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%I #22 Mar 10 2023 10:13:49
%S 1,1,4,10,24,52,125,253,549,1126,2290,4525,8987,17259,33174,62669,
%T 117425,217295,399904,726984,1314257,2354807,4191671,7405590,13009916,
%U 22696115,39384232,67937488,116584833,199001304,338076500,571507377,961855945,1611567819
%N G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(i*j).
%H Vaclav Kotesovec, <a href="/A280541/b280541.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 + x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - _Ilya Gutkovskiy_, Aug 26 2018
%F Conjecture: log(a(n)) ~ 3 * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - _Vaclav Kotesovec_, Aug 29 2018
%t nmax = 50; CoefficientList[Series[Product[(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]*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[s, x] (* _Vaclav Kotesovec_, Aug 27 2018 *)
%Y Cf. A000005, A006171, A038040, A061256, A107742, A192065, A280540.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Jan 05 2017
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