A383837 a(n) = (3*n)!/n! * [x^(3*n)] sinh(x)^n.

1, 1, 16, 820, 87296, 15857205, 4390088704, 1721255653656, 907673633095680, 619593964021650475, 531571294549842067456, 559896149105493602658256, 710322778732936488128872448, 1068386732538408106621063668220, 1879866814874817967233600382304256

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

a(n) = [x^n] 1/Product_{k=0..floor(n/2)} (1 - (n-2*k)^2*x).

a(n) = (1/(2^n*n!)) * Sum_{k=0..n} (-1)^k * (n-2*k)^(3*n) * binomial(n,k).

a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.35572032955623014748562257137412853926900571707993382361... and c = 0.81034327454108346293530087910356437429774959841653144433... - Vaclav Kotesovec, May 13 2025

In closed form, a(n) ~ r^(r*n) * (1 + 2*r)^(3*n+1) * exp(n) * n^(2*n - 1/2) / (sqrt(Pi*(1 - 8*r - 8*r^2)) * 2^(n - 1/2) * (1+r)^((1+r)*n)), where r = 0.002562299585216598238663221142585901101711497682846... is the positive real root of the equation exp(2*arctanh(1 + 2*r) - 6/(1 + 2*r)) = -1. - Vaclav Kotesovec, May 17 2025

Mathematica

Join[{1}, Table[Sum[(-1)^k * (n-2*k)^(3*n) * Binomial[n, k] / (2^n*n!), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 13 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*(n-2*k)^(3*n)*binomial(n, k))/(2^n*n!);

Crossrefs

Main diagonal of A381512.

Cf. A298851, A381459.

Sequence in context: A159872 A220750 A166187 * A363988 A223113 A145243

Adjacent sequences: A383834 A383835 A383836 * A383838 A383839 A383840

Keyword

nonn

Author

Seiichi Manyama, May 11 2025

Status

approved