A328713 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z)^n.

1, 1, 7, 19, 127, 511, 3301, 16297, 103279, 570367, 3595177, 21167917, 133789789, 818625133, 5207248879, 32649752779, 209258291599, 1333828204303, 8612806088761, 55546469634733, 361143420408337, 2349709451702737, 15370341546766939, 100695951740818903, 662213750028892429

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 cubic lattice where each step changes at most one component by -1 or by +1. - Alois P. Heinz, Oct 26 2019

Examples

			(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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1189

Crossrefs

Row 3 of A328718.

Cf. A002426, A002896, A201805.

Programs

PARI

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

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