A353664 Expansion of e.g.f. exp((exp(x) - 1)^3).

1, 0, 0, 6, 36, 150, 900, 9366, 101556, 1031190, 10995300, 134640726, 1844184276, 26656678230, 400614423300, 6347263038486, 106960986110196, 1905688502565270, 35546025523227300, 691014283378745046, 13999772792477879316, 295570215436360196310

Offset

0,4

Links

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

Formula

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(k! * Product_{j=1..3*k} (1 - j * x)).

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,3) * a(n-k).

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

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^3)))
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(k!*prod(j=1, 3*k, 1-j*x))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 2)*v[i-j+1])); v;
(PARI) a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/k!);

Crossrefs

Cf. A000110, A052859, A353665.

Cf. A327504, A353344, A353774.

Sequence in context: A210322 A056268 A001117 * A353774 A357010 A357087

Adjacent sequences: A353661 A353662 A353663 * A353665 A353666 A353667

Keyword

nonn

Author

Seiichi Manyama, May 07 2022

Status

approved