Is there an expression for the sum of $\binom nr^2$ for each $n$? Is there a standard expression for
$$\sum_{r=0}^{n}\binom nr^2$$
 A: Yes, $\sum_{r=0}^n\binom{n}{r}^2=\binom{2n}{n}$.
The proof is simple:
There are $\binom{2n}{n}$ paths from $(0,0)$ to $(n,n)$ going right or up.
On the other hand each path crosses the diagonal $(x,n-x)$ exactly once. 
How many paths cross at $(r,n-r)$? Exactly $\binom{n}{r}\binom{n}{n-r}=\binom{n}{r}^2$.
Therefore $\sum\limits_{r=0}^n\binom{n}{r}^2=\binom{2n}{n}$
A: In fact, we have something slightly more general. Let $a,k$ be integers less than $n$. Then we have $$\sum_{i=0}^k \binom{a}{i} \binom{n-a}{k-i} = \binom{n}{k}.$$
The easiest way to see why this is true is to ask how many ways of picking $k$ elements from a set of size $n$ are there? On one hand, the answer is obviously $\binom{n}{k}$. On the other hand, we can split our set into two halves - one of size $a$, and one of size $n-a$ (how we do this is irrelevant). Then to get $k$ elements, you have to choose a certain amount, $i$, from the first set, and then $n-i$ from the second set. But we can do this for all $i$! Summing over $i$, we see that the two sides are equal. 
A: Use Chu-Vandermonde's identity :
$$
\sum_r {n \choose r}^2 = \sum_r {n \choose r} {n \choose n-r}={2n \choose n}
$$
