Concrete mathematics : a foundation for computer science

366 GENERATING FUNCTIONS
53 The sequence of pentagonal numbers (1,5,12,22,. . . ) generalizes the
triangular and square numbers in an obvious way:
Let the nth triangular number be T,, = n(n+1)/2; let the nth pentagonal
number be P, = n(3n - 1)/2; and let Ll,, be the 3 x n domino-tiling
number defined in (7.38). Prove that the triangular number TIuq,+Lml i,z
is also a pentagonal number. Hint: 3Ui, � = (Vznml + Vln+l)’ + 2.
54 Consider the following curious construction:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 . . .
12 3 4 6 7 8 9 11 12 13 14 16 . . .
1 3 610 16 23 31 40 51 63 76 90 106 . . .
13 6 16 23 31 51 63 76 106 . . .
1 4 10 26 49 80 131 194 270 376 . . .
14 26 49 131 194 376 . . .
15 31 80 211 405 781 . . .
1 31 211 781 . . .
1 32 243 1024 . . .
(Start with a row containing all the positive integers. Then delete every
mth column; here m = 5. Then replace the remaining entries by partial
sums. Then delete every (m - 1 )st column. Then replace with partial
sums again, and so on.) Use generating functions to show that the final
result is the sequence of mth powers. For example, when m = 5 we get
(15,25,35,45 ,...) asshown.
55 Prove that if the power series F(z) and G(z) are differentiably finite (as
defined in exercise 20), then so are F(z) + G(z) and F(z)G(z).
Research problems
56 Prove that there is no “simple closed form” for the coefficient of Z” in
(1 + z + z~)~, as a function of n, in some large class of “simple closed
forms!’
5’7 Prove or disprove: If all the coefficients of G(z) are either 0 or 1, and if
all the coefficients of G (2)’ are less than some constant M, then infinitely
many of the coefficients of G(z)’ are zero.