A391544 Expansion of e.g.f. exp(g^4 - 1), where g = 1+x*g^2 is the g.f. of A000108.

1, 4, 44, 688, 13864, 340544, 9841216, 326498176, 12214596032, 508259328256, 23269655140096, 1161912882351104, 62816589188575744, 3654517810229321728, 227597447625495486464, 15105030195892500373504, 1064087464840967635111936, 79290918998401751752196096

Offset

0,2

Links

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

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A391555.

a(n) = 2 * n! * exp(-1) * Sum_{k>=0} binomial(2*n+4*k+4,n)/((n+2*k+2) * k!) for n > 0.

Mathematica

Table[Factorial[n] SeriesCoefficient[Exp[(Sum[Binomial[2 k, k]/(k+1) x^k, {k, 0, 20}])^4-1], {x, 0, n}], {n, 0, 20}] (* 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^4-1)))
(Magma) N := 20; R<x> := PowerSeriesRing(Rationals(), 2*N); [Coefficient(Exp((&+[Binomial(2*k, k)/(k+1) * x^k : k in [0..n]])^4-1), n)*Factorial(n): n in [0..N]]; // Vincenzo Librandi, Dec 21 2025

Crossrefs

Cf. A250916, A391543.

Cf. A000108, A391555.

Sequence in context: A371680 A056063 A218224 * A177749 A291198 A384523

Adjacent sequences: A391541 A391542 A391543 * A391545 A391546 A391547

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved