A289297 Expansion of (q*j(q))^(1/12) where j(q) is the elliptic modular invariant (A000521).

1, 62, -4735, 651070, -103766140, 17999397756, -3292567703035, 624659270035130, -121698860487451255, 24194029851560118900, -4886913657541566648179, 999849040331683393909232, -206741394604073327046805355

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), this sequence (k=2), A289298 (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)^(1/4) / (2*QPochhammer[-1, x])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/12) + O[q]^13 //
CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/12).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = 0.200236163401945306105645017761063156355568043417672219092096121424... = 3^(1/4) * Gamma(1/4) * Gamma(1/3)^(3/2) / (2^(11/4) * exp(Pi/(4 * sqrt(3))) * Pi^2). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018

a(n) * A299826(n) ~ -exp(2*sqrt(3)*n*Pi) / (2^(5/2)*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018