A289302 Expansion of (q*j(q))^(7/24) where j(q) is the elliptic modular invariant (A000521).

1, 217, 245, 231350, -27293420, 4017072017, -643057897118, 109259930443485, -19377905432572925, 3549922504344871655, -666990037937425724641, 127890778891452935279096, -24934077008209243436961385

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), A289299 (k=4), A289300 (k=5), A289301 (k=6), this sequence (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

Programs

Mathematica

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(7/8) / (2*QPochhammer[-1, x])^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(7/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(7*A192731(n)/24).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(15/8), where c = 0.108789720644871714449969800661839212719879897088563371823367481878... = 7 * 3^(7/8) * sqrt(2 - sqrt(2)) * Gamma(1/3)^(21/4) * Gamma(7/8) / (2^(39/8) * exp(7 * Pi / (8 * sqrt(3))) * Pi^(9/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018