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? $$\frac{(1/2)_j^2}{j!^2}\sum_{i=0}^{j-1}\frac4{2i+1}=\sum_{i=0}^{j-1}\frac{(1/2)_i^2}{i!^2}\frac1{j-i}.$$

I found this amusing in view of the factor that "goes in and out of the sum".