A330200 Expansion of e.g.f. Product_{k>=1} exp(x^k) / (1 - x^k).

1, 2, 9, 52, 389, 3366, 34477, 392624, 5035977, 70674634, 1085687921, 17982460332, 321298513549, 6121639481582, 124336400707989, 2674237637496616, 60799325536137617, 1454405117742700434, 36556297436871331417, 961899014831786663204

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of A000522.

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)^(phi(k)/k + 1), where phi = A000010.

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) * A000262(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 = 19; 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, 19}]
Table[n!*Sum[LaguerreL[k, -1, -1]*PartitionsP[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 09 2021 *)

Crossrefs

Cf. A000010, A000203, A000262, A000522, A053529, A330201, A346964.

Sequence in context: A143508 A052882 A248440 * A143922 A305304 A369090

Adjacent sequences: A330197 A330198 A330199 * A330201 A330202 A330203

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 05 2019

Status

approved