Proving $ \sum_{k=0}^{\infty} \frac{k! \, (2k)! \, (25k - 3)}{(3k)! \, 2^k} = \frac{\pi}{2} $

Question

Wikipedia claims that

$$ \sum_{k=0}^{\infty} \frac{k! \, (2k)! \, (25k - 3)}{(3k)! \, 2^k} = \frac{\pi}{2} $$

but does not give a citation. How can this be proved?

Edit: This was correctly closed as a duplicate, but the linked question in the duplicate message does not provide an answer. One is here.