This came out of some work on the digamma function.

Let $(x)_k=x(x+1)\cdots(x+k-1)$ denote the Pochhammer symbol. Then,

Question. Can you prove/disprove this identity? $$\color{red}{\frac{(\frac12)_j^2}{j!^2}}\color{blue}{\sum_{i=0}^{j-1}}\frac4{2i+1}=\color{blue}{\sum_{i=0}^{j-1}}\color{red}{\frac{(\frac12)_i^2}{i!^2}}\frac1{j-i}.$$

I found this fascinating in view of fact that the $\color{red}{\text{red factor}}$ "goes in and out" of $\color{blue}{\text{the sum}}$.