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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?

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