0,3

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 135.

W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 113.

N. J. A. Sloane, Table of n, a(n) for n = 0..500

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

G. Nebe and N. J. A. Sloane, Home page for lattice

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, Seven Staggering Sequences.

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

Eric Weisstein's World of Mathematics, Theta Series

The simplest way to obtain this is to take the cube of the theta series for E_8 (A004009) and subtract 720 times the g.f. for the Ramanujan numbers (A000594).

with(numtheory); f := 1+240*add(sigma[ 3 ](m)*q^(2*m), m=1..50); t := q^2*mul((1-q^(2*m))^24, m=1..50); series(f^3-720*t, q, 51);

(MAGMA)//Theta series of the Leech lattice, from John Cannon, Dec 29 2006

A008408Q := function(prec) M12 := ModularForms(Gamma0(1), 12); t1 := Basis(M12)[1]; T := PowerSeries(t1, prec); return Coefficients(T); end function; Q := A008408Q(1000); Q[678];

(PARI) {a(n)=if(n<1, n==0, polcoeff( 1+(sum(k=1, n, sigma(k, 11)*x^k)-x*eta(x+O(x^n))^24)*65520/691, n))} /* Michael Somos Oct 19 2006 */

(PARI) {a(n)=if(n<1, n==0, polcoeff( sum(k=1, n, 240*sigma(k, 3)*x^k, 1+x*O(x^n))^3 -720*x*eta(x+O(x^n))^24, n))} /* Michael Somos Oct 19 2006 */

Cf. A004009, A108093, A000594, A108093 (24-th root), A034597, A034598.

Sequence in context: A069305 A129486 A074388 this_sequence A001942 A034597 A037148

Adjacent sequences: A008405 A008406 A008407 this_sequence A008409 A008410 A008411

nonn,easy,nice

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