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 $T$.

Edit2: I have a particular interest in the case $r=1/2$:

$$T(1/2) = 4 \cdot {}_3F_2(\;\;\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\;;\;\frac{3}{2},\;\frac{3}{2}\;;\;\frac{1}{4}\;\;)$$

$$S(1/2)= 2\cdot {}_3F_2(\;\;\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\;;\;1,\;\frac{3}{2}\;;\;1\;\;) = \frac{8G}{\pi}$$

where $G$ is the Catalan constant

$$G = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^2}$$

$G$ arises in my analysis of $T$, but the connection isn't clear (to me).