A034598 Second coefficient of extremal theta series of even unimodular lattice in dimension 24n.

1, 16773120, 39007332000, 15281788354560, 2972108280960000, 406954241261568000, 45569082381053868000, 4499117081888292864000, 408472720963469499617280, 34975479259332252426240000

Offset

0,2

Comments

Although these initially increase, they eventually go negative at about term 1700 (i.e. dimension about 40800) - see references.

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

For Maple program see A034597.

Mathematica

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 *)

Crossrefs

Cf. A034597 (leading coefficient).

Sequence in context: A255164 A330247 A365853 * A326706 A011574 A022540

Adjacent sequences: A034595 A034596 A034597 * A034599 A034600 A034601

Keyword

sign

Author

N. J. A. Sloane.

Status

approved