Using WolframAlpha, I could informally come up with the following result:
$$ \lim_{n \rightarrow \infty} \frac{H_n^{(-\frac{1}{2})}}{n\sqrt{n}} = \frac{2}{3} $$
Allowing me to infer that $H_n^{(-\frac{1}{2})} \in \Theta(n\sqrt{n})$.
How to demonstrate/elaborate this limit ?