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.

Using the fact that $I_0 (2x) = \sum_{j=0}^\infty \frac{x^{2j}}{(j!)^2}$
\begin{equation}
e^x I_{0}(2x) = \sum_{i = 0}^\infty \sum_{j=0}^\infty \frac{x^{i+2j}}{(i!)(j!)^2}
\end{equation}
We can group the factors with the same exponent $i+2j $, obtaining:
\begin{equation}
e^x I_{0}(2x) = \sum_{n = 0}^\infty \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \frac{x^n}{(k!)^2 (n-2k)!} = 
\sum_{n=0}^\infty \frac{x^n}{n!}\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \binom{n}{2k} \binom{2k}{k}
= \sum_{n=0}^\infty \frac{x^n}{n!} T_n
\end{equation}
