The probability for a random walk that starts at the origin to return to the origin for the first time after exactly $2n$ steps has been solved on this site a couple times for the $1$d random walk.
The most direct proof is just to count the number of relevant paths that return to the origin for the first time after $2n$ steps, which is the $n-1$th Catalan number, which yields a probability of
$$P^{(1)}_{2n} = \frac{C_{n-1}}{2^{2n}} = \frac{\binom{2n-2}{n-1}}{n 2^{2n}}$$
My aim is to find the generalization of this result to the $2d$ random walk. That is, what is the probability that the random walk returns to the origin for the first time after $2n$ steps? To be specific, the $2d$ random walk I have in mind is the simple one where each step is a single step to one of the north, south, east, or west with equal probability.
I've been thinking about an appropriate generalization of the Catalan numbers or of Dyck words. I usually view Catalan numbers in terms of monotonic paths above a line from which there follows a simple enumeration via the difference of all paths and reflected paths. The playing field for the requisite generalization of the Catalan numbers will be a $3d$ space where "bad" paths cross a single line; I'm unsure how to generalize the reflection argument in that case.
Numerically, I find the number of the relevant paths for the random walk is $4, 20, 176, 1876, 22064,...$ for $n=1,2,3,4,5...$. This is a known series in OEIS defined exactly as I have. That link provides generating functions and asymptotics for the number of relevant paths. However, it is not clear to me how to find these generating functions or the asymptotics at large $n$.
My question is: how can I find the explicit, ordinary generating function for the number of relevant paths in $2d$ and the large-$n$ asymptotic behavior of the number of relevant paths in $2d$?