A045834 Half of theta series of cubic lattice with respect to edge.

1, 4, 5, 4, 8, 8, 5, 12, 8, 4, 16, 12, 9, 12, 8, 12, 16, 16, 8, 16, 17, 8, 24, 8, 8, 28, 16, 12, 16, 20, 13, 24, 24, 8, 16, 16, 16, 28, 24, 12, 32, 16, 13, 28, 8, 20, 32, 32, 8, 20, 24, 16, 40, 16, 16, 32, 25, 20, 24, 24, 24, 28, 24, 8, 32, 36, 16, 44, 16, 12, 40, 32, 17, 36, 32

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 4*x + 5*x^2 + 4*x^3 + 8*x^4 + 8*x^5 + 5*x^6 + 12*x^7 + 8*x^8 + ...
G.f. = q + 4*q^5 + 5*q^9 + 4*q^13 + 8*q^17 + 8*q^21 + 5*q^25 + 12*q^29 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 107.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A005876.

Programs

Maple

S:= series((1/8)*JacobiTheta2(0, sqrt(q))^2*(JacobiTheta3(0, q^(1/4))+JacobiTheta4(0, q^(1/4)))/q^(1/4), q, 1001):
seq(coeff(S,q,j),j=0..1000); # Robert Israel, Nov 13 2016

Mathematica

s = EllipticTheta[3, 0, q]^2*EllipticTheta[2, 0, q]/(2*q^(1/4)) + O[q]^75; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from 5th formula *)
QP = QPochhammer; s = QP[q^2]^9/(QP[q]^4*QP[q^4]^2) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 +A)^9 / (eta(x + A)^4 * eta(x^4 + A)^2), n))}; /* Michael Somos, Feb 28 2006 */

Formula

Euler transform of period 4 sequence [ 4, -5, 4, -3,...]. - Michael Somos, Feb 28 2006

Expansion of theta_2(q^2)^2 * (theta_3(q) + theta_4(q)) / (8*q) in powers of q^4. - Michael Somos, Feb 28 2006

Expansion of q^(-1/4) * eta(q^2)^9 / (eta(q)^4 * eta(q^4)^2) in powers of q. - Michael Somos, Feb 28 2006

G.f.: Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^3 / (1 + x^(2*k))^2. - Michael Somos, Feb 28 2006

Expansion of phi(x)^2 * psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006

A005876(n) = 2*a(n).