A289209 - OEIS

%I #29 Mar 04 2018 11:41:21

%S 1,1728,1700352,1332930816,939690602496,624182333927040,

%T 399031077924476928,248370528839869094400,151578005556161702559744,

%U 91116938989182168182098368,54119528875319902426524072960,31833210323194251819350736777984

%N Coefficients in expansion of E_4^3/E_6^2.

%H Seiichi Manyama, <a href="/A289209/b289209.txt">Table of n, a(n) for n = 0..365</a>

%F G.f.: 1 + 1728 * q * Product_{k>=1} (1-q^k)^24 / E_6^2.

%F G.f.: (E_4*E_8)/(E_6*E_6) = (E_8*E_8)/(E_6*E_10). - _Seiichi Manyama_, Jun 29 2017

%F a(n) = 1728 * A289417(n - 1) for n > 0. - _Seiichi Manyama_, Jul 08 2017

%F a(n) ~ c * exp(2*Pi*n) * n, where c = 256 * Pi^6 / (3 * Gamma(1/4)^8) = 2.747700206704861755142526128354171788550012833617513654955480535522... - _Vaclav Kotesovec_, Jul 08 2017, updated Mar 04 2018

%F a(0) = 1, a(n) = (288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Feb 26 2018

%t nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 08 2017 *)

%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), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), this sequence (k=288).

%Y Cf. A004009 (E_4), A013973 (E_6), A289063, A289210, A289417, A300025.

%Y E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jun 28 2017