Concrete mathematics : a foundation for computer science

6
Special Numbers
SOME SEQUENCES of numbers arise so often in mathematics that we recognize
them instantly and give them special names. For example, everybody
who learns arithmetic knows the sequence of square numbers (1,4,9,16, . . ).
In Chapter 1 we encountered the triangular numbers (1,3,6,10, . . . ); in Chapter
4 we studied the prime numbers (2,3,5,7,. . .); in Chapter 5 we looked
briefly at the Catalan numbers (1,2,5,14, . . . ).
In the present chapter we’ll get to know a few other important sequences.
First on our agenda will be the Stirling numbers {t} and [L] , and the Eulerian
numbers (i); these form triangular patterns of coefficients analogous to the
binomial coefficients (i) in Pascal’s triangle. Then we’ll take a good look
at the harmonic numbers H,, and the Bernoulli numbers B,; these differ
from the other sequences we’ve been studying because they’re fractions, not
integers. Finally, we’ll examine the fascinating Fibonacci numbers F, and
some of their important generalizations.
6.1 STIRLING NUMBERS
We begin with some close relatives of the binomial coefficients, the
Stirling numbers, named after James Stirling (1692-1770). These numbers
come in two flavors, traditionally called by the no-frills names “Stirling numbers
of the first and second kind!’ Although they have a venerable history
and numerous applications, they still lack a standard notation. We will write
{t} for Stirling numbers of the second kind and [z] for Stirling numbers of
the first kind, because these symbols turn out to be more user-friendly than
the many other notations that people have tried.
Tables 244 and 245 show what {f;} and [L] look like when n and k are
small. A problem that involves the numbers “1, 7, 6, 1” is likely to be related
to {E}, and a problem that involves “6, 11, 6, 1” is likely to be related to
[;I, just as we assume that a problem involving “1, 4, 6, 4, 1” is likely to be
related to (c); these are the trademark sequences that appear when n = 4.
243