$a_{1,n} = a_{n,1} = 1 \, \forall n$

$a_{i+1,j+1} = a_{i,j} + a_{i,j+1} + a_{i+1,j}$

Let $A$ be the matrix of order $k$ made by using $a_{i,j}$ as the element in the $i$th row and $j$th column, in other words, $A = [a_{i,j}]_{k \times k}$.

Find the determinant of matrix $A$ in terms of $k$.

For example, for $k=4$,

\[A = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 3 & 7 & 15 \\ 1 & 5 & 13 & 25 \\ 1 & 7 & 17 & 63 \end{bmatrix}\]