\[\begin{align*} &\sum_{i=1}^n i(n-i)\dbinom ni \\ =&n\sum_{i=1}^n i\dbinom ni-\sum_{i=1}^n i^2\dbinom ni \\ =&n^2\sum_{i=1}^n \dbinom{n-1}{i-1}-n\sum_{i=1}^n i\dbinom{n-1}{i-1} \\ =&n\sum_{i=1}^n (n-i)\dbinom{n-1}{n-i} \\ =&n(n-1)\sum_{i=1}^n \dbinom{n-2}{n-i-1} \\ =&n(n-1)\sum_{i=1}^n \dbinom{n-2}{i-1} \\ =&n(n-1)2^{n-2} \end{align*}\] 标签：ni,begin,dbinom,align,草稿纸,sum From： https://www.cnblogs.com/JerryTcl/p/16909123.html