A356402 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k!) )^(1/(1-x)).

1, 1, 3, 16, 86, 626, 5267, 50793, 543279, 6544805, 86503762, 1242678141, 19259416827, 321457169151, 5736414618209, 108931865485750, 2191495621647324, 46604972526167314, 1043844453093239627, 24555321244430950299, 605239630722584461955, 15600222966916650541099

Offset: 0

Author

Seiichi Manyama, Aug 05 2022

Keywords

• nonn

Crossrefs

Cf. A298906, A356025, A356392, A356401.

Programs

PARI

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

PARI

a356401(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356401(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356401(k) * binomial(n-1,k-1) * a(n-k).