A076835 Coefficients in expansion of Eisenstein series -q*E'_2.

24, 144, 288, 672, 720, 1728, 1344, 2880, 2808, 4320, 3168, 8064, 4368, 8064, 8640, 11904, 7344, 16848, 9120, 20160, 16128, 19008, 13248, 34560, 18600, 26208, 25920, 37632, 20880, 51840, 23808, 48384, 38016, 44064, 40320, 78624, 33744, 54720, 52416, 86400

Offset: 1

Author

N. J. A. Sloane, Feb 28 2009

Keywords

• nonn

Examples

			G.f. = 24*q + 144*q^2 + 288*q^3 + 672*q^4 + 720*q^5 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• Masanobu Kaneko and Don Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
• Eric Weisstein's World of Mathematics, Eisenstein Series.

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A064987.

Cf. this sequence (-q*E'_2), A145094 (q*E'_4), A145095 (-q*E'_6).

Programs

Maple

with(numtheory); E:=proc(k) series(1-(2*k/bernoulli(k))*add( sigma[k-1](n)*q^n, n=1..60),q,61); end; -diff(E(2),q);

Mathematica

terms = 41;
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}];
-(E2[x]^2 - E4[x])/12 + O[x]^terms // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 23 2018 *)
nmax = 40; Rest[CoefficientList[Series[24*x*Sum[k^2*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 01 2025 *)

PARI

a(n) = 24 * n * sigma(n); \\ Amiram Eldar, Jan 07 2025

Formula

q*E'_2 = (E_2^2-E_4)/12.

a(n) = 24*A064987(n).

G.f.: 24*x*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017