A289298 Expansion of (q*j(q))^(1/8) where j(q) is the elliptic modular invariant (A000521).

1, 93, -5661, 741532, -113207799, 19015433748, -3390166183729, 629581913929419, -120437982238038210, 23564574046009042869, -4692899968498921291530, 948024211601180444075739, -193775768073341380441728322

Offset: 0

Author

Seiichi Manyama, Jul 02 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..424

Crossrefs

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), this sequence (k=3), A289299 (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[(65536 + x*QPochhammer[-1, x]^24)^(3/8) / (2*QPochhammer[-1, x])^3, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/8) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

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

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(11/8), where c = 0.2541876595230750963327533839122695596555059904123327336821622582369... = 3^(11/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(9/4) * Gamma(3/8) / (2^(35/8) * exp(sqrt(3) * Pi/8) * Pi^(5/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018

a(n) * A299827(n) ~ -3*2^(1/4)*sqrt(1+sqrt(2)) * exp(2*sqrt(3)*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018