A377975 Expansion of the 6048th root of the series 2*E_6(x) - E_6(x)^2, where E_6 is the Eisenstein series of weight 6.

1, 0, -42, -2772, -5399688, -704781084, -943173698460, -180121119486672, -188146584694894350, -46293152603021155692, -40574254265781269371884, -11963000065787771567311500, -9221266403646163252100062068, -3107813621461888912485774582588, -2176998806586925223600540321844120

Offset: 0

Author

Peter Bala, Nov 14 2024

Keywords

• sign
• easy

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_6(x) lies in P(12) (Heninger et al.).

We claim that the series 2*E_6(x) - E_6(x)^2 belongs to P(6048).

Proof.

E_6(x) = 1 - 504*Sum_{n >= 1} sigma_5(n)*x^n. Hence,

2*E_6(x) - E_6(x)^2 = 1 - (504^2)*( Sum_{n >= 1} sigma_5(n)*x^n )^2 is in R.

Hence, 2*E_6(x) - E_6(x)^2 == 1 (mod 504^2) == 1 (mod (2^6)*(3^4)*(7^2)).

It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_6(x) - E_6(x)^2 belongs to P((2^5)*(3^3)*7) = P(6048). End Proof.

Crossrefs

Cf. A013973 (E_6), A109817 ( (E_6)^1/12 ), A280869 (E_6)^2, A341871 - A341875, A377973, A377974, A377976, A377977.

Programs

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(6) - E(6)^2)^(1/6048), q = 0, n),n = 0..20);

Mathematica

terms = 20; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E6[x] - E6[x]^2)^(1/6048), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

Formula

a(n) ~ c / (r^n * n^(6049/6048)), where r = 0.0018674427317079888144302129348270303934228050024753171993815386383179351229... is the root of the equation Sum_{k>=1} sigma_5(k) * r^k = 1/504 and c = -0.0001653486643613776568861731992670297686378824546... - Vaclav Kotesovec, Aug 03 2025