# Definition:Stirling's Triangles

## Definition

Stirling's Triangles are the arrays formed by arranging Stirling's Numbers of the first and second kind, as follows:

### Stirling's Triangle of the First Kind (Unsigned)

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

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

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

### Stirling's Triangle of the Second Kind

$\begin{array}{r|rrrrrrrrrr} n & {n \brace 0} & {n \brace 1} & {n \brace 2} & {n \brace 3} & {n \brace 4} & {n \brace 5} & {n \brace 6} & {n \brace 7} & {n \brace 8} & {n \brace 9} \\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 1 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 0 & 1 & 7 & 6 & 1 & 0 & 0 & 0 & 0 & 0 \\ 5 & 0 & 1 & 15 & 25 & 10 & 1 & 0 & 0 & 0 & 0 \\ 6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 & 0 & 0 & 0 \\ 7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 & 0 & 0 \\ 8 & 0 & 1 & 127 & 966 & 1701 & 1050 & 266 & 28 & 1 & 0 \\ 9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36 & 1 \\ \end{array}$

## Source of Name

This entry was named for James Stirling.