Concrete mathematics : a foundation for computer science

series explicitly for future reference:
= .qy)& = 1-y.
k
5.4 GENERATING FUNCTIONS 2~1
(5.68)
(5%)
(5.70)
(5.71)
(5.72)
(5.73)
The coefficients (y) $ of BZ (z) are called the Catalan numbers C,, because
Eugene Catalan wrote an influential paper about them in the 1830s [46]. The
sequence begins as follows:
n 0 ’ 2 3 4 5 6 7 8 9 10
G 1 1 2 5 14 42 ‘32 429 ‘430 4862 ‘6796
The coefficients of B-1 (z) are essentially the same, but there’s an extra 1 at the
beginning and the other numbers alternate in sign: (1, 1, -1,2, -5,14,. . . ).
Thus BP1 (z) = 1 + zBz(-z). We also have !B 1 (z) = %2(-z) ‘.
Let’s ClOSe this section by deriving an important consequence of (5.72)
and (5.73), a relation that shows further connections between the functions
L!L, (z) and ‘Bz(-z):
B-1 (z)n+’ - (-Z)n+‘B~(-Z)n+’
VTFG
= x (yk)z,
k