OFFSET
0,2
COMMENTS
From Vaclav Kotesovec, Jun 10 2018: (Start)
For k > 0, if mod(k,8) <> 0 then (q*j(q))^(k/24) is asymptotic to -(-1)^n * sin(k*Pi/8) * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * Gamma(k/8) * exp(Pi*sqrt(3)*n) / (Pi^(k/2 + 1) * 2^(k/8 + 3) * exp(k*Pi/(8*sqrt(3))) * n^(k/8 + 1)). Equivalently, is asymptotic to -(-1)^n * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * exp(Pi*sqrt(3)*(n - k/24)) / (Pi^(k/2) * 2^(k/8 + 3) * Gamma(1 - k/8) * n^(k/8 + 1)).
For k > 0, if mod(k,8) = 0 then (q*j(q))^(k/24) is asymptotic to exp(Pi*sqrt(2*k*n/3)) * k^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).
(End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..424
FORMULA
This is essentially the eighth root of the theta series of E_8 (A108091), divided by the Dedekind eta function. - N. J. A. Sloane, Aug 08 2005
G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/24). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 0.11364889078525240958152388212499254894082832445224690827436413842337... = 3^(1/8) * sqrt(2 - sqrt(2)) * Gamma(1/8) * Gamma(1/3)^(3/4) / (2^(33/8) * exp(Pi/(8 * sqrt(3))) * Pi^(3/2)). - Vaclav Kotesovec, Jul 02 2017, updated Mar 06 2018
a(n) * A289397(n) ~ c * exp(2*Pi*sqrt(3)*n) / n^2, where c = -sqrt(2-sqrt(2)) / (16*Pi). - Vaclav Kotesovec, Mar 06 2018
EXAMPLE
1 + 31*q - 2848*q^2 + 413823*q^3 - 68767135*q^4 + 12310047967*q^5 - 2309368876639*q^6 + ...
MATHEMATICA
CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(1/8) / (2*QPochhammer[-1, x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)
PROG
(PARI) {a(n)=if(n<0, 0, polcoeff( (ellj(x+x^2*O(x^n))*x)^(1/24), n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Apr 25 2005
STATUS
approved