OFFSET
0,3
COMMENTS
a(n) is the number of n-step closed walks (from origin to origin) in cubic 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..1189
FORMULA
From Vaclav Kotesovec, Oct 26 2019: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) + (n-1)*(34*n^2 - 68*n + 41)*a(n-2) - 38*(n-2)*(n-1)*(2*n - 3)*a(n-3) - 105*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 7^(n + 3/2) / (8 * Pi^(3/2) * n^(3/2)). (End)
E.g.f.: exp(x) * BesselI(0,2*x)^3. - Ilya Gutkovskiy, Oct 26 2019
EXAMPLE
(1+x+y+z+1/x+1/y+1/z)^2 = x^2 + 1/x^2 + y^2 + 1/y^2 + z^2 + 1/z^2 + 2 * (xy + 1/(xy) + yz + 1/(yz) + zx + 1/(zx) + x/y + y/x + y/z + z/y + z/x + x/z + x + 1/x + y + 1/y + z + 1/z) + 7. So a(2) = 7.
PROG
(PARI) {a(n) = polcoef(polcoef(polcoef((1+x+y+z+1/x+1/y+1/z)^n, 0), 0), 0)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 26 2019
STATUS
approved