Definition:Extended Pascal's Triangle
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Theorem
Pascal's triangle can be extended for binomial coefficients of negative integers as follows:
$\begin{array}{r|rrrrrrrrrr}
n & \binom n 0 & \binom n 1 & \binom n 2 & \binom n 3 & \binom n 4 & \binom n 5 & \binom n 6 & \binom n 7 & \binom n 8 & \binom n 9 & \binom n {10} & \binom n {11} & \binom n {12} \\
\hline
-3 & 1 & -3 & 6 & -10 & 15 & -21 & 28 & -36 & 45 & -55 & 66 & -78 & 91 \\
-2 & 1 & -2 & 3 & -4 & 5 & -6 & 7 & -8 & 9 & -10 & 11 & -12 & 13 \\
-1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 1 & 3 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 1 & 4 & 6 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 5 & 1 & 5 & 10 & 10 & 5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 6 & 1 & 6 & 15 & 20 & 15 & 6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}$
Construction
The numbers for negatively indexed binomial coefficients can be found by application of Pascal's Rule:
- $\dbinom n k = \dbinom {n + 1} k - \dbinom n {k - 1}$
From Binomial Coefficient with Zero, we have:
- $\forall n \in \Z: \dbinom n 0 = 1$
Thus:
| \(\ds \dbinom {-1} 1\) | \(=\) | \(\ds \dbinom 0 1 - \dbinom {-1} 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0 - 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -1\) |
| \(\ds \dbinom {-1} 2\) | \(=\) | \(\ds \dbinom 0 2 - \dbinom {-1} 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0 - \left({-1}\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
and so on.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $6$