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

0,4

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).

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

Example

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

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A252406 A145307 A232254 * A229508 A249204 A249326

Adjacent sequences: A008771 A008772 A008773 * A008775 A008776 A008777

Keyword

nonn

Author

N. J. A. Sloane

Status

approved