I would like to ask about a certain inequality that I need and which came out of some work in here.
Question. For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof? $$6\prod_{j=1}^k(3n+j)\geq k!\,(nk+3)(nk+2)(nk+1).$$
Caveat. I'm not particularly interested in asymptotic analysis because for fixed $n$ it is rather clear what happens when $k$ is large.