Two dimensional recurrence

Question

My actual problem is a bit more complicated, but let us first assume that we have a recurrence relation of the following type

$a_{n,l} = a_{n-1,l} + a_{n-1,l-1} + a_{n,l-1}$

The boundary is given, let's say $a_{n,0} = n$ and $a_{0,n}=n$ for each $n\geq0$.

I would like to create a table to list the values of $a_{n,l}$. I try the following

RecurrenceTable[{a[n + 1, k + 1] == 
a[n, k] + a[n + 1, k] + a[n, k + 1], a[n, 0] == n, a[0, k] == k}, 
a, {n, 0, 6}, {k, 0, 6}] // Grid

Why does this produce the same output as input? Why can Mathematica not compute this?

Answer 1

I figured that it can be done in an alternative way:

a[n_, k_] := a[n, k] = a[n - 1, k - 1] + a[n, k - 1] + a[n - 1, k]
a[n_, 0] := n
a[0, k_] := k
Grid[Table[a[i, j], {i, 0, 6}, {j, 0, 6}]]

Still, this does not answer the question as to why RecurrenceTable does not produce any output. I am still curious about this.