Concrete mathematics : a foundation for computer science

184 BINOMIAL COEFFICIENTS
We can’t quite replace the inner sum using the result of Problem 6,
because it has the extra condition k > j - n + 1. But this extra condition
is superfluous unless j - n + 1 > 0; that is, unless j > n. And when j 3 n,
the first binomial coefficient of the inner sum is zero, because its upper index
is between 0 and k - 1, thus strictly less than the lower index 2k. We may
therefore place the additional restriction j < n on the outer sum, without
affecting which nonzero terms are included. This makes the restriction k 3
j - n + 1 superfluous, and we can use the result of Problem 6. The double
sum now comes tumbling down:
I&) x ~+k;l-i)~;)%
, k>j-n+l
k>O
= t (,:,)In-1-j=O] = (:I:).
06j 0, we can do the same thing if we expand 1 /(k + 1 + m) into
absorbable terms. And our luck still holds: We proved a suitable identity
-1
r+l integer m 3 0,
= r+l-m’ 7-g {O,l,..., m-l}.
(5.33)