%I #21 Sep 30 2024 19:26:24
%S 1,1,4,8,20,36,86,150,314,564,1088,1902,3557,6085,10902,18506,32124,
%T 53584,91133,149749,249315,405121,662582,1063152,1714580,2719842,
%U 4327302,6797316,10686005,16622003,25861855,39866017,61422891,93910783,143406552,217537696
%N G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l)).
%H Vaclav Kotesovec, <a href="/A280486/b280486.txt">Table of n, a(n) for n = 0..10000</a>
%H 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.
%F G.f.: Product_{k>=1} (1 + x^k)^tau_4(k), where tau_4() = A007426. - _Ilya Gutkovskiy_, May 22 2018
%t nmax = 50; CoefficientList[Series[Product[(1+x^(i*j*k*l)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}], {x, 0, nmax}], x]
%t nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#] * DivisorSigma[0, #] &], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[tau4[[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] (* _Vaclav Kotesovec_, Sep 08 2018 *)
%Y Cf. A000009, A007426, A107742, A280473, A280487, A219561.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Jan 04 2017