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A289300 Expansion of (q*j(q))^(5/24) where j(q) is the elliptic modular invariant (A000521).

%I A289300 #22 Mar 06 2018 11:25:51
%S A289300 1,155,-4630,601265,-83644610,13148835656,-2223584717035,
%T A289300 395257299676190,-72843145114522035,13796578308407774725,
%U A289300 -2669652272250261922223,525556527400692937755655,-104937908072571416700653120
%N A289300 Expansion of (q*j(q))^(5/24) where j(q) is the elliptic modular invariant (A000521).
%H A289300 Seiichi Manyama, <a href="/A289300/b289300.txt">Table of n, a(n) for n = 0..425</a>
%F A289300 G.f.: Product_{n>=1} (1-q^n)^(5*A192731(n)/24).
%F A289300 a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(13/8), where c = 0.251632947646443757912747944865268710111059274679945447776728146817... = 5 * 3^(5/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(15/4) * Gamma(5/8) / (2^(37/8) * exp(5 * Pi / (8 * sqrt(3))) * Pi^(7/2)). - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 06 2018
%F A289300 a(n) * A299829(n) ~ -5*sqrt(2 + sqrt(2)) * exp(2*sqrt(3)*Pi*n) / (16*Pi*n^2). - _Vaclav Kotesovec_, Feb 20 2018
%t A289300 CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(5/8) / (2*QPochhammer[-1, x])^5, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)
%t A289300 (q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(5/24) + O[q]^13 // CoefficientList[#, q]& (* _Jean-François Alcover_, Nov 02 2017 *)
%Y A289300 (q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), this sequence (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).
%Y A289300 Cf. A000521, A192731.
%K A289300 sign
%O A289300 0,2
%A A289300 _Seiichi Manyama_, Jul 02 2017