p4345-solution

P4345 Solution

link

\(p=2333\)

\(f(n,k) (all \bmod p \;below)\)

\(\displaystyle=\sum\limits_{i=0}^k{\text{C}_n^i}\)

\(\displaystyle=\sum\limits_{i=0}^k{\text{C}_{n/p}^{i/p} \times \text{C}_{n\%p}^{i\%p}}\)

\(\displaystyle=\sum\limits_{i=0}^{p-1}({\text{C}_{n\%p}^i \times \sum\limits_{j=0}^{k/p-1}\text{C}_{n/p}^j)}+\text{C}_{n/p}^{k/p}\sum\limits_{i=0}^{k\%p}\text{C}_{n\%p}^i\)

\(\displaystyle=\sum\limits_{i=0}^{p-1}{\text{C}_{n\%p}^i \times f(n/p,k/p-1)}+\text{C}_{n/p}^{k/p}f(n\%p,k\%p)\)

\(= f(n\%p,p-1) \times f(n/p,k/p-1)+\text{C}_{n/p}^{k/p}f(n\%p,k\%p)\)

预处理C+小数据f+Lucas递归求解即可。