Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Aug 09 2021 08:36:18
%S 1,2,9,52,389,3366,34477,392624,5035977,70674634,1085687921,
%T 17982460332,321298513549,6121639481582,124336400707989,
%U 2674237637496616,60799325536137617,1454405117742700434,36556297436871331417,961899014831786663204
%N Expansion of e.g.f. Product_{k>=1} exp(x^k) / (1 - x^k).
%F E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of A000522.
%F E.g.f.: exp(Sum_{k>=1} (sigma(k) / k + 1) * x^k), where sigma = A000203.
%F E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(phi(k)/k + 1), where phi = A000010.
%F a(0) = 1; a(n) = (n - 1)! * Sum_{k=1..n} (sigma(k) + k) * a(n-k) / (n - k)!.
%F a(n) = Sum_{k=0..n} binomial(n,k) * A000262(k) * A053529(n-k).
%F 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
%t nmax = 19; CoefficientList[Series[Product[Exp[x^k]/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t 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}]
%t Table[n!*Sum[LaguerreL[k, -1, -1]*PartitionsP[n-k],{k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 09 2021 *)
%Y Cf. A000010, A000203, A000262, A000522, A053529, A330201, A346964.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 05 2019