51nod1778
题意
\[\sum_{i=0}^{n}\binom{n}{i}(i^k-(n-i)^k)^2 \]
做法
\[\begin{aligned}
\sum\limits_{i=0}^{n}\binom{n}{i}(i^k-(n-i)^k)^2&=\sum\limits_{b=0}^{m-1}{n\%m\choose b}\sum\limits_{a=0}^{\frac{n-b}{m}}{n\choose am+b}(b^k-(n-b)^k)^2\\
&=\sum\limits_{b=0}^{min(m-1,n\%m)}{n\%m\choose b}\sum\limits_{a=0}^{\frac{n-b}{m}}{n\choose am+b}(b^k-(n-b)^k)^2\\
&=\sum\limits_{a=0}^{\frac{n}{m}}{\frac{n}{m}\choose a}\sum\limits_{b=0}^{n\%m}{n}{n\%m \choose b}(b^k-(n-b)^k)\\
\end{aligned}\]
