Higher Factorial-Moment Identity

Higher Factorial-Moment Identity

## Formula

$$\sum_{r=0}^{n}r(r-1)\cdots(r-k+1){}^nC_r=n(n-1)\cdots(n-k+1)2^{n-k}$$

Therefore:

$$\sum_{r=0}^{n}{}^rP_k{}^nC_r={}^{n}P_k2^{n-k}$$

## Usage

- Used in advanced summation problems involving powers or falling factorials of the index.