Vieta's triangular array and a related family of polynomials

Abstract

If n≥1, let the nth row of an infinite triangular array consist of entries B(n,j)=nn−j(jn−j), where 0≤j≤[12n].We develop some properties of this array, which was discovered by Vieta. In addition, we prove some irreducibility properties of the family of polynomials Vn(x)=∑j=0[12n](−1)jB(n,j)xn−2j.These polynomials, which we call Vieta polynomials, are related to Chebychev polynomials of the first kind.