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

%I A076835 #33 Aug 01 2025 09:41:34
%S A076835 24,144,288,672,720,1728,1344,2880,2808,4320,3168,8064,4368,8064,8640,
%T A076835 11904,7344,16848,9120,20160,16128,19008,13248,34560,18600,26208,
%U A076835 25920,37632,20880,51840,23808,48384,38016,44064,40320,78624,33744,54720,52416,86400
%N A076835 Coefficients in expansion of Eisenstein series -q*E'_2.
%H A076835 Seiichi Manyama, <a href="/A076835/b076835.txt">Table of n, a(n) for n = 1..1000</a>
%H A076835 Masanobu Kaneko and Don Zagier, <a href="http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
%H A076835 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series</a>.
%F A076835 q*E'_2 = (E_2^2-E_4)/12.
%F A076835 a(n) = 24*A064987(n).
%F A076835 G.f.: 24*x*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Aug 31 2017
%e A076835 G.f. = 24*q + 144*q^2 + 288*q^3 + 672*q^4 + 720*q^5 + ...
%p A076835 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);
%t A076835 terms = 41;
%t A076835 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A076835 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A076835 -(E2[x]^2 - E4[x])/12 + O[x]^terms // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Feb 23 2018 *)
%t A076835 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 *)
%o A076835 (PARI) a(n) = 24 * n * sigma(n); \\ _Amiram Eldar_, Jan 07 2025
%Y A076835 Cf. A006352 (E_2), A004009 (E_4), A064987.
%Y A076835 Cf. this sequence (-q*E'_2), A145094 (q*E'_4), A145095 (-q*E'_6).
%K A076835 nonn
%O A076835 1,1
%A A076835 _N. J. A. Sloane_, Feb 28 2009