二項係数付き級数

$$\begin{array}{rl}\sum _{n>0}\frac{{(\genfrac{}{}{0ex}{}{2n}{n})}^3}{2^{6n}n}& =\sum _{n>0}\frac{{\left(\frac{1}{2}\right)}_n^3}{(1{)}_n^{3}n}\\ & =\sum _{n\ge 0}\frac{{\left(\frac{1}{2}\right)}_{n+1}^3}{(1{)}_{n+1}^3(n+1)}\\ & =\frac{1}{8}\sum _{n\ge 0}\frac{{\left(\frac{3}{2}\right)}_n^3(1{)}_n^2}{(2{)}_n^3(2{)}_{n}n!}\\ & =\frac{1}{8}{}_5{F}_4[\begin{array}{c}\frac{3}{2},\frac{3}{2},\frac{3}{2},1,1\\ 2,2,2,2\end{array};1]\end{array}$$

は？。