Counting shortest paths on a triangular grid Background

An Eisenstein integer is a complex number of the form \$ z = a + b\omega \$ where \$a, b\$ are integers and \$\omega\$ is the third root of unity \$\frac{1-\sqrt3i}{2}\$. The Eisenstein integers can be viewed as the triangular lattice points, as shown in the image below (from Wikipedia):



Following the triangular grid, one step of movement can be done in six directions:

$$
\begin{array}{r|r}
\text{Direction} &amp; \text{Step} \\ \hline
E  &amp; 1 \\
NE &amp; 1 + \omega \\
NW &amp; \omega \\
W  &amp; -1 \\
SW &amp; -1 - \omega \\
SE &amp; -\omega
\end{array}
$$

Task

Given an Eisenstein integer \$z\$, count all shortest paths from the origin (\$0\$) to the point equivalent to \$z\$ on the triangular grid.

Since \$z = a + b \omega\$ can be represented by two integers \$ a, b \$, you can take the input as two integers \$a, b\$ in any consistent order and structure of your choice.

One way to compute this is (thanks to @xnor):

Take the absolute values of [a, b, a-b], and call it L
Calculate binomial(max(L), any other value in L)


Test cases

 a  b  ans
 0  0    1
 2  0    1
 5  3   10
 4  4    1
 2  3    3
 0  2    1
-2  2    6
-4  0    1
-5 -3   10
-1 -1    1
-3 -5   10
 0 -3    1
 4 -1    5
-4 -9  126
 7 -4  330
 8  1    8
 3 -3   20
Jelly, 7 bytes

;IAṀc$Ṁ


Try it online!

How?

;IAṀc$Ṁ - Link: list of two integers   e.g. [-4, -9]
 I      - incremental differences           [-5]           (since -9 - -4 = -5)
;       - concatenate                       [-4, -9, -5]
  A     - absolute values                   [4, 9, 5]
     $  - last two links as a monad:
   Ṁ    -   maximum                         9
    c   -   choose (vectorises)             [126, 1, 126]  (9c4=9c5=126 and 9c9=1)
      Ṁ - maximum                           126