TITLE: Binomial summation $\sum\binom nkk^2$
QUESTION [7 upvotes]: Question:

Find $$\sum_{k=0}^n\binom nkk^2$$

I know how to do $$\sum_{k=0}^n\binom nkk=n\space2^{n-1}$$
I tried applying same thing and reached $$\sum_{k=0}^n\binom nkk^2=n\sum_{k=0}^n\binom{n-1}{k-1}k$$
How to proceed?

REPLY [9 votes]: There are $n$ people.  You  want to select some of them to form a committee.  Amongst the committee members, you choose a person to be the president.  You also pick from the committee members one person (not necessarily different from the president) to be the treasury.  Thus, if the committee consists of $k$ members, then the number of ways to pick the president and the treasury is $k^2$.  That is, the total number of ways to pick the committee and the two extra positions is $$\sum_{k=0}^n\,k^2\,\binom{n}{k}\,.$$
On the other hand, you can pick the president and the treasury first.  In the case that the president and the treasury are different, there are $n$ ways to choose the president, $n-1$ ways to pick the treasury, and $2^{n-2}$ ways to select the remaining committee members, whence there are $n(n-1)\,2^{n-2}$ ways in total.  If the president is the same as the treasury, then there are $n$ ways to choose this person for two positions, and $2^{n-1}$ ways to complete the committee, making $n\,2^{n-1}$ ways in total.  
Combining the two ways of counting, we conclude that
$$\sum_{k=0}^n\,k^2\,\binom{n}{k}=n(n-1)\,2^{n-2}+n\,2^{n-1}=n(n+1)\,2^{n-2}\,.$$
The claim is now proven.