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
Keywords
Examples
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.
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).
Crossrefs
Cf. A034597 (leading coefficient).
Programs
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Maple
For Maple program see A034597.
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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 *)
Comments