login
Coefficients in expansion of (E_4/E_2^4)^(1/8).
3

%I #19 Jun 03 2018 09:25:32

%S 1,42,-2268,395304,-64600914,11644170552,-2188350306072,

%T 424652412357696,-84326944950450972,17044476557469661986,

%U -3493525880987663047128,724189608821718233434296,-151528575864988356484968840,31955212589107172812017247992

%N Coefficients in expansion of (E_4/E_2^4)^(1/8).

%C Also coefficients in expansion of (E_8/E_2^8)^(1/16).

%F Convolution inverse of A294974.

%F G.f.: Product_{n>=1} (1-q^n)^(-A294626(n)).

%F a(n) ~ -(-1)^n * Pi^(5/4) * exp(Pi*sqrt(3)*n) / (2^(19/8) * 3^(9/8) * Gamma(2/3)^(9/4) * Gamma(7/8) * n^(9/8)). - _Vaclav Kotesovec_, Jun 03 2018

%t terms = 14;

%t E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];

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

%t (E4[x]/E2[x]^4)^(1/8) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)

%Y Cf. A004009 (E_4), A006352 (E_2), A108091, A289565, A294626, A294974.

%K sign

%O 0,2

%A _Seiichi Manyama_, Feb 12 2018