Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers

The Aztec diamond of order n is a certain configuration of 2n(n+1) unit squares. We give a new proof of the fact that the number Πn of tilings of the Aztec diamond of order n with dominoes equals 2n(n+1)/2. We determine a sign-nonsingular matrix of order n(n+1) whose determinant gives Πn. We reduce the calculation of this determinant to that of a Hankel matrix of order n whose entries are large Schröder numbers. To calculate that determinant we make use of the J-fraction expansion of the generating function of the Schröder numbers.