A288840 - OEIS

%I #42 Mar 01 2018 02:45:43

%S 1,984,574488,307081056,164453203992,88062998451984,47157008244215904,

%T 25252184242734325440,13522333949728177520664,

%U 7241096993206804017918456,3877547016709833498690361488,2076394071353012138642420600352

%N Coefficients in expansion of E_8/E_6.

%D Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004.

%H Seiichi Manyama, <a href="/A288840/b288840.txt">Table of n, a(n) for n = 0..366</a>

%F From _Seiichi Manyama_, Jun 27 2017: (Start)

%F Let j_0 = 1 and j_1 = j - 744. Define j_m by j_m = j1 | T_0(m), where T_0(m) = mT_{m, 0} is the normalized m-th weight zero Hecke operator. a(n) = j_n(i).

%F G.f.: Sum_{n >= 0} j_n(i)*q^n. (End)

%F a(n) ~ 2 * exp(2*Pi*n). - _Vaclav Kotesovec_, Jun 28 2017

%F G.f.: -q*j'/(j-1728) where j is the elliptic modular invariant (A000521). - _Seiichi Manyama_, Jul 12 2017

%e G.f.: 1 + 984*q + 574488*q^2 + 307081056*q^3 + 164453203992*q^4 + 88062998451984*q^5 + 47157008244215904*q^6 + ...

%e From _Seiichi Manyama_, Jun 27 2017: (Start)

%e a(0) = j_0(i) = 1,_

%e a(1) = j_1(i) = -744 + 1728^1 = 984,

%e a(2) = j_2(i) = 159768 - 1488*1728^1 + 1728^2 = 574488. (End)

%t nmax = 20; CoefficientList[Series[(1 + 480*Sum[DivisorSigma[7, k]*x^k, {k, 1, nmax}])/(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 28 2017 *)

%t terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[8]/Ei[6] + O[x]^terms, x] (* _Jean-François Alcover_, Mar 01 2018 *)

%Y Cf. A013973 (E_6), A008410 (E_8).

%Y Cf. A288261 (E_6/E_4).

%Y Cf. A000521 (j), A035230 (-q*j'), A289141, A289417.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jun 17 2017