  In this paper we work with a series whose coefficients are the Euler
characteristic of Chow varieties of a given projective variety. For varieties
where the Cox ring is defined, it is easy to see that in this case the ring
associated to the series is the Cox ring. If this ring is noetherian then the
series is rational. It is an open question whether the converse holds. In this
paper we give an example showing the converse fails. However we conjecture that
it holds when the variety is rationally connected. As an evidence of this
conjecture, It is proved that the series is not rational, and in a sense
defined, not algebraic, in the case of the blowup of the projective plane at
nine or more points in general position. Furthermore, we also treat some other
examples of varieties with infinitely generated Cox ring, studied by Mukai and
Hassett-Tschinkel. These are the first examples known where the series is not
rational. We also compute the series for Del Pezzo surfaces.
