「省选联考 2020 A 卷」组合数问题

随手录一道普通幂转下降幂板子题

一句话题意：给定 \(n, x, p, m\) 和一个 \(m\) 次多项式 \(f(k)\), \(1 \leq n, x, p \leq 10^9\), \(0 \leq a_i \leq 10^9\), \(1 \leq m \leq \min(n, 1000)\) ，求

\[\left( \sum_{k = 0}^n f(k) \times x^k \times \binom{n}{k} \right)\mod p \]

为简洁，省略取模

\[\begin{aligned} \sum_{k = 0}^n f(k) \times x^k \times \binom{n}{k} &= \sum_{i = 0}^m a_i \sum_{k = 0}^n k^i \times x^k \times \binom{n}{k} \\ &= \sum_{i = 0}^m a_i \sum_{k = 0}^n \sum_{j = 0}^i {i \brace j} k^{\underline{j}} \times x^k \times \binom{n}{k} \\ &= \sum_{i = 0}^m a_i \sum_{k = 0}^n \sum_{j = 0}^i {i \brace j} n^{\underline{j}} \times x^k \times \binom{n - j}{k - j} \\ &= \sum_{i = 0}^m a_i \sum_{j = 0}^i {i \brace j} n^{\underline{j}} \sum_{k = 0}^n x^k \times \binom{n - j}{k - j} \\ &= \sum_{i = 0}^m a_i \sum_{j = 0}^i {i \brace j} n^{\underline{j}} \times x^j \sum_{k = 0}^n x^{k - j} \times 1^j \times \binom{n - j}{k - j}\\ &= \sum_{i = 0}^m a_i \sum_{j = 0}^i {i \brace j} n^{\underline{j}} \times x^j \times (x + 1)^{n - j} \end{aligned} \]

时间复杂度 \(\text{O}(m^2 \log_2{n})\)