CF932E Team Work

题意

\[\sum_{i=1}^n\binom{n}{i}i^k \]

\(n\le 10^9,k\le 5000\)

思路

\[\sum_{i=0}^n\binom{n}{i}i^k \]

\[\sum_{i=0}^n\binom{n}{i}\sum_{j=1}^k \begin{Bmatrix}k\\j\\\end{Bmatrix}i^{\underline{j}} \]

\[\sum_{j=1}^k \begin{Bmatrix}k\\j\\\end{Bmatrix}\sum_{i=0}^n\binom{n}{i}i^{\underline{j}} \]

\[\sum_{j=1}^k \begin{Bmatrix}k\\j\\\end{Bmatrix}j!\sum_{i=0}^n \binom{n}{i}\binom{i}{j} \]

\[\sum_{j=1}^k \begin{Bmatrix}k\\j\\\end{Bmatrix}j!\binom{n}{j}\sum_{i=0}^n \binom{n-j}{i-j} \]

\[\sum_{j=1}^k \begin{Bmatrix}k\\j\\\end{Bmatrix}j!\binom{n}{j}\sum_{i=0}^{n-j}\binom{n-j}{i} \]

\[\sum_{j=1}^k \begin{Bmatrix}k\\j\\\end{Bmatrix}j!\binom{n}{j}2^{n-j} \]