A293493 Expansion of e.g.f. exp(x/(1 - x^3)).

1, 1, 1, 1, 25, 121, 361, 5881, 62161, 365905, 5145841, 84831121, 812318761, 11450816521, 243450591385, 3371056121161, 51784222133281, 1275605799044641, 23531635543402081, 410559590525890465, 11089633716053137081, 256375957896260034841, 5161258224477109736521

Offset

0,5

Links

Robert Israel, Table of n, a(n) for n = 0..445

Formula

E.g.f.: Product_{k>0} exp(x^(3*k-2)).

a(n) ~ exp(2*sqrt(3*n)/3 - n + 1/6) * n^(n-1/4) / (sqrt(2) * 3^(1/4)). - Vaclav Kotesovec, Oct 10 2017

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k+1)! * a(n-3*k-1). - Ilya Gutkovskiy, Feb 24 2022

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k)/(n-3*k)!. - Seiichi Manyama, Jun 08 2024

D-finite with recurrence: n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*a(n) - 2*(n + 5)*(n + 4)*(n + 3)*a(n + 2) - 2*(n + 5)*(n + 4)*(n + 3)*a(n + 3) - a(n + 5) + a(n + 6) = 0. - Robert Israel, Feb 22 2026

Maple

f:= gfun:-rectoproc({n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*a(n) - 2*(n + 5)*(n + 4)*(n + 3)*a(n + 2) - 2*(n + 5)*(n + 4)*(n + 3)*a(n + 3) - a(n + 5) + a(n + 6), a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=25, a(5)=121}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 22 2026

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^3))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-2)))))

Crossrefs

Cf. A293494.

Sequence in context: A274783 A354553 A367719 * A356629 A362571 A198040

Adjacent sequences: A293490 A293491 A293492 * A293494 A293495 A293496

Keyword

nonn

Author

Seiichi Manyama, Oct 10 2017

Status

approved