If I'm not in error, the number of walks on a 2-dimensional integer lattice of length 2n steps from the origin to a point (x,y) has a nice closed form:

${2n \choose n + (x + y)/2}{2n \choose n + (x - y)/2}$

Question: Is there a similar closed form for the number of walks from the origin to (x,y), where a chosen "forbidden" point ($x_f$, $y_f$) is disallowed from ever appearing anywhere in a walk?

More generally, note that without the forbidden point, for a fixed length, a uniformly randomly chosen walk of that length is asymptotically Gaussian distributed. This sort of gives us a way of seeing how the $\ell_2$ norm on the euclidean plane arises as we "zoom out" while looking at the lattice and let the lattice size go to zero; that is, since the Gaussian distribution gives equal probabilities for equal $\ell_2$ distances from the origin.

A further, vaguer question is: how does forbidding a point --- removing it from the lattice --- affect the "effective" notion of distance that arises in the analogous way from the combinatorics of paths?