This should be a simple question, but I am unfortunately unable to find a closed-form expression for the following quantity: the number of integer-valued paths of a certain length $k$ from $(0,\cdots,0)$ to $(P_1,\cdots,P_n)$ on $\mathbf{Z}^n$.
When $n=1$ this is trivial, but even when $n=2$ I am stuck. Is there a general closed-form expression for the above quantity? Is there an exposition somewhere?
Let $r=|a_1|+|a_2|+\dots +|a_n|-k$ .Clearly if $r<0$ or if $r$ is odd there are no answers, in the second case by chequerboard colouring argument.
Otherwise we obtain the following recursion: $$f_k(a_1,a_2\dots a_n)=\sum\limits_{j=0}^{r/2}f_{k-a_n-2j}(a_1,a_2\dots ,a_{n-1})\binom{k}{a_n+2j}\binom{a_n+2j}{j}$$