contest2590-c-solution

Contest2590 C Solution

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\[\begin{aligned} \sum_{i=0}^ni^{\overline m}\binom n i &=\sum_{i=0}^n\frac{(i+m-1)!}{(i-1)!}\binom n i\\ &=m!\sum_{i=0}^n\binom{i+m-1}{m}\binom n i\\ \end{aligned}\]

考虑范德蒙德卷积，将 \(\binom{i+m-1}{m}\) 展开。

\[\begin{aligned} \sum_{i=0}^ni^{\overline m}\binom n i &=m!\sum_{i=0}^n\binom n i\sum_{j=0}^m\binom i j\binom{m-1}{m-j}\\ &=m!\sum_{j=0}^m\binom{m-1}{m-j}\sum_{i=0}^n\binom n i\binom i j\\ &=m!\sum_{j=0}^m\binom{m-1}{m-j}\sum_{i=0}^n\frac{n!i!}{i!j!(n-i)!(i-j)!}\\ &=m!\sum_{j=0}^m\binom{m-1}{m-j}\sum_{i=0}^n\frac{n!(n-j)!}{j!(n-i)!(i-j)!(n-j)!}\\ &=m!\sum_{j=0}^m\binom{m-1}{m-j}\binom n j\sum_{i=0}^n\binom{n-j}{n-i}\\ &=m!\sum_{j=0}^m\binom{m-1}{m-j}\binom n j2^{n-j}\\ \end{aligned}\]

\(\mathcal O(m)\) 计算即可。