How can I write the equivalent of this formula for non integer n and k in terms of gamma functions?

Question

I am wondering how can I write the following formula for non integer parameters?

$$\sqrt{\frac{\pi^2}{12}+\sum_{k=0}^{n-1}\frac{n!n!}{k!(2n-k)!}.\frac{(-1)^{(n-k)}}{(n-k)}}$$

I have searched some pages like Particular values of the Gamma function, especially between those formulas that have $\sum_{k=0}^{+\infty}$ and also Gamma function properties. But I have no idea

Answer 1

You can always replace factorials by the Gamma function by using $\Gamma(x) = (x-1)!$.

A problem I see is figuring out what the summation means when $n$ is not an integer.