Search: id:A008408
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%I A008408
%S A008408 1,0,196560,16773120,398034000,4629381120,34417656000,187489935360,
%T A008408 814879774800,2975551488000,9486551299680,27052945920000,70486236999360,
%U A008408 169931095326720,384163586352000,820166620815360,1668890090322000
%N A008408 Theta series of Leech lattice.
%D A008408 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.
%D A008408 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 135.
%D A008408 W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 113.
%H A008408 N. J. A. Sloane, <a href="b008408.txt">Table of n, a(n) for n = 0..500</
               a>
%H A008408 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/
               abs/math.NT/0509316">On the Integrality of n-th Roots of Generating 
               Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H A008408 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/Leech.html">Home page for lattice</a>
%H A008408 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A008408 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">
               Seven Staggering Sequences</a>.
%H A008408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LeechLattice.html">Link to a section of The World of Mathematics.</
               a>
%H A008408 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ThetaSeries.html">Theta Series</a>
%F A008408 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).
%p A008408 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);
%o A008408 (MAGMA)//Theta series of the Leech lattice, from John Cannon, Dec 29 
               2006
%o A008408 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 A008408 (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 A008408 (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 */
%Y A008408 Cf. A004009, A108093, A000594, A108093 (24-th root), A034597, A034598.
%Y A008408 Sequence in context: A069305 A129486 A074388 this_sequence A001942 A034597 
               A037148
%Y A008408 Adjacent sequences: A008405 A008406 A008407 this_sequence A008409 A008410 
               A008411
%K A008408 nonn,easy,nice
%O A008408 0,3
%A A008408 N. J. A. Sloane (njas(AT)research.att.com).

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