<s>[INST] Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ $\displaystyle \sum _{ n=1 }^{ \infty  }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! }  }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this infinite series that involves the TriGamma and  Zeta functions but it is not unique. I found one candidate result which is:
$$ \frac{ \pi \sqrt{3}}{72} \Big[3 \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{5}{6} \right) \Big]- \frac{4}{3} \zeta(3) $$
My friend (tormentor) has challenged me to find other such expressions. I know how to transform the look of such expressions if the Gamma function is involved by using functional equations like $\Gamma(x) = \Gamma(1+x)/x$ so I suspect he knows a similar result for the PolyGamma functions. I just can not find the information.  By trial and error, I did find that $3 \psi_{1}(1)=\psi_{1}(1/2)$
 [/INST] The series can be expressed in the form
\begin{align}
\sum_{n=1}^{\infty} \frac{(n!)^2}{n^3 \, (2n)!} &= \frac{1}{2} \, {}_{4}F_{3}\left(1,1,1,1; 2,2,\frac{3}{2}; \frac{1}{4} \right) \\
&= \frac{1}{2} \, \left[ - \frac{8}{3} \, \zeta(3) + \frac{\pi}{4 \sqrt{3}} \, \psi^{(1)}\left(\frac{1}{3}\right) - \frac{\pi}{12 \sqrt{3}} \, \psi^{(1)}\left(\frac{5}{6}\right) \right]
\end{align}
</s>