A354503 Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^(1/k) )^exp(x).

1, 1, 3, 14, 67, 424, 3093, 26060, 233917, 2427224, 27565317, 339002146, 4450167269, 63343680802, 964189902141, 15769859929260, 270255218753593, 4913097747513800, 94513145955643993, 1904990351069631390, 40153307898034641361, 893402292594225679438

Offset: 0

Author

Seiichi Manyama, Aug 15 2022

Keywords

• nonn

Crossrefs

Cf. A347915, A354504.

Cf. A354506, A356392.

Programs

PARI

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

PARI

a354506(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354506(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354506(k) * binomial(n-1,k-1) * a(n-k).