Aztec diamonds and digraphs, and Hankel determinants of Schrِder numbers

The Aztec diamond of order n is a certain configuration of 2 n ( n + 1 ) unit squares. We give a new proof of the fact that the number Π n of tilings of the Aztec diamond of order n with dominoes equals 2 n ( n + 1 ) / 2 . We determine a sign-nonsingular matrix of order n ( n + 1 ) whose determinant gives Π n . 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.