A356393 Expansion of e.g.f. ( Product_{k>0} (1+x^k) )^(1/(1-x)).

1, 1, 4, 27, 188, 1730, 18234, 220206, 2958416, 44470296, 729675720, 13002636240, 249986061192, 5154030469848, 113360272804128, 2648908519611480, 65477559553098240, 1707034986277780800, 46798324479957887424, 1345365460101611611584

Offset: 0

Author

Seiichi Manyama, Aug 05 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..434

Crossrefs

Cf. A356392, A356394.

Cf. A000009, A356335, A356390.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+x^k)^(1/(1-x))))

PARI

a356390(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356390(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356390(k) * binomial(n-1,k-1) * a(n-k).