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A341842 Coefficients of the series whose 12th power equals E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009.

%I A341842 #9 Aug 25 2021 16:28:08
%S A341842 1,18,-2088,301296,-50784174,9174627360,-1734603719472,
%T A341842 338286925650240,-67486440186470016,13697820033167444178,
%U A341842 -2818359890320927630320,586296297186462310481424,-123077156275866375661524864,26034142700316716015964656544
%N A341842 Coefficients of the series whose 12th power equals E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009.
%C A341842 The g.f. is the 12th root of the g.f. of A282019.
%C A341842 It is easy to see that E_2(x)*E_4(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 is always divisible by 3. Hence, E_2(x)*E_4(x) == 1 (mod 72). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x))^(1/12) = 1 + 18*x - 2088*x^2 + 301296*x^3 - 50784174*x^4 + ... has integer coefficients.
%H A341842 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H A341842 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_series">Eisenstein series</a>
%p A341842 E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
%p A341842 E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
%p A341842 with(gfun): series((E(2,x)*E(4,x))^(1/12), x, 20):
%p A341842 seriestolist(%);
%Y A341842 A004009, A006352, A108091, A282019, A289392.
%K A341842 sign,easy
%O A341842 0,2
%A A341842 _Peter Bala_, Feb 21 2021