Definition:Stirling's Triangle of the Second Kind
Definition
Stirling's Triangle of the Second Kind is formed by arranging Stirling numbers of the second kind as follows:
$\begin{array}{r|rrrrrrrrrr}
n & {n \brace 0} & {n \brace 1} & {n \brace 2} & {n \brace 3} & {n \brace 4} & {n \brace 5} & {n \brace 6} & {n \brace 7} & {n \brace 8} & {n \brace 9} \\
\hline
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
3 & 0 & 1 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 0 & 1 & 7 & 6 & 1 & 0 & 0 & 0 & 0 & 0 \\
5 & 0 & 1 & 15 & 25 & 10 & 1 & 0 & 0 & 0 & 0 \\
6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 & 0 & 0 & 0 \\
7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 & 0 & 0 \\
8 & 0 & 1 & 127 & 966 & 1701 & 1050 & 266 & 28 & 1 & 0 \\
9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36 & 1 \\
\end{array}$
This sequence is A008277 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Also see
- Definition:Stirling's Triangle of the First Kind (Unsigned)
- Definition:Stirling's Triangle of the First Kind (Signed)
Source of Name
This entry was named for James Stirling.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Table $2$