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 A377976 #15 Aug 03 2025 10:18:04
%S A377976 1,-6,-894,-174420,-38431614,-9048710040,-2221653118116,
%T A377976 -561444889080960,-144914324838755910,-38011797621225586602,
%U A377976 -10098281618881696696392,-2710458654395655881518356,-733711171629600485187568404,-200033609249999737396399900920,-54867682197669353983111639906656
%N A377976 Expansion of the 48th root of the series 2*E_2(x) - E_4(x), where E_2(x) and E_4(x) are the Eisenstein series of weight 2 and 4.
%C A377976 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) and E_4(x) lies in P(8) (Heninger et al.).
%C A377976 We claim that the series 2*E_2(x) - E_4(x) belongs to P(48).
%C A377976 Proof.
%C A377976 E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.
%C A377976 E_4(x) = 1 + 240*Sum_{n >= 1} sigma_3(n)*x^n.
%C A377976 Hence,
%C A377976 2*E_2(x) - E_4(x) = 1 - (288)*Sum_{n >= 1} ((1/6)*sigma_1(n) + (5/6)*sigma_3(n))*x^n belongs to the set R, since the polynomial (1/6)*k + (5/6)*k^3 has integer values for integer k. See A004068.
%C A377976 Hence, 2*E_2(x) - E_4(x) == 1 (mod 288) == 1 (mod (2^5)*(3^2)).
%C A377976 It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_2(x) - E_4(x) belongs to P((2^4)*3) = P(48). End Proof.
%C A377976 In a similar way we find that the series 3*E_2(x) - E_6(x) - 1 belongs to P(72) and the three series 3*E_4(x) - 2*E_6(x), 5*E_4(x) - 2*E_10(x) - 2 and 5*E_6(x) - 3*E_10(x) - 1 belong to P(288).
%H A377976 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 A377976 a(n) ~ c * d^n / n^(49/48), where d = 295.8669385406700495308385233671383399895922733900742171390678012914822364544611... and c = -0.0205882497833853345146399243734199945444083043388859856935627869352251231763... - _Vaclav Kotesovec_, Aug 03 2025
%p A377976 with(numtheory):
%p A377976 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 A377976 seq(coeftayl((2*E(2) - E(4))^(1/48), q = 0, n),n = 0..20);
%t A377976 terms = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E2[x] - E4[x])^(1/48), {x, 0, terms}], x] (* _Vaclav Kotesovec_, Aug 03 2025 *)
%Y A377976 Cf. A004068, A006352 (E_2), A004009 (E_4), A108091 ((E_4)^1/8), A289392 ((E_2)^(1/4)), A341871 - A341875, A377973, A377974, A377975, A377977.
%K A377976 sign,easy
%O A377976 0,2
%A A377976 _Peter Bala_, Nov 14 2024