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%I #6 Feb 06 2016 10:53:52
%S 1,2,0,8,-2,8,16,8,8,10,80,-8,72,-24,144,128,134,40,224,120,232,688,
%T 176,696,32,1194,-96,1840,1144,2248,288,2968,800,4160,752,5104,6438,
%U 4984,5104,5488,10960,4856,14080,3480,24408,15448,26832,7080,42120,11178
%N Expansion of Product_{k>=1} ((1 + 3*x^k) / (1 + x^k)).
%C In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 + x^k)) then a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt((m+1)*Pi) * n^(3/4)), where c = Pi^2/3 + 2*log(m)^2 + 4*polylog(2, -1/m).
%H Vaclav Kotesovec, <a href="/A268499/b268499.txt">Table of n, a(n) for n = 0..5000</a>
%F a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(3)^2 + 4*polylog(2, -1/3) = 4.467633549370382939364... .
%t nmax = 100; CoefficientList[Series[Product[(1+3*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A032308, A266821, A268498, A268500.
%K sign
%O 0,2
%A _Vaclav Kotesovec_, Feb 06 2016
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