0,2

D. Bump, Automorphic Forms..., Cambridge Univ. Press, 1997, p. 29.

W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.

R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.

Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978

Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994

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.

T. D. Noe, Table of n, a(n) for n=0..1000

R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.

H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to Eisenstein series

E6(q) = 1 - 504 sum_{i=1}^infinity sigma_5(i)q^i where sigma_5(n) is A001160, the sum of fifth powers of the divisors of n. It can also be expressed as E6(q) = 1 - 504 sum_{i=1}^infinity i^5 q^i/(1-q^i). - Gene Ward Smith (genewardsmith(AT)gmail.com), Aug 22 2006

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u^2*v -8*u^2*w -66*u*v^2 +592*u*v*w -512*u*w^2 +121*v^3 -4224*v^2*w +4096*v*w^2. - Michael Somos, Apr 10 2005

Expansion of Ramanujan's function R(q)= 216*g3 (Weierstrass invariant).

Expansion of (eta(q)^8 + 32 * eta(q^4)^8) * (eta(q)^16 - 512 * eta(q)^8 * eta(q^4)^8 - 8192 * eta(q^4)^16) / eta(q^2)^12 in powers of q. - Michael Somos Dec 30 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / t) = - (t/i)^6 * f(t) where q = exp(2 pi i t). - Michael Somos Dec 30 2008

1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + ...

E := proc(k) local n, t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..60); series(t1, q, 60); end; E(6);

(PARI) a(n)=if(n<1, n==0, -504*sigma(n, 5))

(PARI) {a(n) = local(A, A1, A4); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A4 = 32 * x * eta(x^4 + A)^8; polcoeff( (A1 + A4) * (A1^2 - 16 * A1 * A4 - 8 * A4^2) / eta(x^2 + A)^12, n))} /* Michael Somos Dec 30 2008 */

Cf. A004009, A008410, A013974, A145095.

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).

Cf. A001160.

Sequence in context: A141145 A166763 A012829 this_sequence A012744 A145095 A035293

Adjacent sequences: A013970 A013971 A013972 this_sequence A013974 A013975 A013976

sign,easy

N. J. A. Sloane (njas(AT)research.att.com).