A391557 Expansion of e.g.f. exp((g^3 - 1)/3), where g = 1+x*g^3 is the g.f. of A001764.

1, 1, 9, 135, 2865, 78981, 2684721, 108655779, 5106054465, 273406908105, 16438153575321, 1096777967659551, 80433633647877489, 6431883559148239245, 557038051659369897345, 51947423242094451402171, 5190379048351466059523841, 553206686039306266970265489

Offset

0,3

Links

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

Formula

a(n) = n! * exp(-1/3) * Sum_{k>=0} binomial(3*n+3*k+3,n)/((3*n+3*k+3) * 3^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]^3-1)/3], {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^3-1)/3)))
(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]])^3-1)/3), n):n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A001764, A391546.

Sequence in context: A235339 A306848 A034723 * A188685 A052137 A003376

Adjacent sequences: A391554 A391555 A391556 * A391558 A391559 A391560

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved