A005879 - OEIS

%I M4509 #52 Mar 12 2021 22:24:41

%S 8,32,48,64,104,96,112,192,144,160,256,192,248,320,240,256,384,384,

%T 304,448,336,352,624,384,456,576,432,576,640,480,496,832,672,544,768,

%U 576,592,992,768,640,968,672,864,960,720,896,1024,960,784,1248,816,832,1536

%N Theta series of D_4 lattice with respect to deep hole.

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

%C 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

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005879/b005879.txt">Table of n, a(n) for n = 0..10000</a>

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html">Home page for this lattice</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>

%F Expansion of Jacobi theta_2(q)^4/(2q) in powers of q^2. - _Michael Somos_, Apr 11 2004

%F Expansion of q^(-1/2) * 8 * (eta(q^2)^2 / eta(q))^4 in powers of q. - _Michael Somos_, Apr 11 2004

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

%F Expansion of (phi(q)^4 - phi(-q)^4) / (2 * q) in powers of q^2. - _Michael Somos_, May 23 2012

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

%F a(n) = 8 * A008438(n) = 4 * A005880(n) = A000118(2*n + 1) = - A096727(2*n + 1). - _Michael Somos_, Nov 01 2006

%e 8 + 32*x + 48*x^2 + 64*x^3 + 104*x^4 + 96*x^5 + 112*x^6 + 192*x^7 + ...

%e 8*q + 32*q^3 + 48*q^5 + 64*q^7 + 104*q^9 + 96*q^11 + 112*q^13 + ...

%e .

%e 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.

%e [-2,-1,0,0], [-2,0,-1,0], [-2,0,0,-1], [-2,0,0,1], [-2,0,1,0], [-2,1,0,0],

%e [-1,-2,0,0], [-1,0,-2,0], [-1,0,0,-2], [-1,0,0,2], [-1,0,2,0], [-1,2,0,0],

%e [0,-2,-1,0], [0,-2,0,-1], [0,-2,0,1], [0,-2,1,0], [0,-1,-2,0], [0,-1,0,-2],

%e [0,-1,0,2], [0,-1,2,0], [0,0,-2,-1], [0,0,-2,1], [0,0,-1,-2], [0,0,-1,2],

%e [0,0,1,-2], [0,0,1,2], [0,0,2,-1], [0,0,2,1], [0,1,-2,0], [0,1,0,-2],

%e [0,1,0,2], [0,1,2,0], [0,2,-1,0], [0,2,0,-1], [0,2,0,1], [0,2,1,0],

%e [1,-2,0,0], [1,0,-2,0], [1,0,0,-2], [1,0,0,2], [1,0,2,0], [1,2,0,0],

%e [2,-1,0,0], [2,0,-1,0], [2,0,0,-1], [2,0,0,1], [2,0,1,0], [2,1,0,0].

%e - _Peter Luschny_, Nov 03 2015

%p S:= series(JacobiTheta2(0,q)^4/(2*q), q, 202):

%p seq(coeff(S,q,2*j),j=0..100); # _Robert Israel_, Nov 03 2015

%t (* 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 *)

%t 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_ *)

%o (PARI) {a(n) = if( n<0, 0, 8 * sigma(2*n + 1))} /* _Michael Somos_, Apr 11 2004 */

%o (PARI) q='q+O('q^66); Vec(8*(eta(q^2)^2/eta(q))^4) \\ _Joerg Arndt_, Nov 03 2015

%Y Cf. A000118, A005880, A008438, A096727.

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_