Theorem: define the Abel-sum type sequence by n ----- \ (3 k - 1) (n - k + 3) A[n](r, s) = ) binomial(n, k) (r + 2 k) (s - 5 k) / ----- k = 0 and in Maple notation A[n](r,s) = Sum(binomial(n,k)*(r+2*k)^(3*k-1)*(s-5*k)^(n-k+3),k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s (3 n s - 5 n + 3 s - 5) A[n](r, s) - ---------------------------------- 3 n + 2 /d \ (5 n r + 2 n s + 5 r + 2 s) |-- A[n](r, s)| \dr / + ------------------------------------------- + A[n + 1](r, s) 3 n + 2 /d \ (2 n + r + 2) |-- A[n + 1](r, s)| \dr / - --------------------------------- = 0 3 n + 2 2 (-5/3 n + 1/3 s n - 25/3 n + 1/3 s - 20/3) A[n](r, s) + A[n + 1](r, s) /5 n \ /d \ + |--- - s/3 + 5/3| |-- A[n + 1](r, s)| = 0 \ 3 / \ds / and in Maple notation -(3*n*s-5*n+3*s-5)/(3*n+2)*A[n](r,s)+(5*n*r+2*n*s+5*r+2*s)/(3*n+2)*diff(A[n](r, s),r)+A[n+1](r,s)-(2*n+r+2)/(3*n+2)*diff(A[n+1](r,s),r) = 0 (-5/3*n^2+1/3*s*n-25/3*n+1/3*s-20/3)*A[n](r,s)+A[n+1](r,s)+(5/3*n-1/3*s+5/3)* diff(A[n+1](r,s),s) = 0 ------------------------------------------------- This took, 0.143, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ A[n](r, s) = ) / ----- k = 0 2 (3 k) (3 k - 1) (n - k + 3) binomial(n, k) (r + 5 k) (r + 2 k) (s - 5 k) and in Maple notation A[n](r,s) = Sum(binomial(n,k)^2*(r+5*k)^(3*k)*(r+2*k)^(3*k-1)*(s-5*k)^(n-k+3),k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 21 8 3 8 2 2 7 3 2 - -- (71938125000 n r s + 50356687500 n r s - 115101000000 n r s 50 7 2 3 6 5 6 4 2 - 80570700000 n r s - 19348875000 n r s - 46019137500 n r s 6 3 3 6 2 4 5 6 + 46328152500 n r s + 48342420000 n r s + 30214012500 n r s 5 5 2 5 4 3 5 3 4 + 94437393750 n r s + 111014064000 n r s + 23382768150 n r s 5 2 5 4 7 4 6 2 - 12891312000 n r s + 16297706250 n r s + 23604564375 n r s 4 5 3 4 4 4 4 3 5 - 15444897750 n r s - 44821222425 n r s - 17781953490 n r s 4 2 6 3 8 3 7 2 + 1289131200 n r s + 2083725000 n r s - 6935827500 n r s 3 6 3 3 5 4 3 4 5 - 24314944500 n r s - 19123176600 n r s - 613369260 n r s 3 3 6 2 9 2 8 2 + 2616475932 n r s - 49612500 n r s - 2437391250 n r s 2 7 3 2 6 4 2 5 5 - 3964727250 n r s + 88650450 n r s + 3072073230 n r s 2 4 6 9 2 8 3 + 1323990360 n r s - 99225000 n r s + 194197500 n r s 7 4 6 5 5 6 + 901651500 n r s + 880154100 n r s + 264732300 n r s 10 2 9 3 8 4 7 5 + 2835000 r s + 38434500 r s + 83689200 r s + 66279060 r s 6 6 8 3 8 2 + 17889984 r s - 59948437500 n r + 22739062500 n r s 8 2 7 3 7 2 2 + 105840000000 n r s + 1103051250000 n r s + 668611125000 n r s 7 3 6 5 6 4 - 169344000000 n r s + 16124062500 n r + 24050250000 n r s 6 3 2 6 2 3 6 4 - 1505992500000 n r s - 1014575625000 n r s + 101606400000 n r s 5 6 5 5 5 4 2 - 25178343750 n r - 176049956250 n r s - 302415828750 n r s 5 3 3 5 2 4 5 5 + 744772556250 n r s + 618531133500 n r s - 27095040000 n r s 4 7 4 6 4 5 2 - 13581421875 n r + 327146596875 n r s + 978397683750 n r s 4 4 3 4 3 4 4 2 5 + 1019853858375 n r s + 117476447175 n r s - 170411642100 n r s 4 6 3 8 3 7 + 2709504000 n r s - 1736437500 n r + 141296400000 n r s 3 6 2 3 5 3 3 4 4 + 185559963750 n r s - 169770775500 n r s - 390329605350 n r s 3 3 5 3 2 6 2 9 - 139644278640 n r s + 16783222680 n r s + 41343750 n r 2 8 2 7 2 2 6 3 + 14867212500 n r s - 42102568125 n r s - 149719158000 n r s 2 5 4 2 4 5 2 3 6 - 110716636725 n r s + 4296733830 n r s + 18862586802 n r s 9 8 2 7 3 - 16537500 n r s - 9518985000 n r s - 14766104250 n r s 6 4 5 5 4 6 + 2963366100 n r s + 14497245810 n r s + 5949234648 n r s 10 9 2 8 3 7 4 - 2362500 r s - 186806250 r s + 517225500 r s + 2054347650 r s 6 5 5 6 8 2 + 1946282580 r s + 579210588 r s - 68906250000 n r 8 8 2 7 3 - 145687500000 n r s + 64181250000 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25512704360000 n r s + 12188749259250 n r s 2 5 2 6 7 - 2786716500000 n r s + 44153184000 n s - 484502625000 n r 6 5 2 4 3 + 3417274250000 n r s + 9129966570000 n r s + 6125678844000 n r s 3 4 2 5 6 - 1931784001400 n r s - 2220023938800 n r s + 147438048000 n r s 8 7 6 2 - 14182250000 r + 352405662500 r s + 355265502500 r s 5 3 4 4 3 5 - 617444722000 r s - 951897898700 r s - 233027483900 r s 2 6 7 6 + 80846822640 r s + 437500000000 n + 1394531250000 n r 6 5 2 5 - 10294531250000 n s - 20230000000000 n r - 44376687500000 n r s 5 2 4 3 4 2 + 30673737500000 n s - 37871250000000 n r + 34367218750000 n r s 4 2 4 3 3 4 + 110340593750000 n r s - 24208203750000 n s - 2327875000000 n r 3 3 3 2 2 + 87455015000000 n r s + 20946839000000 n r s 3 3 3 4 2 5 - 72001645500000 n r s + 8631840300000 n s - 5804565000000 n r 2 4 2 3 2 - 9136494375000 n r s - 47286228600000 n r s 2 2 3 2 4 2 5 - 15240551575000 n r s + 23343045100000 n r s - 1325533360000 n s 6 5 4 2 - 3273351500000 n r + 75958050000 n r s + 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79746875000000 n s 3 2 3 3 2 - 58292500000000 n r - 130779500000000 n r s + 113078800000000 n s 2 3 2 2 2 2 - 44678025000000 n r + 45769725000000 n r s + 146887920000000 n r s 2 3 4 3 - 50044420000000 n s - 6115450000000 n r + 27458030000000 n r s 2 2 3 4 - 5762100000000 n r s - 45978658000000 n r s + 9509857200000 n s 5 4 3 2 - 2801317500000 r - 4236049000000 r s - 5755699300000 r s 2 3 4 5 + 1884391400000 r s + 5904848400000 r s - 528763200000 s 4 3 3 + 19375000000000 n + 13650000000000 n r - 110270000000000 n s 2 2 2 2 2 - 48545000000000 n r - 110113000000000 n r s + 105116200000000 n s 3 2 2 - 22143000000000 n r + 21170000000000 n r s + 72532300000000 n r s 3 4 3 - 30653820000000 n s - 2514900000000 r + 3077100000000 r s 2 2 3 4 - 3918572000000 r s - 11328268000000 r s + 3037539200000 s 3 2 2 + 26000000000000 n + 11287500000000 n r - 93967500000000 n s 2 2 - 22750000000000 n r - 52175000000000 n r s + 54665000000000 n s 3 2 2 - 4885200000000 r + 3698300000000 r s + 14895460000000 r 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2 3 4 3 - 202848146989400 n r s - 159569782188400 n r s 3 3 4 2 8 2 7 + 35800595956800 n r s + 4441352192500 n r - 36387116116000 n r s 2 6 2 2 5 3 - 63118530448700 n r s - 17641829173800 n r s 2 4 4 9 8 + 15502927608720 n r s - 559835897500 n r - 5623240081000 n r s 7 2 6 3 5 4 - 5658098015900 n r s + 1566487398400 n r s + 2939146975296 n r s 10 9 8 2 - 40964998750 r - 215152845500 r s - 56686206950 r s 7 3 6 4 9 + 290064001400 r s + 198127821528 r s + 309475000000000 n 8 8 7 2 + 3772511625000000 n r - 379550625000000 n s + 8445914587500000 n r 7 7 2 6 3 - 4203406050000000 n r s + 185998200000000 n s + 3981983136250000 n r 6 2 6 2 - 8569390623750000 n r s + 1944796779750000 n r s 6 3 5 4 5 3 - 40709870000000 n s + 1095486408250000 n r - 2909120304500000 n r s 5 2 2 5 3 + 4115463407700000 n r s - 402823308800000 n r s 5 4 4 5 4 4 + 3108712000000 n s + 1091051865062500 n r + 1193840549650000 n r s 4 3 2 4 2 3 + 2162193481312500 n r s - 887105105000000 n r s 4 4 3 6 3 5 + 27947048800000 n r s + 513681840425000 n r + 503870761630000 n r s 3 4 2 3 3 3 - 42685162731000 n r s - 626867560660000 n r s 3 2 4 2 7 2 6 + 58439139280000 n r s + 79046195412500 n r - 49286768530000 n r s 2 5 2 2 4 3 - 239645583775500 n r s - 159592127344000 n r s 2 3 4 8 7 + 41606705897600 n r s + 1716138080000 n r - 28460701426000 n r s 6 2 5 3 4 4 - 44122217371200 n r s - 9696994652800 n r s + 11472209051360 n r s 9 8 7 2 - 233099822500 r - 1972921267000 r s - 1804825665300 r s 6 3 5 4 8 + 722210928960 r s + 1048851977600 r s + 1113728125000000 n 7 7 6 2 + 9686399750000000 n r - 1222152500000000 n s + 16292198181250000 n r 6 6 2 5 3 - 9491796875000000 n r s + 544768712500000 n s + 6463060697500000 n r 5 2 5 2 - 13970811885000000 n r s + 3946066974000000 n r s 5 3 4 4 - 107786220000000 n s + 1932657475375000 n r 4 3 4 2 2 - 3084925656750000 n r s + 5978138406675000 n r s 4 3 4 4 3 5 - 727006096000000 n r s + 7161320000000 n s + 1409695599500000 n r 3 4 3 3 2 + 1495527716600000 n r s + 2159715574660000 n r s 3 2 3 3 4 - 1128811783280000 n r s + 42469340800000 n r s 2 6 2 5 + 438978819475000 n r + 343778168700000 n r s 2 4 2 2 3 3 - 174868667091000 n r s - 553570076304000 n r s 2 2 4 7 6 + 59317189824000 n r s + 39733799520000 n r - 48123520864000 n r s 5 2 4 3 - 151988779672800 n r s - 86986210769600 n r s 3 4 8 7 + 26335738028800 n r s + 126347160000 r - 8962081734000 r s 6 2 5 3 4 4 - 12627973862000 r s - 2064364066720 r s + 3468510186560 r s 7 6 6 + 2826131250000000 n + 17969176625000000 n r - 2729893750000000 n s 5 2 5 + 22848567575000000 n r - 15124151000000000 n r s 5 2 4 3 + 1084082600000000 n s + 7561208225000000 n r 4 2 4 2 - 15901467117500000 n r s + 5489887661000000 n r s 4 3 3 4 - 188476200000000 n s + 2256155631500000 n r 3 3 3 2 2 - 2023419869000000 n r s + 5812082333900000 n r s 3 3 3 4 2 5 - 863937137600000 n r s + 10439200000000 n s + 1129044056500000 n r 2 4 2 3 2 + 1134121598000000 n r s + 1320389196100000 n r s 2 2 3 2 4 6 - 903451147680000 n r s + 39723607680000 n r s + 209133652300000 n r 5 4 2 + 118641101360000 n r s - 151102779884000 n r s 3 3 2 4 7 - 274471040688000 n r s + 33504131712000 n r s + 8259859340000 r 6 5 2 4 3 - 16378509648000 r s - 40102188293600 r s - 19949698000000 r s 3 4 6 5 + 6983765388800 r s + 5182887500000000 n + 24259660500000000 n r 5 4 2 - 4313595000000000 n s + 23139437575000000 n r 4 4 2 - 16992193900000000 n r s + 1483247650000000 n s 3 3 3 2 + 6254115100000000 n r - 12371262380000000 n r s 3 2 3 3 + 5166275888000000 n r s - 217421040000000 n s 2 4 2 3 + 1671727091000000 n r - 717288520000000 n r s 2 2 2 2 3 + 3616933384800000 n r s - 651142057600000 n r s 2 4 5 4 + 9384032000000 n s + 510936172000000 n r + 475513653600000 n r s 3 2 2 3 + 440558599280000 n r s - 412848378880000 n r s 4 6 5 + 20855459840000 n r s + 42560492600000 r + 14000162680000 r s 4 2 3 3 2 4 - 45246462824000 r s - 58498018000000 r s + 8048214848000 r s 5 4 4 + 6918625000000000 n + 23651205500000000 n r - 4818725000000000 n s 3 2 3 + 16518156800000000 n r - 13182170400000000 n r s 3 2 2 3 + 1376583200000000 n s + 3481246300000000 n r 2 2 2 2 - 6258131760000000 n r s + 3144220356000000 n r s 2 3 4 3 - 159339040000000 n s + 711413454000000 n r - 81539812000000 n r s 2 2 3 4 + 1299722211600000 n r s - 282056665600000 n r s + 4742016000000 n s 5 4 3 2 + 99864633000000 r + 84203040000000 r s + 59127311080000 r s 2 3 4 4 - 82067553280000 r s + 4687311360000 r s + 6669450000000000 n 3 3 2 2 + 16235912000000000 n r - 3727120000000000 n s + 7897245900000000 n r 2 2 2 3 - 6720195600000000 n r s + 828629400000000 n s + 1171277720000000 n r 2 2 3 - 1849943520000000 n r s + 1116460512000000 n r s - 67205440000000 n s 4 3 2 2 + 132283282000000 r + 12679164000000 r s + 204591560400000 r s 3 4 3 - 53414553600000 r s + 1026944000000 s + 4526300000000000 n 2 2 2 + 7448886000000000 n r - 1899780000000000 n s + 2272698800000000 n r 2 3 - 2024854400000000 n r s + 291852800000000 n s + 179877040000000 r 2 2 3 - 241689400000000 r s + 175497328000000 r s - 12403840000000 s 2 + 2052200000000000 n + 2050792000000000 n r - 574000000000000 n s 2 2 + 298011600000000 r - 273032000000000 r s + 45631200000000 s + 558000000000000 n + 256264000000000 r - 77840000000000 s / 2 \ |d | + 68800000000000) |--- A[n + 2](r, s)|/((5 r + 2 s) (r + s) %3 %2) + 1/50 | 2 | \dr / 6 5 5 4 2 4 3 2 (5000 n + 7000 n r - 2000 n s + 2950 n r - 3500 n r s - 2360 n r s 2 4 2 3 5 4 6 5 - 295 n r - 763 n r s - 70 n r - 118 n r s - 5 r - 7 r s 5 4 4 3 2 3 + 50000 n + 56000 n r - 18000 n s + 16700 n r - 25200 n r s 2 3 2 2 4 3 5 4 - 1400 n r - 12780 n r s - 1280 n r - 2772 n r s - 140 r - 216 r s 4 3 3 2 2 2 + 205000 n + 175000 n r - 64000 n s + 32850 n r - 67200 n r s 3 2 4 3 3 2 - 4900 n r - 22800 n r s - 1330 r - 2492 r s + 440000 n + 266000 n r 2 2 3 2 - 112000 n s + 25400 n r - 78400 n r s - 4200 r - 13360 r s 2 2 + 520000 n + 196000 n r - 96000 n s + 5800 r - 33600 r s + 320000 n / 3 \ |d | / + 56000 r - 32000 s + 80000) |--- A[n + 2](r, s)| / ( | 3 | / \dr / 2 2 (5 r + 7 r s + 2 s ) %2 %1) = 0 2 2 %1 := 50 n - 20 n s - 5 r - 7 r s + 200 n - 40 s + 200 4 3 2 %2 := n + 6 n + 13 n + 12 n + 4 8 2 7 2 6 4 %3 := 1141875000 n r - 1827000000 n r s - 421312500 n r 6 3 6 2 2 5 5 - 610087500 n r s + 1096200000 n r s + 479587500 n r 5 4 5 3 2 5 2 3 + 1300320000 n r s + 1061356500 n r s - 292320000 n r s 4 6 4 5 4 4 2 + 289406250 n r + 264060000 n r s - 398175750 n r s 4 3 3 4 2 4 3 7 - 490392900 n r s + 29232000 n r s - 14883750 n r 3 6 3 5 2 3 4 3 - 277026750 n r s - 447831450 n r s - 148482810 n r s 3 3 4 2 8 2 7 + 65385720 n r s - 26656875 n r - 96359625 n r s 2 6 2 2 5 3 2 4 4 9 - 67695075 n r s + 39750525 n r s + 41447322 n r s - 3307500 n r 8 7 2 6 3 5 4 - 4079250 n r s + 12988350 n r s + 24050250 n r s + 10589814 n r s 10 9 8 2 7 3 6 4 + 78750 r + 1122750 r s + 3033450 r s + 2972250 r s + 993888 r s 8 7 2 7 + 1312500000 n r + 15986250000 n r - 2100000000 n r s 6 3 6 2 6 2 - 468750000 n r - 23066250000 n r s + 1260000000 n r s 5 4 5 3 5 2 2 - 3252375000 n r - 3867975000 n r s + 13165290000 n r s 5 3 4 5 4 4 - 336000000 n r s + 5169206250 n r + 12383077500 n r s 4 3 2 4 2 3 4 4 + 8649254250 n r s - 3386394000 n r s + 33600000 n r s 3 6 3 5 3 4 2 + 2100262500 n r + 1234597500 n r s - 3941410500 n r s 3 3 3 3 2 4 2 7 - 3835117500 n r s + 315103200 n r s - 140548125 n r 2 6 2 5 2 2 4 3 - 1743292125 n r s - 2591925975 n r s - 706413255 n r s 2 3 4 8 7 + 439084440 n r s - 100878750 n r - 346728750 n r s 6 2 5 3 4 4 9 - 195370650 n r s + 212546070 n r s + 175931088 n r s - 4886250 r 8 7 2 6 3 5 4 - 1488000 r s + 35102700 r s + 53117520 r s + 21979182 r s 8 7 7 6 2 + 312500000 n + 18375000000 n r - 500000000 n s + 96953125000 n r 6 6 2 5 3 - 25881250000 n r s + 300000000 n s - 4098750000 n r 5 2 5 2 5 3 - 122097750000 n r s + 14048250000 n r s - 80000000 n s 4 4 4 3 4 2 2 - 7766062500 n r - 4030537500 n r s + 64379880000 n r s 4 3 4 4 3 5 - 3393450000 n r s + 8000000 n s + 21796650000 n r 3 4 3 3 2 3 2 3 + 46483395000 n r s + 26813634000 n r s - 15135259000 n r s 3 4 2 6 2 5 + 295660000 n r s + 5668000000 n r + 1287765000 n r s 2 4 2 2 3 3 2 2 4 - 13723233000 n r s - 11072890600 n r s + 1201132200 n r s 7 6 5 2 - 358875000 n r - 3559330000 n r s - 4897242000 n r s 4 3 3 4 8 7 - 1011649200 n r s + 972802120 n r s - 92203750 r - 299498750 r s 6 2 5 3 4 4 7 - 112345650 r s + 268034470 r s + 186007088 r s + 4375000000 n 6 6 5 2 + 111562500000 n r - 6125000000 n s + 332490000000 n r 5 5 2 4 3 - 135187500000 n r s + 3300000000 n s - 13550625000 n r 4 2 4 2 4 3 - 349926000000 n r s + 64829625000 n r s - 790000000 n s 3 4 3 3 3 2 2 + 350250000 n r + 24856950000 n r s + 164277090000 n r s 3 3 3 4 2 5 - 13623450000 n r s + 68000000 n s + 45127387500 n r 2 4 2 3 2 2 2 3 + 85967347500 n r s + 38468374500 n r s - 32882845500 n r s 2 4 6 5 + 969360000 n r s + 6747725000 n r - 1158915000 n r s 4 2 3 3 2 4 - 20294205000 n r s - 13958880200 n r s + 1958774400 n r s 7 6 5 2 4 3 - 273202500 r - 2358215500 r s - 3007478700 r s - 385723860 r s 3 4 6 5 5 + 712324240 r s + 26562500000 n + 383250000000 n r - 31875000000 n s 4 2 4 4 2 + 704793750000 n r - 387281250000 n r s + 15075000000 n s 3 3 3 2 3 2 - 19492500000 n r - 582948000000 n r s + 158434500000 n r s 3 3 2 4 2 3 - 3110000000 n s + 29423250000 n r + 82913100000 n r s 2 2 2 2 3 2 4 + 230736870000 n r s - 27173400000 n r s + 216000000 n s 5 4 3 2 + 46001100000 n r + 78312780000 n r s + 23985756000 n r s 2 3 4 6 - 34887132000 n r s + 1404720000 n r s + 2995300000 r 5 4 2 3 3 - 1816470000 r s - 10855650000 r s - 6460921600 r s 2 4 5 4 4 + 1166133600 r s + 91250000000 n + 813750000000 n r - 91250000000 n s 3 2 3 3 2 + 945190000000 n r - 655750000000 n r s + 36600000000 n s 2 3 2 2 2 2 - 7140000000 n r - 559128000000 n r s + 216156000000 n r s 2 3 4 3 - 6100000000 n s + 44274000000 n r + 96693000000 n r s 2 2 3 4 + 169025760000 n r s - 26922600000 n r s + 304000000 n s 5 4 3 2 2 3 + 18505350000 r + 28104330000 r s + 4182426000 r s - 14496890000 r s 4 4 3 + 759680000 r s + 193750000000 n + 1092000000000 n r 3 2 2 2 - 155000000000 n s + 783060000000 n r - 654450000000 n r s 2 2 3 2 + 49800000000 n s + 9630000000 n r - 281172000000 n r s 2 3 4 + 156018000000 n r s - 5960000000 n s + 20802000000 r 3 2 2 3 4 + 40348200000 r s + 50375160000 r s - 10596000000 r s + 160000000 s 3 2 2 + 260000000000 n + 903000000000 n r - 156000000000 n s 2 2 + 366600000000 n r - 355200000000 n r s + 36000000000 n s 3 2 2 3 + 7530000000 r - 55344000000 r s + 46518000000 r s - 2320000000 s 2 2 + 215000000000 n + 420000000000 n r - 86000000000 n s + 74380000000 r 2 - 80500000000 r s + 10800000000 s + 100000000000 n + 84000000000 r - 20000000000 s + 20000000000 4 3 2 2 3 2 2 2 2 1/5 (25 n - 10 n s + n s + 250 n - 60 n s + 2 n s + 825 n - 90 n s + s 4 + 1000 n - 40 s + 400) A[n](r, s)/((5 n - s + 5) (5 n - s + 17)) - (25 n 3 2 2 3 2 2 2 2 - 10 n s + n s + 250 n - 60 n s + 2 n s + 825 n - 90 n s + s /d \ + 1000 n - 40 s + 400) |-- A[n](r, s)|/((5 n - s + 5) (5 n - s + 17)) \ds / (5 n - s + 15) A[n + 1](r, s) /d \ + 9/5 ----------------------------- + |-- A[n + 1](r, s)| (5 n - s + 5) (5 n - s + 17) \ds / / 2 \ 2 2 |d | (25 n - 10 n s + s + 125 n - 25 s + 100) |--- A[n + 1](r, s)| | 2 | \ds / + 1/5 --------------------------------------------------------------- = 0 5 n - s + 17 and in Maple notation -21/50*(71938125000*n^8*r^3*s+50356687500*n^8*r^2*s^2-115101000000*n^7*r^3*s^2-\ 80570700000*n^7*r^2*s^3-19348875000*n^6*r^5*s-46019137500*n^6*r^4*s^2+ 46328152500*n^6*r^3*s^3+48342420000*n^6*r^2*s^4+30214012500*n^5*r^6*s+ 94437393750*n^5*r^5*s^2+111014064000*n^5*r^4*s^3+23382768150*n^5*r^3*s^4-\ 12891312000*n^5*r^2*s^5+16297706250*n^4*r^7*s+23604564375*n^4*r^6*s^2-\ 15444897750*n^4*r^5*s^3-44821222425*n^4*r^4*s^4-17781953490*n^4*r^3*s^5+ 1289131200*n^4*r^2*s^6+2083725000*n^3*r^8*s-6935827500*n^3*r^7*s^2-24314944500* n^3*r^6*s^3-19123176600*n^3*r^5*s^4-613369260*n^3*r^4*s^5+2616475932*n^3*r^3*s^ 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20802000000*r^4+40348200000*r^3*s+50375160000*r^2*s^2-10596000000*r*s^3+ 160000000*s^4+260000000000*n^3+903000000000*n^2*r-156000000000*n^2*s+ 366600000000*n*r^2-355200000000*n*r*s+36000000000*n*s^2+7530000000*r^3-\ 55344000000*r^2*s+46518000000*r*s^2-2320000000*s^3+215000000000*n^2+ 420000000000*n*r-86000000000*n*s+74380000000*r^2-80500000000*r*s+10800000000*s^ 2+100000000000*n+84000000000*r-20000000000*s+20000000000)/(n^4+6*n^3+13*n^2+12* n+4)*diff(diff(A[n+2](r,s),r),r)+1/50*(5000*n^6+7000*n^5*r-2000*n^5*s+2950*n^4* r^2-3500*n^4*r*s-2360*n^3*r^2*s-295*n^2*r^4-763*n^2*r^3*s-70*n*r^5-118*n*r^4*s-\ 5*r^6-7*r^5*s+50000*n^5+56000*n^4*r-18000*n^4*s+16700*n^3*r^2-25200*n^3*r*s-\ 1400*n^2*r^3-12780*n^2*r^2*s-1280*n*r^4-2772*n*r^3*s-140*r^5-216*r^4*s+205000*n ^4+175000*n^3*r-64000*n^3*s+32850*n^2*r^2-67200*n^2*r*s-4900*n*r^3-22800*n*r^2* s-1330*r^4-2492*r^3*s+440000*n^3+266000*n^2*r-112000*n^2*s+25400*n*r^2-78400*n* r*s-4200*r^3-13360*r^2*s+520000*n^2+196000*n*r-96000*n*s+5800*r^2-33600*r*s+ 320000*n+56000*r-32000*s+80000)/(5*r^2+7*r*s+2*s^2)/(n^4+6*n^3+13*n^2+12*n+4)/( 50*n^2-20*n*s-5*r^2-7*r*s+200*n-40*s+200)*diff(diff(diff(A[n+2](r,s),r),r),r) = 0 1/5*(25*n^4-10*n^3*s+n^2*s^2+250*n^3-60*n^2*s+2*n*s^2+825*n^2-90*n*s+s^2+1000*n -40*s+400)/(5*n-s+5)/(5*n-s+17)*A[n](r,s)-(25*n^4-10*n^3*s+n^2*s^2+250*n^3-60*n ^2*s+2*n*s^2+825*n^2-90*n*s+s^2+1000*n-40*s+400)/(5*n-s+5)/(5*n-s+17)*diff(A[n] (r,s),s)+9/5*(5*n-s+15)/(5*n-s+5)/(5*n-s+17)*A[n+1](r,s)+diff(A[n+1](r,s),s)+1/ 5*(25*n^2-10*n*s+s^2+125*n-25*s+100)/(5*n-s+17)*diff(diff(A[n+1](r,s),s),s) = 0 ------------------------------------------------- This took, 6.391, seconds.