A328725 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

1, 1, 15, 115, 1255, 13671, 160461, 1936425, 24071895, 305313415, 3939158905, 51521082405, 681635916325, 9105864515125, 122657982366375, 1664151758259915, 22720725637684215, 311933068664333175, 4303704125389134825, 59640225721889127525, 829774531966386480705

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..854

Formula

a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.

From Vaclav Kotesovec, Oct 28 2019: (Start)

Recurrence: n^3*a(n) = (2*n - 1)^3*a(n-1) + (n-1)*(94*n^2 - 188*n + 93)*a(n-2) + 80*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 75*(n-3)*(n-2)*(n-1)*a(n-4).

a(n) ~ 15^(n + 3/2) / (2^(11/2) * Pi^(3/2) * n^(3/2)). (End)

Prog

(PARI) {a(n) = polcoef(polcoef(polcoef((-1+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}
(PARI) {a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

Crossrefs

Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A002426 (m=2), A172634 (m=3), this sequence (m=4), A328750 (m=5).

Cf. A002899, A005260, A328713.

Sequence in context: A091347 A125352 A126510 * A331211 A183475 A253804

Adjacent sequences: A328722 A328723 A328724 * A328726 A328727 A328728

Keyword

nonn

Author

Seiichi Manyama, Oct 26 2019

Status

approved