login
Expansion of e.g.f. 1/(1 - x^3 * (exp(x) - 1)).
5

%I #15 Aug 26 2024 14:45:15

%S 1,0,0,0,24,60,120,210,40656,363384,2117520,9980190,520250280,

%T 9496208436,109522054824,982593614730,28426015541280,762523155318000,

%U 14192088961120416,204618562767970614,4906638448867994040,154037798077765359660,4000484484370905087480

%N Expansion of e.g.f. 1/(1 - x^3 * (exp(x) - 1)).

%F a(0) = 1; a(n) = n! * Sum_{k=4..n} 1/(k-3)! * a(n-k)/(n-k)!.

%F a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(n-3*k)!.

%t With[{nn=30},CoefficientList[Series[1/(1-x^3 (Exp[x]-1)),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 26 2024 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*(exp(x)-1))))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;

%o (PARI) a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(n-3*k)!);

%Y Cf. A052848, A358013.

%Y Cf. A292891, A351504, A353999.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Oct 24 2022