OIMU 2006 Problem 6

Let $x_0(t)=1$ , $x_{k+1}(t)=(1+t^{k+1})x_k(t)$ for all $k\geq 0$ ; $y_{n,0}(t)=1$ , $y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)$ for all $n\geq 0$ , $1\leq k \leq n$ .

Prove that $\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}$ for all $n\geq 1$ .