Matrices Related to the Pascal Triangle

for 0 ≤ i, j ∈ N. The matrix P is hence the famous Pascal triangle yielding the binomial coefficients and can be recursively constructed by the rules p0,i = pi,0 = 1 for i ≥ 0 and pi,j = pi−1,j + pi,j−1 for 1 ≤ i, j. In this paper we are interested in (sequences of determinants of finite) matrices related to P . The present section deals with some minors (determinants of submatrices) of the above Pascal triangle P , perhaps slightly perturbed. Sections 2-6 are devoted to the study of matrices satisfying the Pascal recursion rule mi,j = mi−1,j +mi,j−1 for 1 ≤ i, j < n (with various choices for the first row m0,j and column mi,0). Our main result is the experimental observation (Conjecture 3.3 and Remarks 3.4) that given such an infinite matrix whose first row and column satisfy linear recursions (like for instance the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, . . .), then the determinants of a suitable sequence of submatrices seem also to satisfy a linear recursion. We give a proof if all linear recursions are of length at most 2 (Theorem 3.1). Section 7 is seemingly unrelated since it deals with matrices which are “periodic” along strips parallel to the diagonal. If such a matrix consists only of a finite number of such strips, then an appropriate sequence of determinants satisfies a linear recursion (Theorem 7.1). Section 8 is an application of section 7. It deals with matrices which are periodic on the diagonal and off-diagonal coefficients satisfy a different kind of Pascal-like relation. We come now back to the Pascal triangle P with coefficients pi,j = (i+j i )