I have tried to find a differential equation for such $S(x)$ but I could not. I also tried to express it in binomial form but I could not.
I was finding expected value of some other problem when I got this. So, I don't know if it has a closed form.
2026-04-12 22:54:04.1776034444
How to find $\sum_{n=1}^{\infty}{\binom{2n-1}{n} x^n}, |x| < \frac{1}{4} ?$
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HINT:
$$\binom{2n-1}{n}=\frac12 \binom{2n}{n}$$
Now, look at the series expansion of $(1-t)^{-1/2}$.