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

0,5

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

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A280202 A061485 A136335 * A063926 A239834 A337148

Adjacent sequences: A008419 A008420 A008421 * A008423 A008424 A008425

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved