I accidentally noticed that:
$$(7)_4=7 \cdot 8 \cdot 9 \cdot 10=2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7=7!$$
Here $(n)_k$ is the Pochhammer symbol.
I wonder, are there any other non-trivial integer solutions $(n,k)$?
Among the ones I consider trivial we have $(0,0),(1,0),(1,1),(2,1)$. Somehow, I'm sure that I will get a lot of comments with these four solutions.
This is the implicit plot of the equivalent equation:
$$\Gamma (n+k)=n \Gamma^2 (n)$$