范德蒙德卷积

范德蒙德卷积

(displaystyle C _{x+y}^{k}= sum _{i=0}^{k}C_{x}^{i} imes C_{y}^{k-i});

证明：过菜已隐藏。

意会：在(x+y)中选k个，相当于在(x)和(y)中分别选(i)个和(k-i)个；

所以…… (C _{x+y}^{k}= sum _{i=0}^{k}C_{x}^{a+i} imes C_{y}^{k-a-i})?

(eg:)

求 (displaystyle sum _{i=0}^{x} C_{x}^{i} C_{y}^{z+i} ~mod~998244353); ((0<=x,y,z<=1000000,且保证y>=z))

(displaystyle 原式=sum _{i=0}^{x}C_{x}^{x-i}C_{y}^{z+i}=C_{x+y}^{x+z})

(ex:)

1. (displaystyle C_{2n}^{n-1}= sum _{i=0}^{n-1}C_{n}^{i}C_{n}^{i-1}= sum _{i=1}^{n}C_{n}^{i}C_{n}^{i+1})

   证：

   (displaystyle C_{2n}^{n}= sum _{i=0}^{n-1}C_{n}^{i}C_{n}^{n-(n-1-i)}= sum _{i=0}^{n-1}C_{n}^{i}C_{n}^{i+1}= sum _{i=1}^{n} C_{n}^{i}C_{n}^{i-1})

   毕；

2. (displaystyle C_{2n}^{n}= sum _{i=0}^{n}(C_{n}^{i})^2)

3. (C_{n+m}^{m}= sum _{i=0}^{m}C_{n}^{i}C_{m}^{i})