A005879 Theta series of D_4 lattice with respect to deep hole.

8, 32, 48, 64, 104, 96, 112, 192, 144, 160, 256, 192, 248, 320, 240, 256, 384, 384, 304, 448, 336, 352, 624, 384, 456, 576, 432, 576, 640, 480, 496, 832, 672, 544, 768, 576, 592, 992, 768, 640, 968, 672, 864, 960, 720, 896, 1024, 960, 784, 1248, 816, 832, 1536

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

The D_4 lattice is the set of all integer quadruples [a, b, c, d] where a + b + c + d is even. The deep holes are quadruples [a, b, c, d] where each coordinate is half an odd integer and where a + b + c + d is even. - Michael Somos, May 23 2012

Examples

			8 + 32*x + 48*x^2 + 64*x^3 + 104*x^4 + 96*x^5 + 112*x^6 + 192*x^7 + ...
8*q + 32*q^3 + 48*q^5 + 64*q^7 + 104*q^9 + 96*q^11 + 112*q^13 + ...
.
For n = 2 the objects counted are the ways to represent the integer 5 = (2*n+1) as a sum of 4 squares, 0 and negative numbers allowed.
[-2,-1,0,0], [-2,0,-1,0], [-2,0,0,-1], [-2,0,0,1], [-2,0,1,0], [-2,1,0,0],
[-1,-2,0,0], [-1,0,-2,0], [-1,0,0,-2], [-1,0,0,2], [-1,0,2,0], [-1,2,0,0],
[0,-2,-1,0], [0,-2,0,-1], [0,-2,0,1], [0,-2,1,0], [0,-1,-2,0], [0,-1,0,-2],
[0,-1,0,2], [0,-1,2,0], [0,0,-2,-1], [0,0,-2,1], [0,0,-1,-2], [0,0,-1,2],
[0,0,1,-2], [0,0,1,2], [0,0,2,-1], [0,0,2,1], [0,1,-2,0], [0,1,0,-2],
[0,1,0,2], [0,1,2,0], [0,2,-1,0], [0,2,0,-1], [0,2,0,1], [0,2,1,0],
[1,-2,0,0], [1,0,-2,0], [1,0,0,-2], [1,0,0,2], [1,0,2,0], [1,2,0,0],
[2,-1,0,0], [2,0,-1,0], [2,0,0,-1], [2,0,0,1], [2,0,1,0], [2,1,0,0].
- _Peter Luschny_, Nov 03 2015
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for sequences related to D_4 lattice

Crossrefs

Cf. A000118, A005880, A008438, A096727.

Programs

Maple

S:= series(JacobiTheta2(0,q)^4/(2*q), q, 202):
seq(coeff(S,q,2*j),j=0..100); # Robert Israel, Nov 03 2015

Mathematica

(* a(n) gives the number of ways to represent the integer 2n+1 as a sum of 4 squares *) a[n_] := SquaresR[4, 2n+1]; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Nov 03 2015 *)
terms = 53; QP = QPochhammer; s = 8 QP[q^2]^8/QP[q]^4 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *)

PARI

{a(n) = if( n<0, 0, 8 * sigma(2*n + 1))} /* Michael Somos, Apr 11 2004 */

PARI

q='q+O('q^66); Vec(8*(eta(q^2)^2/eta(q))^4) \\ Joerg Arndt, Nov 03 2015

Formula

Expansion of Jacobi theta_2(q)^4/(2q) in powers of q^2. - Michael Somos, Apr 11 2004

Expansion of q^(-1/2) * 8 * (eta(q^2)^2 / eta(q))^4 in powers of q. - Michael Somos, Apr 11 2004

Expansion of 8 * psi(x)^4 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, May 23 2012

Expansion of (phi(q)^4 - phi(-q)^4) / (2 * q) in powers of q^2. - Michael Somos, May 23 2012

G.f.: 8 * (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Apr 11 2004

a(n) = 8 * A008438(n) = 4 * A005880(n) = A000118(2*n + 1) = - A096727(2*n + 1). - Michael Somos, Nov 01 2006