Is there a generating function for $\sqrt{n}$? 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.

You could try to use the integral formulas for fractional integrals and fractional derivatives (i.e., like half-weight derivatives of the power function; see for example semi-derivatives) in the theory of fractional calculus to express a generating function for this sequence. Note that the resulting generating function will not, however, be "''nice''" in the sense of it being P-recursive. 
