Vieta's triangular array and a related family of polynomials (Q1175429)

Let \(B(n,j)={n\over n-j}{n-j\choose j}\) for \(n\geq 1\) and \(0\leq j\leq[{n\over 2}]\). The author establishes several properties of these numbers, and proves some irreducibility properties of the sequence \[ 2\;T_ n\left({x \over 2}\right)=\sum_{j=0}^{[n/2]} (-1)^ j B(n,j) x^{n-2j}, \] where \(\{T_ n(x)\}\) are the Chebyshev polynomials of the first kind.