A328875 - OEIS

%I #34 Oct 30 2019 08:04:47

%S 1,0,80,2160,121200,6136800,356570960,21225304800,1321586558320,

%T 84398804078400,5518934916677280,367489108030524480,

%U 24852668879410144080,1702677155195779963200,117960677109321028039200,8251450286371615261498560,582087494621171173360817520

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

%C Also number of n-step closed walks (from origin to origin) in 4-dimensional lattice, using steps (t_1,t_2,t_3,t_4) (t_k = -1, 1 or 0 for 1 <= k <= 4) except for (0,0,0,0).

%C For fixed m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m ~ (3^m - 1)^(n + m/2) / (2^m * 3^(m*(m-1)/2) * Pi^(m/2) * n^(m/2)). - _Vaclav Kotesovec_, Oct 30 2019

%H Seiichi Manyama, <a href="/A328875/b328875.txt">Table of n, a(n) for n = 0..528</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^4.

%F a(n) ~ 5 * 80^(n+1) / (729 * Pi^2 * n^2). - _Vaclav Kotesovec_, Oct 30 2019

%t Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^4, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 30 2019 *)

%o (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^4)}

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

%Y Cf. A326920.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 29 2019