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%I #32 Mar 22 2018 06:01:44
%S 1,0,1,0,2,-1,3,-1,5,-2,6,-3,10,-5,13,-7,19,-11,25,-15,35,-22,45,-30,
%T 62,-41,79,-55,105,-75,134,-98,175,-130,220,-168,284,-219,355,-280,
%U 451,-360,561,-455,705,-578,870,-725,1085,-910,1331,-1132,1644,-1410
%N Expansion of Product_{k>=1} (1-x^(5*k))/(1-x^(2*k)).
%C In general, if m >= 1 and g.f. = Product_{k>=1} (1-x^((2*m+1)*k))/(1-x^(2*k)), then a(n) ~ (-1)^n * exp(Pi*sqrt((4*m+1)*n/(6*(2*m+1)))) * (4*m+1)^(1/4) / (2^(7/4) * 3^(1/4) * (2*m+1)^(3/4) * n^(3/4)).
%H Vaclav Kotesovec, <a href="/A262364/b262364.txt">Table of n, a(n) for n = 0..5000</a>
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016, p. 15.
%F a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n/10)) / (2^(7/4) * 5^(3/4) * n^(3/4)).
%e G.f. = 1 + x^2 + 2*x^4 - x^5 + 3*x^6 - x^7 + 5*x^8 - 2*x^9 + ...
%t nmax = 60; CoefficientList[Series[Product[(1-x^(5*k))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x]
%t CoefficientList[Series[QPochhammer[x^5]/QPochhammer[x^2], {x, 0, 60}], x]
%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^5)/eta(q^2))} \\ _Altug Alkan_, Mar 21 2018
%Y Cf. A035959, A147783, A262346.
%K sign
%O 0,5
%A _Vaclav Kotesovec_, Sep 23 2015
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