A373940 - OEIS

%I #14 Aug 27 2024 12:59:14

%S 1,0,0,0,0,120,1800,16800,126000,834120,8731800,229191600,6352632000,

%T 143603580120,2736395461800,47283190718400,860150574738000,

%U 20236134851478120,614854122909391800,19930647062659477200,615406024970593164000,17883373100352330768120

%N Expansion of e.g.f. 1/(1 - (exp(x) - 1)^5).

%F G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/Product_{j=1..5*k} (1 - j * x).

%F a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).

%F a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k).

%F a(n) ~ n! / (10 * log(2)^(n+1)). - _Vaclav Kotesovec_, Aug 27 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^5)))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*stirling(j, 5, 2)*v[i-j+1])); v;

%o (PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2));

%Y Cf. A000670, A052841, A353774, A353775.

%Y Cf. A353200.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Aug 27 2024