A330201 Expansion of e.g.f. Product_{k>=1} exp(-x^k) / (1 - x^k).

1, 0, 1, 2, 21, 44, 1045, 2694, 74473, 421784, 8776521, 52518410, 1843753021, 11476952772, 387068115421, 4277646186254, 125796357803985, 1343857519264304, 53205974734877713, 621203524858308114, 25357790175078682981, 388778926109137187420

Offset

0,4

Links

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

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of A000166.

E.g.f.: exp(Sum_{k>=1} (sigma(k) / k - 1) * x^k), where sigma = A000203.

E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(cototient(k)/k), where cototient = A051953.

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

a(n) = Sum_{k=0..n} binomial(n,k) * A293116(k) * A053529(n-k).

a(n) ~ sqrt(-1/Pi + Pi/6) * n^(n - 1/2) / (2 * exp(n - 1/2 - sqrt(2*(-6 + Pi^2)*n/3))). - Vaclav Kotesovec, Aug 09 2021

Mathematica

nmax = 21; CoefficientList[Series[Product[Exp[-x^k]/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (n - 1)! Sum[(DivisorSigma[1, k] - k) a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 21}]

Crossrefs

Cf. A000166, A000203, A051953, A053529, A293116, A330200.

Sequence in context: A041049 A224346 A005484 * A042455 A245546 A074875

Adjacent sequences: A330198 A330199 A330200 * A330202 A330203 A330204

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 05 2019

Status

approved