For partial binomial sums such as $\sum_{k\le\Delta} \binom{n}{k}$ we don't tend to have closed forms. However we still know asymptotic expansions that are easy to work with.
Can we do something similar in two dimensions, for a sum like the following guarded by a number of linear inequalities?
$$\sum_{\substack{0\ \le\ i,\ j\\i+j\ \le\ \Delta\\r-i+j\ \le\ \Delta}} \binom{n}{i}\binom{m}{j}$$
For a start we could consider triangles, and then compose an answer, but that doesn't seem a lot easier. It doesn't even seem easy to identify which term might be the largest of the sum. Any ideas?
What about working with continuous approximation ? Replacing the binomial by Gaussians, and the sum by integrals ? The integral of a multidimensional Gaussian is reasonably easy to compute over rectangle domains (product of $erf$), and you may get the simplex using a symmetry argument.