A005876 - OEIS

%I M1824 #28 Mar 12 2021 22:24:41

%S 2,8,10,8,16,16,10,24,16,8,32,24,18,24,16,24,32,32,16,32,34,16,48,16,

%T 16,56,32,24,32,40,26,48,48,16,32,32,32,56,48,24,64,32,26,56,16,40,64,

%U 64,16,40,48,32,80,32,32,64,50,40,48,48,48,56,48,16,64,72,32,88,32,24

%N Theta series of cubic lattice with respect to edge.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

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

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

%H G. C. Greubel, <a href="/A005876/b005876.txt">Table of n, a(n) for n = 0..1000</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>

%F a(n) = 2*A045834(n).

%F Expansion of 2*phi(q)*psi(q)^2 in powers of q where phi(),psi() are Ramanujan theta functions. - _Michael Somos_, Feb 21 2006

%F Expansion of theta_2(q^2)^2(theta_3(q)+theta_4(q))/(4q) in powers of q^4. - _Michael Somos_, Feb 21 2006

%F Expansion of 2q^(-1/4)eta(q^2)^9/(eta(q)^4*eta(q^4)^2) in powers of q. - _Michael Somos_, Feb 21 2006

%F G.f.: 2*Product_{k>0} (1+x^k)^4*(1-x^(2k))^3/(1+x^(2k))^2. - _Michael Somos_, Feb 21 2006

%t s = EllipticTheta[3, 0, q]^2*EllipticTheta[2, 0, q]/q^(1/4) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 04 2015, from 1st formula *)

%t s = (2*(QPochhammer[q^2]^9/(QPochhammer[q]^4*QPochhammer[q^4]^2))) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 09 2015, from 3rd formula *)

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); 2*polcoeff( eta(x^2+A)^9/ eta(x+A)^4/eta(x^4+A)^2, n))} /* _Michael Somos_, Feb 21 2006 */

%Y Cf. A045834.

%K nonn

%O 0,1

%A _N. J. A. Sloane_