I tried to come up with a closed form for the ordinary generating function for the sequence $\{\sqrt{n}\}_0^{\infty}$ but I could not. Is there a way to derive it using the recurrence relation $$a_{n+1} = \sqrt{a_n^2+1}. $$ Because if there is, it is no obvious to me how to do so.
I noticed that the task is trivial if we use a Dirichelt series generating function, namely $\zeta(s-\frac{1}{2})$ but this seems less in interesting to me than having a closed form for the ogf or perhaps even the exponential generating function.
It does exist, defined as $g(z) = \sum_{n \ge 0} \sqrt{n} z^n$. It is even a nice function, in that it is analytic in a region around the origin (apply your favorite test). It doesn't have a representation in terms of elementary functions, however.