First, we note all the terms of the sum under cosideration are positive. Second, we take one of them, namely
Floor[n/2]^(n - Floor[n/2]), and consider its limit as $n$ approaches $\infty$:
Limit[Floor[n/2]^(n - Floor[n/2]), n -> Infinity]$\infty$.
This implies the limit under consideration is infinite too. Of course, there may exist a generalised sum of the series under consideration, but this is a math stuff, not a Mathematica matter.