OFFSET
0,3
COMMENTS
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).
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
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..528
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^4.
a(n) ~ 5 * 80^(n+1) / (729 * Pi^2 * n^2). - Vaclav Kotesovec, Oct 30 2019
MATHEMATICA
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 *)
PROG
(PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^4)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 29 2019
STATUS
approved