1. Generating Function and Applications
Chapter 7
Generating Function and Applications

3. 7.2 Generating Function

4. Definition of Generating Function G(x):
e.g.
vs.

5. e.g. 1.2

6. e.g. 1.4 =?

7. e.g. 2.2 (NCKU)
Show that
Constant term is

8. Considering
constant term is

9. 7.3 Application(1): Solving Recurrence

10. Example 1
Solve with and
From and , we have
From , we have

11. From original recurrence and boundary conditions, we further have

12. We have
Final solution is

13. Reading Assignment:
Solve nonhomogeneous recurrence
By using generating function.

14. 7.4 Application(2): Combinatorial Counting

15. Given n nodes, how many rooted binary trees that we can build up?
Example 4
Given n nodes, how many rooted binary trees that we can build up?
When n=5, there are five trees.

16. Recurrence:
We still need the boundary conditions:
, , and

17. We have
and
It yields to

19. Derivation: