A generating function for Blattner's formula

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.