A391559 - OEIS

%I #13 Dec 22 2025 12:36:04

%S 1,1,10,184,5002,180946,8194876,446499040,28458981244,2078421338332,

%T 171172926117496,15697577655564256,1586608160726540920,

%U 175250235244856579224,21003839031922634413072,2714857174269749946558976,376464923925271242105926416,55750171957389243732994627600

%N Expansion of e.g.f. exp((g^2 - 1)/2), where g = 1+x*g^4 is the g.f. of A002293.

%H Vincenzo Librandi, <a href="/A391559/b391559.txt">Table of n, a(n) for n = 0..300</a>

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

%t nmax=20; g[x_]:=Sum[Binomial[4*k,k]/(3*k+1) x^k,{k,0,nmax}];

%t Table[n! SeriesCoefficient[Exp[(g[x]^2-1)/2],{x,0,n}],{n,0,nmax}] (* _Vincenzo Librandi_, Dec 22 2025 *)

%o (PARI) my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); Vec(serlaplace(exp((g^2-1)/2)))

%o (Magma) N:=20; R<x>:=PowerSeriesRing(Rationals(),2*N+1); [Factorial(n)*Coefficient(Exp(((&+[Binomial(4*k,k)/(3*k+1)*x^k:k in [0..N]])^2-1)/2),n):n in [0..N]]; // _Vincenzo Librandi_, Dec 22 2025

%Y Cf. A002293, A391548.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 13 2025