A391546 - OEIS

A391546

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

%I #15 Dec 21 2025 09:44:45

%S 1,3,33,573,13617,411183,15072489,650252529,32284539009,1813630146651,

%T 113752664510769,7880888252411397,597800960080948017,

%U 49282909533714860487,4388007245760196544313,419685340277699761142457,42916708159524954104566401,4672893727892359767041996211

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

%H Vincenzo Librandi, <a href="/A391546/b391546.txt">Table of n, a(n) for n = 0..300</a>

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

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

%t 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 *)

%o (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)))

%o (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

%Y Cf. A391545, A391547.

%Y Cf. A001764, A391557.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 13 2025