Determinants of matrices related to the Pascal triangle (Q1396430)

A generalized Pascal triangle \(P\) is an infinite matrix \((p_{i,j})_{i,j\geq 0}\) whose entries satisfy the recursion \(p_{i,j}=p_{i-1,j} +p_{i,j-1}\) for all \(i\geq 1\) and \(j\geq 1\). Thus, \(P\) is determined by its first row and column, i.e., by the two sequences \((p_{0,j})_{j\geq 0}\) and \((p_{i,0})_{i\geq 0}\). When the above two sequences are constant \(1\), the resulting matrix \(P\) is just the Pascal triangle. This paper contains several interesting results and conjectures about the behavior of the sequence of subdeterminants \((P(n))_{n\geq 1}\) formed by the first \(n\) rows and columns of \(P\) provided that the starting sequences \((p_{0,j})_{j\geq 0}\) and \((p_{i,0})_{i\geq 0}\) satisfy certain properties. The most interesting conjecture is Conjecture 3.3 which asserts that if the above two sequences satisfy linear recurrences, then the sequence \((P(n))_{n\geq 1}\) also satisfies a linear recurrence. The most difficult result of the paper is Theorem 3.1, which shows that the above conjecture holds if the two starting sequences are linearly recurrent of order \(2\). The paper also contains results about the case of symmetric matrices (when \(p_{0,n}=p_{n,0}\) for all \(n\geq 0\)), skew-symmetric matrices (when \(p_{0,n}=-p_{n,0}\) for all \(n\geq 0\)), and several examples.