hemotropS7A

2022-11-24

How to find the Direct Discrete Laplace Transform of $\left(\genfrac{}{}{0}{}{2n}{n}\right)$

Kenyon Valencia

Expert

The generating function of Catalan's numbers ${y}_{n}=\frac{1}{n+1}\left(\genfrac{}{}{0}{}{2n}{n}\right)$ is
$\sum _{n⩾0}\frac{1}{n+1}\left(\genfrac{}{}{0}{}{2n}{n}\right){x}^{n}=\frac{1-\sqrt{1-4x}}{2x}$
Multplying by x and differentitating gives that
$\sum _{n⩾0}\left(\genfrac{}{}{0}{}{2n}{n}\right){x}^{n}=\frac{d}{dx}\frac{1-\sqrt{1-4x}}{2}=\frac{1}{\sqrt{1-4x}}$
One can also try to prove this directly, by noting that
$\left(\genfrac{}{}{0}{}{-1/2}{k}\right)=\left(-1{\right)}^{k}{4}^{-k}\left(\genfrac{}{}{0}{}{2k}{k}\right)$
and that
$\left(1+x{\right)}^{\alpha }=\sum _{n⩾0}\left(\genfrac{}{}{0}{}{\alpha }{n}\right){x}^{n}$
At any rate, any claim begs for a proof.

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