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

1, 196560, 52416000, 6218175600, 565866362880, 45792819072000, 3486157968384000, 256206274225902000, 18422726047165440000, 1305984407917646096640, 91692325887531393024000

Offset

0,2

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..100

C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane, Upper bounds for modular forms, lattices and codes, J. Alg., 36 (1975), 68-76.

C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Example

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.

Maple

# Extremal theta series:
with(numtheory): B := 1:
# set mu:
for mu from 1 to 10 do
   # set max deg:
   md := mu+3;
   f := 1+240*add(sigma[3](i)*x^i, i=1..md);
   f := series(f, x, md);
   f := series(f^3, x, md);
   g := series(x*mul((1-x^i)^24, i=1..md), x, md);
   W0 := series(f^mu, x, md):
   h := series(g/f, x, md):
   A := series(W0, x, md):
   Z := A:
   for i from 1 to mu do
      Z := series(Z*h, x, md);
      A := series(A-coeff(A, x, i)*Z, x, md);
   od:
   B := B, coeff(A, x, mu+1);
od:
lprint(B);

Mathematica

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 *)

Crossrefs

Cf. A034598 (second coefficient, which eventually becomes negative), A034414, A034415.

Sequence in context: A008408 A305920 A001942 * A037148 A323282 A175744

Adjacent sequences: A034594 A034595 A034596 * A034598 A034599 A034600

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved