OFFSET
0,2
COMMENTS
It is easy to see that E_2(x)^7/E_14(x) == 1 - 24*Sum_{k >= 1} (7*k - 11*k^13)*x^k/(1 - x^k) (mod 144), and also that the integer 7*k - k^13 is always divisible by 6. Hence, E_2(x)^7/E_14(x) == 1 (mod 144). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^7/E_14(x))^(1/24) = 1 - 6*x + 8118*x^2 + 1740636*x^3 + 937783902*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, 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(14, x) := 1 - 24*add(k^13*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2, x)^7/E(14, x))^(1/24), x, 20):
seriestolist(%);
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Feb 23 2021
STATUS
approved