Integral polynomial sequences arising from matrix powers of order 2

In this paper we study a recursive system of integers {chi (n, k) : n > k >= 0}; {chi(n + 2, k) = chi(n + 1, k) - chi(n + 1, k - 1) +2 chi (n, k - 1) chi(k + 1, k) = -chi(k, k - 1) which is uniquely determined by the initial values {chi (n, 0)}(n=1)(infinity). We show under the constant initial dates chi (n, 0) = chi (1, 0) for all n that the polynomial chi(n) (x) = Sigma(n-1)(k=0) x (n, k)x(k) of degree n - 1 is (anti) palindromic. Several explicit formulae for x (n, k) via Vander-monde matrix, mirrored Gamma-matrix, weighed Delannoy number, Riordan array, hypergeometric function, Jacobi polynomial, and some combinatorial identities are derived. (C) 2012 Elsevier Inc. All rights reserved.