A391561 - OEIS

%I #12 Dec 23 2025 15:27:51

%S 1,1,12,244,7084,268716,12618616,707542312,46164369504,3437762234464,

%T 287830529920576,26772093907478976,2739431829975444352,

%U 305869238654653788544,37010335831481152199424,4824682223713815167769856,674162053956687816905529856,100526605452826479060450949632

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

%H Vincenzo Librandi, <a href="/A391561/b391561.txt">Table of n, a(n) for n = 0..300</a>

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

%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]])^4-1)/4),n):n in [0..N]]; // _Vincenzo Librandi_, Dec 22 2025

%Y Cf. A002293, A391550.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 13 2025