A356337 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^k )^(1/(1-x)).

1, 1, 8, 63, 644, 7610, 107994, 1713726, 30671024, 603160344, 12974475240, 301879678320, 7561610279112, 202437968475288, 5769455216675136, 174234738889383480, 5556311629901103360, 186482786151757707840, 6568881383985687359424, 242221409390815100812224

Offset: 0

Author

Seiichi Manyama, Aug 04 2022

Keywords

• nonn

Crossrefs

Cf. A000219, A001157, A356298, A356335, A356336.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Product[1/((1-x^k)^k)^(1/(1-x)),{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2023 *)

PARI

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

PARI

a356298(n) = n!*sum(k=1, n, sigma(k, 2)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356298(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356298(k) * binomial(n-1,k-1) * a(n-k).