%I A014969
%S A014969 1,8,32,96,256,624,1408,3008,6144,12072,22976,42528,76800,
%T A014969 135728,235264,400704,671744,1109904,1809568,2914272,4640256,
%U A014969 7310592,11404416,17626944,27009024,41047992,61905088,92681664
%N A014969 Expansion of (theta_3 / theta_4)^2.
%D A014969 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 102.
%D A014969 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; Eq. (34.3).
%D A014969 R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 
               1922, Vol. 2, see p. 375. Eq. (17)
%H A014969 T. D. Noe, <a href="b014969.txt">Table of n, a(n) for n=0..1000</a>
%F A014969 Euler transform of period 4 sequence [8, -4, 8, 0, ...]. - Michael Somos, 
               Jul 07 2005
%F A014969 G.f.: (theta_3/theta_4)^2 = (Sum_{k} x^k^2)/(Sum_{k} (-x)^k^2)^2 = (Product_{k>
               0} (1-x^(4k-2))/((1-x^(4k-1))(1-x^(4k-3)))^2)^4.
%F A014969 Expansion of Fricke tau_8(omega)/2+1 in powers of q = exp(2*pi*i*z).
%F A014969 Expansion of elliptic 1/sqrt(1-lambda(z))=1/k' in powers of nome q = 
               exp(pi*i*z).
%F A014969 G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v) = (1+u)^2 -4*u*v^2 
               . - Michael Somos Nov 14 2006
%F A014969 Expansion of (phi(q) / phi(-q))^2 in powers of q where phi() is a Ramanujan 
               theta function.
%e A014969 1 + 8*q + 32*q^2 + 96*q^3 + 256*q^4 + 624*q^5 + 1408*q^6 + 3008*q^7 + 
               ...
%o A014969 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)^3/
               eta(x+A)^2/eta(x^4+A))^4, n))} /* Michael Somos Jul 07 2005 */
%Y A014969 8 * A107035(n) = a(n) unless n=0. 2 * A131126(n) = a(n) unless n=0.
%Y A014969 Sequence in context: A159941 A053348 A019256 this_sequence A139820 A071345 
               A100312
%Y A014969 Adjacent sequences: A014966 A014967 A014968 this_sequence A014970 A014971 
               A014972
%K A014969 nonn,nice
%O A014969 0,2
%A A014969 N. J. A. Sloane (njas(AT)research.att.com).