Definition:Stirling's Triangle of the Second Kind
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Definition
Stirling's Triangle of the Second Kind is formed by arranging Stirling numbers of the second kind as follows:
$\begin{array}{r|rrrrrrrrrr}
n & \left\{{n \atop 0}\right\} & \left\{{n \atop 1}\right\} & \left\{{n \atop 2}\right\} & \left\{{n \atop 3}\right\} & \left\{{n \atop 4}\right\} & \left\{{n \atop 5}\right\} & \left\{{n \atop 6}\right\} & \left\{{n \atop 7}\right\} & \left\{{n \atop 8}\right\} & \left\{{n \atop 9}\right\} \\
\hline
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
3 & 0 & 1 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 0 & 1 & 7 & 6 & 1 & 0 & 0 & 0 & 0 & 0 \\
5 & 0 & 1 & 15 & 25 & 10 & 1 & 0 & 0 & 0 & 0 \\
6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 & 0 & 0 & 0 \\
7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 & 0 & 0 \\
8 & 0 & 1 & 127 & 966 & 1701 & 1050 & 266 & 28 & 1 & 0 \\
9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36 & 1 \\
\end{array}$
This sequence is A008277 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
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Source of Name
This entry was named for James Stirling.
Sources
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.6$: Table $2$
