OIMU 1998 Problem 5

A sequence of polynomials $\{f_n\}_{n=0}^{\infty}$ is defined recursively by $f_0(x)=1$ , $f_1(x)=1+x$ , and $$(k+1)f_{k+1}(x)-(x+1)f_k(x)+(x-k)f_{k-1}(x)=0, k=1,2,\ldots$$ Prove that $f_k(k)=2^k$ for all $k\geq 0$ .