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A145095 Coefficients in expansion of Eisenstein series -q*E'_6.

%I A145095 #19 Aug 01 2025 09:35:55
%S A145095 504,33264,368928,2130912,7877520,24349248,59298624,136382400,
%T A145095 268953048,519916320,892872288,1559827584,2432718288,3913709184,
%U A145095 5766344640,8728481664,12165343344,17750901168,23711133600,33306154560,43406592768,58929571008
%N A145095 Coefficients in expansion of Eisenstein series -q*E'_6.
%H A145095 Seiichi Manyama, <a href="/A145095/b145095.txt">Table of n, a(n) for n = 1..1000</a>
%H A145095 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 A145095 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series</a>
%F A145095 q*E'_6 = (E_2*E_6-E_4^2)/2.
%e A145095 G.f. = 504*q + 33264*q^2 + 368928*q^3 + 2130912*q^4 + 7877520*q^5 + ...
%t A145095 terms = 23;
%t A145095 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A145095 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A145095 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A145095 -(E2[x]*E6[x] - E4[x]^2)/2 + O[x]^terms // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Feb 23 2018 *)
%t A145095 nmax = 40; Rest[CoefficientList[Series[504*x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Aug 01 2025 *)
%Y A145095 Cf. A076835 (-q*E'_2), A145094 (q*E'_4), this sequence (-q*E'_6).
%K A145095 nonn
%O A145095 1,1
%A A145095 _N. J. A. Sloane_, Feb 28 2009