- Title Pages
- Dedication
- Preface
- Symbols
- 1 Binomial Coefficients
- 2 The Central Binomial Coefficient
- 3 The Central Binomial Coefficient Revisited
- 4 Binomial Coefficients Revisited
- 5 Catalan Numbers
- 6 The Ubiquity of Catalan Numbers I
- 7 The Ubiquity of Catalan Numbers II
- 8 Tress and Catalan Numbers
- 9 Lattice Paths and Catalan Numbers
- 10 Partitions and Catalan Numbers
- 11 Algebra, Sports, and Catalan Numbers
- 12 Pascal' Triangle and Catalan Numbers
- 13 Divisibility Properties
- 14 A Catalan Triangle
- 15 A Family of Binary Words
- 16 Tribinomial Coefficients
- 17 Generalized Catalan Numbers
- Appendix A
- Appendix B The First 100 Catalan Numbers
- References
- Index of Mathematicians
- <i>Index</i>

# Catalan Numbers

# Catalan Numbers

- Chapter:
- (p.103) 5 Catalan Numbers
- Source:
- Catalan Numbers with Applications
- Author(s):
### Thomas Koshy

- Publisher:
- Oxford University Press

This chapter presents Catalan numbers in an interesting, historical perspective, beginning with the well-known triangulation problem. In addition, it develops three recursive formulas for C_{n}, an approximate value for C_{n}, a relationship with the gamma function, and Shapiro's identity. It also shows how the parenthesization and triangulation problems are closely related.

*Keywords:*
triangulation problem, gamma function, reflection formula, Shapiro's identity, superfactorials

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- Title Pages
- Dedication
- Preface
- Symbols
- 1 Binomial Coefficients
- 2 The Central Binomial Coefficient
- 3 The Central Binomial Coefficient Revisited
- 4 Binomial Coefficients Revisited
- 5 Catalan Numbers
- 6 The Ubiquity of Catalan Numbers I
- 7 The Ubiquity of Catalan Numbers II
- 8 Tress and Catalan Numbers
- 9 Lattice Paths and Catalan Numbers
- 10 Partitions and Catalan Numbers
- 11 Algebra, Sports, and Catalan Numbers
- 12 Pascal' Triangle and Catalan Numbers
- 13 Divisibility Properties
- 14 A Catalan Triangle
- 15 A Family of Binary Words
- 16 Tribinomial Coefficients
- 17 Generalized Catalan Numbers
- Appendix A
- Appendix B The First 100 Catalan Numbers
- References
- Index of Mathematicians
- <i>Index</i>