Prove that $\displaystyle \forall n \in \mathbb{N}: \ \ \prod_{k=1}^n k^k < (n+1)^{\frac{n(n+1)}{2}} e^{-\frac{(n-1)^2}{4}}.$
Using logarithm, we have to show that $\displaystyle \frac{\left(n-1\right)^2}4\leq \sum_{k=1}^n k\left(\ln \left(n+1\right)-\ln \left(k\right)\right)$.
We have $\displaystyle \sum_{k=1}^n k\left(\ln \left(n+1\right)-\ln \left(k\right)\right) = \sum_{k=1}^n k\ln \left( 1+\frac{n+1}k-1\right)$ and $\displaystyle \frac{n+1}k-1\geq 0$ so using $\displaystyle \ln\left(1+x\right) \geq x-\frac{x^2}2$ we should have the result.