A045819 Theta series of E_8 lattice with respect to midpoint of edge.

2, 56, 252, 688, 1514, 2664, 4396, 7056, 9828, 13720, 19264, 24336, 31502, 40880, 48780, 59584, 74592, 86688, 101308, 123088, 137844, 159016, 190764, 207648, 235986, 275184, 297756, 335664, 384160, 410760, 453964, 520816, 553896, 601528

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			2*q^(1/2) + 56*q^(3/2) + 252*q^(5/2) + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 123.

Links

• Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.

Crossrefs

Cf. A001158, A008438, A013662, A045823.

Programs

Mathematica

a[n_] := 2 DivisorSigma[3, 2 n + 1]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jul 06 2017, after Benoit Cloitre *)

Formula

G.f.: (1/2)*(theta_2^2*theta_3^6 + theta_2^6*theta_3^2).

a(n) = 2*sigma_3(2n+1). - Benoit Cloitre, Apr 12 2003

a(n) = 2 * A045823(n). - Alois P. Heinz, Mar 21 2021

Sum_{k=0..n} a(k) ~ (15*zeta(4)/4) * n^4. - Amiram Eldar, Dec 12 2023

Extensions

More terms from Benoit Cloitre, Apr 12 2003