We have a class of probability distributions (parameterized by $\alpha =\frac{1}{2},1,\frac{3}{2},2,\ldots$) over the interval $[-2^{-12} \cdot 3^{-3},2^{-16}]$ having as their $n$-th moments \begin{equation} \frac{2^{-12 n} (2 n)! (\alpha +1)_{2 n} (2 \alpha +1)_{2 n} \, _4F_3\left(-n,-4 n-5 \alpha -1,\alpha ,\alpha +\frac{1}{2};\frac{1}{2}-n,-2 n-2 \alpha ,-2 n-\alpha ;1\right)}{\left(3 \alpha +\frac{3}{2}\right)_{2 n} \left(6 \alpha +\frac{5}{2}\right)_{4 n}}, \end{equation} where generalized hypergeometric and Pochhammer function notation is employed.
Of particular interest is the $\alpha =1$ (standard quantum-mechanical) case \begin{equation} \frac{2^{-12 n} (2 n)! (2)_{2 n} (3)_{2 n} \, _4F_3\left(1,\frac{3}{2},-4 n-6,-n;-2 n-2,-2 n-1,\frac{1}{2}-n;1\right)}{\left(\frac{9}{2}\right)_{2 n} \left(\frac{17}{2}\right)_{4 n}}. \end{equation} It appears remarkably (http://arxiv.org/pdf/1301.6617v4.pdf [J. Phys. A-Math. Theor., Vol. 46 (2013), pp. 445302]) and http://arxiv.org/pdf/1109.2560.pdf [ J. Phys. A: Math. Theor. 45 (2012) 095305]) that the cumulative ("separability") probabilities over the negative subinterval $[0,2^{-16}]$ are expressible as \begin{equation} \label{Hou1} P(\alpha) =\Sigma_{i=0}^\infty f(\alpha+i), \end{equation} where \begin{equation} \label{Hou2} f(\alpha) = P(\alpha)-P(\alpha +1) = \frac{ q(\alpha) 2^{-4 \alpha -6} \Gamma{(3 \alpha +\frac{5}{2})} \Gamma{(5 \alpha +2})}{3 \Gamma{(\alpha +1)} \Gamma{(2 \alpha +3)} \Gamma{(5 \alpha +\frac{13}{2})}}, \end{equation} and \begin{equation} \label{Hou3} q(\alpha) = 185000 \alpha ^5+779750 \alpha ^4+1289125 \alpha ^3+1042015 \alpha ^2+410694 \alpha +63000 . \end{equation} For (the real, complex and quaternionic cases of) $\alpha = \frac{1}{2}, 1, 2$, this formula yields the "separability probabilities" of $\frac{29}{64}, \frac{8}{33}$ and $\frac{26}{323}$, respectively.
Can one construct the full probability distributions over $[-2^{-12} \cdot 3^{-3},2^{-16}]$ or some transformation of that interval? (There is a close variant [see references] of this problem having moments involving $5F4$ hypergeometric functions and the interval $[-2^{-4},2^{-8}]$.)
In response to the first two comments, I was hoping (probably rather naively) for some sophisticated (presumably Mathematica) implementation of the inverse Mellin transform, such as in eqs. (4)-(5) in the Penson/Zyczkowski article Phys. Rev. E 92, 012121 (2015) (couldn't provide another arXiv link because of insufficient "reputation"). One clear obstacle to doing so is that the desired probability densities extend into negative regions, so some changes-of-variables seem in order. I tried using the Mathematica FindSequenceFunction command to convert the moment formula to one over [0,1], in order to further proceed, but without success.