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A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

%I A305127 #21 Sep 30 2024 21:10:11
%S A305127 1,1,5,23,179,1279,13699,135085,1764377,22527521,344625461,5283739471,
%T A305127 94562354875,1685808248383,33947023942259,694786150879829,
%U A305127 15613612524749489,357353282848083265,8880505496901812197,224851013929747732231,6106205671049245677251,169523515381173773551871
%N A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.
%C A305127 a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].
%H A305127 Vaclav Kotesovec, <a href="/A305127/b305127.txt">Table of n, a(n) for n = 0..438</a>
%H A305127 Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://doi.org/10.1007/s44007-024-00134-w">A unified treatment of families of partition functions</a>, La Matematica (2024). Preprint available as <a href="https://arxiv.org/abs/2303.02240">arXiv:2303.02240</a> [math.CO], 2023.
%H A305127 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305127 E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).
%F A305127 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).
%F A305127 log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - _Vaclav Kotesovec_, Sep 04 2018
%p A305127 with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # _Vaclav Kotesovec_, Sep 04 2018
%t A305127 nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A305127 nmax = 21; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A305127 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
%t A305127 nmax = 21; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 03 2018 *)
%Y A305127 Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.
%K A305127 nonn
%O A305127 0,3
%A A305127 _Ilya Gutkovskiy_, May 26 2018