Concrete mathematics : a foundation for computer science

350 GENERATING FUNCTIONS
Chapter 5 also mentions the generalized exponential series
which is said in (5.60) to hzve an equally remarkable property:
[z”] Et(=)’ = etn +-,w
We can prove this as a limiting case of the formulas for ‘BBt (z), because it is
not difficult to show that
7.6 EXPONEN’I’IAL GF’S
Sometimes a sequence (gn) has a generating function whose properties
are quite complicated, while the related sequence (g,/n!) has a generating
function that’s quite simple. In such cases we naturally prefer to work with
(gJn!) and then multiply by n! at the end. This trick works sufficiently
often that we have a special name for it: We call the power series
(7.71)
the exponential generating function or ‘O n>l
zn
G-l j&y = w-1 - ;
n!
n>O
this is the egf of (0, go,Zgl, . . .) = (ng,-1).
Differentiating the egf of (go, 91, g2, . . . ) with respect to z gives Are we having
fun yet?
(7.72)