A377975 - OEIS

%I #10 Nov 26 2024 16:28:44

%S 1,0,-42,-2772,-5399688,-704781084,-943173698460,-180121119486672,

%T -188146584694894350,-46293152603021155692,-40574254265781269371884,

%U -11963000065787771567311500,-9221266403646163252100062068,-3107813621461888912485774582588,-2176998806586925223600540321844120

%N 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.

%C 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.).

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

%C Proof.

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

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

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

%C 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.

%p with(numtheory):

%p 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:

%p seq(coeftayl((2*E(6) - E(6)^2)^(1/6048), q = 0, n),n = 0..20);

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

%K sign,easy

%O 0,3

%A _Peter Bala_, Nov 14 2024