A391547 Expansion of e.g.f. exp(g^4 - 1), where g = 1+x*g^3 is the g.f. of A001764.

1, 4, 52, 1024, 26968, 888544, 35136256, 1620888448, 85448232256, 5067327054592, 333892475942656, 24201538373054464, 1913830001496212992, 163975129736104044544, 15132238166103973470208, 1496471873412346550247424, 157887765054017375739252736, 17703128954188217292480790528

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 A391558.

a(n) = 4 * n! * exp(-1) * Sum_{k>=0} binomial(3*n+4*k+4,n)/((3*n+4*k+4) * k!) for n > 0.

Mathematica

Table[Factorial[n] SeriesCoefficient[Exp[(Sum[Binomial[3*k, k]/(2*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(3*k, k)/(2*k+1)*x^k)); Vec(serlaplace(exp(g^4-1)))
(Magma) N := 20; R<x> := PowerSeriesRing(Rationals(), 2*N); [Coefficient(Exp((&+[Binomial(3*k, k)/(2*k+1) * x^k : k in [0..n]])^4-1), n)*Factorial(n): n in [0..N]]; // Vincenzo Librandi, Dec 21 2025

Crossrefs

Cf. A380512, A391545, A391546.

Cf. A001764, A391558.

Sequence in context: A247020 A059580 A379206 * A091463 A354253 A354262

Adjacent sequences: A391544 A391545 A391546 * A391548 A391549 A391550

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved