A023924 Theta series of A*_12 lattice.

1, 0, 0, 0, 0, 0, 26, 0, 0, 0, 0, 156, 0, 156, 0, 572, 0, 0, 1430, 1716, 2574, 3432, 0, 0, 5746, 0, 4290, 0, 13182, 0, 0, 22308, 26052, 29744, 33462, 0, 0, 54912, 0, 36036, 0, 89232, 0, 0, 123708, 143000, 156156, 178464, 0, 0, 234234, 0, 141726, 0, 348374, 0

Offset

0,7

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.

Links

Table of n, a(n) for n=0..55.

Kok Seng Chua, Circular Summation of the 13th powers of Ramanujan Theta Functions, Ramanujan J., 5 (2001), 353-354.

Formula

G.f.: y^13/x + 13*x*y^11 + 78*x^3*y^9 + 260*x^5*y^7 + 494*x^7*y^5 + 468*x^9*y^3 + 169*x^11*y + 13*x^13/y where x=eta(13z) and y=eta(z).

Example

1 + 26*q^6 + 156*q^11 + 156*q^13 + 572*q^15 + 1430*q^18 + 1716*q^19 + ...

Mathematica

a[n_] := Module[{A, B}, A = x*O[x]^n; B = x*(QPochhammer[x^13 + A] / QPochhammer[x + A])^2; SeriesCoefficient[(QPochhammer[x + A]^13 / QPochhammer[x^13 + A])*(1 + 13*B*(1 + 6*B + 20*B^2 + 38*B^3 + 36*B^4 + 13*B^5 + B^6)), {x, 0, n}]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 05 2015, translated from PARI *)

Prog

(PARI) {a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = x * (eta(x^13 + A) / eta(x + A))^2; polcoeff( eta(x + A)^13 / eta(x^13 + A) * (1 + 13*B * (1 + 6*B + 20*B^2 + 38*B^3 + 36*B^4 + 13*B^5 + B^6)), n))} /* Michael Somos, Jun 24 2013 */

Crossrefs

Sequence in context: A237522 A347810 A116197 * A022066 A291551 A217232

Adjacent sequences: A023921 A023922 A023923 * A023925 A023926 A023927

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 31 2001

Status

approved