|
| |
|
|
A008408
|
|
Theta series of Leech lattice.
|
|
11
|
|
|
|
1, 0, 196560, 16773120, 398034000, 4629381120, 34417656000, 187489935360, 814879774800, 2975551488000, 9486551299680, 27052945920000, 70486236999360, 169931095326720, 384163586352000, 820166620815360, 1668890090322000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
REFERENCES
|
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.
|
|
|
LINKS
|
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
K. Ono, S. Robins and P. T. Wahl, On the Representation of Integers as Sums of Triangular Numbers, p. 12.
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, Leech Lattice.
Eric Weisstein's World of Mathematics, Theta Series
|
|
|
FORMULA
|
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).
This theta series is thus also the q-expansion of 7/12 E_4(z)^3 + 5/12 E_6(z)^2. Cf. A013973. - Daniel Briggs, Nov 25 2011
|
|
|
MAPLE
|
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);
|
|
|
MATHEMATICA
|
max = 17; f = 1 + 240*Sum[ DivisorSigma[3, m]*q^(2m), {m, 1, max}]; t = q^2*Product[(1 - q^(2m))^24, {m, 1, max}]; Partition[ CoefficientList[ Series[f^3 - 720t, {q, 0, 2 max}], q], 2][[All, 1]] (* Jean-François Alcover, Oct 14 2011, after Maple *)
(* From version 6 on *) f[q_] = LatticeData["Leech", "ThetaSeriesFunction"][x] /. x -> -I*Log[q]/Pi; Series[f[q], {q, 0, 32}] // CoefficientList[#, q^2]& (* Jean-François Alcover, May 15 2013 *)
|
|
|
PROG
|
(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 */
|
|
|
CROSSREFS
|
Cf. A004009, A108093, A000594, A108093 (24-th root), A034597, A034598, A198343.
Sequence in context: A074388 A202434 A179253 * A001942 A034597 A037148
Adjacent sequences: A008405 A008406 A008407 * A008409 A008410 A008411
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
