login
Coefficients in expansion of (E_4^3/E_6^2)^(1/6).
19

%I #17 Mar 04 2018 12:37:21

%S 1,288,76032,33042816,14318032896,6651157620672,3146793694792704,

%T 1522045714678435584,745464270665241870336,369134048335617435664800,

%U 184269983601798163049283072,92610644166133510115124717696

%N Coefficients in expansion of (E_4^3/E_6^2)^(1/6).

%H Seiichi Manyama, <a href="/A300052/b300052.txt">Table of n, a(n) for n = 0..367</a>

%F Convolution inverse of A299860.

%F a(n) ~ 2^(4/3) * Pi * exp(2*Pi*n) / (3^(1/6) * Gamma(1/4)^(4/3) * Gamma(1/3) * n^(2/3)). - _Vaclav Kotesovec_, Mar 04 2018

%F a(n) * A299860(n) ~ -exp(4*Pi*n) / (2*sqrt(3)*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018

%t terms = 12;

%t E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];

%t E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];

%t (E4[x]^3/E6[x]^2)^(1/6) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 28 2018 *)

%Y (E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), this sequence (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

%Y Cf. A004009 (E_4), A013973 (E_6), A299860.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 23 2018