Definition:Stirling's Triangle of the First Kind (Signed)

Definition
Stirling's Triangle of the First Kind (Signed) is formed by arranging signed Stirling numbers of the first kind as follows:

$$\begin{array}{r|rrrrrrrrrr} n & s(n,0) & s(n,1) & s(n,2) & s(n,3) & s(n,4) & s(n,5) & s(n,6) & s(n,7) & s(n,8) & s(n,9) \\ [4pt] \hline 0 & 1 &    0 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 1 & 0 &     1 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 2 & 0 &    -1 &       1 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 3 & 0 &     2 &      -3 &      1 &      0 &     0 &     0 &   0 &   0  & 0 \\ 4 & 0 &    -6 &      11 &     -6 &      1 &     0 &     0 &   0 &   0  & 0 \\ 5 & 0 &    24 &     -50 &     35 &    -10 &     1 &     0 &   0 &   0  & 0 \\ 6 & 0 &  -120 &     274 &   -225 &     85 &   -15 &     1 &   0 &   0  & 0 \\ 7 & 0 &   720 &   -1764 &   1624 &   -735 &   175 &   -21 &   1 &   0  & 0 \\ 8 & 0 & -5040 &   13068 & -13132 &   6769 & -1960 &   322 & -28 &   1  & 0 \\ 9 & 0 & 40320 & −109584 & 118124 & −67284 & 22449 & −4536 & 546 & −36  & 1 \\ \end{array}$$

Also see

 * Definition:Stirling's Triangle of the First Kind (Unsigned)
 * Definition:Stirling's Triangle of the Second Kind

Compare with

 * Pascal's Triangle