欧拉数 (Eulerian Number)

\[\newcommand{\e}{\mathrm{e}} \begin{aligned} \left\langle\begin{matrix}n \\ i\end{matrix} \right\rangle&=\sum_{j=i}^{n-1}(-1)^{j-i} \binom ji \times n ^ {n - j}\\ &=\sum_{j=i}^{n-1}(-1)^{j-i}\binom ji \times n! [x^n] \sum_{k=0}^{n-j}\e^{kx}(-1)^k \binom{n-j}k \\ &=\sum_{j=i}^{n-1}(-1)^{j-i}\binom ji \sum_{k=0}^{n-j} k^n (-1)^{n-j-k} \binom{n-j}k\\ &=\sum_{k=1}^{n-i} k^n (-1)^{n-i-k} \sum_{j=0}^{n-k} \binom{n-j}k \binom ji \\ &=\sum_{k=1}^{n-i} k^n (-1)^{n-i-k}\binom{n+1}{k+i+1} \\ &=\sum_{k=1}^{i+1} k^n (-1)^{n-(n-1-i)-k} \binom{n+1}{k+(n-1-i)+1} \\ &=\sum_{k=1}^{i+1} k^n (-1)^{i+1-k} \binom{n+1}{i+1-k} \\ &=\sum_{k=0}^i (k+1)^n (-1)^{i-k} \binom{n+1}{i-k} \\ &=\sum_{k=0}^i (i+1-k)^n (-1)^k \binom{n+1}{k} \end{aligned} \]