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%I #10 Mar 08 2021 16:39:13
%S 1,30,5310,2453220,910100190,409796742600,181276113779460,
%T 84362079365838960,39636500385830239350,18986938020443181757410,
%U 9186944625290601368703000,4491611148118819794144792660
%N Coefficients of the series whose 24th power equals E_2(x)*E_4(x)/E_6(x), where E_2(x), E_4(x) and E_6(x) are the Eisenstein series A006352, A004009 and A013973.
%C Since E_2(x)*E_4(x)/E_6(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3 - 21*k^5)*x^k/(1 - x^k) (mod 144), and since the integer k - 10*k^3 - 21*k^5 is always divisible by 6 it follows that E_2(x)*E_4(x)/E_6(x) == 1 (mod 144). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x)/E_6(x))^(1/24) = 1 + 30*x + 5310*x^2 + 2453220*x^3 + 910100190*x^4 + ... has integer coefficients.
%H 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_series">Eisenstein series</a>
%F a(n) ~ c * exp(2*Pi*n) / n^(23/24), where c = 0.0431061156115657949750305669836959595841497962033916083447436... - _Vaclav Kotesovec_, Mar 08 2021
%p E(2,x) := 1 - 24*add(k*x^k/(1-x^k), k = 1..20):
%p E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
%p E(6,x) := 1 - 504*add(k^5*x^k/(1-x^k), k = 1..20):
%p with(gfun): series((E(2,x)*E(4,x)/E(6,x))^(1/24), x, 20):
%p seriestolist(%);
%Y Cf. A006352, A004009, A013973, A341871 - A341874.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Feb 23 2021
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