Define Pascal's triangle as follows: it is an array $(T_{m,n})_{m, n \in \mathbf{Z}}$ of integers, satisfying

if $m<0$, then $T_{m,n}=0$.

$T_{0,0}=1$ and if $n \neq 0$, then $T_{0,n}=0$.

if $m>0$, then $T_{m,j} = T_{m-1,j-1} + T_{m-1,j+1}$.

The last item makes the indexing perhaps a little unusual, but if you ignore entries $T_{i,j}$ where $i$ and $j$ have different parity (because those entries are always zero), you get Pascal's triangle.

Now consider a variation on this by truncating it to the left of column 0 and to the right of some column $N$: fix $N$ and replace the last condition with:

- if $m>0$, then for $0 \leq j \leq N$, $T_{m,j} = T_{m-1,j-1} + T_{m-1,j+1}$. If $j<0$ or $j>N$, then $T_{m,j}=0$.

For example, if $N=3$, we get this:

```
1
1
1 1
2 1
2 3
5 3
5 8
13 8
```

The entries are Fibonacci numbers, and the sums of the rows give the Fibonacci sequence.

If $N=5$, we get this:

```
1
1
1 1
2 1
2 3 1
5 4 1
5 9 5
14 14 5
14 28 19
```

The row sums are 1, 1, 2, 3, 6, 10, 19, 33, 61, 108, 197, ... (presumably this sequence).

I'm interested in what happens as $N$ varies, and in particular when $N$ is two less than a prime. For example, what are the sums of the rows? What is the limit of the $n$th root of the $n$th row? Has anyone seen this sort of thing in the literature?