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A034597 Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.

%I A034597 #23 Jul 08 2025 21:19:28
%S A034597 1,196560,52416000,6218175600,565866362880,45792819072000,
%T A034597 3486157968384000,256206274225902000,18422726047165440000,
%U A034597 1305984407917646096640,91692325887531393024000
%N A034597 Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.
%D A034597 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
%H A034597 N. J. A. Sloane, <a href="/A034597/b034597.txt">Table of n, a(n) for n = 0..100</a>
%H A034597 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 A034597 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 A034597 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 A034597 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 A034597 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 A034597 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 A034597 # Extremal theta series:
%p A034597 with(numtheory): B := 1:
%p A034597 # set mu:
%p A034597 for mu from 1 to 10 do
%p A034597    # set max deg:
%p A034597    md := mu+3;
%p A034597    f := 1+240*add(sigma[3](i)*x^i, i=1..md);
%p A034597    f := series(f, x, md);
%p A034597    f := series(f^3, x, md);
%p A034597    g := series(x*mul((1-x^i)^24, i=1..md), x, md);
%p A034597    W0 := series(f^mu, x, md):
%p A034597    h := series(g/f, x, md):
%p A034597    A := series(W0, x, md):
%p A034597    Z := A:
%p A034597    for i from 1 to mu do
%p A034597       Z := series(Z*h, x, md);
%p A034597       A := series(A-coeff(A, x, i)*Z, x, md);
%p A034597    od:
%p A034597    B := B, coeff(A,x,mu+1);
%p A034597 od:
%p A034597 lprint(B);
%t A034597 terms = 11; Reap[For[mu = 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+1}]; Print[an]; Sow[an]]][[2,1]] (* _Jean-François Alcover_, Jul 08 2017, adapted from Maple *)
%Y A034597 Cf. A034598 (second coefficient, which eventually becomes negative), A034414, A034415.
%K A034597 nonn
%O A034597 0,2
%A A034597 _N. J. A. Sloane_