Concrete mathematics : a foundation for computer science

450 ASYMPTOTIC3
a rough estimate and plugging it into the recurrence; in this way we can often
derive better and better estimates, “pulling ourselves up by our bootstraps.”
Here’s another problem that illustrates bootstrapping nicely: What is the
asymptotic value of the coefficient g,, = [zn] G(z) in the generating function
G(z) = exp(t $) ,
k>l
as n + oo? If we differentiate this equation with respect to z, we find
G’(z) = F ngnznpl = (1 y) G(z) ;
n=O k>l
equating coefficients of zn-’ on both sides gives the recurrence
wh =
O