A391555 - OEIS

%I #15 Dec 22 2025 12:40:23

%S 1,1,8,94,1468,28676,673096,18456208,578989664,20451955168,

%T 803281220416,34725507491456,1638442250105728,83783085867535744,

%U 4615534061313644288,272513023397196560896,17167327855852316678656,1149361306938635404179968,81495563955795289474269184

%N Expansion of e.g.f. exp((g^4 - 1)/4), where g = 1+x*g^2 is the g.f. of A000108.

%H Vincenzo Librandi, <a href="/A391555/b391555.txt">Table of n, a(n) for n = 0..300</a>

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

%F From _Vaclav Kotesovec_, Dec 21 2025: (Start)

%F Recurrence: (n^2 + 11*n - 70)*a(n) = (7*n^3 + 76*n^2 - 556*n + 415)*a(n-1) - (n-1)*(13*n^3 + 137*n^2 - 1161*n + 1685)*a(n-2) + (n-2)*(n-1)*(4*n^3 + 42*n^2 - 347*n + 545)*a(n-3) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n^2 + 16*n + 10)*a(n-4) + 2*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 5)*(n^2 + 13*n - 58)*a(n-5).

%F a(n) ~ 2^(2*n + 7/2) * n^(n-1) / exp(n - 15/4). (End)

%t nmax=20; g[x_]:=Sum[Binomial[2*k,k]/(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(2*k, k)/(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(2*k,k)/(k+1)*x^k:k in [0..N]])^4-1)/4),n):n in [0..N]]; // _Vincenzo Librandi_, Dec 22 2025

%Y Cf. A000108, A391544.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 13 2025