TITLE: Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$
QUESTION [13 upvotes]: $\displaystyle \sum _{ n=1 }^{ \infty  }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! }  }$ 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 $3 \psi_{1}(1)=\psi_{1}(1/2)$

REPLY [2 votes]: I am French and I research expressions of trigamma zeta 3 Catalan ...
it's very difficult but for your example, I propose $\frac{\pi\sqrt{3}\Psi\left(1,\frac{1}{3}\right)}{9}-\frac{2\sqrt{3}\pi^3}{27}-4\frac{\zeta(3)}{3}$. So I confirm the expression with MAPLE.