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: $$ \Large \exp \small \left(\begin{bmatrix} . & . & . & . \\\ 1 & . & . & . \\\ . & 2 & . & . \\\ . & . & 3 & . \\\ \end{bmatrix} \right)= \begin{bmatrix} 1 & . & . & . \\\ 1 & 1 & . & . \\\ 1 & 2 & 1 & . \\\ 1 & 3 & 3 & 1 \\\ \end{bmatrix}$$

With a bit explanation which might be useful for other guests http://go.helms-net.de/math/binomial/index-Dateien/image008.png