  Let G be a connected, semisimple Lie group with finite center and let K be a
maximal compact subgroup. We investigate a method to compute multiplicities of
K-types in the discrete series using a rational expression for a generating
function obtained from Blattner's formula. This expression involves a product
with a character of an irreducible finite dimensional representation of K and
is valid for any discrete series system. Other results include a new proof of a
symmetry of Blattner's formula, and a positivity result for certain low rank
examples. We consider in detail the situation for G of type split G_2. The
motivation for this work came from an attempt to understand pictures coming
from Blattner's formula, some of which we include in the paper.
