i am trying to find the expected value of $$a_{n}=\frac{1+i}{2+n}$$ where the pmf is given by $$\binom {n}{i}\frac{(2i-1)!!((2(n-i)-1)!!}{(2n)!!}$$
$$E(a_{n})=\sum\limits_{i=0}^{n}\frac{1+i}{2+n}\binom {n}{i}\frac{(2i-1)!!((2(n-i)-1)!!}{(2n)!!}$$ $$=\sum\limits_{i=0}^{n}\frac{1+i}{2+n}\frac{1}{4^{n}}\binom{2i}{i}\binom{2(n-i)}{n-i}$$ this is what i did so far and got stuck
Hint:$$\sum_{i+j=n}i\binom{2i}i\binom{2j}j=\sum_{i+j=n}j\binom{2i}i\binom{2j}j$$and:$$\sum_{i+j=n}i\binom{2i}i\binom{2j}j+\sum_{i+j=n}j\binom{2i}i\binom{2j}j=n\sum_{i+j=n}\binom{2i}i\binom{2j}j$$
So that $$\sum_{i+j=n}i\binom{2i}i\binom{2j}j=\frac12n\sum_{i+j=n}\binom{2i}i\binom{2j}j$$