A377973 - OEIS

%I #15 Nov 26 2024 16:27:14

%S 1,0,-6,-36,-1812,-20748,-773340,-12237456,-386587650,-7368446268,

%T -211914644940,-4517757977820,-123221458979940,-2814502962357420,

%U -74551748141034552,-1778129476480366320,-46377354051910716180,-1137191336376638407704,-29438532048777299115090,-735051729258136807204140

%N 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.

%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_2(x) lies in P(4) (Heninger et al.). Hence E_2(x)^2 lies in P(8).

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

%C Proof.

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

%C Hence,

%C 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.

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

%C 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.

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%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(2) - E(2)^2)^(1/96), q = 0, n),n = 0..20);

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

%K sign,easy

%O 0,3

%A _Peter Bala_, Nov 13 2024