A391556 Expansion of e.g.f. exp((g^2 - 1)/2), where g = 1+x*g^3 is the g.f. of A001764.

1, 1, 8, 112, 2266, 60266, 1991416, 78760648, 3630762332, 191265470236, 11339034426496, 747343586057696, 54219279828815608, 4294310822951749432, 368738274887057206496, 34122945513257911827616, 3385693640030179229760016, 358570966134764272331164688, 40375297920588635043870781312

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

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

Mathematica

nmax=20; g[x_]:=Sum[Binomial[3*k, k]/(2*k+1) x^k, {k, 0, nmax}];
Table[n! SeriesCoefficient[Exp[(g[x]^2-1)/2], {x, 0, n}], {n, 0, nmax}] (* Vincenzo Librandi, Dec 22 2025 *)

Prog

(PARI) my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(3*k, k)/(2*k+1)*x^k)); Vec(serlaplace(exp((g^2-1)/2)))
(Magma) N:=20; R<x>:=PowerSeriesRing(Rationals(), 2*N+1); [Factorial(n)*Coefficient(Exp(((&+[Binomial(3*k, k)/(2*k+1)*x^k:k in [0..N]])^2-1)/2), n):n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A001764, A391545.

Sequence in context: A302104 A219184 A034689 * A010041 A099703 A296467

Adjacent sequences: A391553 A391554 A391555 * A391557 A391558 A391559

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved