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.

1, 18, -2088, 301296, -50784174, 9174627360, -1734603719472, 338286925650240, -67486440186470016, 13697820033167444178, -2818359890320927630320, 586296297186462310481424, -123077156275866375661524864, 26034142700316716015964656544

Offset: 0

Author

Peter Bala, Feb 21 2021

Keywords

• sign
• easy

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

Crossrefs

A004009, A006352, A108091, A282019, A289392.

Programs

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(%);