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%I #20 Mar 10 2023 10:14:47
%S 1,1,5,11,33,67,180,366,871,1782,3927,7885,16637,32763,66469,128938,
%T 253871,484034,930959,1747304,3292730,6092664,11282364,20596790,
%U 37568653,67736175,121886533,217261372,386216073,681119439,1197524035,2091091902,3639519280
%N G.f.: Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(i*j).
%H Vaclav Kotesovec, <a href="/A280540/b280540.txt">Table of n, a(n) for n = 0..10000</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 G.f.: Product_{k>=1} 1/(1 - x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - _Ilya Gutkovskiy_, Aug 26 2018
%F log(a(n)) ~ (3/2)^(2/3) * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3). - _Vaclav Kotesovec_, Aug 28 2018
%t nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[k*DivisorSigma[0, 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] (* _Vaclav Kotesovec_, Aug 27 2018 *)
%Y Cf. A000005, A006171, A038040, A061256, A107742, A192065, A280541.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Jan 05 2017
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