A354623 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^x.

1, 0, 2, 9, 44, 390, 2754, 32760, 310064, 4244184, 54887400, 818615160, 12909921672, 225872515440, 4045885572624, 79360837887240, 1649832369335040, 35666417240193600, 822291935260976064, 19830352438530840960, 501144432316767688320, 13229590606682042436480

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

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

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k)^x))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Crossrefs

Cf. A355064, A356554.

Cf. A000203, A053529, A066166, A356335, A356565.

Sequence in context: A322613 A318913 A327940 * A088369 A356588 A303938

Adjacent sequences: A354620 A354621 A354622 * A354624 A354625 A354626

Keyword

nonn

Author

Seiichi Manyama, Aug 12 2022

Status

approved