A391553 Expansion of e.g.f. exp((g^2 - 1)/2), where g = 1+x*g^2 is the g.f. of A000108.

1, 1, 6, 58, 778, 13386, 281476, 7000876, 201163068, 6559979068, 239428935496, 9672134739576, 428510225570296, 20661898792071928, 1077289606360647408, 60398988138132738256, 3623792843060787912976, 231683722216692321900816, 15725444708263745576296288

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(2*n+2*k+2,n)/((2*n+2*k+2) * 2^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]^2-1)/2], {x, 0, n}], {n, 0, nmax}] (* Vincenzo Librandi, Dec 21 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^2-1)/2)))
(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]])^2-1)/2), n):n in [0..N]]; // Vincenzo Librandi, Dec 21 2025

Crossrefs

Cf. A000108, A250916.

Sequence in context: A259612 A305599 A316653 * A302598 A302922 A370908

Adjacent sequences: A391550 A391551 A391552 * A391554 A391555 A391556

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved