A045828 One fourth of theta series of cubic lattice with respect to face.

1, 2, 2, 4, 3, 2, 6, 4, 4, 6, 4, 4, 7, 8, 2, 8, 8, 4, 10, 4, 4, 10, 10, 8, 9, 4, 6, 12, 8, 6, 10, 12, 4, 14, 8, 4, 16, 10, 8, 8, 9, 10, 12, 12, 8, 12, 12, 4, 20, 10, 6, 20, 8, 6, 10, 12, 8, 20, 18, 8, 11, 12, 12, 16, 8, 6, 20, 16, 12, 14, 8, 12, 20, 14, 6, 12, 20, 8, 26, 12, 8, 22, 8, 12, 15

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).

Number of solutions to n = t1 + t2 + 2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006

The cubic lattice is the set of triples [a, b, c] where the entries are all integers. A face is centered at a triple where one entry is an integer and the other two are one half an odd integer. - Michael Somos, Jun 29 2012

Examples

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 2*x^5 + 6*x^6 + 4*x^7 + 4*x^8 + 6*x^9 + ...
G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + 3*q^9 + 2*q^11 + 6*q^13 + 4*q^15 + 4*q^17 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A005877, A005884, A033761, A010054, A033763, A212885.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 1/4 EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x]^2, {x, 0, n + 1/2}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, x^2] EllipticTheta[ 2, 0, x]^2, {x, 0, 2 n + 1}]; (* Michael Somos, Jun 29 2012 *)
QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^2)/QP[q]^2 + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^2, n))}; /* Michael Somos, Oct 25 2006 */

Formula

Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^2) / eta(q)^2 in powers of q. - Michael Somos, Jan 02 2006

Expansion of phi(x) * psi(x^2)^2 = psi(x)^2 * psi(x^2) = psi(x)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 29 2012

Euler transform of period 4 sequence [2, -1, 2, -3, ...]. - Michael Somos, Mar 05 2003

Convolution of A033761 and A010054. - Michael Somos, Jun 29 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A212885. - Michael Somos, Sep 08 2018

Extensions

Edited by Michael Somos, Mar 05 2003