A005879 Theta series of D_4 lattice with respect to deep hole. (Formerly M4509)

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

0,1

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

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000118, A005880, A008438, A096727.

Sequence in context: A127988 A371454 A129749 * A067519 A253295 A290960

Adjacent sequences: A005876 A005877 A005878 * A005880 A005881 A005882

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved