A391554 Expansion of e.g.f. exp((g^3 - 1)/3), where g = 1+x*g^2 is the g.f. of A000108.

1, 1, 7, 75, 1089, 20001, 444711, 11614107, 348509505, 11816697249, 446808836871, 18641920151691, 850777764125697, 42162606476192385, 2254918424892662439, 129452195262941386011, 7940526673653916461441, 518303074153507655696577, 35871572721037024546635015

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(2*n+3*k+3,n)/((2*n+3*k+3) * 3^k * k!) for n > 0.

Mathematica

nmax=20; g[x_]:=Sum[Binomial[2*k, k]/(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(2*k, k)/(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(2*k, k)/(k+1)*x^k:k in [0..N]])^3-1)/3), n):n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A000108, A391543.

Sequence in context: A384622 A243692 A258293 * A127190 A121316 A220215

Adjacent sequences: A391551 A391552 A391553 * A391555 A391556 A391557

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved