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%I #13 Sep 16 2017 06:46:31
%S 1,1,4,90,272,1275,49284,124901,536640,1620648,104040100,223290012,
%T 880969104,2485978170,7454471332,592164776475,1138401673472,
%U 4109108002310,10877348160900,30962024560494,72270337440400,7523649856001916,13202150810778116,44577985082575400
%N a(n) = [x^n] Product_{k>=1} (1 + n^2*x^k).
%H Vaclav Kotesovec, <a href="/A292304/b292304.txt">Table of n, a(n) for n = 0..5000</a>
%H Vaclav Kotesovec, <a href="/A292304/a292304.jpg">Graph - The asymptotic ratio</a>
%F Conjecture: a(n) ~ exp(2*sqrt((Pi^2/6 + 2*log(n)^2)*n)) * (Pi^2/6 + 2*log(n)^2)^(1/4) / (2 * sqrt(Pi) * n^(7/4)).
%t nmax = 30; Table[SeriesCoefficient[Product[(1+n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
%t Table[SeriesCoefficient[QPochhammer[-n^2, x, 1 + n]/(1 + n^2), {x, 0, n}], {n, 0, 30}]
%Y Cf. A092484, A291698.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Sep 14 2017