The CreativeTelescoping(f, k=m..n) command computes a closed form of the definite sum of f over the specified range of k using creative telescoping method.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{DefiniteSum}\right]\right)\:$

$F≔{\mathrm{binomial}\left(2n\,2k\right)}^2$

$F≔{\left(\genfrac{}{}{0ex}{}{2n}{2k}\right)}^2$

$\mathrm{Sum}\left(F\,k=0..n\right)=\mathrm{CreativeTelescoping}\left(F\,k=0..n\right)$

$\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{2n}{2k}\right)}^2=\frac{{\left(\mathrm{-1}\right)}^n\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}{2}+\frac{\left(\genfrac{}{}{0ex}{}{4n}{2n}\right)}{2}$

van Hoeij, M. "Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations." Journal of Pure and Applied Algebra. Vol. 139. (1999): 109-131.

Zeilberger, D. "The Method of Creative Telescoping." Journal of Symbolic Computing. Vol. 11. (1991): 195-204.