I'm investigating a function that has led me to this series:

$$T(r) = \sum_{k=0}^{\infty}{2k \choose k}\frac{1}{(k+r)^2 16^k}={}_3F_2(\;\;\frac{1}{2},\;r,\;r\;;\;1+r,\;1+r\;;\;\frac{1}{4}\;\;)\frac{1}{r^2}$$

This paper ("A certain series associated with Catalan's constant") by Victor Adamchik spotlights a tantalizingly similar series ...

$$S(r)=\sum_{k=0}^{\infty}{2k\choose k}^2 \frac{1}{(k+r) \;\; 16^k}={}_3F_2(\;\;\frac{1}{2},\;\frac{1}{2},\;r\;;\;1,\;1+r\;;\;1\;\;)\frac{1}{r}$$

... where the square is on the "wrong" factor.

I'm not very familiar with hypergeometric series (yet). Is there an identity that relates $S$ and $T$?

Edit: I've added the "${}_3F_2$" representations of $S$ and $T$.