Let $A_n=\sum_{n=0}^{N-1} \binom{2n}{n}^2$

Question

Let $A_n=\sum_{n=0}^{N-1} \binom{2n}{n}^2$ and we have the following expresions:

$\sum_{n=0}^{N-1} \binom{2n}{n} \binom{2n}{n+1}, \ \cdots (1) \\ \sum_{n=0}^{N-1} \binom{2n}{n+1}^2, \ \cdots (2)$

My target is to write $(1)$ and $(2)$ in terms of $A_n$.

I tried but failed.

Can someone help me with any kind of manipulation ?

Answer 1

We have $$ \binom{2n}{n+1} = \frac{2n}{(n-1)!(n+1)!} = \frac{2n}{n!n!}\frac{n}{n+1} = \binom{2n}{n}\frac{n}{n+1} $$ so $(1)$ rewrites $$ A_n-\sum_{n=0}^{N-1}\binom{2n}{n}^2\frac{1}{n+1} $$ and $(2)$ rewrites $$ A_n-2\sum_{n=0}^{N-1}\binom{2n}{n}^2\frac{1}{n+1}+\sum_{n=0}^{N-1}\binom{2n}{n}^2\frac{1}{(n+1)^2} $$