Denominators for One Variable Poincar\'e Series of Generic Matrices

We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we presume but have not been able to prove to be in lowest terms, in the case of $3\times3$ matrices; and a conjectured denominator in the general case.