\(
\begin{aligned}
a_{i, j}
& = a_{i - 1, j} + a_{i, j - 1} \cr
& = a_{i - 1, j} + a_{i - 1, j - 1} + a_{i, j - 2} \cr
& = a_{i - 1, j} + a_{i - 1, j - 1} + a_{i - 1, j - 2} + a_{i, j - 3} \cr
& = \sum_{k = 1}^{j}{a_{i - 1, k}} + a_{i, 0} \cr
& = \sum_{k = 1}^{j}{a_{i - 1, k}}
\end{aligned}
\)

令 \(F_i = \begin{pmatrix}
a_{i, 1} & a_{i, 2} & a_{i, 3} & \dots & a_{i, m}
\end{pmatrix}\)

且 \(A = \begin{pmatrix}
1 & 1 & 1 & \cdots & 1 \cr
0 & 1 & 1 & \cdots & 1 \cr
0 & 0 & 1 & \cdots & 1 \cr
\vdots & \vdots & \vdots & \ddots & \vdots \cr
0 & 0 & 0 & \cdots & 1
\end{pmatrix}\)

那么有 \(F_i \times A = F_{i + 1}\)

所以 \(F_n = F_1 \times A^{n - 1}\)