A391543 - OEIS

%I #16 Dec 21 2025 05:37:22

%S 1,3,27,357,6201,133083,3395619,100313181,3365491761,126353541843,

%T 5247146593899,238762376683317,11812298088897513,631226977556646123,

%U 36232167030479087859,2223141931879664490477,145203861075389109473121,10058197199128695920525859

%N Expansion of e.g.f. exp(g^3 - 1), where g = 1+x*g^2 is the g.f. of A000108.

%H Vincenzo Librandi, <a href="/A391543/b391543.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 A391554.

%F a(n) = 3 * n! * exp(-1) * Sum_{k>=0} binomial(2*n+3*k+3,n)/((2*n+3*k+3) * k!) for n > 0.

%t Table[Factorial[n] SeriesCoefficient[Exp[(Sum[Binomial[2 k,k]/(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(2*k, k)/(k+1)*x^k)); Vec(serlaplace(exp(g^3-1)))

%o (Magma) N := 20; R<x> := PowerSeriesRing(Rationals(), 3*N); [Coefficient(Exp((&+[Binomial(2*k, k)/(k+1) * x^k : k in [0..n]])^3-1), n)*Factorial(n): n in [0..N]]; // _Vincenzo Librandi_, Dec 21 2025

%Y Cf. A250916, A391544.

%Y Cf. A000108, A391554.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 13 2025