Whilst playing around on Wolfram Alpha, I typed in the sum $$\sum_{x=0}^\infty \frac{1}{\binom{2x}{x}}=\frac{2}{27}(18+\pi\sqrt 3)$$ I'm not sure how to derive the answer. My first instinct was to expand the binomial coefficient to get $$\sum_{x=0}^\infty \frac{x!^2}{(2x)!}$$ and then to try using a Taylor Series to get the answer. I thought that if I could find a function $f(n)$ with $$f(n)=\sum_{x=0}^\infty \frac{x!^2n^x}{(2x)!}$$ Then my sum would be equal to $f(1)$. How do I find such a function?
EDIT: I continued on this path and realized that I can use this to set up a recurrence relation for $f^{(x)}(0)$:
$$f^{(0)}(0)=1$$ $$f^{(x)}(0)=\frac{x^2}{2x(2x-1)}f^{(x-1)}(0)$$
However, I'm not sure how this helps me find $f(1)$...
Am I on the right track? Can somebody help me finish what I started, or point me towards a better method of calculating this sum?
Thanks!