OFFSET
0,2
COMMENTS
Also the theta series for the Niemeier lattice of type E_8 D_16. - Ben Mares, Jul 17 2022
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
FORMULA
This series is the q-expansion of E_4(z)^3. Cf. A004009. - Daniel D. Briggs, Nov 25 2011
EXAMPLE
G.f. = 1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + ...
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 *)
PROG
(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 */
(Magma) A := Basis( ModularForms( Gamma1(1), 12), 19); A[1] + 720*A[2]; /* Michael Somos, Jan 28 2017 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved