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.