I have a non symmetric random walk on a 2 dimensional lattice with transition probabilities $\pi_l, \pi_r, \pi_u, \pi_d$ such that $\pi_l + \pi_r + \pi_u + \pi_d = 1$. What is the distribution of the displacement of the random walk after $n$ steps along just the vertical or horizonal dimension.
I am aware of a number of potential methods for answering this question in the limit (CLT) or with an analytically intractable sum (multinomial distribution) but I was hoping for something closed form. I had been trying to work with the Pascal's triangle visualization for the heads minus tails distribution, but using the trinomial triangle instead of Pascal's. I had been assuming $\pi_l = \pi_r$ or $\pi_d = \pi_u$ for simplicity but I can't get it to work even then.
If anyone has a solution, it would be much appreciated.