0,2

Eisenstein series E_8(q) (alternate convention E_4(q)); theta series of direct sum of 2 copies of E_8 lattice.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.

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.

S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.

S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.

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

Index entries for sequences related to Eisenstein series

Equivalently, g.f. = (theta2^16+theta3^16+theta4^16)/2.

G.f. Sum{k>=0} a(k)q^(2k) = (theta2^16+theta3^16+theta4^16)/2.

Expansion of ((eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8)^2 in powers of q. - Michael Somos Dec 30 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^8 * f(t) where q = exp(2 pi i t). - Michael Somos Dec 30 2008

1 + 480*q + 61920*q^2 + 1050240*q^3 + 7926240*q^4 + 37500480*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(8);

(PARI) a(n)=if(n<1, n==0, 480*sigma(n, 7))

(PARI) {a(n)=local(A, e1, e2, e4); if(n<0, 0, n*=2; A=x*O(x^n); e1=eta(x+A)^16; e2=eta(x^2+A)^16; e4=eta(x^4+A)^16; polcoeff( (e1*e2^3 +256*x^2*e4*(e2^3+e1^2*e4))/(e1*e2*e4), n))} /* Michael Somos Jun 29 2005 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8)^2, n))} /* Michael Somos Dec 30 2008 */

Cf. A013973.

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).

Convolution square of A004009.

Sequence in context: A035314 A022047 A107511 this_sequence A020286 A064909 A051980

Adjacent sequences: A008407 A008408 A008409 this_sequence A008411 A008412 A008413

nonn

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