calculate $\sum_{1\le i\le r}\frac{i+1} { r+1}{2r-i\choose r-i}{s+i-2\choose i}+\frac{1}{r+1}{2r\choose r}$? prove the following equation :
$$\sum_{1\le i\le r}\frac{i+1} { r+1}{2r-i\choose r-i}{s+i-2\choose i}+\frac{1}{r+1}{2r\choose r}={s+2r-1\choose r} - {s+2r-1\choose r-1}$$

I use two known properties of binomial coefficients:  $${n+k \choose k} = (-1)^k {-n-1 \choose k}\tag{$i$}$$ and $$\sum_{i}{a \choose i} {b\choose c-i} = {a+b \choose c}\tag{$ii$}.$$
First,
$$
\sum_{0\le i\le r}\frac{r-i} {r+1}{2r-i\choose r-i}{s+i-2\choose i} \overset{(i)}{=} 
(-1)^r \sum_{0\le i\le r-1}\frac{r-i} {r+1}{-r-1\choose r-i}{-s+1\choose i}  \\
= (-1)^{r+1}\sum_{0\le i\le r-1}{-r-2\choose r-i-1}{-s+1\choose i}  \\\overset{(ii)}{=}  
(-1)^{r+1}{-r-s-1\choose r-1} \overset{(i)}{=}  {2r+s-1\choose r-1}.\tag{1}
$$
Similarly, 
$$
\sum_{0\le i\le r}{2r-i\choose r-i}{s+i-2\choose i} \overset{(i)}{=}  
(-1)^r \sum_{0\le i\le r}{-r-1\choose r-i}{-s+1\choose i}  \\
\overset{(ii)}{=}  (-1)^r{-r-s\choose r} \overset{(i)}{=}  {2r+s-1\choose r} \tag{2}.
$$
Subtracting (1) from (2), 
$$
\sum_{0\le i\le r}\frac{i+1} {r+1}{2r-i\choose r-i}{s+i-2\choose i} = {2r+s-1\choose r-1} - {2r+s-1\choose r}.
$$
