A023916 Theta series of A*_4 lattice.

1, 0, 10, 20, 0, 20, 0, 60, 50, 0, 30, 0, 60, 120, 0, 60, 0, 160, 70, 0, 60, 0, 120, 220, 0, 120, 0, 200, 180, 0, 40, 0, 210, 240, 0, 180, 0, 360, 200, 0, 150, 0, 120, 420, 0, 140, 0, 460, 220, 0, 130, 0, 360, 520, 0, 240, 0, 400, 300, 0, 180, 0, 320, 420, 0, 360, 0, 660, 480, 0, 120

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			1 + 10*x^2 + 20*x^3 + 20*x^5 + 60*x^7 + 50*x^8 + 30*x^10 + 60*x^12 + ...
1 + 10*q^4 + 20*q^6 + 20*q^10 + 60*q^14 + 50*q^16 + 30*q^20 + 60*q^24 + 120*q^26 + 60*q^30 + 160*q^34 + 70*q^36 + 60*q^40 + 120*q^44 + 220*q^46 + 120*q^50 + 200*q^54 + 180*q^56 + 40*q^60 + O(q^62).
		

References

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

Links

• John Cannon, Table of n, a(n) for n = 0..5000
• S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_5(q).
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Crossrefs

Cf. A008444.

Programs

Magma

L:=Lattice("A",4); D:=Dual(L); T1 := ThetaSeries(D,60);

Mathematica

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[QPochhammer[x+A]^5 / QPochhammer[x^5+A] + 5*x*(QPochhammer[x^5+A]^5 / QPochhammer[x+A]), {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from PARI *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^5 + A) + 5 * x * eta(x^5 + A)^5 / eta(x + A), n))}; /* Michael Somos, Jan 29 2011 */

Formula

Expansion of f(-x)^5 / f(-x^5) + 5 * x * f(-x^5)^5 / f(-x) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 29 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^(1/2) (t/i)^2 g(t) where q = exp(2*Pi*i*t) and g() is g.f. for A008444. - Michael Somos, Jan 29 2011

a(5*n) = A008444(n). a(5*n + 1) = a(5*n + 4) = 0. - Michael Somos, Jan 29 2011

Extensions

More terms from N. J. A. Sloane, Dec 24 2006