Definition:Stirling Numbers of the Second Kind

Definition
Stirling Numbers of the Second Kind 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\} + k \left\{{n-1 \atop k}\right\} & : \text{otherwise} \\ \end{cases}$$

where:
 * $$\delta_{nk}$$ is the Kronecker delta;
 * $$n$$ and $$k$$ are always non-negative integers.

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

Compare with
Stirling Numbers of the First Kind

Notation
The notation given here is that proposed by Jovan Karamata and publicised by Knuth.

Other notations exist, but usage is inconsistent in the literature.