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%I #84 Nov 03 2023 10:44:40
%S 1,0,196560,16773120,398034000,4629381120,34417656000,187489935360,
%T 814879774800,2975551488000,9486551299680,27052945920000,
%U 70486236999360,169931095326720,384163586352000,820166620815360,1668890090322000
%N Theta series of Leech lattice.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Third Edition, Springer-Verlag,1993, pp. 51, 134-135.
%D W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 113.
%D 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.
%H Seiichi Manyama, <a href="/A008408/b008408.txt">Table of n, a(n) for n = 0..10000</a> (first 501 terms from N. J. A. Sloane)
%H Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, <a href="http://arxiv.org/abs/1603.06518">The sphere packing problem in dimension 24</a>, arXiv:1603.06518 [math.NT], 2016.
%H Henry Cohn and Stephen D. Miller, <a href="http://arxiv.org/abs/1603.04759">Some properties of optimal functions for sphere packing in dimensions 8 and 24</a>, arXiv:1603.04759 [math.MG], 2016
%H D de Laat and F Vallentin, <a href="http://arxiv.org/abs/1607.02111">A Breakthrough in Sphere Packing: The Search for Magic Functions</a>, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Leech.html">Home page for lattice</a>
%H K. Ono, S. Robins and P. T. Wahl, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the Representation of Integers as Sums of Triangular Numbers</a>, (see p. 12), Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/g4g7.pdf">Seven Staggering Sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeechLattice.html">Leech Lattice</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ThetaSeries.html">Theta Series</a>.
%F 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).
%F 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
%F 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
%e G.f. = 1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + ...
%p 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);
%t 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 *)
%t (* 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 *)
%t 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 *)
%o (Magma) // Theta series of the Leech lattice, from _John Cannon_, Dec 29 2006
%o 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];
%o (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 */
%o (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 */
%o (Sage) A = ModularForms( Gamma0(1), 12, prec=30) . basis() ; A[1] - 65520/691*A[0] # _Michael Somos_, Jun 09 2014
%o (Magma) Basis( ModularForms( Gamma0(1), 12), 30) [1] ; /* _Michael Somos_, Jun 09 2014 */
%o (Python)
%o from sympy import divisor_sigma
%o 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
%Y Cf. A004009, A108093, A000594, A108093 (24th root), A034597, A034598, A198343, A013959.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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