Evaluate in closed form

$$\sum_{k=1}^{n}\binom{n}{k} k^{2}$$

Note:

$$\binom{n}{k}=\frac{n(n-1) \cdots(n-k+1)}{1 \cdot 2 \cdots k}$$Please provide the value when \(n = 3\). 

Let's think step by step and output the final answer within \boxed{}.
user