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%I #16 May 29 2022 18:00:20
%S 1,-6,558,-88884,15433662,-2864048616,552921962724,-109731286565040,
%T 22220439670517814,-4569456313225317114,951159953810624453208,
%U -199945837161334089352548,42373766861587365894611604
%N Coefficients of the series whose 48th power equals E_2(x)^2/E_4(x), where E_2(x) and E_4(x) are the Eisenstein series A006352 and A004009.
%C It is easy to see that E_2(x)^2/E_4(x) == 1 - 48*Sum_{k >= 1} (k + 5*k^3)*x^k/(1 - x^k) (mod 288), and also that the integer k + 5*k^3 is always divisible by 6. Hence, E_2(x)^2/E_4(x) == 1 (mod 288). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^2/E_4(x))^(1/48) = 1 - 6*x + 558*x^2 - 88884*x^3 + 15433662*x^4 - ... has integer coefficients.
%C Note that (E_2(x)^2/E_4(x))^(1/48) = (E_2(x)^4/E_8(x))^(1/96).
%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>
%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 with(gfun): series((E(2,x)^2/E(4,x))^(1/48), x, 20):
%p seriestolist(%);
%Y Cf. A004009, A006352, A108091, A289392, A341872, A341873, A341874, A341875.
%K sign,easy
%O 0,2
%A _Peter Bala_, Feb 22 2021
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