Definition:Stirling's Triangle of the First Kind (Unsigned)

Definition

Stirling's Triangle of the First Kind (Unsigned) is formed by arranging unsigned Stirling numbers of the first kind as follows:

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

This sequence is A094216 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see

• Definition:Stirling's Triangle of the First Kind (Signed)
• Definition:Stirling's Triangle of the Second Kind

Compare with

• Pascal's Triangle

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 $2$