A289301 Expansion of (q*j(q))^(1/4) where j(q) is the elliptic modular invariant (A000521).

1, 186, -2673, 430118, -56443725, 8578591578, -1411853283028, 245405765574252, -44373155962556475, 8266332741845429800, -1576306833508315403544, 306275559567641721838494, -60432437032381794135586069

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), this sequence (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)^(3/4) / (64 * QPochhammer[-1, x]^6), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/4) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/4).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(7/4), where c = 0.1955865990744763088634116856422381013939034554805874572099292810179... = 3^(7/4) * Gamma(1/3)^(9/2) / (2^(11/4) * exp(sqrt(3) * Pi/4) * Pi^3 * Gamma(1/4)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018

a(n) * A299830(n) ~ -3*exp(2*sqrt(3)*Pi*n) / (2^(5/2)*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018