Concrete mathematics : a foundation for computer science

302 SPECIAL NUMBERS
61 Prove the identity
What is ~~=, 1 /FJ.2k?
integer n 3 1.
62 Let A, = 4” + @-” and B, = 4” - a-“.
a Find constants OL and B such that A,, = aA,-1 + @An-2 and B, =
OLB~-I + BBn-2 for all n 3 0.
b Express A,, and B, in terms of F, and L, (see exercise 28).
C Prove that xE=, 1 ,/(Fzk+l + 1) = B,/A,+l.
d Find a closed form for EL=, l/(F~k+, - 1).
Bonus problems Bogus problems
63 How many permutations 7~1~2.. . rrn of {1,2,. . . , n} have exactly k indices
j such that
a rri < 7Cj for all i < j? (Such j are called “left-to-right maxima!‘)
b nj > j? (Such j are called “excedances!‘)
64 What is the denominator of [,j/f,], when this fraction is reduced to
lowest terms?
65 Prove the identity
1
. . .
1
s 0 s 0
n f(k)
f(lx, +...+x,])dx, . ..dx. = x k nl.
k 0 ’
66 Show that ((y)) = 2(y), and find a closed form for ((y)).
67 Find a closed form for Et=, k’H,,+k.
68 Show that the generalized harmonic numbers of exercise 22 have the
power series expansion
H, = x(-l)nHL)zn-‘.
n>2
69 Prove that the generalized factorial of equation (5.83) can be written
by considering the limit as n + 00 of the first n factors of this infinite
product. Show that -&(z!) is related to the general harmonic numbers of
exercise 22. .