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%I #16 Mar 10 2023 10:14:03
%S 1,1,4,16,106,658,6088,51952,592828,6577948,88213744,1173121024,
%T 18663391096,289030343704,5157010548064,92428084599232,
%U 1848308567352592,37038307949425168,822602470902709312,18285742807660340992,444405771941314880416,10883864256927386369056,286778106663948874858624
%N Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(1/(i*j)).
%H Vaclav Kotesovec, <a href="/A318695/b318695.txt">Table of n, a(n) for n = 0..440</a>
%H Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://arxiv.org/abs/2303.02240">A unified treatment of families of partition functions</a>, arXiv:2303.02240 [math.CO], 2023.
%F E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau(k)/k), where tau = number of divisors (A000005).
%F E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} tau(d) ) * x^k/k).
%p seq(n!*coeff(series(mul(1/(1-x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # _Paolo P. Lava_, Jan 09 2019
%t nmax = 22; CoefficientList[Series[Product[Product[1/(1 - x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
%Y Cf. A000005, A006171, A007425, A028342, A280540, A305127, A318696, A318977.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 31 2018
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