A005129 Theta series of {E_6}* lattice. (Formerly M5309)

1, 0, 54, 72, 0, 432, 270, 0, 918, 720, 0, 2160, 936, 0, 2700, 2160, 0, 5184, 2214, 0, 5616, 3600, 0, 9504, 4590, 0, 9180, 6552, 0, 15120, 5184, 0, 14742, 10800, 0, 21600, 9360, 0, 19548, 12240, 0, 30240, 13500, 0, 28080, 17712, 0, 39744, 14760, 0, 32454

Offset

0,3

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. 127.

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

Links

Table of n, a(n) for n=0..50.

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 b(q)^3 + c(q)^3 / 3 in power of q where b(), c() are cubic AGM theta functions.

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

Example

G.f. = 1 + 54*x^2 + 72*x^3 + 432*x^5 + 270*x^6 + 918*x^8 + 720*x^9 + 2160*x^11 + ...

Mathematica

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x+A]^3 / QPochhammer[x^3+A])^3 + 9*x*(QPochhammer[x^3+A]^3 / QPochhammer[x+A])^3, {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from 1st PARI script *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3 + 9 q (QPochhammer[ q^3]^3 /QPochhammer[ q])^3 , {q, 0, n}]; ( Michael Somos, Dec 28 2015 )

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 + 9 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))}; /* Michael Somos, Feb 28 2012 */
(PARI) {a(n) = my(A, a1, p3); if( n<0, 0, A = x * O(x^n); a1 = sum( k=1, n, 6 * sumdiv(k, d, kronecker( d, 3)) * x^k, 1 + A); p3 = sum( k=1, n\3, -24 * sigma(k) * x^(3*k), 1 + A); polcoeff( (a1^3 + a1 * p3 - 4 * x * a1') / 2, n))}; /* Michael Somos, Feb 28 2012 */
(Magma) A := Basis( ModularForms( Gamma1(3), 3), 51); A[1]; /* Michael Somos, Dec 28 2015 */

Crossrefs

Cf. A004007 (E_6).

Sequence in context: A335035 A350886 A281920 * A039532 A007244 A365262

Adjacent sequences: A005126 A005127 A005128 * A005130 A005131 A005132

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved