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.

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

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

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

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