A377973 Expansion of the 96th root of the series 2*E_2(x) - E_2(x)^2, where E_2 is the Eisenstein series of weight 2.

1, 0, -6, -36, -1812, -20748, -773340, -12237456, -386587650, -7368446268, -211914644940, -4517757977820, -123221458979940, -2814502962357420, -74551748141034552, -1778129476480366320, -46377354051910716180, -1137191336376638407704, -29438532048777299115090, -735051729258136807204140

Offset

0,3

Comments

Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Let P(n) = {g^n, g in R}. The Eisenstein series E_2(x) lies in P(4) (Heninger et al.). Hence E_2(x)^2 lies in P(8).

We claim that the series 2*E_2(x) - E_2(x)^2 belongs to P(96).

Proof.

E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.

Hence,

2*E_2(x) - E_2(x)^2 = 1 - (24^2)*(Sum_{n >= 1} sigma_1(n)*x^n )^2 is in the set R.

Hence, 2*E_2(x) - E_2(x)^2 == 1 (mod 24^2) == 1 (mod (2^6)*(3^2)).

It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_2(x) - E_2(x)^2 belongs to P((2^5)*3) = P(96). End Proof.

Links

Table of n, a(n) for n=0..19.

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.

Maple

with(numtheory):
E := proc (k) local n, t1; t1 := 1 - 2*k*add(sigma[k-1](n)*q^n, n = 1..30)/bernoulli(k); series(t1, q, 30) end:
seq(coeftayl((2*E(2) - E(2)^2)^(1/96), q = 0, n), n = 0..20);

Crossrefs

Cf. A006352 (E_2), A281374 (E_2)^2, A289392 ((E_2)^(1/4)), A341801, A341871 - A341875, A377974, A377975, A377976, A377977.

Sequence in context: A351796 A069031 A061234 * A061584 A197191 A208655

Adjacent sequences: A377970 A377971 A377972 * A377974 A377975 A377976

Keyword

sign,easy

Author

Peter Bala, Nov 13 2024

Status

approved