Need explanation on the solution below:
Stirlings Formula: $$n! = \sqrt{2\pi n}\left(\frac n e\right)^n\left(1+\frac {1}{12n} + \frac{1}{288n^2}- \frac{1}{51840n^3}+O\left(\frac{1}{n^4}\right)\right)$$
We have the formula:
$$\log(C_n)=\log[(2n)!]-\log[n!]-\log[(n+1)!] $$
The author proceed to use Stirling three times and with Taylor series to the above equation to get: $$\log(C_n)=n \log (4)-\frac{1}{2} \log \left({\pi n^3}\right)-\frac{9}{8 n}+\frac{1}{2 n^2}-\frac{21}{64 n^3}+O\left(\frac{1}{n^4}\right)$$
Could anyone explain how the author came to this solution for $\log(C_n)$ ?