A328715 Constant term in the expansion of (1 + v + w + x + y + z + 1/v + 1/w + 1/x + 1/y + 1/z)^n.

1, 1, 11, 31, 331, 1451, 15101, 85961, 876331, 5917531, 59415961, 450749861, 4481629021, 36869221741, 364723196891, 3177413896031, 31389891383531, 284948206851691, 2818704750978761, 26367817118386661, 261622144605718681, 2502704635436220281, 24932548891897186991

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 5-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..967

Crossrefs

Row 5 of A328718.

Programs

PARI

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

Formula

E.g.f.: exp(x) * BesselI(0,2*x)^5. - Ilya Gutkovskiy, Oct 26 2019

From Vaclav Kotesovec, Oct 27 2019: (Start)

Recurrence: n^5*a(n) = (2*n - 1)*(n^2 - n + 1)*(3*n^2 - 3*n + 1)*a(n-1) + (n-1)*(125*n^4 - 500*n^3 + 903*n^2 - 806*n + 289)*a(n-2) - 2*(n-2)*(n-1)*(2*n - 3)*(135*n^2 - 405*n + 419)*a(n-3) - (n-3)*(n-2)*(n-1)*(3319*n^2 - 13276*n + 14637)*a(n-4) + 3867*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 5)*a(n-5) + 10395*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).

a(n) ~ 11^(n + 5/2) / (32 * Pi^(5/2) * n^(5/2)). (End)

Extensions

a(13)-a(22) from Alois P. Heinz, Oct 26 2019