  We study the enumeration of off-diagonally symmetric domino tilings of
odd-order Aztec diamonds in two directions: (1) with one boundary defect, and
(2) with maximally-many zeroes on the diagonal. In the first direction, we
prove a symmetry property which states that the numbers of off-diagonally
symmetric domino tilings of the Aztec diamond of order $2n-1$ are equal when
the boundary defect is at the $k$th position and the $(2n-k)$th position on the
boundary, respectively. This symmetry property proves a special case of a
recent conjecture by Behrend, Fischer, and Koutschan.
  In the second direction, a Pfaffian formula is obtained for the number of
"nearly" off-diagonally symmetric domino tilings of odd-order Aztec diamonds,
where the entries of the Pfaffian satisfy a simple recurrence relation. The
numbers of domino tilings mentioned in the above two directions do not seem to
have a simple product formula, but we show that these numbers satisfy simple
matrix equations in which the entries of the matrix are given by Delannoy
numbers. The proof of these results involves the method of non-intersecting
lattice paths and a modification of Stembridge's Pfaffian formula for families
of non-intersecting lattice paths. Finally, we propose conjectures concerning
the log-concavity and asymptotic behavior of the number of off-diagonally
symmetric domino tilings of odd-order Aztec diamonds.
