A361703 Constant term in the expansion of (1 + w + x + y + z + 1/(w*x*y*z))^n.

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 15121, 143641, 1302841, 7579441, 32586841, 113753641, 509068561, 3599319361, 25076993761, 142188273361, 662296228561, 2933770097881, 15581813723281, 99333170493481, 623696622059281, 3466773281312881, 17406784944114721

Offset

0,6

Comments

Diagonal of the rational function 1/(1 - (v^5 + w^5 + x^5 + y^5 + z^5 + v*w*x*y*z)). - Seiichi Manyama, Jul 04 2025

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..floor(n/5)} (4*k)!/k!^4 * binomial(5*k,4*k) * binomial(n,5*k) = Sum_{k=0..floor(n/5)} (5*k)!/k!^5 * binomial(n,5*k).

a(n) ~ 9 * 6^n / (sqrt(5) * Pi^2 * n^2). - Vaclav Kotesovec, Mar 25 2023

a(n) = hypergeom([-n/5, (1 - n)/5, (2 - n)/5, (3 - n)/5, (4 - n)/5], [1, 1, 1, 1], -5^5). - Peter Luschny, Oct 22 2025

Maple

a := n -> hypergeom([-n/5, (1 - n)/5, (2 - n)/5, (3 - n)/5, (4 - n)/5], [1, 1, 1, 1], -3125): seq(simplify(a(n)), n = 0..25);  # Peter Luschny, Oct 22 2025

Prog

(PARI) a(n) = sum(k=0, n\5, (4*k)!/k!^4*binomial(5*k, 4*k)*binomial(n, 5*k));

Crossrefs

Column 5 of A389641.

Cf. A361675, A361704, A361705, A002426, A344560, A361637, A082489, A361636.

Sequence in context: A020229 A141350 A235408 * A190877 A361636 A354554

Adjacent sequences: A361700 A361701 A361702 * A361704 A361705 A361706

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 21 2023

Status

approved