A328714 Constant term in the expansion of (1 + w + x + y + z + 1/w + 1/x + 1/y + 1/z)^n.

1, 1, 9, 25, 217, 921, 7761, 41889, 345465, 2162617, 17605249, 121120209, 980612161, 7174425025, 58079091513, 442755733065, 3595708057785, 28197440412345, 230133477721665, 1841288167473105, 15113407062476817, 122714906949538257, 1013127345082389513

Offset: 0

Author

Seiichi Manyama, Oct 26 2019

Keywords

• nonn

Comments

a(n) is the number of n-step closed walks (from origin to origin) in 4-dimensional lattice where each step changes at most one component by -1 or by +1. - Alois P. Heinz, Oct 26 2019

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1054

Crossrefs

Row 4 of A328718.

Cf. A039699.

Programs

PARI

{a(n) = polcoef(polcoef(polcoef(polcoef((1+w+x+y+z+1/w+1/x+1/y+1/z)^n, 0), 0), 0), 0)}

Formula

From Vaclav Kotesovec, Oct 26 2019: (Start)

Recurrence: n^4*a(n) = (5*n^4 - 10*n^3 + 10*n^2 - 5*n + 1)*a(n-1) + (n-1)^2*(70*n^2 - 140*n + 113)*a(n-2) - (n-2)*(n-1)*(230*n^2 - 690*n + 583)*a(n-3) - 789*(n-3)*(n-2)^2*(n-1)*a(n-4) + 945*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).

a(n) ~ 9^(n+2) / (16 * Pi^2 * n^2). (End)

E.g.f.: exp(x) * BesselI(0,2*x)^4. - Ilya Gutkovskiy, Oct 26 2019

Extensions

a(19)-a(22) from Alois P. Heinz, Oct 26 2019