# Simple recurrence relation in three dimensions

I have the following recurrence relation:

$$f[i,j,k] = f[i-1,j,k] + f[i,j-1,k] + f[i,j,k-1],\quad \mbox{for } i \geq j+k,$$

starting with $f[0,0,0]=1$, for $i$, $j$, and $k$ non-negative.

Is there any way to find a closed form expression for $f[i,j,k]$?

Note that this basically is a three dimensional version of the Catalan triangle, for which $f[i,j] = f[i-1,j] + f[i,j-1]$, for $i \geq j$, starting with $f[0,0]=1$. For this, a closed form expression is known: $f[i,j] = \frac{(i+j)!(i-j+1)}{j!(i+1)!}$.