TITLE: Evaluate a finite sum with four factorials
QUESTION [9 upvotes]: Given positive integers $k, m, n$ such that $1 \leq  k \leq  m \leq  n$. Evaluate 
$$ \sum^{n}_{i\mathop{=}0}\frac{1}{n+k+i}\cdot\frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$

Any hints? I'm stuck on this one.

REPLY [0 votes]: I think you can find this
 $$\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$
can see  http://www.artofproblemsolving.com/Forum/viewtopic.php?p=239364&sid=2fbf367cb9fab8240df03e632a41085a#p239364
{"set_name": "stack_exchange", "score": 9, "question_id": 686356}