A326920 Constant term in the expansion of (-1 + Product_{k=1..n} (1 + x_k + 1/x_k))^n.

1, 0, 8, 264, 121200, 332810400, 7753173594200, 1440193875113407680, 2250630808138439243100640, 29565964235758317208187044137600, 3307988125501026209547184198622507128848, 3165738749695300492286911657015518806826344524560

Offset: 0

Author

Seiichi Manyama, Oct 29 2019

Keywords

• nonn

Comments

Also number of n-step closed walks (from origin to origin) in n-dimensional lattice, using steps (t_1,t_2, ... ,t_n) (t_k = -1, 1 or 0 for 1 <= k <= n) except for (0,0, ... ,0) (t_k = 0 for 1 <= k <= n).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..47

Crossrefs

Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), A328874 (m=3), A328875 (m=4).

Programs

Mathematica

Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^n, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 30 2019 *)

PARI

{a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^n)}

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^n.

a(n) ~ 3^(n^2 + n/2) / (exp(3/16) * 2^n * Pi^(n/2) * n^(n/2)). - Vaclav Kotesovec, Oct 30 2019