\[C\left(n\right)=\frac{1}{n+1}\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{1}{2n+1}%
\genfrac{(}{)}{0.0pt}{}{2n+1}{n}=\genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}%
{0.0pt}{}{2n}{n-1}=\genfrac{(}{)}{0.0pt}{}{2n-1}{n}-\genfrac{(}{)}{0.0pt}{}{2n%
-1}{n+1}.\]