A289305 Expansion of (q*j(q))^(11/24) where j(q) is the elliptic modular invariant (A000521).

1, 341, 21527, -244112, 50791235, -6177875286, 883458515093, -136541356378141, 22354744100161913, -3821528558157433970, 675604462786881129711, -122689458583136157060647, 22774615293799045532223797

Offset: 0

Author

Seiichi Manyama, Jul 02 2017

Keywords

• sign

Comments

In general, the expansion of (q*j(q))^m, where j(q) is the elliptic modular invariant (A000521) and m <> 0, is asymptotic to exp(4*Pi*sqrt(m*n)) * m^(1/4) / (sqrt(2) * n^(3/4)) if 3*m is the positive integer, else is asymptotic to (-1)^n * 3^(3*m) * Gamma(1/3)^(18*m) * exp(Pi*sqrt(3)*(n-m)) / (2^(3*m) * Pi^(12*m) * Gamma(-3*m) * n^(3*m + 1)). - Vaclav Kotesovec, Mar 06 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..426

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), A289303 (k=9), A289304 (k=10), this sequence (k=11), A161361 (k=12).

Cf. A000521, A192731.

Programs

Mathematica

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(11/8) / (2*QPochhammer[-1, x])^11, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(11/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(11*A192731(n)/24).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(19/8), where c = 0.3243413869190225417519777558769755719610962636852073820509897134587... = 11 * 3^(19/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(33/4) * Gamma(3/8) / (2^(67/8) * exp(11 * Pi / (8 * sqrt(3))) * Pi^(13/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018