A354263 Expansion of e.g.f. 1/(1 + 3 * log(1-x)).

1, 3, 21, 222, 3132, 55242, 1169262, 28873800, 814870584, 25871762016, 912684973968, 35416732159872, 1499286521185776, 68757945743286576, 3395829155786528976, 179693346163010491008, 10142543588881013369856, 608262031900883147262336

Offset: 0

Author

Seiichi Manyama, May 21 2022

Keywords

• nonn

Crossrefs

Column k=3 of A320079.

Cf. A335531.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

PARI

a(n) = sum(k=0, n, 3^k*k!*abs(stirling(n, k, 1)));

Formula

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

a(n) = Sum_{k=0..n} 3^k * k! * |Stirling1(n, k)|.

a(n) ~ n! * exp(n/3) / (3 * (exp(1/3) - 1)^(n+1)). - Vaclav Kotesovec, Jun 04 2022