Combinatorics of lattice walks with forbidden points 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?
 A: First, let me confirm your formula for walks $(0,0) \xrightarrow{m}(x,y)$ with $x+y\equiv m ~\mod 2$, $p_m(x,y)$, which you stated for $m=2n$.
Choose $(m+x+y)/2$ out of $m$ steps $s_i$ to be $(+1/2,+1/2)$ versus $(-1/2,-1/2)$.
Choose $(m+x-y)/2$ out of $m$ steps $t_i$ to be $(+1/2,-1/2)$ versus $(-1/2,+1/2)$.
Then letting $w_i = s_i + t_i$, $(w_i)$ are the steps of a walk $(0,0) \xrightarrow{m}(x,y)$ with steps of $(\pm 1,0)$ or  $(0,\pm 1)$. 

Second, let me fill in some details for the generating function approach.
Any walk $(0,0) \xrightarrow m (x,y)$ either avoids $(x_f,y_f)$, or it can be broken into $3$ pieces: 
$(0,0) \xrightarrow t (x_f,y_f)$ which first visits $(x_f,y_f)$ at the end,
$(x_f,y_f) \xrightarrow {2u} (x_f,y_f)$ of size ${2u \choose u}^2$.
$(x_f,y_f) \xrightarrow {m-t-2u} (x,y)$ which last visits $(x_f,y_f)$ at the start.
The first and third pieces have the same form reversed in time. Let $q_n(a,b)$ be the number of paths  $(0,0)\xrightarrow n (a,b)$ which only visit (a,b) at the end. 
$$\sum_n p_n(a,b)x^n = \bigg(\sum_n q_n(a,b)x^n\bigg)\bigg(\sum_n \sideset{}{^{^{^2}}}{2n \choose n} x^{2n}\bigg)$$
$$\sum_n q_n(a,b)x^n = \sum_{n} p_n(a,b)x^n \bigg/\sum_n \sideset{}{^{^{^2}}}{2n \choose n} x^{2n}$$
So, a generating function for walks $(0,0) \to (x,y)$ which visit $(x_f,y_f)$ is 
$$\bigg(\sum_n q_n(x_f,y_f)x^n\bigg)\bigg( \sum_nq_n(x-x_f,y-y_f)x^n\bigg)\bigg(\sum_n \sideset{}{^{^{^2}}}{2n \choose n} x^{2n}\bigg)$$
$$=\bigg(\sum_n p_n(x_f,y_f)x^n\bigg)\bigg( \sum_nq_n(x-x_f,y-y_f)x^n\bigg)$$
$$=\bigg(\sum_n p_n(x_f,y_f)x^n\bigg)\bigg( \sum_np_n(x-x_f,y-y_f)x^n\bigg)\bigg/\sum_n \sideset{}{^{^{^2}}}{2n \choose n} x^{2n}$$
You want to subtract this from $\sum_n p_n(x,y)x^n$.
I don't see a closed form expression for the coefficients.
