Integral polynomial sequences arising from matrix powers of order 2

In this paper we study a recursive system of integers {χ(n,k):n>k⩾0};χ(n+2,k)=χ(n+1,k)-χ(n+1,k-1)+2χ(n,k-1)χ(k+1,k)=-χ(k,k-1)which is uniquely determined by the initial values {χ(n,0)}n=1∞. We show under the constant initial dates χ(n,0)=χ(1,0) for all n that the polynomial χn(x)=∑k=0n-1χ(n,k)xk of degree n-1 is (anti) palindromic. Several explicit formulae for χ(n,k) via Vandermonde matrix, mirrored Γ-matrix, weighed Delannoy number, Riordan array, hypergeometric function, Jacobi polynomial, and some combinatorial identities are derived.