Recently, someone asked a question involving the expression$$ \sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$
At first glance, I knew that the expression was the value of a hypergeometric function, and anticipated difficulty in determining its exact value. Surprisingly, the value obtained using Mathematica was $\frac{2}{\pi}$. He couldn't recall the context in which the formula appeared well. (He mentioned seeing the expression in a comic book!) However, at this point, I suspect this may be a series studied by Ramanujan.Interestingly, when variables are introduced into the expression, as the following$$ \sum_{n=0}^{\infty} x^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$a very complicated hypergeometric function emerges, Nonetheless, Mathematica was unable to simplify the expression obtained by substituting $x=-1$ to revert to the original form.
Therefore, I believe that the result might be derived from something like a complex contour integral or a similar method, and might not be readily analyzed as a special value of a hypergeometric function.
Please help in understanding the reasoning behind the evaluation of this series.