A356409 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^(1/(1-x)).

1, 1, 5, 28, 203, 1756, 17802, 205010, 2644287, 37669096, 586855058, 9914829508, 180429770402, 3516313661706, 73029591042943, 1609531482261375, 37504691293842367, 920966310015565936, 23764054962685200642, 642681497080268685092, 18174504398294667649782

Offset: 0

Author

Seiichi Manyama, Aug 05 2022

Keywords

• nonn

Crossrefs

Cf. A005651, A356025, A356407, A356408.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k!)^(1/(1-x))))

PARI

a356407(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356407(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356407(k) * binomial(n-1,k-1) * a(n-k).