Summary
Summary: Define an infinite lower triangular matrix $D(e,h) = [d_{n,k}]_{n,k \ge 0}$ by the recurrence $d_{0,0} = d_{1,0} = d_{1,1} = 1, d_{n,k} = d_{n-1,k-1} + ed_{n-1,k} + hd_{n-2,k-1}$, where $e, h$ are both nonnegative and $d_{n,k} = 0$ unless $n \ge k \ge 0$. We call $D(e, h)$ the Delannoy-like triangle. The entries $d_{n,k}$ count lattice paths from (0, 0) to $(n - k, k)$ using the steps (0, 1), (1, 0) and (1, 1) with assigned weights 1, $e$, and $h$. Some well-known combinatorial triangles are such matrices, including the Pascal triangle $D$(1, 0), the Fibonacci triangle $D$(0, 1), and the Delannoy triangle $D$(1, 1). Futhermore, the Schröder triangle and Catalan triangle also arise as inverses of Delannoy-like triangles. Here we investigate the total positivity of Delannoy-like triangles. In addition, we show that each row and diagonal of Delannoy-like triangles are all PF sequences.
