%I #18 Jan 31 2019 19:35:13
%S 1,720,911520,1301011200,1958042030400,3036508587993600,
%T 4800606078996518400,7692266556998230118400,
%U 12448466349673022940816000,20299324957420186505619072000,33301542920000564787678367257600
%N Coefficients in expansion of E_2*E_4/(E_6*j) in powers of 1/j.
%D R. A. Rankin, The zeros of Eisenstein series, Publ. Ramanujan Institute 1 (1969), 137-144. (On page 139).
%H Vaclav Kotesovec, <a href="/A030185/b030185.txt">Table of n, a(n) for n = 1..300</a>
%H Oscar E. González, <a href="https://faculty.math.illinois.edu/~oscareg2/resources/publications/rankinDeterminantsV11.pdf">An observation of Rankin on Hankel determinants</a>, Department of Mathematics, University of Illinois at Urbana-Champaign, 2018.
%H M. Kaneko and D. Zagier, <a href="http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, AMS/IP Studies in Advanced Mathematics, vol. 7, 97--126, (1998). See esp. p. 110.
%F a(n) ~ Pi^(3/2) * 1728^n / (72 * Gamma(1/4)^4 * sqrt(3*n)). - _Vaclav Kotesovec_, Apr 07 2018
%t nmax = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); A000521x = Normal[Series[(1728 E4[x]^3/(E4[x]^3 - E6[x]^2)), {x, 0, nmax}]]; expansion = CoefficientList[Series[E2[x]*E4[x]/(E6[x]*(1728 E4[x]^3/(E4[x]^3 - E6[x]^2))), {x, 0, nmax}], x]; A[x_] := Sum[c[k]/A000521x^k, {k, 0, nmax}]; Array[c, nmax] /. Solve[CoefficientList[Series[A[x], {x, 0, nmax}], x] == expansion][[1]] (* _Vaclav Kotesovec_, Apr 07 2018 *)
%Y Cf. A000521, A006352, A004009, A013973, A145200.
%K nonn
%O 1,2
%A _N. J. A. Sloane_.
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010