Two dimensional recurrence

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?

• The relevant examples in the documentation have a[n + 1, k] or a[n, k + 1], but not both, on the right side of the recurrence equation. However, I do not see why that should matter. Jan 27, 2016 at 14:45
• Indeed. It feels like those examples are pretty much the only working examples. You only have to change a little bit and it breaks down... Jan 27, 2016 at 15:13
• Perhaps, a bug. I encourage you to report your findings to Wolfram, Inc. Jan 27, 2016 at 15:18

a[n_, k_] := a[n, k] = a[n - 1, k - 1] + a[n, k - 1] + a[n - 1, k]