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A341872 Coefficients of the series whose 72nd power equals E_2(x)^3/E_6(x), where E_2(x) and E_6(x) are the Eisenstein series A006352 and A013973.

%I A341872 #10 Mar 08 2021 16:47:17
%S A341872 1,6,1998,722484,291762942,125454173544,56146411655460,
%T A341872 25832836404319152,12128921727745915062,5783583949613172902394,
%U A341872 2791762868052719757442008,1360988846025232489401029220,668925190887642335984231235348,331039288912491308442251418152952
%N A341872 Coefficients of the series whose 72nd power equals E_2(x)^3/E_6(x), where E_2(x) and E_6(x) are the Eisenstein series A006352 and A013973.
%C A341872 It is easy to see that E_2(x)^3/E_6(x) == 1 - 72*Sum_{k >= 1} (k - 7*k^5)*x^k/(1 - x^k) (mod 432), and also that the integer k - 7*k^5 is always divisible by 6. Hence, E_2(x)^3/E_6(x) == 1 (mod 432). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^2/E_6(x))^(1/72) = 1 + 6*x + 1998*x^2 + 722484*x^3 + 291762942* x^4 + ... has integer coefficients.
%H A341872 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H A341872 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_series">Eisenstein series</a>
%F A341872 a(n) ~ c * exp(2*Pi*n) / n^(71/72), where c = 0.013960369132490470055158573616810629626490780934389076244815126342923645628... - _Vaclav Kotesovec_, Mar 08 2021
%p A341872 E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
%p A341872 E(6,x) := 1 - 504*add(k^5*x^k/(1-x^k), k = 1..20):
%p A341872 with(gfun): series((E(2,x)^3/E(6,x))^(1/72), x, 20):
%p A341872 seriestolist(%);
%Y A341872 Cf. A006352, A013973, A341871 - A341875.
%K A341872 nonn,easy
%O A341872 0,2
%A A341872 _Peter Bala_, Feb 22 2021