This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377973 #18 Aug 03 2025 09:04:19
%S A377973 1,0,-6,-36,-1812,-20748,-773340,-12237456,-386587650,-7368446268,
%T A377973 -211914644940,-4517757977820,-123221458979940,-2814502962357420,
%U A377973 -74551748141034552,-1778129476480366320,-46377354051910716180,-1137191336376638407704,-29438532048777299115090,-735051729258136807204140
%N 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.
%C A377973 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 A377973 We claim that the series 2*E_2(x) - E_2(x)^2 belongs to P(96).
%C A377973 Proof.
%C A377973 E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.
%C A377973 Hence,
%C A377973 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 A377973 Hence, 2*E_2(x) - E_2(x)^2 == 1 (mod 24^2) == 1 (mod (2^6)*(3^2)).
%C A377973 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 A377973 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.
%F A377973 a(n) ~ c / (r^n * n^(97/96)), where r = A211342 = 0.03727681029645165815098... and c = -0.0104397599261506010365791466642760245638473040812140699981294533624... - _Vaclav Kotesovec_, Aug 03 2025
%p A377973 with(numtheory):
%p A377973 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 A377973 seq(coeftayl((2*E(2) - E(2)^2)^(1/96), q = 0, n),n = 0..20);
%t A377973 terms = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E2[x] - E2[x]^2)^(1/96), {x, 0, terms}], x] (* _Vaclav Kotesovec_, Aug 03 2025 *)
%Y A377973 Cf. A006352 (E_2), A281374 (E_2)^2, A289392 ((E_2)^(1/4)), A341801, A341871 - A341875, A377974, A377975, A377976, A377977.
%K A377973 sign,easy
%O A377973 0,3
%A A377973 _Peter Bala_, Nov 13 2024