A008422 Theta series of D*_5 lattice.

1, 0, 0, 0, 10, 32, 0, 0, 40, 0, 0, 0, 80, 160, 0, 0, 90, 0, 0, 0, 112, 320, 0, 0, 240, 0, 0, 0, 320, 480, 0, 0, 200, 0, 0, 0, 250, 800, 0, 0, 560, 0, 0, 0, 560, 992, 0, 0, 400, 0, 0, 0, 560, 1120, 0, 0, 800, 0, 0, 0, 960, 1760, 0, 0, 730, 0, 0, 0, 480, 1920, 0, 0, 1240, 0, 0, 0, 1520, 1920

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120, Eq. 96.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..800
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

terms = 78; phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q])*InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, q^(1/2)]; s = phi[q^4]^5 + (2*q*psi[q^8])^5 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017, after Michael Somos *)

PARI

N=66;  q='q+O('q^N);
T3(q) = eta(q^2)^5 / ( eta(q)^2 * eta(q^4)^2 );
T2(q) = eta(q^4)^2 / eta(q^2);
Vec( T3(q^4)^5 + (2 * q * T2(q^4))^5 )
\\ Joerg Arndt, Mar 30 2018

Formula

Theta series in terms of Jacobi theta series: (theta_2)^5 + (theta_3)^5.

Expansion of phi(q^4)^5 + ( 2 * q * psi(q^8) )^5 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 17 2007