A289304 - OEIS

%I #20 Mar 06 2018 11:34:02

%S 1,310,14765,-232770,40539830,-5199871688,765038308115,

%T -121140033966330,20242157273780710,-3521886754264327670,

%U 632344647471171938140,-116428917411726531951590,21883035176258955622401245

%N Expansion of (q*j(q))^(5/12) where j(q) is the elliptic modular invariant (A000521).

%H Seiichi Manyama, <a href="/A289304/b289304.txt">Table of n, a(n) for n = 0..425</a>

%F G.f.: Product_{n>=1} (1-q^n)^(5*A192731(n)/12).

%F a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(9/4), where c = 0.232272469556851820006346410170543574844213494230850435863953522617... = 5 * 3^(5/4) * Gamma(1/4) * Gamma(1/3)^(15/2) / (2^(23/4) * exp(5 * Pi / (4 * sqrt(3))) * Pi^6). - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 06 2018

%t CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(5/4) / (2*QPochhammer[-1, x])^10, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)

%t (q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(5/12) + O[q]^13 //

%t 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), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), this sequence (k=10), A289305 (k=11), A161361 (k=12).

%Y Cf. A000521, A192731.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jul 02 2017