A328750 - OEIS

A328750

Constant term in the expansion of (-1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

1, 1, 31, 391, 8071, 161671, 3634921, 84109201, 2032357111, 50355327991, 1277302604521, 32983865502721, 864982811998801, 22976755021842961, 617140285389771391, 16735405610179740151, 457647302453165769751, 12607719926638032161431, 349620344754345216824041

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..18.

FORMULA

a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.

From Vaclav Kotesovec, Oct 28 2019: (Start)

Recurrence: n^4*(440*n^2 - 2728*n + 3723)*a(n) = (6600*n^6 - 54120*n^5 + 147013*n^4 - 174348*n^3 + 102442*n^2 - 29260*n + 3108)*a(n-1) + (194920*n^6 - 1988184*n^5 + 7650713*n^4 - 14588908*n^3 + 14793198*n^2 - 7658420*n + 1601964)*a(n-2) + (n-2)*(690800*n^5 - 7046160*n^4 + 26712814*n^3 - 47822370*n^2 + 40779795*n - 13361628)*a(n-3) + (n-3)*(n-2)*(975480*n^4 - 8974416*n^3 + 28602923*n^2 - 37477643*n + 16905924)*a(n-4) + (n-4)*(n-3)*(n-2)*(622600*n^3 - 4482720*n^2 + 9455173*n - 5628497)*a(n-5) + 341*(n-5)*(n-4)*(n-3)*(n-2)*(440*n^2 - 1848*n + 1435)*a(n-6).

a(n) ~ 31^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)

MATHEMATICA

Table[Sum[(-1)^(n - i)*Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)

PROG

(PARI) {a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

CROSSREFS

Column k=5 of A328747.

Cf. A005261, A328751.

Sequence in context: A108268 A214886 A091348 * A179465 A142829 A300869

Adjacent sequences: A328747 A328748 A328749 * A328751 A328752 A328753

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 27 2019

STATUS

approved