We learn in calculus how to obtain a sum of binomial coefficients \frac{(2d)!}{(d!)^2} in terms of a generating function

$\sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d$

by the Taylor series of $(1-4x)^{-1/2}$ at $x=0$.

**My question**: is there a way to do this for **trinomial** coefficients? In particular what is

$\sum_{d \geq 1} \frac{(3d-1)!}{(d!)^3} x^d =?$

I can't imagine this not being studied before, but can not find a specific answer after a few futile hours of searching.

Thanks!