Is it true that $\sum_{k=0}^{n}k\cdot \left(\begin{array}{l}{n}\\{k}\end{array}\right)=O\left(2 ^ {n\log _{3}n}\right)?$
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?