cf932e-solution

CF932E Solution

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\(\begin{aligned} \sum_{i=0}^n\binom n i i^k &=\sum_{i=0}^n\binom n i\sum_{j=1}^k{k\brace j}i^{\underline j}\\ &=\sum_{j=1}^k{k\brace j}\sum_{i=0}^n\binom n ii^{\underline j}\\ &=\sum_{j=1}^k{k\brace j}\sum_{i=0}^n\frac{n!}{i!(n-i)!}\frac{i!}{(i-j)!}\\ &=\sum_{j=1}^k{k\brace j}j!\sum_{i=0}^n\frac{n!}{i!(n-i)!}\frac{i!}{j!(i-j)!}\\ &=\sum_{j=1}^k{k\brace j}j!\sum_{i=0}^n\frac{n!(n-j)!}{(n-i)!j!(i-j)!(n-j)!}\\ &=\sum_{j=1}^k{k\brace j}j!\sum_{i=0}^n\binom n j\binom{n-j}{i-j}\\ &=\sum_{j=1}^k{k\brace j}j!\binom n j\sum_{i=0}^{n-j}\binom{n-j}{i}\\ &=\sum_{j=1}^k{k\brace j}n^{\underline j}2^{n-j} \end{aligned}\)

\(\mathcal O(k^2)\) 预处理 \(\displaystyle{k\brace j}\) 即可。