There is a well known triangle scheme for generating Bell Numbers as explained on the Wikipedia page
The sequence of Bell Numbers that are generated starts $1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147,...$
In playing around I have found a different triangle scheme for generating Bell Numbers
$$\newcommand\cn[3]{\llap{#1}#2\rlap{#3}} \begin{array}{c} &&&&&&\cn{}{1}{}\\ &&&&&\cn{}{1}{}&&\cn{}{1}{}\\ &&&&\cn{}{1}{}&&\cn{}{3}{}&&\cn{}{1}{}\\ &&&\cn{}{1}{}&&\cn{}{7}{}&&\cn{}{6}{}&&\cn{}{1}{}\\ &&\cn{}{1}{}&&\cn{}{15}{}&&\cn{}{25}{}&&\cn{}{10}{}&&\cn{}{1}{}\\ &\cn{}{1}{}&&\cn{}{31}{}&&\cn{9}{}{0}&&\cn{6}{}{5}&&\cn{}{15}{}&&\cn{}{1}{}\\ \cn{}{1}{}&&\cn{}{63}{}&&\cn{30}{}{1}&&\cn{35}{}{0}&&\cn{14}{}{0}&&\cn{}{21}{}&&\cn{}{1}{} \end{array}$$
where each new row adds up to the next Bell Number in the sequence
Considering diagonals from top right to bottom left each new number in the triangle is generated by
[number above right] $\times$ [diagonal number] $+$ [number above left]
For example $25$ on the third row is given $6\times3+7$
The numbers in each row $n$ appear in the polynomials generated in Mathematica by
$$\frac{1}{e^x}\sum _{k=0}^{\infty } \frac{k^n x^k}{k!} $$
Do these polynomials and the triangle generated by the coefficients have a name? If not can anyone point to study of them in the mathematical literature with perhaps a combinatorial interpretation?