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Theta series of (probably nonexistent) exceptionally good 16-dimensional sphere packing.
3

%I #18 Sep 11 2019 05:03:10

%S 1,0,0,7680,4320,276480,61440,2903040,522720,16896000,2211840,

%T 68774400,8960640,221460480,23224320,603325440,67154400,1448202240,

%U 135168000,3154982400,319809600,6359654400,550195200,12016788480,1147643520

%N Theta series of (probably nonexistent) exceptionally good 16-dimensional sphere packing.

%H G. C. Greubel, <a href="/A008774/b008774.txt">Table of n, a(n) for n = 0..1000</a>

%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 190, Equation (47).

%F Expansion of ( E_4(q) * 2 * (E_4(q^2) - E_4(q^4)) + E_4(q^2) * (32 * E_4(q^4) - 17 * E_4(q^2)) ) / 15 in powers of q. - _Michael Somos_, Nov 29 2007

%e 1 + 7680*q^3 + 4320*q^4 + 276480*q^5 + 61440*q^6 + 2903040*q^7 + ...

%t QP = QPochhammer; a[n_] := Module[{A, A1, A2, A4}, A = x*O[x]^n; A1 = QP[x+ A]^8; A2 = QP[x^2+A]^8; A4 = QP[x^4+A]^8; SeriesCoefficient[(A1*(A2^6 + x^2*32*A2^3*A4^3 + x^4*4096*A4^6) + x^3*3840*A4^4*(A1^2*A4 + A2^3)) / (A1*A2^2*A4^2), n]]; a[0] = 1; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A, A1, A2, A4); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^8; A4 = eta(x^4 + A)^8; polcoeff( ( A1 * (A2^6 + x^2 * 32 * A2^3 * A4^3 + x^4 * 4096 * A4^6) + x^3 * 3840 * A4^4 * ( A1^2 * A4 + A2^3 ) ) / (A1 * A2^2 * A4^2 ), n))} /* _Michael Somos_, Nov 29 2007 */

%Y A008409(n) = a(2*n). 7680 * A135828(n) = a(2*n+3).

%K nonn

%O 0,4

%A _N. J. A. Sloane_