Concrete mathematics : a foundation for computer science

7.6 EXPONENTIAL GENERATING FUNCTIONS 353
And the latter sumI is a geometric progression, so there’s a closed form
S(z,n) = $+. (7.77)
All we need to do is figure out the coefficients of this relatively simple function,
and we’ll know S,i:n), because S,(n) = m! [z”‘]S(z,n).
Here’s where 13ernoulli numbers come into the picture. We observed a
moment ago that t.he egf for Bernoulli numbers is
hence we can write
enz-1
S(z) = B(z) - -
z
= Bo~.+B,~+Bz~+...)(n~+n2~+n3~+-..)
(
The sum S,(n) is m! times the coefficient of z”’ in this product. For example,
So(n) = O! (h3&)
nL n
S(n) = 1! Elom+Blm (
>
f%(n) = .2! Bo$ + B1 & + B2 &) =
( . . . . * .
n;
= .!n2-d-n;
in3-tn2+in.
We have therefore derived the formula 0, = Sz(n) = $n(n - i)(n - 1) for
the umpteenth time, and this was the simplest derivation of all: In a few lines
we have found the general behavior of S,(n) for all m.
The general fo:rmula can be written
%-l(n) = &EL,(n) - B,(O)) ,
where B,(x) is the Bernoulli polynomial defined by
B,(x) = t (;)BkX-‘.
k
(7.78)
(7.79)
Here’s why: The Bernoulli polynomial is the binomial convolution of the
sequence (Bo, B1, B;r, . . . ) with (1, x,x2,. . . ); hence the exponential generating