# 生成函数trick

### 生成函数trick

#### 牛顿二项式定理

${(1+x)^m}=\sum_{i\geq 0} {m\choose i}x^i$

\begin{aligned} (1-x)^{-n} &=\sum_{i \geq 0}\left(\begin{array}{c} -n \\ i \end{array}\right)(-x)^{i} \\ &=\sum_{i\geq 0}\frac{-n^{\underline{i}}}{i!} (-1)^i x^i\\ &=\sum_{i\geq 0}\frac{n^{\overline{i}}}{i!}x^i \\&=\sum_{i\geq 0}\frac{(n+i-1)^{\underline{i}}}{i!}x^i \\&=\sum_{i \geq 0}\left(\begin{array}{c} n-1+i \\ i \end{array}\right) x^{i} \end{aligned}

#### 组合数恒等式：顶变

$\sum_{i=0}^n {i\choose m}={n+1\choose m+1}$

$\sum_{i=0}^n {m+i-1\choose i}={m+n\choose m}={m+n\choose n}$

$\sum_{i=0}^n {m+i-1\choose i}= \sum_{i=0}^n {m+i-1\choose m-1} =\sum_{i=0}^{m-1+n} {i-1\choose m-1}={m+n\choose m}$

#### 斯特林数拆下降幂

$m^{k}=\sum_{i=0}^{k}\left\{\begin{array}{l} k \\ i \end{array}\right\} i!{m\choose i}=\sum_{i=0}^{k}\left\{\begin{array}{l} k \\ i \end{array}\right\} m^{\underline{i}}$

#### 小等式

${n\choose k}{k\choose i}={n\choose i}{n-i\choose k-i}$

${n\choose i}{n\choose k}={n\choose i+k}{i+k\choose i}$

posted @ 2020-11-02 17:30  lcyfrog  阅读(244)  评论(0编辑  收藏  举报