A353774 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^3).

1, 0, 0, 6, 36, 150, 1260, 16926, 197316, 2286150, 32821020, 548528046, 9515702196, 174531124950, 3521913283980, 76969474578366, 1777400236160676, 43405229295464550, 1126972561394470140, 30949983774936839886, 893095888222540548756, 27035433957000465352950

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..424

Formula

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

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

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

a(n) ~ n! / (6 * log(2)^(n+1)). - Vaclav Kotesovec, May 08 2022

Prog

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

Crossrefs

Cf. A000670, A052841, A353775.

Cf. A143815, A346894, A353118, A353664.

Sequence in context: A056268 A001117 A353664 * A357010 A357087 A357025

Adjacent sequences: A353771 A353772 A353773 * A353775 A353776 A353777

Keyword

nonn

Author

Seiichi Manyama, May 07 2022

Status

approved