3*r^2-138*n^3*r*s-208*n^3*s^2+3*n^2*r^2*s+396*n^4+1660*n^3*r-28* n^3*s-42*n^2*r^2-697*n^2*r*s-1002*n^2*s^2+17*n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2* s^2+2*r*s^3+630*n^3+4601*n^2*r-798*n^2*s-92*n*r^2-1567*n*r*s-2142*n*s^2+28*r^2* s-50*r*s^2-42*s^3-854*n^2+6116*n*r-2400*n*s-60*r^2-1356*r*s-1740*s^2-3828*n+ 3048*r-2232*s-3312)/(n+4)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+ 46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s^2+3*n^2*r^2*s+76*n^4+ 696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11*n*r^2*s-16*n*r*s^2-12 *n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26*n*r^2-547*n*r*s-698*n* s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-988*n*s-4*r^2-358*r*s-408*s^2-\ 1440*n+144*r-576*s-648)*a(n+1,k,r,s)+(r+s)*(n+2)*(12*n^10+94*n^9*r+74*n^9*s+180 *n^8*r^2+320*n^8*r*s+84*n^8*s^2+190*n^7*r^2*s+72*n^7*r*s^2-110*n^7*s^3-40*n^6*r ^2*s^2-264*n^6*r*s^3-180*n^6*s^4-50*n^5*r^2*s^3-110*n^5*r*s^4-48*n^5*s^5+260*n^ 9+1954*n^8*r+1384*n^8*s+3687*n^7*r^2+5883*n^7*r*s+1086*n^7*s^2-54*n^6*r^3+3162* n^6*r^2*s+348*n^6*r*s^2-2552*n^6*s^3-57*n^5*r^3*s-976*n^5*r^2*s^2-4757*n^5*r*s^ 3-3078*n^5*s^4+12*n^4*r^3*s^2-742*n^4*r^2*s^3-1616*n^4*r*s^4-704*n^4*s^5+15*n^3 *r^3*s^3+9*n^3*r^2*s^4+2384*n^8+17280*n^7*r+10418*n^7*s+32473*n^6*r^2+44997*n^6 *r*s+3286*n^6*s^2-702*n^5*r^3+21755*n^5*r^2*s-5807*n^5*r*s^2-24332*n^5*s^3-552* n^4*r^3*s-8733*n^4*r^2*s^2-35175*n^4*r*s^3-21670*n^4*s^4+184*n^3*r^3*s^2-4511*n ^3*r^2*s^3-9623*n^3*r*s^4-4158*n^3*s^5+134*n^2*r^3*s^3-116*n^2*r*s^5-36*n^2*s^6 +10*n*r^3*s^4+16*n*r^2*s^5+6*n*r*s^6+11780*n^7+83980*n^6*r+37806*n^6*s+160011*n ^5*r^2+182811*n^5*r*s-18882*n^5*s^2-3696*n^4*r^3+78218*n^4*r^2*s-66278*n^4*r*s^ 2-124582*n^4*s^3-1909*n^3*r^3*s-39460*n^3*r^2*s^2-137137*n^3*r*s^3-80352*n^3*s^ 4+952*n^2*r^3*s^2-14066*n^2*r^2*s^3-28878*n^2*r*s^4-12200*n^2*s^5+391*n*r^3*s^3 -259*n*r^2*s^4-716*n*r*s^5-222*n*s^6+26*r^3*s^4+44*r^2*s^5+18*r*s^6+31904*n^6+ 239858*n^5*r+47568*n^5*s+480195*n^4*r^2+411675*n^4*r*s-177934*n^4*s^2-10014*n^3 *r^3+151251*n^3*r^2*s-287119*n^3*r*s^2-370722*n^3*s^3-2652*n^2*r^3*s-97793*n^2* r^2*s^2-297603*n^2*r*s^3-165020*n^2*s^4+2024*n*r^3*s^2-22599*n*r^2*s^3-43689*n* r*s^4-17658*n*s^5+392*r^3*s^3-466*r^2*s^4-1056*r*s^5-342*s^6+33968*n^5+389050*n ^4*r-126358*n^4*s+893070*n^3*r^2+460330*n^3*r*s-597704*n^3*s^2-14562*n^2*r^3+ 137908*n^2*r^2*s-630536*n^2*r*s^2-640342*n^2*s^3-802*n*r^3*s-127698*n*r^2*s^2-\ 340704*n*r*s^3-177084*n*s^4+1556*r^3*s^2-14992*r^2*s^3-26928*r*s^4-10188*s^5-\ 59068*n^4+275928*n^3*r-614860*n^3*s+996080*n^2*r^2+90096*n^2*r*s-1033552*n^2*s^ 2-10572*n*r^3+20892*n*r^2*s-697312*n*r*s^2-592368*n*s^3+756*r^3*s-69252*r^2*s^2 -161100*r*s^3-77220*s^4-264744*n^3-103776*n^2*r-1062432*n^2*s+600336*n*r^2-\ 282432*n*r*s-914160*n*s^2-2880*r^3-33408*r^2*s-307920*r*s^2-225072*s^3-406512*n ^2-298224*n*r-882144*n*s+145728*r^2-188064*r*s-325728*s^2-304992*n-138240*r-\ 292032*s-93312)/(n+4)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+46*n^ 5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s^2+3*n^2*r^2*s+76*n^4+696*n^ 3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11*n*r^2*s-16*n*r*s^2-12*n*s^3 +2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26*n*r^2-547*n*r*s-698*n*s^2+14 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4-72*n^3*s^5+14*n^2*r^3*s^3+26*n^2*r^2*s^4+12*n^2*r*s^5+3784*n^7+24858*n^6*r+ 17438*n^6*s+45629*n^5*r^2+80843*n^5*r*s-3658*n^5*s^2-1070*n^4*r^3+46227*n^4*r^2 *s-13564*n^4*r*s^2-58960*n^4*s^3-1089*n^3*r^3*s-33915*n^3*r^2*s^2-76284*n^3*r*s ^3-41010*n^3*s^4+755*n^2*r^3*s^2-796*n^2*r^2*s^3-2040*n^2*r*s^4-648*n^2*s^5+78* n*r^3*s^3+150*n*r^2*s^4+72*n*r*s^5+15168*n^6+94736*n^5*r+29088*n^5*s+160441*n^4 *r^2+199261*n^4*r*s-88078*n^4*s^2-3568*n^3*r^3+75312*n^3*r^2*s-122569*n^3*r*s^2 -196042*n^3*s^3-1152*n^2*r^3*s-71719*n^2*r^2*s^2-157918*n^2*r*s^3-82490*n^2*s^4 +1342*n*r^3*s^2-2758*n*r^2*s^3-5726*n*r*s^4-1932*n*s^5+104*r^3*s^3+212*r^2*s^4+ 108*r*s^5+26248*n^5+207486*n^4*r-58254*n^4*s+332540*n^3*r^2+240934*n^3*r*s-\ 353396*n^3*s^2-5788*n^2*r^3+37898*n^2*r^2*s-335884*n^2*r*s^2-368756*n^2*s^3+368 *n*r^3*s-83134*n*r^2*s^2-176924*n*r*s^3-88344*n*s^4+960*r^3*s^2-2592*r^2*s^3-\ 5244*r*s^4-1908*s^5-19704*n^4+240992*n^3*r-375164*n^3*s+392044*n^2*r^2+40272*n^ 2*r*s-681624*n^2*s^2-4224*n*r^3-43628*n*r^2*s-419924*n*r*s^2-367056*n*s^3+1032* r^3*s-41784*r^2*s^2-85176*r*s^3-40176*s^4-182216*n^3+92336*n^2*r-745720*n^2*s+ 232336*n*r^2-198224*n*r*s-658800*n*s^2-944*r^3-47552*r^2*s-204864*r*s^2-150768* s^3-353056*n^2-71344*n*r-691632*n*s+47904*r^2-142464*r*s-254880*s^2-310944*n-\ 59520*r-251136*s-107136)/(n+4)/(n+5)/(6*n^6+20*n^5*r+4*n^5*s-10*n^4*r*s-16*n^4* s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r*s-144*n^3*s^2+3*n^2*r^2*s+76*n ^4+696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474*n^2*s^2+11*n*r^2*s-16*n*r*s^2 -12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-574*n^2*s-26*n*r^2-547*n*r*s-698 *n*s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n*r-988*n*s-4*r^2-358*r*s-408*s^2-\ 1440*n+144*r-576*s-648)*a(n+3,k+2,r-2,2+s)+2*(2*n+9)*(n+r+4)/(n+5)*a(n+4,k,r,s) -(18*n^8+72*n^7*r-6*n^7*s+40*n^6*r^2-82*n^6*r*s-60*n^6*s^2-20*n^5*r^2*s-2*n^5*r *s^2+48*n^5*s^3+270*n^7+1177*n^6*r-98*n^6*s+564*n^5*r^2-1303*n^5*r*s-886*n^5*s^ 2-12*n^4*r^3-269*n^4*r^2*s-24*n^4*r*s^2+640*n^4*s^3+6*n^3*r^3*s-9*n^3*r^2*s^2+ 1438*n^6+7790*n^5*r-818*n^5*s+3074*n^4*r^2-8484*n^4*r*s-5256*n^4*s^2-144*n^3*r^ 3-1457*n^3*r^2*s-139*n^3*r*s^2+3258*n^3*s^3+70*n^2*r^3*s-48*n^2*r^2*s^2+62*n^2* r*s^3+36*n^2*s^4+4*n*r^3*s^2-2*n*r^2*s^3-6*n*r*s^4+2260*n^5+26836*n^4*r-4398*n^ 4*s+8410*n^3*r^2-28780*n^3*r*s-16092*n^3*s^2-436*n^2*r^3-3823*n^2*r^2*s-528*n^2 *r*s^2+7822*n^2*s^3+204*n*r^3*s-199*n*r^2*s^2+242*n*r*s^3+210*n*s^4+32*r^3*s^2+ 14*r^2*s^3-18*r*s^4-8166*n^4+50580*n^3*r-15312*n^3*s+12018*n^2*r^2-54120*n^2*r* s-27292*n^2*s^2-424*n*r^3-5045*n*r^2*s-1235*n*r*s^2+8730*n*s^3+224*r^3*s-280*r^ 2*s^2+216*r*s^3+306*s^4-44362*n^3+48387*n^2*r-32300*n^2*s+8066*n*r^2-54497*n*r* s-25326*n*s^2-64*r^3-2866*r^2*s-1212*r*s^2+3546*s^3-85258*n^2+15878*n*r-36804*n *s+1644*r^2-23550*r*s-10728*s^2-77232*n-3360*r-17208*s-27432)/(n+5)/(6*n^6+20*n ^5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r *s-144*n^3*s^2+3*n^2*r^2*s+76*n^4+696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-\ 474*n^2*s^2+11*n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r -574*n^2*s-26*n*r^2-547*n*r*s-698*n*s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n *r-988*n*s-4*r^2-358*r*s-408*s^2-1440*n+144*r-576*s-648)*a(n+4,k+1,r-1,s+1)-(6* n^8+38*n^7*r+28*n^7*s+60*n^6*r^2+82*n^6*r*s-30*n^5*r^2*s-88*n^5*r*s^2-64*n^5*s^ 3+112*n^7+615*n^6*r+346*n^6*s+827*n^5*r^2+1014*n^5*r*s-188*n^5*s^2-18*n^4*r^3-\ 445*n^4*r^2*s-1212*n^4*r*s^2-864*n^4*s^3+9*n^3*r^3*s+12*n^3*r^2*s^2+786*n^6+ 4075*n^5*r+1290*n^5*s+4505*n^4*r^2+4549*n^4*r*s-2522*n^4*s^2-114*n^3*r^3-2423*n ^3*r^2*s-6440*n^3*r*s^2-4500*n^3*s^3+54*n^2*r^3*s+2*n^2*r^2*s^2-130*n^2*r*s^3-\ 48*n^2*s^4+6*n*r^3*s^2+14*n*r^2*s^3+8*n*r*s^4+2314*n^5+14075*n^4*r-666*n^4*s+ 12135*n^3*r^2+8000*n^3*r*s-13104*n^3*s^2-246*n^2*r^3-6327*n^2*r^2*s-16790*n^2*r *s^2-11406*n^2*s^3+119*n*r^3*s-204*n*r^2*s^2-716*n*r*s^3-300*n*s^4+14*r^3*s^2+ 42*r^2*s^3+28*r*s^4+484*n^4+26551*n^3*r-17186*n^3*s+16491*n^2*r^2+179*n^2*r*s-\ 32990*n^2*s^2-194*n*r^3-8261*n*r^2*s-22146*n*r*s^2-14422*n*s^3+98*r^3*s-298*r^2 *s^2-942*r*s^3-456*s^4-14650*n^3+25314*n^2*r-47996*n^2*s+9870*n*r^2-15276*n*r*s -40420*n*s^2-28*r^3-4450*r^2*s-12240*r*s^2-7596*s^3-38420*n^2+8124*n*r-56248*n* s+1400*r^2-12964*r*s-19488*s^2-40632*n-1896*r-24624*s-15984)/(n+5)/(6*n^6+20*n^ 5*r+4*n^5*s-10*n^4*r*s-16*n^4*s^2+46*n^5+191*n^4*r+10*n^4*s-6*n^3*r^2-98*n^3*r* s-144*n^3*s^2+3*n^2*r^2*s+76*n^4+696*n^3*r-108*n^3*s-24*n^2*r^2-343*n^2*r*s-474 *n^2*s^2+11*n*r^2*s-16*n*r*s^2-12*n*s^3+2*r^2*s^2+2*r*s^3-254*n^3+1167*n^2*r-\ 574*n^2*s-26*n*r^2-547*n*r*s-698*n*s^2+14*r^2*s-34*r*s^2-30*s^3-1098*n^2+830*n* r-988*n*s-4*r^2-358*r*s-408*s^2-1440*n+144*r-576*s-648)*a(n+4,k+2,r-2,2+s)-a(n+ 5,k+1,r-1,s+1)+a(n+5,k+2,r-2,2+s) = 0 The proof of this identity is routine (divide by a(n,k,r,s), simplify each t\ erm,and now each term is a rational function. Now add them all up and \ verify that they add up to zero.) Now sum it from k=0 to k=n, which is the same as from k=-infinity to k=infin\ ity (since it vanishes for k<0 and k>n). Since the above recurrence is \ free from k, we get the claimed recurrence. QED. ------------------------------------------------- This took, 286134.379, seconds. -------------------------