Calculating number of ways two persons can meet each other in one cell on a N x N board

Question

Person A stands in the upper left cell. Person B stands in the bottom right cell.

Person A can only go down or right. Person B can only go up and left.

Persons move completely independent of each other.

In how many ways can Person A meet Person B in one cell?

Example for $N = 2$:

Answer 1

Here's a start: Let $M_{i,j}$ denote the cell in the $i$th row and $j$th column of board.

Call $A_{i,j}$ the number of ways person $A$ can get to $M_{i,j}$.
You can think of that like a word of length $i+j$ consisting of the letters $d$ for down and $r$ for right, containing $d$ exactly $i$ times and $r$ exactly $j$ times. For example, to get from the upper left corner $M_{0,0}$ to $M_{2,1}$ you can go $ddr$, $drd$, $rdd$. Can you find the function for this?

Now, if players $A$ and $B$ move independently, you can represent the number of ways that they can meet at a particular spot as the number of ways that one player can get there times the number of ways that the other player can get there.