A008411 Theta series of direct sum of 3 copies of E_8 lattice (the Niemeier lattice of type E_8^3).

1, 720, 179280, 16954560, 396974160, 4632858720, 34413301440, 187477879680, 814940600400, 2975469665040, 9486467837280, 27053330840640, 70485969919680, 169930679355360, 384163875688320, 820167497170560, 1668890801059920, 3249626139960480, 6096884624994960

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also the theta series for the Niemeier lattice of type E_8 D_16. - Ben Mares, Jul 17 2022

Examples

			G.f. = 1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123, 407.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000594, A004009, A029828, A280869.

Programs

Magma

A := Basis( ModularForms( Gamma1(1), 12), 19); A[1] + 720*A[2]; /* Michael Somos, Jan 28 2017 */

Mathematica

a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, (t2^2 + 14 t2 t3 + t3^2)^3 ], {q, 0, n}]; (* Michael Somos, Jan 28 2017 *)
terms = 19; QP = QPochhammer; s = (QP[x]^24 + 256*x*QP[x^2]^24)^3 / (QP[x]*QP[x^2])^24 + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 07 2017, adapted from PARI *)
terms = 19; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E4[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24)^3 / (eta(x + A) * eta(x^2 + A))^24, n))}; /* Michael Somos, Jan 28 2017 */

Formula

This series is the q-expansion of E_4(z)^3. Cf. A004009. - Daniel D. Briggs, Nov 25 2011

691*a(n) - A029828(n) = 432000*A000594(n). - Seiichi Manyama, Jan 28 2017