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Coefficients in expansion of Eisenstein series -q*E'_6.
9

%I #16 Feb 23 2018 03:40:19

%S 504,33264,368928,2130912,7877520,24349248,59298624,136382400,

%T 268953048,519916320,892872288,1559827584,2432718288,3913709184,

%U 5766344640,8728481664,12165343344,17750901168,23711133600,33306154560,43406592768,58929571008

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

%H Seiichi Manyama, <a href="/A145095/b145095.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series</a>

%F q*E'_6 = (E_2*E_6-E_4^2)/2.

%e G.f. = 504*q + 33264*q^2 + 368928*q^3 + 2130912*q^4 + 7877520*q^5 + ...

%t terms = 23;

%t E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];

%t E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];

%t E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];

%t -(E2[x]*E6[x] - E4[x]^2)/2 + O[x]^terms // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Feb 23 2018 *)

%Y Cf. A076835 (-q*E'_2), A145094 (q*E'_4), this sequence (-q*E'_6).

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Feb 28 2009