Some Nice Sums are Almost as Nice if you turn them Upside Down

We represent the sums ${\sum }_{k=0}^{n-1}{\left(\genfrac{}{}{0px}{}{n}{k}\right)}^{-2}$, ${\sum }_{k=0}^m{\left(\genfrac{}{}{0px}{}{m}{k}\right)}^{-1}{\left(\genfrac{}{}{0px}{}{a}{n-k}\right)}^{-1}$, ${\sum }_{k=0}^{n-1}\frac{q^{-k(k-1)}}{{\left[\genfrac{}{}{0px}{}{n}{k}\right]}_q}$, and the sum of the reciprocals of the summands in Dixon’s identity, each as a product of an {\it indefinite hypergeometric sum} times a (closed form) {\it hypergeometric sequence}