A356588 Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k)^(1/k) )^x.

1, 0, 2, 9, 44, 450, 2754, 45360, 340304, 6481944, 81801000, 1370631240, 21731534472, 511117017840, 8113055559504, 193958323289640, 4765385232157440, 108183734293844160, 2754467397591689664, 80416694712647352960, 2132862160676063137920, 67803682111729108433280

Offset: 0

Author

Seiichi Manyama, Aug 14 2022

Keywords

• nonn

Crossrefs

Cf. A055225, A355064, A356439, A356587.

Programs

PARI

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

PARI

a055225(n) = sumdiv(n, d, d^(n/d));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a055225(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * A055225(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).