A391546 Expansion of e.g.f. exp(g^3 - 1), where g = 1+x*g^3 is the g.f. of A001764.

1, 3, 33, 573, 13617, 411183, 15072489, 650252529, 32284539009, 1813630146651, 113752664510769, 7880888252411397, 597800960080948017, 49282909533714860487, 4388007245760196544313, 419685340277699761142457, 42916708159524954104566401, 4672893727892359767041996211

Offset

0,2

Links

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A391557.

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

Mathematica

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

Crossrefs

Cf. A391545, A391547.

Cf. A001764, A391557.

Sequence in context: A395576 A002916 A009659 * A367427 A361212 A144756

Adjacent sequences: A391543 A391544 A391545 * A391547 A391548 A391549

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved