A008408 Theta series of Leech lattice.
1, 0, 196560, 16773120, 398034000, 4629381120, 34417656000, 187489935360, 814879774800, 2975551488000, 9486551299680, 27052945920000, 70486236999360, 169931095326720, 384163586352000, 820166620815360, 1668890090322000
Offset: 0
Examples
G.f. = 1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + ...
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.
Programs
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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];
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Magma
Basis( ModularForms( Gamma0(1), 12), 30) [1] ; /* Michael Somos, Jun 09 2014 */
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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);
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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 *) -
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 */ -
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
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Sage
A = ModularForms( Gamma0(1), 12, prec=30) . basis() ; A[1] - 65520/691*A[0] # Michael Somos, Jun 09 2014
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