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

## Definition

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

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

## Source of Name

This entry was named for James Stirling.