A289303 Expansion of (q*j(q))^(3/8) where j(q) is the elliptic modular invariant (A000521).

1, 279, 8964, -129885, 23406255, -3128904747, 473738861853, -76824787699971, 13098300010462845, -2318947179364181165, 422782870045511526012, -78914282330756685655485, 15016013710284896513279286

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), A289302 (k=7), A007245 (k=8), this sequence (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

Programs

Mathematica

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(9/8) / (2*QPochhammer[-1, x])^9, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(3/8) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(3*A192731(n)/8).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(17/8), where c = 0.1186486859763112993214522284920488979797011156387080809639905476634... = 3^(25/8) * sqrt(2 - sqrt(2)) * Gamma(1/8) * Gamma(1/3)^(27/4) / (2^(65/8) * exp(3 * sqrt(3) * Pi/8) * Pi^(11/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018