Generating Functions

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# Generating Functions ## 4.1 Intuition: Encoding Sequences as Power Series A generating function wraps an entire sequence into a single algebraic expression. Operations on the series translate to operations on the sequence.

Generating Functions

4.1 Intuition: Encoding Sequences as Power Series

A generating function wraps an entire sequence into a single algebraic expression. Operations on the series translate to operations on the sequence.

🔑 Key Insight: Generating functions turn combinatorial problems into algebra problems.

4.2 Ordinary Generating Functions (OGF)

For sequence

\{a_n\}

G(x) = \sum_{n=0}^\infty a_n x^n

Key OGFs

$\sum_{n=0}^\infty x^n = \frac{1}{1-x}$
$\sum_{n=0}^\infty \binom{n+k-1}{k-1} x^n = \frac{1}{(1-x)^k}$
$\sum_{n=0}^\infty n x^n = \frac{x}{(1-x)^2}$
$\sum_{n=0}^\infty \binom{m}{n} x^n = (1+x)^m$

4.3 Using OGF to Solve Recurrences

Example: Fibonacci

F_n = F_{n-1} + F_{n-2}

Let

G(x) = \sum F_n x^n

G(x) = F_0 + F_1 x + \sum_{n=2}^\infty (F_{n-1} + F_{n-2}) x^n

= 0 + x + x\sum_{n=1}^\infty F_n x^n + x^2\sum_{n=0}^\infty F_n x^n

= x + x G(x) + x^2 G(x)

G(x)(1 - x - x^2) = x

G(x) = \frac{x}{1-x-x^2}

4.4 Exponential Generating Functions (EGF)

E(x) = \sum_{n=0}^\infty a_n \frac{x^n}{n!}

EGF is natural for labeled structures (permutations, sets).

✅ Practice Questions

Q1: Find the OGF for

a_n = 2^n

Solution

$G(x) = \sum 2^n x^n = \sum (2x)^n = \frac{1}{1-2x}$ . Q2: Find the coefficient of $x^5$ in $(1-x)^{-3}$ . Solution

$(1-x)^{-3} = \sum \binom{n+2}{2} x^n$ . Coefficient of $x^5$ : $\binom{7}{2} = 21$ . Join Discord PreviousRecurrence ApplicationsNextGraph Theory Basics