# Sum of the infinite series $\sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!}$ [duplicate]

I need to show that the sum of the infinite series $$\sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!} = \frac{2}{27}(18+\sqrt{3}\pi)$$. However, I don't know how to start, since this series does not fit a power series, and I can't find a rearrangement that makes the series a known power series. How can I do this?

• The absurdly specific result makes me think finding a way to manipulate known series involving $\pi$ might help. What is the context here? – The Count Apr 26 '19 at 21:56
• See this paper esp if you're familiar with generating functions. – Calvin Lin Apr 26 '19 at 22:05