A005129 -id:A005129 - cp's OEIS frontend

A109144 G.f.: cube root of theta series of E*_6 lattice (cf. A005129).

1, 0, 18, 24, -324, -720, 9234, 33696, -307926, -1614000, 10860480, 78618672, -385935492, -3854459520, 13226566500, 189101997360, -408244450788, -9247223107008, 9292147273170, 449330421257568, 41307157825920, -21633699079357296, -25580513164295232, 1028997388914059808

Offset: 0

N. J. A. Sloane and Nadia Heninger, Aug 18 2005

sign

Vaclav Kotesovec, 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.

terms = 24; QP = QPochhammer; s = ((QP[q]^3/QP[q^3])^3 + 9 q (QP[q^3]^3 / QP[q])^3)^(1/3) + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 08 2017, after Michael Somos *)

A004007 Theta series of E_6 lattice.

Original entry on oeis.org

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

N. J. A. Sloane

nonn

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

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).

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

Cf. A005129 (dual lattice).

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 *)

{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 */

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

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A109144 G.f.: cube root of theta series of E*_6 lattice (cf. A005129).

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A004007 Theta series of E_6 lattice.

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