Definition:Stirling Numbers of the First Kind

Definition
Stirling Numbers of the First Kind come in two forms.

In the below, $$n$$ and $$k$$ are always non-negative integers.

Unsigned Stirling Numbers of the First Kind
These are defined recursively by:


 * $$\left[{n \atop k}\right] = \begin{cases}

\delta_{n k} & : k = 0 \text{ or } n = 0 \\ \left[{n-1 \atop k-1}\right] + \left({n-1}\right) \left[{n-1 \atop k}\right] & : \text{otherwise} \\ \end{cases}$$

where $$\delta_{nk}$$ is the Kronecker delta.

Signed Stirling Numbers of the First Kind
These are defined recursively by:


 * $$s(n,k) = \begin{cases}

\delta_{n k} & : k = 0 \text{ or } n = 0 \\ s(n-1,k-1) - \left({n-1}\right) s(n-1,k) & : \text{otherwise} \\ \end{cases}$$

where $$\delta_{nk}$$ is the Kronecker delta.

Stirling's Triangle of the First Kind
Arranging the values into a table, we obtain Stirling's Triangle of the First Kind:

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

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

Compare with

 * Stirling Numbers of the Second Kind
 * Pascal's Triangle

Notation
The notation given here for the unsigned type is that proposed by Jovan Karamata and publicised by Donald E. Knuth.

The notation given here for the signed type is similar to alternative versions of the unsigned. Usage is inconsistent in the literature.