A289299 Expansion of (q*j(q))^(1/6) where j(q) is the elliptic modular invariant (A000521).

1, 124, -5626, 715000, -104379375, 16966161252, -2946652593626, 535467806605000, -100554207738307500, 19359037551684042500, -3800593180746056684372, 757968936254309704500248, -153133996443087103652605627

Offset: 0

Author

Seiichi Manyama, Jul 02 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..425

Crossrefs

(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).

Cf. A000521, A192731.

Programs

Mathematica

CoefficientList[Series[Sqrt[65536 + x*QPochhammer[-1, x]^24] / (2*QPochhammer[-1, x])^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/6) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/6).

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

a(n) * A299828(n) ~ -exp(2*sqrt(3)*Pi*n) / (2*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018