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A034597 Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n. 5
1, 196560, 52416000, 6218175600, 565866362880, 45792819072000, 3486157968384000, 256206274225902000, 18422726047165440000, 1305984407917646096640, 91692325887531393024000 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified January 20 02:57 EST 2019. Contains 319323 sequences. (Running on oeis4.)