A004033 Theta series of lattice A_2 tensor E_8 (dimension 16, det. 6561, min. norm 4). Also theta series of Eisenstein version of E_8 lattice. (Formerly M5472)

1, 0, 720, 13440, 97200, 455040, 1714320, 4821120, 12380400, 29043840, 58980960, 114076800, 219310320, 367338240, 621878400, 1037727360, 1583679600, 2401816320, 3747180240, 5232470400, 7551983520, 10938261120, 14715224640, 19930775040, 28073386800, 35727920640

Offset

0,3

Comments

Also theta series of 16-dimensional lattice (SL(2,9) Y SL(2,9)).(C2 x C2). - John Cannon, Jan 10 2007

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

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Walter Feit, Some lattices over Q(sqrt(-3)), J. Algebra 52 (1978), no. 1, 248-263.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Theta series is x^8-48*x^2*y, x = phi_0(z) (see A004016), y = Delta_12(z) (see A007332) in the notation of SPLAG, Chap. 4. See A037150 for Maple code.

Expansion of a(x)^2 * (a(x)^6 - 48*x * f(-x)^6 * f(-x^3)^6) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Feb 01 2017

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 01 2017

Example

G.f. = 1 + 720*x^2 + 13440*x^3 + 97200*x^4 + 455040*x^5 + 1714320*x^6 + 4821120*x^7 + ...
G.f. = 1 + 720*q^4 + 13440*q^6 + 97200*q^8 + 455040*q^10 + 1714320*q^12 + 4821120*q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ With[ {a1 = (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3) / QPochhammer[ x^3]}, a1^2 (a1^6 - 48 x QPochhammer[ x]^6 QPochhammer[ x^3]^6)], {x, 0, n}]; (* Michael Somos, Feb 01 2017 *)

Prog

(Magma)
// Definition for lattice (SL(2, 9) Y SL(2, 9)).(C2 x C2), from John Cannon
LatticeWithBasis(16, \[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], MatrixRing(IntegerRing(), 16) ! \[
4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 1, 2,
1, -1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 0, 1, 2, 2, 1, 1, 1, 2, 1, 0, 1,
1, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 4,
1, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2,
0, 2, 1, 2, 1, 2, 2, 1, 1, 1, 4, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1,
2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 1, 1, 0, 1, -1, 1, 0, 1, 2, 0, 1, 4, 1,
1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 2, 1, 4, 2, 2, 1, 0, 0, -1, 2,
1, 2, 1, 0, 2, 2, 2, 1, 2, 4, 2, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2,
2, 2, 2, 4, 1, 0, -1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 1, 4, 1, 1,
1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1,
1, 1, 1, 0, 1, -1, 1, 2, 4, 1, 1, 1, 2, 0, 2, 2, 1, 0, 1, -1, 1, 1, 1,
1, 1, 4 ])
(Magma)
// Definition for lattice A_2 tensor E_8, from John Cannon
A := Lattice("A", 2);
B := Lattice("E", 8);
L := TensorProduct(A, B);
T<q> := ThetaSeries(L, 16);
(Magma) A := Basis( ModularForms( Gamma0(3), 8), 26); A[1] + 720*A[3]; /* Michael Somos, Feb 01 2017 */
(PARI) {a(n) = if( n<0, 0, my(A, a1); A = x * O(x^n); a1 = (eta(x + A)^3 + 9*x * eta(x^9 + A)^3) / eta(x^3 + A); polcoeff( a1^2 * (a1^6 - 48*x * eta(x + A)^6 * eta(x^3 + A)^6), n))}; /* Michael Somos, Feb 01 2017 */

Crossrefs

Cf. A004016, A007332, A037150.

Sequence in context: A052785 A052783 A112002 * A056271 A000920 A052779

Adjacent sequences: A004030 A004031 A004032 * A004034 A004035 A004036

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved