Does there exist any closed form formula for the following exponential generating function:
$$ F_k(x) = \sum_{n=0}^{\infty} \frac{\binom{2(k+n)-1}{n}x^n}{(k+n)!} $$
The coefficients in the numerator are vertical binomial coefficients in pascal's triangle.
For example, in the image, for $k=2$ we have,
$$F_2(x) = \frac{1x^0}{2!}+\frac{5x^1}{3!}+\frac{21x^2}{4!}+\frac{84x^3}{5!}+...$$