Do you know that Pascal’s triangle extends upward? The extended triangle gives the coefficients of binomial series for negative exponents.
In Figure 5.1, Pascal’s identity is used to calculate binomial coefficients( n k
) with negative values of n. (In the figure, the triangle is left-justified and
some entries are padded with 0s to aid in the calculation.) The recurrence relation is (
( n+ 1
) − (
k − 1 ) , k ≥ 0,
and we define ( n −1
) = 0, for all n. Try to verify some of the entries in
the extended Pascal’s triangle. Do you recognize the values? They are the numbers (−1)k(n+k−1k ), given by the binomial series theorem for the coefficients of x in the expansion of (1 + x)−n.