A008774 Theta series of (probably nonexistent) exceptionally good 16-dimensional sphere packing.

1, 0, 0, 7680, 4320, 276480, 61440, 2903040, 522720, 16896000, 2211840, 68774400, 8960640, 221460480, 23224320, 603325440, 67154400, 1448202240, 135168000, 3154982400, 319809600, 6359654400, 550195200, 12016788480, 1147643520

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 190, Equation (47).

Crossrefs

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

Programs

Mathematica

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 *)

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 */

Formula

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