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A186209 Coefficients of modular function denoted g_5(tau) by Atkin.

%I A186209 #16 Jan 13 2020 11:28:57
%S A186209 1,12,90,520,2535,10908,42614,153960,521235,1669720,5098938,14931060,
%T A186209 42124236,114944700,304344780,784057428,1969912725,4836549432,
%U A186209 11624458120,27391979940,63368714904,144094899520,322411353540
%N A186209 Coefficients of modular function denoted g_5(tau) by Atkin.
%H A186209 G. C. Greubel, <a href="/A186209/b186209.txt">Table of n, a(n) for n = 5..2500</a>
%H A186209 A. O. L. Atkin, <a href="https://doi.org/10.1017/S0017089500000045">Proof of a conjecture of Ramanujan</a>, Glasgow Math. J., 8 (1967), 14-32.
%H A186209 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.
%F A186209 Expansion of (eta(q^11) / eta(q))^12 in powers of q.
%F A186209 Euler transform of period 11 sequence [12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 0, ...].
%F A186209 G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^-6 / f(t) where q = exp(2 Pi i t).
%F A186209 G.f.: x^5 * (Product_{k>0} (1 - x^(11*k)) / (1 - x^k))^12.
%F A186209 a(n) ~ 5^(1/4) * exp(4*Pi*sqrt(5*n/11)) / (sqrt(2) * 11^(25/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e A186209 G.f. = q^5 + 12*q^6 + 90*q^7 + 520*q^8 + 2535*q^9 + 10908*q^10 + 42614*q^11 + ...
%t A186209 nmax=40; Drop[CoefficientList[Series[x^5*Product[((1-x^(11*k)) / (1-x^k))^12,{k,1,nmax}],{x,0,nmax}],x], 5] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%t A186209 eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q^11] /eta[q])^12, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 14 2018 *)
%o A186209 (PARI) {a(n) = my(A); if( n<5, 0, n-=5; A = x * O(x^n); polcoeff( (eta(x^11 + A) / eta(x + A))^12, n))};
%K A186209 nonn
%O A186209 5,2
%A A186209 _Michael Somos_, Feb 15 2011