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A263065
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Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one or more components by one.
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3
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1, 541, 2244361, 14638956721, 117029959485121, 1050740615666453461, 10169807398958450670001, 103746115308050354021387521, 1100327453912286201909924526081, 12024609569670508078686022988554381, 134565509066155510620216211257550349401
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OFFSET
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0,2
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COMMENTS
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Also, the number of alignments for 5 sequences of length n each (Slowinski 1998).
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LINKS
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FORMULA
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Recurrence: (n-1)^2*n^4*(1484375*n^10 - 35625000*n^9 + 380459375*n^8 - 2379691875*n^7 + 9648658375*n^6 - 26483049785*n^5 + 49802335614*n^4 - 63319196190*n^3 + 52054873977*n^2 - 24970596338*n + 5304283784)*a(n) = (n-1)^2*(20417578125*n^14 - 530857031250*n^13 + 6241768390625*n^12 - 43891361846875*n^11 + 205684605242500*n^10 - 677508063221175*n^9 + 1612085561506345*n^8 - 2803601202034769*n^7 + 3564158318615391*n^6 - 3277539874902099*n^5 + 2131595379572790*n^4 - 942888582994608*n^3 + 265378603877984*n^2 - 41959963867392*n + 2757380659200)*a(n-1) + (5551562500*n^16 - 166546875000*n^15 + 2299855640625*n^14 - 19385476578125*n^13 + 111504090473125*n^12 - 463446487931900*n^11 + 1437445134614810*n^10 - 3387090699899014*n^9 + 6112545662650711*n^8 - 8450360220919608*n^7 + 8884444155685163*n^6 - 6993443486776441*n^5 + 4013714078940498*n^4 - 1609501825795072*n^3 + 420362394759120*n^2 - 62905338995616*n + 3977994685824)*a(n-2) + (1870312500*n^16 - 59850000000*n^15 + 882904671875*n^14 - 7959826712500*n^13 + 49013510712500*n^12 - 218196359173225*n^11 + 724960516804615*n^10 - 1829325996659659*n^9 + 3532586966500778*n^8 - 5219142662751755*n^7 + 5853433612256896*n^6 - 4902859151966701*n^5 + 2984426972027036*n^4 - 1263868309818152*n^3 + 346653353359072*n^2 - 54082532707344*n + 3532661100864)*a(n-3) - (n-3)^2*(7421875*n^14 - 207812500*n^13 + 2631125000*n^12 - 19925368750*n^11 + 100603166250*n^10 - 357324371050*n^9 + 917934587470*n^8 - 1726295128861*n^7 + 2377475157009*n^6 - 2372287254911*n^5 + 1675297653876*n^4 - 803613605640*n^3 + 244104664208*n^2 - 41262015600*n + 2856959424)*a(n-4) + (n-4)^4*(n-3)^2*(1484375*n^10 - 20781250*n^9 + 126631250*n^8 - 440391875*n^7 + 962896500*n^6 - 1373591410*n^5 + 1282871689*n^4 - 765049709*n^3 + 274306866*n^2 - 52342548*n + 3936312)*a(n-5). - Vaclav Kotesovec, Mar 22 2016
a(n) ~ c * d^n / (Pi^2 * n^2), where d = 13755.27190241150817120839544215413203... is the real root of the equation -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5 = 0 and c = 0.5698188923151523225906967169329722766951557573868... is the root of the equation -1 - 1600*c^2 - 896000*c^4 - 204800000*c^6 - 16384000000*c^8 + 52428800000*c^10 = 0. - Vaclav Kotesovec, Mar 22 2016
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MATHEMATICA
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With[{k = 5}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 15}]] (* Vaclav Kotesovec, Mar 22 2016 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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