A391559 Expansion of e.g.f. exp((g^2 - 1)/2), where g = 1+x*g^4 is the g.f. of A002293.

1, 1, 10, 184, 5002, 180946, 8194876, 446499040, 28458981244, 2078421338332, 171172926117496, 15697577655564256, 1586608160726540920, 175250235244856579224, 21003839031922634413072, 2714857174269749946558976, 376464923925271242105926416, 55750171957389243732994627600

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(4*n+2*k+2,n)/((4*n+2*k+2) * 2^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]^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(4*k, k)/(3*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(4*k, k)/(3*k+1)*x^k:k in [0..N]])^2-1)/2), n):n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A002293, A391548.

Sequence in context: A239764 A364987 A179521 * A211102 A121973 A223146

Adjacent sequences: A391556 A391557 A391558 * A391560 A391561 A391562

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved