A328751 Constant term in the expansion of (-2 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

1, 0, 30, 300, 6690, 124920, 2778600, 61790400, 1452751650, 34806097200, 855836532180, 21393889763400, 543342862524000, 13972938142363200, 363356617578926400, 9538720137580233600, 252510537115100657250, 6733792260826534332000, 180751978201192700659500

Offset

0,3

Links

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

Formula

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.

From Vaclav Kotesovec, Mar 20 2023: (Start)

Recurrence: n^4*(22*n^2 - 198*n + 323)*a(n) = (n-1)*(198*n^5 - 1980*n^4 + 4535*n^3 - 2641*n^2 + 119*n + 210)*a(n-1) + (11066*n^6 - 143858*n^5 + 628715*n^4 - 1298438*n^3 + 1394723*n^2 - 756728*n + 165060)*a(n-2) + 4*(n-2)*(19096*n^5 - 248248*n^4 + 1086158*n^3 - 2156993*n^2 + 2004912*n - 708435)*a(n-3) + 40*(n-3)*(n-2)*(5346*n^4 - 64152*n^3 + 242653*n^2 - 363566*n + 182959)*a(n-4) + 400*(n-4)*(n-3)*(n-2)*(682*n^3 - 6820*n^2 + 17955*n - 12432)*a(n-5) + 6000*(n-5)*(n-4)*(n-3)*(n-2)*(22*n^2 - 154*n + 147)*a(n-6).

a(n) ~ 2^(n-6) * 3^(n+2) * 5^(n + 3/2) / (Pi^2 * n^2). (End)

Mathematica

Table[Sum[(-2)^(n-i)*Binomial[n, i] * Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 20 2023 *)

Prog

(PARI) {a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

Crossrefs

Column k=5 of A328748.

Cf. A005261, A328750.

Sequence in context: A061605 A125393 A126551 * A042750 A074994 A134287

Adjacent sequences: A328748 A328749 A328750 * A328752 A328753 A328754

Keyword

nonn

Author

Seiichi Manyama, Oct 27 2019

Status

approved