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%I #24 Mar 06 2018 11:27:51
%S 1,186,-2673,430118,-56443725,8578591578,-1411853283028,
%T 245405765574252,-44373155962556475,8266332741845429800,
%U -1576306833508315403544,306275559567641721838494,-60432437032381794135586069
%N Expansion of (q*j(q))^(1/4) where j(q) is the elliptic modular invariant (A000521).
%H Seiichi Manyama, <a href="/A289301/b289301.txt">Table of n, a(n) for n = 0..425</a>
%F G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/4).
%F a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(7/4), where c = 0.1955865990744763088634116856422381013939034554805874572099292810179... = 3^(7/4) * Gamma(1/3)^(9/2) / (2^(11/4) * exp(sqrt(3) * Pi/4) * Pi^3 * Gamma(1/4)). - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 06 2018
%F a(n) * A299830(n) ~ -3*exp(2*sqrt(3)*Pi*n) / (2^(5/2)*Pi*n^2). - _Vaclav Kotesovec_, Feb 20 2018
%t CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(3/4) / (64 * QPochhammer[-1, x]^6), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)
%t (q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/4) + 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), A289299 (k=4), A289300 (k=5), this sequence (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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