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%I #19 Jul 08 2017 12:17:21
%S 1,16773120,39007332000,15281788354560,2972108280960000,
%T 406954241261568000,45569082381053868000,4499117081888292864000,
%U 408472720963469499617280,34975479259332252426240000
%N Second coefficient of extremal theta series of even unimodular lattice in dimension 24n.
%C Although these initially increase, they eventually go negative at about term 1700 (i.e. dimension about 40800) - see references.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
%H N. J. A. Sloane, <a href="/A034598/b034598.txt">Table of n, a(n) for n = 0..100</a>
%H C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane, <a href="https://doi.org/10.1016/0021-8693(75)90155-6">Upper bounds for modular forms, lattices and codes</a>, J. Alg., 36 (1975), 68-76.
%H C. L. Mallows and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/S0019-9958(73)90273-8">An Upper Bound for Self-Dual Codes</a>, Information and Control, 22 (1973), 188-200.
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (<a href="http://neilsloane.com/doc/self.txt">Abstract</a>, <a href="http://neilsloane.com/doc/self.pdf">pdf</a>, <a href="http://neilsloane.com/doc/self.ps">ps</a>).
%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).
%e When n=1 we get the theta series of the 24-dimensional Leech lattice: 1+196560*q^4+16773120*q^6+... (see A008408). For n=2 we get A004672 and for n=3, A004675.
%p For Maple program see A034597.
%t terms = 10; Reap[For[mu = 1; Print[1]; Sow[1], mu < terms, mu++, md = mu + 3; f = 1 + 240*Sum[DivisorSigma[3, i]*x^i, {i, 1, md}]; f = Series[f, {x, 0, md}]; f = Series[f^3, {x, 0, md}]; g = Series[x*Product[ (1 - x^i)^24, {i, 1, md}], {x, 0, md}]; W0 = Series[f^mu, {x, 0, md}]; h = Series[g/f, {x, 0, md}]; A = Series[W0, {x, 0, md}]; Z = A; For[i = 1, i <= mu, i++, Z = Series[Z*h, {x, 0, md}]; A = Series[A - SeriesCoefficient[A, {x, 0, i}]*Z, {x, 0, md}]]; an = SeriesCoefficient[A, {x, 0, mu+2}]; Print[an]; Sow[an]]][[2,1]] (* _Jean-François Alcover_, Jul 08 2017, adapted from Maple program for A034597 *)
%Y Cf. A034597 (leading coefficient).
%K sign
%O 0,2
%A _N. J. A. Sloane_.
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