and the generating function?

Well, in the previous post we were talking of the infinite sum of the Catalan numbers’  inverses, now let me tell you that the same method, but with $x^n$ included,  gonna give you the generating funcion

$\frac{2\,\left( {\sqrt{4-x}}\,\left(8+x\right)+12\,{\sqrt{x}}\,\arctan (\frac{{\sqrt{x}}}{{\sqrt{4 - x}}}) \right) }{\sqrt{\left( 4 - x \right)^5}}$

Yes, she (the formula) expands about the origin as

$1+x+\frac{x^2}{2}+\frac{x^3}{5}+\frac{x^4}{14}+\frac{x^5}{42}+\frac{x^6}{132}+\frac{x^7}{429}+\cdots$

cool! isn’t it?  By the way do not try to reproduce the expansion without software because just at first derivative you gonna get angry enough to want to pull your hair away! This function is not completely new at all -but strangely- it is hard to find it explicitly in the modern math literature.

A next Frage is: what the heck is it good for? $: )$

Exercise: insert $(-1)^nx^n$ in the beta fuction method explained earlier and tell me what you get… bye