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%I #23 Mar 06 2018 11:23:57
%S 1,124,-5626,715000,-104379375,16966161252,-2946652593626,
%T 535467806605000,-100554207738307500,19359037551684042500,
%U -3800593180746056684372,757968936254309704500248,-153133996443087103652605627
%N Expansion of (q*j(q))^(1/6) where j(q) is the elliptic modular invariant (A000521).
%H Seiichi Manyama, <a href="/A289299/b289299.txt">Table of n, a(n) for n = 0..425</a>
%F G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/6).
%F a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(3/2), where c = 0.27174882346571745439868471841345665496773077910099184617347055088... = sqrt(3) * Gamma(1/3)^3 / (2^(3/2) * exp(Pi/(2 * sqrt(3))) * Pi^(5/2)). - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 06 2018
%F a(n) * A299828(n) ~ -exp(2*sqrt(3)*Pi*n) / (2*Pi*n^2). - _Vaclav Kotesovec_, Feb 20 2018
%t CoefficientList[Series[Sqrt[65536 + x*QPochhammer[-1, x]^24] / (2*QPochhammer[-1, x])^4, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)
%t (q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/6) + O[q]^13 // CoefficientList[#, q]& (* _Jean-François Alcover_, Nov 02 2017 *)
%Y (q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), this sequence (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).
%Y Cf. A000521, A192731.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jul 02 2017
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