Concrete mathematics : a foundation for computer science

230 BINOMIAL COEFFICIENTS
valid for all n # -1. The shifted version of (5.125) is
-1 L+l/(l-2) F
ZZ--l-z
1+1
(5.130)
With a bit of patience, we can compute a few more indefinite summation
identities that are potentially useful:
Exercises
Warmups
a, 2+(1-a)z/(l-z), 1
l+(l-a)z/(l-z),2
a, b,
c+l, (c-ab)/(c-a-b+l), 2
c+l, a+b-c+l
= (c)(c-b-a)
(c - a)(c - b)
F (,,“dI;l,j ‘)k. (5.133)
What is 1 l4 ? Why is this number easy to compute, for a person who
knows binomial coefficients?
For which value(s) of k is (i) a maximum, when n is a given positive
integer? Prove your answer.
Prove the hexagon property, (;I:) (k:,) (nk+‘) = (“i’) (i,‘:) (,“,).
Evaluate (-,‘) by negating (actually un-negating) its upper index.
Let p be prime. Show that (F) mod p = 0 for 0 < k < p. What does this
imply about the binomial coefficients (“i’)?
Fix up the text’s derivation in Problem 6, Section 5.2, by correctly ap- A caseof
plying symmetry. mistaken identity.
Is (5.34) true also when k < O?