A377974 Expansion of the 1920th root of the series 2*E_4(x) - E_8(x), where E_4 and E_8 are the Eisenstein series of weight 4 and weight 8.

1, 0, -30, -540, -867660, -31107300, -33668157900, -1795572812400, -1477793386682970, -103845834995498100, -69550699526934273180, -6017200267937951322660, -3426636160378174348594500, -349303370036461528632524580, -174458882971934188146144343320, -20314204536496741742949242177040

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_4(x) lies in P(8) (Heninger et al.). Since E_8(x) = E_4(x)^2, it follows that E_8(x) lies in P(16).

We claim that the series 2*E_4(x) - E_8(x) belongs to P(1920).

Proof.

E_4(x) = 1 + 240*Sum_{n >= 1} sigma_3(n)*x^n. Hence,

2*E_4(x) - E_8(x) = 2*E_4(x) - E_4(x)^2 = 1 - 240^2*( Sum_{n >= 1} sigma_3(n) )^2 is in the set R.

Hence, 2*E_4(x) - E_8(x) == 1 mod(240^2) == 1 (mod (2^8)*(3^2)*(5^2)).

It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_4(x) - E_8(x) belongs to P((2^7)*3*5) = P(1920). End Proof.

Links

Table of n, a(n) for n=0..15.

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(4) - E(8))^(1/1920), q = 0, n), n = 0..20);

Crossrefs

Cf. A004009 (E_4), A008410 (E_8), A108091 (eighth root of E_4), A341871 - A341875, A377973, A377975, A377976, A377977.

Sequence in context: A286975 A139626 A037961 * A143399 A293103 A075510

Adjacent sequences: A377971 A377972 A377973 * A377975 A377976 A377977

Keyword

sign,easy

Author

Peter Bala, Nov 13 2024

Status

approved