A004007 Theta series of E_6 lattice. (Formerly M5349)

1, 72, 270, 720, 936, 2160, 2214, 3600, 4590, 6552, 5184, 10800, 9360, 12240, 13500, 17712, 14760, 25920, 19710, 26064, 28080, 36000, 25920, 47520, 37638, 43272, 45900, 59040, 46800, 75600, 51840, 69264, 73710, 88560, 62208, 108000, 85176

Offset

0,2

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

G. Nebe and N. J. A. Sloane, Home page for this lattice

Formula

Expansion of eta(q)^9 / eta(q^3)^3 + 81*q * eta(q^3)^9 / eta(q)^3 in powers of q.

Expansion of a(q)^3 + 2*c(q)^3 in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Oct 24 2006

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^9 / QPochhammer[ q^3]^3 + 81 q QPochhammer[ q^3]^9 / QPochhammer[ q]^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
terms = 37; f[q_] = LatticeData["E6", "ThetaSeriesFunction"][-I Log[q]/Pi]; s = Series[f[q], {q, 0, 2 terms}] // Normal // Simplify[#, q > 0]&; (List @@ s)[[1 ;; terms]] /. q -> 1 (* Jean-François Alcover, Jul 04 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^9 / eta(x^3 + A)^3 + 81 * x * eta(x^3 + A)^9 / eta(x + A)^3, n))}; /* Michael Somos, Oct 24 2006 */

Crossrefs

Cf. A005129 (dual lattice).

Sequence in context: A165139 A305222 A316800 * A279272 A378002 A173546

Adjacent sequences: A004004 A004005 A004006 * A004008 A004009 A004010

Keyword

nonn

Author

N. J. A. Sloane

Status

approved