A380512 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 1, 7, 91, 1753, 45001, 1447471, 56041987, 2539200721, 131859347473, 7723214721271, 503787793244011, 36223369111466857, 2846582772323685721, 242741539845295265503, 22325483241906758894611, 2202979676409063904473121, 232158319570869255177386017, 26024052774273208806612761191

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

Formula

E.g.f.: exp(G(x)-1), where G(x) is described above.

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

a(n+1) = n! * LaguerreL(n, 2*n+3, -1).

a(n) = (-1)^(n+1)*U(1-n, 2*(1+n), -1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025

E.g.f.: exp( Series_Reversion( x/(1+x)^3 ) ). - Seiichi Manyama, Mar 15 2025

a(n) ~ 3^(3*n + 1/2) * n^(n-1) / (2^(2*n + 3/2) * exp(n - 1/2)). - Vaclav Kotesovec, Apr 01 2026

Mathematica

Join[{1}, Table[(n-1)! * Sum[Binomial[3*n, k] / (n-k-1)!, {k, 0, n-1}], {n, 1, 20}]] (* Vaclav Kotesovec, Apr 01 2026 *)

Prog

(PARI) a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 2*n+1, -1));

Crossrefs

Cf. A001764, A251569, A380511.

Cf. A000262, A251568, A380516.

Cf. A382033.

Sequence in context: A131940 A008542 A121940 * A177784 A361142 A326266

Adjacent sequences: A380509 A380510 A380511 * A380513 A380514 A380515

Keyword

nonn

Author

Seiichi Manyama, Jan 26 2025

Status

approved