Off-diagonally symmetric domino tilings of the Aztec diamond were introduced by the author in [Electron. J. Combin. 30 (2023), no. 4, Paper No. 4.20]. We further investigate this symmetry class in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetric 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 symmetric property proves a special case of a recent conjecture by Behrend, Fischer and Koutschan. In the second direction, a Pfaffian formula is obtained, 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 product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers.
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