Finding a recursive formula for the number of ways to represent a vector in $\mathbb{Z}^2$ as a sum of vectors in $\mathbb{Z}^2$

Question

Let $z=a+bi$, where $a$ and $b$ are non-negative integers. I need to count how many ways are there to represent $z$ as $$z=\sum\limits_{j}a_j+b_ji$$, where $\forall j \ \ a_j\not=0$ or $b_j\not=0$. $a_j$ and $b_j$ are non-negative integers.

Edit: I have found that the result is OEIS A223544: $$\begin{cases}T(n,1) = n - 1 \\ T(n,k) = T(n,k-1) + n\end{cases}$$ through computation. However, I still do not have a proof for this formula.