OFFSET
0,3
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Third Edition, Springer-Verlag,1993, pp. 51, 134-135.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 113.
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.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from N. J. A. Sloane)
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension 24, arXiv:1603.06518 [math.NT], 2016.
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016
David de Laat and Frank Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; 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, (see p. 12), Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
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 D. Briggs, Nov 25 2011
a(n) = 65520*(A013959(n) - A000594(n))/691, n >= 1. a(0) = 1. Expansion of the Theta series of the Leech lattice in powers of q^2. See the Conway and Sloane reference. - Wolfdieter Lang, Jan 16 2017
EXAMPLE
G.f. = 1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + ...
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 *)
a[ n_] := If[ n < 1, Boole[n == 0], SeriesCoefficient[(1 + 240 Sum[ q^k DivisorSigma[ 3, k], {k, n}])^3 - 720 q QPochhammer[ q]^24, {q, 0, n}]]; (* Michael Somos, Jun 09 2014 *)
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 */
(Sage) A = ModularForms( Gamma0(1), 12, prec=30) . basis() ; A[1] - 65520/691*A[0] # Michael Somos, Jun 09 2014
(Magma) Basis( ModularForms( Gamma0(1), 12), 30) [1] ; /* Michael Somos, Jun 09 2014 */
(Python)
from sympy import divisor_sigma
def A008408(n): return 65520*(divisor_sigma(n, 11)-(n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1, m)))))//691 if n else 1 # Chai Wah Wu, Nov 17 2022
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved