An elegant and rather unexpected formula for the powers of the generating function for [Catalan numbers][1] $C_n = \frac{1}{n+1}\binom{2n}{n}$:
$$\left(\sum_{n=0}^{\infty} \frac{1}{n+1}\binom{2n}{n} \cdot x^n\right)^m = 
\sum_{n=0}^{\infty} \frac{m}{n+m}\binom{2n+m-1}{n} \cdot x^n.$$
The formula can be further continued as $\ldots =\left(\frac{1-\sqrt{1-4x}}{2x}\right)^m$, but this would spoil the beauty of the above identity.


  [1]: https://en.wikipedia.org/wiki/Catalan_number