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A145094 Coefficients in expansion of Eisenstein series q*E'_4.

%I A145094 #25 Aug 01 2025 09:39:11
%S A145094 240,4320,20160,70080,151200,362880,577920,1123200,1635120,2721600,
%T A145094 3516480,5886720,6857760,10402560,12700800,17975040,20049120,29432160,
%U A145094 31281600,44150400,48545280,63296640,67167360,94348800,94506000,123439680,132451200
%N A145094 Coefficients in expansion of Eisenstein series q*E'_4.
%H A145094 Seiichi Manyama, <a href="/A145094/b145094.txt">Table of n, a(n) for n = 1..1000</a>
%H A145094 M. Kaneko and D. 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 A145094 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series</a>
%F A145094 q*E'_4 = (E_2*E_4-E_6)/3.
%F A145094 G.f.: 240*x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Aug 31 2017
%F A145094 Sum_{k=1..n} a(k) ~  8 * Pi^4 * n^5 / 15. - _Vaclav Kotesovec_, May 09 2022
%e A145094 G.f. = 240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + ...
%t A145094 terms = 28;
%t A145094 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A145094 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A145094 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A145094 (E2[x]*E4[x] - E6[x])/3 + O[x]^terms // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Feb 23 2018 *)
%t A145094 nmax = 40; Rest[CoefficientList[Series[240*x*Sum[k^4*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Aug 01 2025 *)
%Y A145094 Cf. A004009, A076835.
%Y A145094 Cf. A076835 (-q*E'_2), this sequence (q*E'_4), A145095 (-q*E'_6).
%K A145094 nonn
%O A145094 1,1
%A A145094 _N. J. A. Sloane_, Feb 28 2009