%I #11 Mar 07 2018 17:50:58

%S 1,-328,-41956,-8596032,-2597408634,-916285828640,-352170121921992,

%T -143129703441671168,-60517599938503137519,-26355020095077489965264,

%U -11743692598044815023990588,-5329748160859504303225598464

%N Coefficients of (q*(j(q)-1728))^(1/3) where j(q) is the elliptic modular invariant.

%C In general, the expansion of (q*(j(q)-1728))^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 2*m is the positive integer, else is asymptotic to 2^(2*m) * 3^(4*m) * Pi^(2*m) * exp(2*Pi*(n-m)) / (Gamma(-2*m) * Gamma(3/4)^(8*m) * n^(2*m + 1)). - _Vaclav Kotesovec_, Mar 07 2018

%F G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/3).

%F a(n) ~ c * exp(2*Pi*n) / n^(5/3), where c = -2^(2/3) * 3^(5/6) * exp(-2*Pi/3) * Gamma(2/3) / (Pi^(1/3) * Gamma(3/4)^(8/3)) = -0.262554753987597280323546158564... - _Vaclav Kotesovec_, Mar 07 2018

%t CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 07 2018 *)

%Y (q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A289339 (k=7), this sequence (k=8), A007242 (k=12), A289063 (k=24).

%Y Cf. A289061.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jul 02 2017