A005876 Theta series of cubic lattice with respect to edge. (Formerly M1824)

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

0,1

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

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

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 *)

Prog

(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 */

Crossrefs

Cf. A045834.

Sequence in context: A036898 A053372 A088155 * A139370 A101532 A032708

Adjacent sequences: A005873 A005874 A005875 * A005877 A005878 A005879

Keyword

nonn

Author

N. J. A. Sloane

Status

approved