Concrete mathematics : a foundation for computer science

238 BINOMIAL COEFFICIENTS
72 Prove that, if m, n, and k are integers and n > 0,
n2k-v(k)
is an integer,
where v(k) is the number of l’s in the binary representation of k.
73 Use the repertoire method to solve the recurrence
X0 = a; x, := p;
Xn = (n-1)(X,-j +X,-2),
for n > 1
Hint: Both n! and ni satisfy this recurrence.
74 This problem concerns a deviant version of Pascal’s triangle in which the
sides consist of the numbers 1, 2, 3, 4, . . . instead of all l’s, although the
interior numbers still satisfy the addition formula:
i
1
i
2 2
343 :
I
S’
4 7 7 4
5 v4 l1 5
G, . ii0 i/., $ lb .b
If ((t)) denotes the kth number in row n, for 1 < k < n, we have
((T)) = ((t)) = n, and ((L)) = ((“,‘)) + ((:I:)) for 1 < k < n. Express
the quantity ((i)) in closed form.
75 Find a relation between the functions
” (n) = ; (31;: 1) ’
S2(n) = 6 (Sk”, 2)
and the quantities 12”/.3J and [2n/31.
76 Solve the following recurrence for n, k 3 0:
Q n,O = 1; Qo,k = [k=Ol;
Q n,k = Qn-l,k + Qn-l,k-, + for n, k > 0.