A375773 Expansion of e.g.f. exp((exp(x) - 1)^5).

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 6917400, 129399600, 3259080000, 72252300120, 1370602233000, 23218349918400, 377834084082000, 6709735404918120, 147369456297228600, 3899127761438053200, 109421543771265852000, 3002806840023201408120

Offset

0,6

Links

Table of n, a(n) for n=0..21.

Formula

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

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

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

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^5)))
(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-1, j-1)*stirling(j, 5, 2)*v[i-j+1])); v;
(PARI) a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/k!);

Crossrefs

Cf. A000110, A052859, A353664, A353665.

Cf. A353404, A373940.

Sequence in context: A053567 A056270 A001118 * A373940 A354230 A354232

Adjacent sequences: A375770 A375771 A375772 * A375774 A375775 A375776

Keyword

nonn

Author

Seiichi Manyama, Aug 27 2024

Status

approved