Question 196
Let $k$ be a positive integer. Prove that there exist POSITIVE integers $a_{0}, \ldots, a_{k}$ such that for all integers $x \geq 0$,
$$
x^{k}=a_{0}\left(\begin{array}{l}
x \\
k
\end{array}\right)+a_{1}\left(\begin{array}{c}
x+1 \\
k
\end{array}\right)+\cdots+a_{k-1}\left(\begin{array}{c}
x+k-1 \\
k
\end{array}\right) \text{.}
$$
$$
x^{k}=a_{0}\left(\begin{array}{l}
x \\
k
\end{array}\right)+a_{1}\left(\begin{array}{c}
x+1 \\
k
\end{array}\right)+\cdots+a_{k-1}\left(\begin{array}{c}
x+k-1 \\
k
\end{array}\right) \text{.}
$$