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 A004007 Theta series of E_6 lattice. (Formerly M5349) 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 A173546 A242534 Adjacent sequences:  A004004 A004005 A004006 * A004008 A004009 A004010 KEYWORD nonn AUTHOR STATUS approved

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Last modified November 13 01:32 EST 2018. Contains 317118 sequences. (Running on oeis4.)