It is an obvious fact that the sum $\sum_{n\geq 0} \binom{2n}{n} x^n$ defines an algebraic function. I am interested in the variation of this sum, namely

$$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n$$

which is not an algebraic function due to the growth of the coefficients (see Enumerative Combinatorics, Vol. 2 from Stanley, for instance).

My question is the following: can we say something about the algebraicity/transcendence of $A(x_0)$, for any real $x_0$ in the region of convergence)? If $x_0$ is algebraic this is quite obvious I guess, but I do not know if there exists an argument saying something when $x_0$ is transcendent.

I assume that this is a very general question, and an answer for any transcendent number should be very difficult. But I would be happy to know if something it can be said for some particular transcendental number ($\pi$ or $e$, for instance).

Maybe the integral representation

$$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n=\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{du}{(1-16x\cos^2 u)^{1/2}}$$

could be helpful, and something concerning elliptic integrals can be used here, but I do not know enough on this topic.