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

504, 33264, 368928, 2130912, 7877520, 24349248, 59298624, 136382400, 268953048, 519916320, 892872288, 1559827584, 2432718288, 3913709184, 5766344640, 8728481664, 12165343344, 17750901168, 23711133600, 33306154560, 43406592768, 58929571008

Offset: 1

Author

N. J. A. Sloane, Feb 28 2009

Keywords

• nonn

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• M. Kaneko and D. 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. A076835 (-q*E'_2), A145094 (q*E'_4), this sequence (-q*E'_6).

Programs

Mathematica

terms = 23;
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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
-(E2[x]*E6[x] - E4[x]^2)/2 + O[x]^terms // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 23 2018 *)
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 *)

Formula

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