This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A318966 #14 Sep 30 2024 21:23:12
%S A318966 1,1,5,21,165,1077,11457,103905,1345257,15834825,237535389,3372509709,
%T A318966 59235634125,979573962429,19224990899865,366788042231193,
%U A318966 8019002662543953,171360055378885905,4132946756763614133,97947895990285022085,2576516749059849502581,67124117357620005459141
%N A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(1/(i*j*k)).
%H A318966 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 A318966 E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau_3(k)/k), where tau_3 = A007425.
%F A318966 E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).
%p A318966 a:=series(mul(mul(mul(1/(1-x^(i*j*k))^(1/(i*j*k)),k=1..21),j=1..50),i=1..50),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # _Paolo P. Lava_, Apr 02 2019
%t A318966 nmax = 21; CoefficientList[Series[Product[Product[Product[1/(1 - x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
%t A318966 nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A318966 a[n_] := a[n] = (n - 1)! Sum[Sum[Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
%Y A318966 Cf. A000005, A007425, A007426, A028342, A174465, A318413, A318695, A318967.
%K A318966 nonn
%O A318966 0,3
%A A318966 _Ilya Gutkovskiy_, Sep 06 2018