2025.03.19 CW 模拟赛 B. 组合数问题

B. 组合数问题

考场乱搞的.

思路

首先 \(f(n, 0) = 1\), 然后

\[\begin{align*} f(n, k) & = \sum_{x_1 = 0}^n \binom{n}{x_1} \sum_{x_2 = 0}^{x_1} \binom{x_1}{x_2} \dots \sum_{x_k}^{x_{k - 1}} \binom{x_{k - 1}}{x_k} \\ & = \sum_{x_1 = 0}^n \binom{n}{x_1} \sum_{x_2 = 0}^{x_1} \binom{x_1}{x_2} \dots \sum_{x_{k - 1} = 0}^{x_{k - 2}} \binom{x_{k - 2}}{x_{k - 1}} 2^{x_{k - 1}} \end{align*} \]

根据二项式定理, \(\displaystyle \sum_{x = 0}^n \binom{n}{x} 2^x = \sum_{x = 0}^n \binom{n}{x} 1^{n - x} 2^x = 3^n\), 一层一层套下去, 可以得到 \(f(n, k) = (k + 1)^n\).

题目所求即为 \(\displaystyle \sum_{i = 1}^n (i + 1)^{(i + 1)^n}\), 快速幂即可.