## The nth Catalan Number Expressed as a Beta Function

John Cook who blogs here, posted the following expression for the nth Catalan number on one of his twitter accounts

C_{n} = \frac{1}{2\pi} \int\limits_{0}^{4} x^{n} \sqrt{\frac{4-x}{x}} \mathrm{d} x
\label{eq:1608121}
\tag{1}

Let us express this as a Beta function
\begin{align}
\int\limits_{0}^{4} x^{n} \sqrt{\frac{4-x}{x}} \mathrm{d} x & = x^{n-\frac{1}{2}} (4-x)^{\frac{1}{2}} \mathrm{d} x \\
& = 4^{n+1} \int\limits_{0}^{1} y^{n-\frac{1}{2}} (1-y)^{\frac{1}{2}} \mathrm{d} y \\
& = 4^{n+1} \, \mathrm{B}\left(n + \frac{1}{2}, \frac{3}{2} \right)
\label{eq:1608122}
\tag{2}
\end{align}
We used the substitution $$y=\frac{x}{4}$$.

Now we have

C_{n} = \frac{1}{2\pi} \int\limits_{0}^{4} x^{n} \sqrt{\frac{4-x}{x}} \mathrm{d} x = \frac{2^{2n+1}}{\pi} \mathrm{B}\left(n + \frac{1}{2}, \frac{3}{2} \right)
\label{eq:1608123}
\tag{3}

We will check this by using the following definition of the nth Catalan number

C_{n} = \frac{(2n)!}{(n+1)! \, n!}
\label{eq:1608124}
\tag{4}

Using the following Gamma function expressions