ABSTRACT
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
k
) =
( n+ 1
k
) − (
n
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.
