Concrete mathematics : a foundation for computer science

7 EXERCISES 359
15 The Bell number b, is the number of ways to partition n things into
subsets. For example, bs = 5 because we can partition {l ,2,3} in the
following ways:
Prove that b,+l = x.k (L)bnpk, and use this recurrence to find a closed
form for the exponential generating function I,, b,z”/n!.
16 Two sequences (a,,) and (b,,) are related by the convolution formula
b, =
k,i-Zkz+...nk,=n
(al+il-1) ((12+:-l) ,., (an+:-‘) ;
also as = 0 a:nd bo = 1. Prove that the corresponding generating functions
satisfy l:nB(z) =A(z) + iA + iA(z3) +....
17 Show that the exponential generating function G(z) of a sequence is related
to the ordinary generating function G(z) by the formula
G(zt)e-‘dt = G(z),
Jm0 if the integral exists.
18 Find the Dirichlet generating functions for the sequences
a sn=@;
b g,, = Inn.;
C gn = [n is squarefree].
Express your answers in terms of the zeta function. (Squarefreeness is
defined in exercise 4.13.)
19 Every power series F(z) = x naO f,z” with fo = 1 defines a sequence of
polynomials f,,(x) by the rule
F(z)' = ~f,(x)z",
II>0
where f,( 1) = f, and f,(O) = [n = 01. In general, f,(x) has degree n.
Show that such polynomials always satisfy the convolution formulas
f fk(X)fn-k(Y) = fn(x +Y) ;
k=O
(x+Y)kkfk(x)fnpk(Y) = Xnf,(X+y).
kzo
(The identities in Tables 202 and 258 are special cases of this trick.)