Definition:Pascal's Triangle/Diagonal
Definition
Consider Pascal's Triangle:
- $\begin{array}{r|rrrrrrrrrr}
n & \binom n 0 & \binom n 1 & \binom n 2 & \binom n 3 & \binom n 4 & \binom n 5 & \binom n 6 & \binom n 7 & \binom n 8 & \binom n 9 & \binom n {10} & \binom n {11} & \binom n {12} \\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 1 & 3 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 1 & 4 & 6 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 5 & 1 & 5 & 10 & 10 & 5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 6 & 1 & 6 & 15 & 20 & 15 & 6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 7 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 & 0 & 0 & 0 & 0 & 0 \\ 8 & 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 & 0 & 0 & 0 & 0 \\ 9 & 1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1 & 0 & 0 & 0 \\ 10 & 1 & 10 & 45 & 120 & 210 & 252 & 210 & 120 & 45 & 10 & 1 & 0 & 0 \\ 11 & 1 & 11 & 55 & 165 & 330 & 462 & 462 & 330 & 165 & 55 & 11 & 1 & 0 \\ 12 & 1 & 12 & 66 & 220 & 495 & 792 & 924 & 792 & 495 & 220 & 66 & 12 & 1 \\ \end{array}$
The $n$th diagonal of Pascal's triangle consists of the entries $\dbinom {n + m} m$ for $m \ge 0$:
- $\dbinom n 0, \dbinom {n + 1} 1, \dbinom {n + 2} 2, \dbinom {n + 3} 3, \ldots$
Hence the diagonal leading down and to the right from $\dbinom 0 0$, containing all $1$s, is identified as the zeroth diagonal, or diagonal $0$.
Edge Cells
The numbers in column $0$ and in diagonal $0$, containing all $1$s, are referred to as the edge cells.
Also see
- Definition:Row of Pascal's Triangle
- Definition:Column of Pascal's Triangle
- Definition:Lesser Diagonal of Pascal's Triangle
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $35$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $35$
- Weisstein, Eric W. "Pascal's Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PascalsTriangle.html