Chapter 4: Calculating Coefficient of Generating Functions
(TB1 Ch.3, Article 2)
1. Introduction to Generating Functions
A generating function is a formal power series in which the coefficients encode a sequence.
If a sequence is [ a_0, a_1, a_2, a_3, \dots ]
its ordinary generating function (OGF) is: [ G(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots ]
In discrete mathematics, we are not concerned with convergence, only with coefficients.
2. Why Generating Functions Are Important
Generating functions allow us to:
- Count combinatorial objects
- Solve recurrence relations
- Model constrained counting problems
- Extract coefficients systematically
In CS, they appear in:
- Algorithm analysis
- Counting paths, strings, trees
- Recurrences and complexity analysis
3. Coefficient Operator
The notation: [ [x^n]G(x) ] means the coefficient of (x^n) in the generating function (G(x)).
Example
If: [ G(x) = 3 + 5x + 7x^2 + 9x^3 ]
Then:
- ([x^0]G(x) = 3)
- ([x^2]G(x) = 7)
4. Standard Generating Functions and Their Coefficients
These are very important for exams.
(a) Geometric Series
[ \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots ]
[ [x^n]\frac{1}{1-x} = 1 ]
(b) Shifted Geometric Series
[ \frac{1}{1 - ax} = 1 + ax + a^2x^2 + a^3x^3 + \cdots ]
[ [x^n]\frac{1}{1-ax} = a^n ]
(c) Power of Geometric Series
[ \frac{1}{(1-x)^k} ]
Coefficient: [ [x^n]\frac{1}{(1-x)^k} = \binom{n+k-1}{k-1} ]
Example
Find the coefficient of (x^4) in: [ \frac{1}{(1-x)^3} ]
[ [x^4] = \binom{4+3-1}{3-1} = \binom{6}{2} = 15 ]
5. Coefficient Extraction Using Algebraic Manipulation
Often, we manipulate generating functions into a known standard form.
Example 1
Find the coefficient of (x^5) in: [ \frac{x^2}{1 - x} ]
Step 1: Rewrite [ \frac{x^2}{1-x} = x^2(1 + x + x^2 + \cdots) ]
Step 2: Expand [ = x^2 + x^3 + x^4 + x^5 + \cdots ]
Answer [ [x^5] = 1 ]
Example 2
Find the coefficient of (x^6) in: [ \frac{x^3}{(1-x)^2} ]
We know: [ \frac{1}{(1-x)^2} = \sum_{n=0}^{\infty} (n+1)x^n ]
So: [ \frac{x^3}{(1-x)^2} = \sum_{n=0}^{\infty} (n+1)x^{n+3} ]
Set: [ n+3 = 6 \Rightarrow n = 3 ]
Coefficient: [ n+1 = 4 ]
6. Coefficient of Product of Generating Functions
If: [ A(x) = \sum a_n x^n,\quad B(x) = \sum b_n x^n ]
Then: [ x^n = \sum_{k=0}^{n} a_k b_{n-k} ]
This is called convolution.
Example
Find the coefficient of (x^3) in: [ (1 + x + x^2)(1 + x + x^2) ]
Expand: [ = 1 + 2x + 3x^2 + 2x^3 + x^4 ]
[ [x^3] = 2 ]
7. Generating Functions for Counting Problems
Example: Number of Solutions
Find the number of non-negative integer solutions to: [ x_1 + x_2 + x_3 = 5 ]
Each variable has generating function: [ 1 + x + x^2 + \cdots = \frac{1}{1-x} ]
Total generating function: [ \left(\frac{1}{1-x}\right)^3 ]
Coefficient: [ [x^5]\frac{1}{(1-x)^3} = \binom{5+3-1}{3-1} = \binom{7}{2} = 21 ]
8. Coefficient with Constraints
Example
Find the number of solutions to: [ x_1 + x_2 + x_3 = 6,\quad x_1 \ge 1 ]
For (x_1 \ge 1): [ x + x^2 + x^3 + \cdots = \frac{x}{1-x} ]
Total generating function: [ \frac{x}{1-x} \cdot \frac{1}{1-x} \cdot \frac{1}{1-x} = \frac{x}{(1-x)^3} ]
Coefficient: [ [x^6]\frac{x}{(1-x)^3} = [x^5]\frac{1}{(1-x)^3} ]
[ = \binom{5+3-1}{2} = \binom{7}{2} = 21 ]
9. General Strategy for Exam Problems
- Write the generating function
- Convert to a known standard form
- Shift powers if needed
- Extract coefficient using formula
- Clearly show steps
10. Common Exam Mistakes
- Forgetting power shifts
- Using wrong binomial coefficient
- Not identifying correct generating function
- Algebraic simplification errors
⭐ Exam-Important Formulas (Memorize)
[ \frac{1}{1-x} = \sum x^n ]
[ \frac{1}{(1-x)^k} = \sum \binom{n+k-1}{k-1}x^n ]
[ x^n = \sum_{k=0}^n a_k b_{n-k} ]