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

240, 4320, 20160, 70080, 151200, 362880, 577920, 1123200, 1635120, 2721600, 3516480, 5886720, 6857760, 10402560, 12700800, 17975040, 20049120, 29432160, 31281600, 44150400, 48545280, 63296640, 67167360, 94348800, 94506000, 123439680, 132451200

Offset: 1

Author

N. J. A. Sloane, Feb 28 2009

Keywords

• nonn

Examples

			G.f. = 240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*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. A004009, A076835.

Cf. A076835 (-q*E'_2), this sequence (q*E'_4), A145095 (-q*E'_6).

Programs

Mathematica

terms = 28;
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]*E4[x] - E6[x])/3 + O[x]^terms // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 23 2018 *)
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 *)

Formula

q*E'_4 = (E_2*E_4-E_6)/3.

G.f.: 240*x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017

Sum_{k=1..n} a(k) ~ 8 * Pi^4 * n^5 / 15. - Vaclav Kotesovec, May 09 2022