I encountered the following series while evaluating an integral
\begin{equation} \sum_{n=1}^\infty\binom{n}{n/2}\frac{(-1)^{n}}{n(2a)^n}, \quad a\in\mathbb{R} \end{equation}
and am looking for a closed form. I have looked up generating functions and found some close candidates such as,
\begin{equation} \sum_{n=1}^\infty\binom{2n}{n}\frac{x^n}{4^nn} = 2\log\left(\frac{2}{1+\sqrt{1-x}}\right) \end{equation}
but I can't figure out how to deal with the $\binom{n}{n/2}$ term. Looking at the power series for $(1+x)^n$, I assume finding such a function may be difficult, but I hope a closed form exists.
I will be happy to add my evaluation of the original integral up to the point where the series appears, if that may be necessary.
Thank you in advance.
Edit:
Here is the integral I was evaluating, $$J(a)=\int_0^{\pi/2}\log\left(1+\frac{\sin x}{a}\right)\ dx,\quad a\in\mathbb{R}$$ which comes from this post, and the derivations of the series are in my answer there. I hope this helps @ClaudeLeibovici.