  The Al-Salam-Chihara polynomials are an important family of orthogonal
polynomials in one variable $x$ depending on 3 parameters $\alpha$, $\beta$ and
$q$. They are closely connected to a model from statistical mechanics called
the partially asymmetric simple exclusion process (PASEP) and they can be
obtained as a specialization of the Askey-Wilson polynomials. We give two
different combinatorial formulas for the coefficients of the (transformed)
Al-Salam-Chihara polynomials. Our formulas make manifest the fact that the
coefficients are polynomials in $\alpha$, $\beta$ and $q$ with positive
coefficients.
