Is there a closed form to this summation?

Question

I cannot get a closed form for $\sum\limits_{r=0}^{m} \frac{(m+r) !}{(m-r)! (2 r)!}$ ‘ Does anyone have any idea on what it is?

Answer 1

This sum is the Fibonacci number $F_{2 m+1}$.

Answer 2

Hint: $$\frac{(m+r)!}{(m-r)!(2r)!}= {{m+r} \choose {2r}}$$