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The Telescoping(f, k=m..n) command computes a closed form of the definite sum of f over the specified range of k using telescoping method, or Newton-Leibniz's formula, that is, it first computes a closed form of the corresponding indefinite sum.

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

$F≔\mathrm{binomial}\left(2n-2k\,n-k\right)2^{4k}{\left(2k\left(2k+1\right)\mathrm{binomial}\left(2k\,k\right)\right)}^{-1}$

$F≔\frac{\left(\genfrac{}{}{0ex}{}{2n-2k}{n-k}\right)2^{4k}}{2k\left(2k+1\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}$

$\mathrm{Sum}\left(F\,k=1..n\right)=\mathrm{Telescoping}\left(F\,k=1..n\right)$

$\sum _{k=1}^n\frac{\left(\genfrac{}{}{0ex}{}{2n-2k}{n-k}\right)2^{4k}}{2k\left(2k+1\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}=-\frac{\left(\genfrac{}{}{0ex}{}{2n-2}{n-1}\right)\left(-16n+8\right)}{2\left(2n+1\right)}$