A361637 Constant term in the expansion of (1 + x + y + z + 1/(x*y*z))^n.

1, 1, 1, 1, 25, 121, 361, 841, 4201, 25705, 118441, 423721, 1628881, 8065201, 41225185, 184416961, 768211081, 3420474121, 16620237001, 79922011465, 364149052705, 1638806098945, 7655390077105, 36739991161105, 174363209490625, 811840219629121, 3790118889635521

Offset

0,5

Comments

Diagonal of the rational function 1/(1 - (x^4 + y^4 + z^4 + w^4 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Links

Winston de Greef, Table of n, a(n) for n = 0..1429

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^4 * (n-4*k)!).

G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(4*k)/(1-x)^(4*k+1).

From Vaclav Kotesovec, Mar 20 2023: (Start)

Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) - (n-1)*(6*n^2 - 12*n + 7)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 255*(n-3)*(n-2)*(n-1)*a(n-4).

a(n) ~ 5^(n + 3/2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). (End)

a(n) = hypergeom([-n/4, (1 - n)/4, (2 - n)/4, (3 - n)/4], [1, 1, 1], 4^4). - Peter Luschny, Oct 22 2025

Maple

a := n -> hypergeom([-n/4, (1 - n)/4, (2 - n)/4, (3 - n)/4], [1, 1, 1], 256):
seq(simplify(a(n)), n = 0..26);  # Peter Luschny, Oct 22 2025

Prog

(PARI) a(n) = n!*sum(k=0, n\4, 1/(k!^4*(n-4*k)!));

Crossrefs

Column 4 of A389641.

Cf. A002426, A344560, A361703, A008977, A328725, A082488, A274783, A274785.

Sequence in context: A016970 A174371 A062938 * A190875 A205800 A330045

Adjacent sequences: A361634 A361635 A361636 * A361638 A361639 A361640

Keyword

nonn

Author

Seiichi Manyama, Mar 19 2023

Status

approved