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

0,2

Comments

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

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

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).

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A005876.

Sequence in context: A201296 A350089 A246954 * A106148 A192038 A046577

Adjacent sequences: A045831 A045832 A045833 * A045835 A045836 A045837

Keyword

nonn

Author

N. J. A. Sloane

Status

approved