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%I #34 Sep 09 2022 15:59:07
%S 1,720,179280,16954560,396974160,4632858720,34413301440,187477879680,
%T 814940600400,2975469665040,9486467837280,27053330840640,
%U 70485969919680,169930679355360,384163875688320,820167497170560,1668890801059920,3249626139960480,6096884624994960
%N Theta series of direct sum of 3 copies of E_8 lattice (the Niemeier lattice of type E_8^3).
%C Also the theta series for the Niemeier lattice of type E_8 D_16. - _Ben Mares_, Jul 17 2022
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123, 407.
%H Seiichi Manyama, <a href="/A008411/b008411.txt">Table of n, a(n) for n = 0..10000</a>
%F This series is the q-expansion of E_4(z)^3. Cf. A004009. - _Daniel D. Briggs_, Nov 25 2011
%F 691*a(n) - A029828(n) = 432000*A000594(n). - _Seiichi Manyama_, Jan 28 2017
%e G.f. = 1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + ...
%t 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 *)
%t 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 *)
%t 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 *)
%o (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 */
%o (Magma) A := Basis( ModularForms( Gamma1(1), 12), 19); A[1] + 720*A[2]; /* _Michael Somos_, Jan 28 2017 */
%Y Cf. A000594, A004009, A029828, A280869.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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