Example 5. Find the value of $\sum_{k=1}^{n} k^{2} C_{n}^{k}$. (3rd Putnam Mathematical Competition, USA)
$\begin{array}{l}\text { Solve } \sum_{k=1}^{n} k^{2} C_{n}^{k}=\sum_{k=1}^{n} k \cdot k C_{n}^{k} \\ =\sum_{k=1}^{n} k \cdot n C_{n-1}^{k-1}=n \sum_{k=1}^{n}[(k-1)+1] C_{n-1}^{k-1} \\ =n \sum_{k=1}^{n} C_{n-1}^{k-1}+n \sum_{k=2}^{n}(n-1) C_{n-2}^{k-2} \\ =n \sum_{k=1}^{n} C_{n-1}^{k-1}+n(n-1) \sum_{k=2}^{n} C_{n-2}^{k-2} \\ =n(1+1)^{n-1}+n(n-1)(1+1)^{n-2} \\ =n \cdot 2^{n-1}+n(n-1) \cdot 2^{n-2} \\ =n(n+1) \cdot 2^{n-2} .\end{array}$