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A318769 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

%I A318769 #23 Sep 30 2024 21:13:47
%S A318769 1,1,3,17,83,639,5749,53227,561273,7216577,94292531,1352253561,
%T A318769 21657812923,359338829407,6460367397093,126124578755939,
%U A318769 2527688612931569,54137820027005697,1236730462664172643,29137619131277727457,725282418459957414051,18981526480933601454911
%N A318769 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.
%C A318769 a(n)/n! is the weigh transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].
%H A318769 Vaclav Kotesovec, <a href="/A318769/b318769.txt">Table of n, a(n) for n = 0..440</a>
%H A318769 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 A318769 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A318769 E.g.f.: Product_{k>=1} (1 + x^k)^(A017665(k)/A017666(k)).
%F A318769 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(j*k*(1 - x^(j*k)))).
%F A318769 log(a(n)/n!) ~ sqrt(n/2) * Pi^2 / 3. - _Vaclav Kotesovec_, Sep 04 2018
%F A318769 a(n)/n! ~ c * exp(sqrt(n/2)*Pi^2/3) / n^(3/4 + log(2)/4), where c = 0.15653645678497413538057076667218805302154965061194080137... - _Vaclav Kotesovec_, Sep 05 2018
%p A318769 with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(-(-1)^(j/d)*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 A318769 nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A318769 nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A318769 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
%t A318769 nmax = 21; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, 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}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 03 2018 *)
%Y A318769 Cf. A000203, A017665, A017666, A168243, A192065, A288417, A305127, A318696, A318814.
%K A318769 nonn
%O A318769 0,3
%A A318769 _Ilya Gutkovskiy_, Sep 03 2018