Evaluation of a family of binomial determinants

Abstract

Motivated by a recent work about finite sequences where the n-th term is bounded by n^2, some classes of determinants are evaluated such as the (n â 2) Ã (n â 2) determinant â_n=\det [ (x_i+j \choose i-1)] for n \geq 1, where n, k, h, i, j are integers, (x_k) is a sequence of indeterminates over C and ( A \choose B ) is the usual binomial coefficient. It is proven that D_n=1 and â_n=(-1)^{ (n-2)(n-3)/2}.