Is there a closed form for sum of chooses? Is there a closed form for $\sum_{i=a}^{b}\frac{{{i} \choose{n}}^2}{i}$?
 A: This is already closed-form (in a certain sense): What you have given is a finite sum, where each terms in the sum is given by a finite number of arithmetic operations.  Assuming you have $n \leqslant a \leqslant b$, you have:
$$\begin{equation} \begin{aligned}
S(n,a,b) \equiv \sum_{i=a}^b \frac{1}{i} {i \choose n}^2
&= \sum_{i=a}^b \frac{1}{i} \cdot \frac{i!}{n!(n-i)!} \cdot \frac{i!}{n!(n-i)!}.
\end{aligned} \end{equation}$$
Each factorial is a finite product of integers, and so each term in the sum can be obtained with a finite number of arithmetic operations on the integers.  Since your function is a finite sum of these terms, it can be calculated in a finite number of arithmetic operations.  Hence, it is already written as a closed-form expression, or what Chow (1999) calls an "elementary function".
From the comments below this answer (between whuber and I) you will see that there are some different ideas of the meaning of "closed form".  I have interpreted this in the above sense.  If you meant it in some other way, then this would lead to a different answer.
