A005876 Theta series of cubic lattice with respect to edge.

2, 8, 10, 8, 16, 16, 10, 24, 16, 8, 32, 24, 18, 24, 16, 24, 32, 32, 16, 32, 34, 16, 48, 16, 16, 56, 32, 24, 32, 40, 26, 48, 48, 16, 32, 32, 32, 56, 48, 24, 64, 32, 26, 56, 16, 40, 64, 64, 16, 40, 48, 32, 80, 32, 32, 64, 50, 40, 48, 48, 48, 56, 48, 16, 64, 72, 32, 88, 32, 24

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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. 107.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A045834.

Programs

Mathematica

s = EllipticTheta[3, 0, q]^2*EllipticTheta[2, 0, q]/q^(1/4) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from 1st formula *)
s = (2*(QPochhammer[q^2]^9/(QPochhammer[q]^4*QPochhammer[q^4]^2))) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, from 3rd formula *)

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); 2*polcoeff( eta(x^2+A)^9/ eta(x+A)^4/eta(x^4+A)^2, n))} /* Michael Somos, Feb 21 2006 */

Formula

a(n) = 2*A045834(n).

Expansion of 2*phi(q)*psi(q)^2 in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, Feb 21 2006

Expansion of theta_2(q^2)^2(theta_3(q)+theta_4(q))/(4q) in powers of q^4. - Michael Somos, Feb 21 2006

Expansion of 2q^(-1/4)eta(q^2)^9/(eta(q)^4*eta(q^4)^2) in powers of q. - Michael Somos, Feb 21 2006

G.f.: 2*Product_{k>0} (1+x^k)^4*(1-x^(2k))^3/(1+x^(2k))^2. - Michael Somos, Feb 21 2006