组合数前缀和

列前缀和

$$\sum _{i=0}^n(\genfrac{}{}{0ex}{}{i}{m})=(\genfrac{}{}{0ex}{}{n+1}{m+1})$$

考虑加上 $(\genfrac{}{}{0ex}{}{0}{m+1})$，然后用 $(\genfrac{}{}{0ex}{}{n}{m})=(\genfrac{}{}{0ex}{}{n-1}{m-1})+(\genfrac{}{}{0ex}{}{n-1}{m})$ 来化。

习题：[ABC154F] Many Many Paths

行前缀和

记 $f(n,k)=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{n}{i})$，我们有：

$$f(n+1,k)=2f(n,k)-(\genfrac{}{}{0ex}{}{n}{k})$$

$$f(n,k+1)=f(n,k)+(\genfrac{}{}{0ex}{}{n}{k+1})$$

莫队即可做到 $\mathcal{O}(q\sqrt{n})$。当然也有不用排序的情况。

习题：CF1194F Crossword Expert