A318580 Expansion of e.g.f. exp(-1 + Product_{k>=1} 1/(1 - x^k)^k).

1, 1, 7, 55, 601, 7561, 116191, 1999327, 39267985, 850964401, 20332107991, 527930427751, 14838001344937, 447653776595065, 14440021169407471, 495398956418435791, 18012260306904120481, 691502230924473978337, 27948692251661337581095, 1185878351946613955122711

Offset: 0

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

• nonn

Crossrefs

Cf. A000219, A001157, A058892, A318250.

Programs

Maple

seq(n!*coeff(series(exp(-1+mul(1/(1-x^k)^k,k=1..100)),x=0,20),x,n),n=0..19); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 19; CoefficientList[Series[Exp[-1 + Product[1/(1 - x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Exp[-1 + Exp[Sum[DivisorSigma[2, k] x^k/k, {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!
p[n_] := p[n] = Sum[DivisorSigma[2, k] p[n - k], {k, n}]/n; p[0] = 1; a[n_] := a[n] = Sum[p[k] k! Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

Formula

E.g.f.: exp(-1 + exp(Sum_{k>=1} sigma_2(k)*x^k/k)).

E.g.f.: A(x) = exp(B(x) - 1), where B(x) = o.g.f. of A000219.

a(0) = 1; a(n) = Sum_{k=1..n} A000219(k)*k!*binomial(n-1,k-1)*a(n-k).