A391560 Expansion of e.g.f. exp((g^3 - 1)/3), where g = 1+x*g^4 is the g.f. of A002293.

1, 1, 11, 213, 5985, 221661, 10218531, 564561201, 36390748353, 2682479586105, 222652398049371, 20554978625001261, 2089547699051558241, 231964689103531639893, 27924323112643836661875, 3623564792426316196908681, 504240479834595078024218241, 74908344255661216879220109681

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(4*n+3*k+3,n)/((4*n+3*k+3) * 3^k * k!) for n > 0.

Mathematica

nmax=20; g[x_]:=Sum[Binomial[4*k, k]/(3*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(4*k, k)/(3*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(4*k, k)/(3*k+1)*x^k:k in [0..N]])^3-1)/3), n):n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A002293, A391549.

Sequence in context: A134069 A137464 A357915 * A187650 A357083 A214687

Adjacent sequences: A391557 A391558 A391559 * A391561 A391562 A391563

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved