One which I like much is $$\exp \left(\begin{bmatrix} . & . & . & . & .\\ 1 & . & . & . & . \\ . & 2 & . & . & . \\ . & . & 3 & . & . \\ . & . & . & 4 & . \\ \end{bmatrix} \right)= \begin{bmatrix} 1 & . & . & . & . \\ 1 & 1 & . & . & . \\ 1 & 2 & 1 & . & . \\ 1 & 3 & 3 & 1 & . \\ 1 & 4 & 6 & 4 & 1 \\ \end{bmatrix}$$ It is practically easier and a bit more iconic if we reduce it a bit - although for me it is not so pleasing, because the immediate remembering of the Pascal-triangle comes with the 1-4-6-4-1-row: $$\Large \exp \small \left(\begin{bmatrix} . & . & . & . \\ 1 & . & . & . \\ . & 2 & . & . \\ . & . & 3 & . \\ \end{bmatrix} \right)= \begin{bmatrix} 1 & . & . & . \\ 1 & 1 & . & . \\ 1 & 2 & 1 & . \\ 1 & 3 & 3 & 1 \\ \end{bmatrix}$$
One which I like much is $$\exp \left(\begin{bmatrix} . & . & . & . & .\\ 1 & . & . & . & . \\ . & 2 & . & . & . \\ . & . & 3 & . & . \\ . & . & . & 4 & . \\ \end{bmatrix} \right)= \begin{bmatrix} 1 & . & . & . & . \\ 1 & 1 & . & . & . \\ 1 & 2 & 1 & . & . \\ 1 & 3 & 3 & 1 & . \\ 1 & 4 & 6 & 4 & 1 \\ \end{bmatrix}$$ It is practically easier and a bit more iconic if we reduce it a bit - although for me it is not so pleasing, because the immediate remembering of the Pascal-triangle comes with the 1-4-6-4-1-row: $$\Large \exp \small \left(\begin{bmatrix} . & . & . & . \\ 1 & . & . & . \\ . & 2 & . & . \\ . & . & 3 & . \\ \end{bmatrix} \right)= \begin{bmatrix} 1 & . & . & . \\ 1 & 1 & . & . \\ 1 & 2 & 1 & . \\ 1 & 3 & 3 & 1 \\ \end{bmatrix}$$