Definition:Stirling's Triangles

Stirling's Triangles are the arrays formed by arranging Stirling's Numbers of the first (usually unsigned) and second kind:

Stirling's Triangle of the First Kind
$$\begin{array}{r|rrrrrrrrrr} n & \left[{n \atop 0}\right] & \left[{n \atop 1}\right] & \left[{n \atop 2}\right] & \left[{n \atop 3}\right] & \left[{n \atop 4}\right] & \left[{n \atop 5}\right] & \left[{n \atop 6}\right] & \left[{n \atop 7}\right] & \left[{n \atop 8}\right] & \left[{n \atop 9}\right] \\ [4pt] \hline 0 & 1 &    0 &      0 &      0 &     0 &     0 &    0 &   0 &  0  & 0 \\ 1 & 0 &     1 &      0 &      0 &     0 &     0 &    0 &   0 &  0  & 0 \\ 2 & 0 &     1 &      1 &      0 &     0 &     0 &    0 &   0 &  0  & 0 \\ 3 & 0 &     2 &      3 &      1 &     0 &     0 &    0 &   0 &  0  & 0 \\ 4 & 0 &     6 &     11 &      6 &     1 &     0 &    0 &   0 &  0  & 0 \\ 5 & 0 &    24 &     50 &     35 &    10 &     1 &    0 &   0 &  0  & 0 \\ 6 & 0 &   120 &    274 &    225 &    85 &    15 &    1 &   0 &  0  & 0 \\ 7 & 0 &   720 &   1764 &   1624 &   735 &   175 &   21 &   1 &  0  & 0 \\ 8 & 0 &  5040 &  13068 &  13132 &  6769 &  1960 &  322 &  28 &  1  & 0 \\ 9 & 0 & 40320 & 109584 & 118124 & 67284 & 22449 & 4536 & 546 & 36  & 1 \\ \end{array}$$

Stirling's Triangle of the Second Kind
$$\begin{array}{r|rrrrrrrrrr} n & \left\{{n \atop 0}\right\} & \left\{{n \atop 1}\right\} & \left\{{n \atop 2}\right\} & \left\{{n \atop 3}\right\} & \left\{{n \atop 4}\right\} & \left\{{n \atop 5}\right\} & \left\{{n \atop 6}\right\} & \left\{{n \atop 7}\right\} & \left\{{n \atop 8}\right\} & \left\{{n \atop 9}\right\} \\ [4pt] \hline 0 & 1 & 0 &  0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 1 & 0 & 1 &   0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 2 & 0 & 1 &   1 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 3 & 0 & 1 &   3 &    1 &    0 &    0 &    0 &   0 &  0  & 0 \\ 4 & 0 & 1 &   7 &    6 &    1 &    0 &    0 &   0 &  0  & 0 \\ 5 & 0 & 1 &  15 &   25 &   10 &    1 &    0 &   0 &  0  & 0 \\ 6 & 0 & 1 &  31 &   90 &   65 &   15 &    1 &   0 &  0  & 0 \\ 7 & 0 & 1 &  63 &  301 &  350 &  140 &   21 &   1 &  0  & 0 \\ 8 & 0 & 1 & 127 &  966 & 1701 & 1050 &  266 &  28 &  1  & 0 \\ 9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36  & 1 \\ \end{array}$$

Compare with

 * Pascal's Triangle