Deriving exponential generating function for central trinomial coefficients A recent question (link) asked for a derivation of the (ordinary) generating function for the central trinomial coefficients $\{T_n\}$. But the OEIS page (A002426) also lists an exponential generating function
$$\sum_{n=0}^\infty T_n \frac{x^n}{n!}=e^x I_0(2x)$$ where $I_0(x)$ is the zeroth Bessel function. How is this derived? I'll take a stab myself at showing this using the tools of analytic combinatorics, but I wanted to open this up to more knowledgeable folks as well.

$$\sum_{n=0}^\infty T_n\frac{x^n}{n!}=\frac{1}{2\pi i}\sum_{n=0}^\infty\frac{x^n}{n!}\oint\frac{(1+z+z^2)^n}{z^{n+1}}\,dz=\frac{e^x}{2\pi i}\oint e^{x(z+1/z)}\frac{dz}{z}=e^x I_0(2x),$$ using $T_n=[z^n](1+z+z^2)^n$, then the exponential series, then the contour integral representation of $I_0$ based on the generating function $e^{z(t+1/t)/2}=\sum_{n\in\mathbb{Z}}I_n(z)t^n$.
