Chapter 1Sequences

1.1 Sequences as Lists1.2 Recurrences1.3 Pascal’s Recurrence1.4 Differences and Partial Sums1.5 Falling Powers1.6 Stirling Numbers: A Preview1.7 Ordinary Generating Functions1.8 Synthesizing Generating Functions1.9 Asymptotic Estimates

1

2 Chapter 1 Sequences

Section 1.1 Sequences as Lists 3

1.1 SEQUENCES AS LISTS

4 Chapter 1 Sequences

Section 1.1 Sequences as Lists 5

Fast-Growing Sequences

6 Chapter 1 Sequences

Slow-Growing Sequences

Section 1.1 Sequences as Lists 7

Bounded Sequences

8 Chapter 1 Sequences

Periodic Sequences

Section 1.1 Sequences as Lists 9

Generalizations

10 Chapter 1 Sequences

Eventual Behavior of Sequences

Fig 1.1.1

Section 1.1 Sequences as Lists 11

Fig 1.1.2

12 Chapter 1 Sequences

1.2 RECURRENCES

Section 1.2 Recurrences 13

Method of Small Cases

14 Chapter 1 Sequences

Tower of Hanoi

Fig 1.2.1

Section 1.2 Recurrences 15

16 Chapter 1 Sequences

Fibonacci Sequence

Section 1.2 Recurrences 17

Catalan Sequence

18 Chapter 1 Sequences

Proving Properties of Sequences

Section 1.2 Recurrences 19

Fig 1.2.2

20 Chapter 1 Sequences

Fig 1.2.3

Section 1.3 Pascal’s Recurrence 21

1.3 PASCAL’S RECURRENCE

Fig 1.3.1

22 Chapter 1 Sequences

Binomial Coefficients

Section 1.3 Pascal’s Recurrence 23

24 Chapter 1 Sequences

Section 1.4 Differences and Sums 25

Table 1.3.1

26 Chapter 1 Sequences

1.4 DIFFERENCES AND SUMS

Section 1.4 Differences and Sums 27

Properties of the Difference Function

28 Chapter 1 Sequences

Section 1.4 Differences and Sums 29

Summation Operator

30 Chapter 1 Sequences

Section 1.4 Differences and Sums 31

32 Chapter 1 Sequences

Fig 1.4.1

Section 1.4 Differences and Sums 33

Fig 1.4.2

34 Chapter 1 Sequences

Growth Rate of Sequences

Section 1.4 Differences and Sums 35

36 Chapter 1 Sequences

1.5 FALLING POWERS

Section 1.5 Falling Powers 37

38 Chapter 1 Sequences

Unimodal Sequences

Fig 1.5.1

Section 1.5 Falling Powers 39

40 Chapter 1 Sequences

Log-Concavity and Log-Convexity

Section 1.5 Falling Powers 41

42 Chapter 1 Sequences

Section 1.5 Falling Powers 43

44 Chapter 1 Sequences

1.6 STIRLING NUMBERS: PREVIEW

Falling into Ordinary Powers

Section 1.6 Stirling Numbers: Preview 45

46 Chapter 1 Sequences

Section 1.6 Stirling Numbers: Preview 47

Ordinary into Falling Powers

48 Chapter 1 Sequences

Section 1.6 Stirling Numbers: Preview 49

50 Chapter 1 Sequences

Partitions

Section 1.6 Stirling Numbers: Preview 51

Stirling Subset Numbers

52 Chapter 1 Sequences

Stirling Cycle Number

Section 1.7 Ordinary Generating Fns 53

1.7 ORDINARY GENERATING FNS

Exponential Generating Functions

54 Chapter 1 Sequences

Direct Counting with OGFs

Section 1.7 Ordinary Generating Fns 55

Direct Counting with EGFs

56 Chapter 1 Sequences

Section 1.7 Ordinary Generating Fns 57

Analyzing a Generating Function

58 Chapter 1 Sequences

Section 1.7 Ordinary Generating Fns 59

Rational Functions

60 Chapter 1 Sequences

Taylor Series

Section 1.7 Ordinary Generating Fns 61

Addition and Scalar Multiplication

62 Chapter 1 Sequences

Section 1.7 Ordinary Generating Fns 63

Products and Convolutions

64 Chapter 1 Sequences

Section 1.7 Ordinary Generating Fns 65

66 Chapter 1 Sequences

Sums and Generating Functions

Section 1.7 Ordinary Generating Fns 67

68 Chapter 1 Sequences

Table 1.7.1

Section 1.7 Ordinary Generating Fns 69

70 Chapter 1 Sequences

1.8 SYNTHESIZING OGF’S

Section 1.8 Synthesizing OGF’s 71

Substitution

Shifting Right and Left

72 Chapter 1 Sequences

Section 1.8 Synthesizing OGF’s 73

Spacing Out

Isolating a Subsequence

74 Chapter 1 Sequences

Section 1.8 Synthesizing OGF’s 75

Differentiation

76 Chapter 1 Sequences

1.9 ASYMPTOTIC ESTIMATES

Section 1.9 Asymptotic Estimates 77

Ratio Method

78 Chapter 1 Sequences

Section 1.9 Asymptotic Estimates 79

Tightening Bounds on Estimates

80 Chapter 1 Sequences

Section 1.9 Asymptotic Estimates 81

82 Chapter 1 Sequences

Asymptotic Dominance

Section 1.9 Asymptotic Estimates 83