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A008408 Theta series of Leech lattice. 12
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

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

N. J. A. Sloane and 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, Maryna Viazovska, The sphere packing problem in dimension 24, arXiv:1603.06518 [math.NT], 2016.

Henry Cohn, Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016

D de Laat, F Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111, 2016

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, (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 */

CROSSREFS

Cf. A004009, A108093, A000594, A108093 (24th root), A034597, A034598, A198343, A013959.

Sequence in context: A234394 A202434 A179253 * A001942 A034597 A037148

Adjacent sequences:  A008405 A008406 A008407 * A008409 A008410 A008411

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 29 21:28 EDT 2017. Contains 287257 sequences.