%I A002448
%S A002448 1,2,0,0,2,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,
%T A002448 0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,
%U A002448 0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0
%V A002448 1,-2,0,0,2,0,0,0,0,-2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,
               0,0,0,2,0,0,0,0,
%W A002448 0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,
               0,0,0,0,0,0,-2,0,
%X A002448 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0
%N A002448 Expansion of Jacobi theta function theta_4(x).
%C A002448 Euler transform of period 2 sequence [ -2,-1,...].
%C A002448 Expansion of eta(q)^2/eta(q^2) in powers of q.
%D A002448 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 103.
%D A002448 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 93, Eq. (34.11).
%D A002448 E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge 
               Univ. Press, 4th ed., 1963, p. 464.
%H A002448 T. D. Noe, <a href="b002448.txt">Table of n, a(n) for n=0..10000</a>
%H A002448 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A002448 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               q-SeriesIdentities.html">q-Series Identities</a>
%F A002448 G.f.: Sum_n (-1)^n*x^(n^2) = Product_{n>0} (1-x^n)/(1+x^n).
%F A002448 a(n)=-2*b(n) where b(n) is multiplicative and b(2^e) = (-1)^(e/2) if 
               e even, b(p^e) = 1 if p>2 and e even, otherwise 0. - Michael Somos 
               Jul 07 2006
%F A002448 a(3n+1)=-2*A089802(n), a(9n)=a(n). - Michael Somos Jul 07 2006
%p A002448 Sum((-x)^(m^2),m=-10..10);
%o A002448 (PARI) a(n)=if(n<0,0,(-1)^n*issquare(n)*2-(n==0))
%o A002448 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^2/eta(X^2),
               n))
%Y A002448 Cf. A000122.
%Y A002448 Sequence in context: A128771 A139380 A000122 this_sequence A033759 A033755 
               A033753
%Y A002448 Adjacent sequences: A002445 A002446 A002447 this_sequence A002449 A002450 
               A002451
%K A002448 sign
%O A002448 0,2
%A A002448 N. J. A. Sloane (njas(AT)research.att.com).