Definition:Extended Pascal's Triangle

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:

and so on.