OFFSET
0,2
COMMENTS
The g.f. is the 12th root of the g.f. of A282019.
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.
LINKS
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Wikipedia, Eisenstein series
MAPLE
E(2, x) := 1 - 24*add(k*x^k/(1-x^k), k = 1..20):
E(4, x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2, x)*E(4, x))^(1/12), x, 20):
seriestolist(%);
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Feb 21 2021
STATUS
approved