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Proof strategy - Stirling numbers formula
Prove inequality:
$ \left\{\begin{array}{c}n\\k-1\end{array}\right\}\left\{\begin{array}{c}n\\k+1\end{array}\right\} \le \left\{\begin{array}{c}n\\k\end{array}\right\}^2$
for $n,k\in\mathbb{N}$, where $\left\{\begin{array}{c}n\\k\end{array}\right\}$ is the Stirling number of the second kind.
I think induction is the last resort. I don't know if it will work but I rather want to avoid it.