Matrix and Lattice Paths

Question

I have a $k\times k$ matrix $$ A_{k}= \begin{pmatrix} 1 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & \cdots & 1 &1 & 0\\ &\vdots & &\vdots \\ 1 & 1 & \cdots & 0 & 0 & 0\\ 1 & 0 & \cdots & 0 & 0 & 0 \end{pmatrix} $$ where $k\ge 1$. I wish that the $(1,1)$-entry of $(A_{k})^{n}$ is counting the something related to the weighted lattice paths which lies between $y=0$ and $y=k$, starting from $(0,0)$ and ends at $(0,l)$, for some $l\ge 0$.

The steps allowed are the following:

1. One unit horizontally to the right
2. One unit diagonally to the upper-right
3. One unit diagonally to the down-right
4. It cannot go below $y=0$ or above $y=k$

If needed, we can associate a weight on each step

1. For each diagonal step, associate it with weight $1$
2. For each step horizontally to the right at top level (i.e. $y=k$), associate it with weight $0$
3. For each step horizontally to the right at non-top level (i.e. $y\not= k$), I am not sure which weight I should associate it with (maybe still $1$?)

**Question: can we realize the $(1,1)$-entry of the matrix $A_{k}^{n}$ as counting (weighted) lattice path? **

I know that the matrix can count the paths in a directed graph, but the relationship between matrix and lattice graph is not very clear to me. Intuitively I felt that "directed graph" is very close to "lattice path", but I cannot make it rigorous. I would appreciate it if someone can give me some ideas or reference, especially on how to use matrix in counting lattice paths.