Is there a known way of expressing the Catalan numbers or the generating function of the Catalan numbers as a function of the Taylor series of $e^{-x}$ i.e.
$e^{-x}=1-x+\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{24}-...$?
I have been looking all over for this, and even though I happen to come across several different representations of the Catalan numbers, I cannot seem to find one involving this formula.
Yes. The exponential generating function of the Catalan numbers is given by $$ e^{2x}(I_0(2x)-I_1(2x)),$$ where $I_n(x)$ is the $n-$th Modified Bessel function.
In theory, you can apply the Borel transform of the usual generating function to get this.