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%I M5349 #29 Jan 11 2018 01:09:04
%S 1,72,270,720,936,2160,2214,3600,4590,6552,5184,10800,9360,12240,
%T 13500,17712,14760,25920,19710,26064,28080,36000,25920,47520,37638,
%U 43272,45900,59040,46800,75600,51840,69264,73710,88560,62208,108000,85176
%N Theta series of E_6 lattice.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A004007/b004007.txt">Table of n, a(n) for n = 0..1000</a>
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E6.html">Home page for this lattice</a>
%F Expansion of eta(q)^9 / eta(q^3)^3 + 81*q * eta(q^3)^9 / eta(q)^3 in powers of q.
%F 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
%t 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 *)
%t 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 *)
%o (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 */
%Y Cf. A005129 (dual lattice).
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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