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Catalan numbers∗ bbukh† 2013-03-21 13:48:14 The Catalan numbers, or Catalan sequence, have many interesting applications in combinatorics. The nth Catalan number is given by: 2n Cn = n n+1 , where nr represents the binomial coefficient. The first several Catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862 ,. . . (see OEIS sequence A000108 for more terms). The Catalan numbers are also generated by the recurrence relation C0 = 1, Cn = n−1 X Ci Cn−1−i . i=0 For example, C3 = 1 · 2 + 1 · 1 + 2 · 1 = 5, C4 = 1 · 5 + 1 · 2 + 2 · 1 + 5 · 1 = 14, etc. The ordinary generating function for the Catalan numbers is √ ∞ X 1 − 1 − 4z Cn z n = . 2z n=0 Interpretations of the nth Catalan number include: 1. The number of ways to arrange n pairs of matching parentheses, e.g.: () (()) ()() ((())) (()()) ()(()) (())() ()()() 2. The number of ways a convex polygon of n + 2 sides can be split into n triangles. ∗ hCatalanNumbersi created: h2013-03-21i by: hbbukhi version: h32724i Privacy setting: h1i hDefinitioni h05A10i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 3. The number of rooted binary trees with exactly n + 1 leaves. The Catalan sequence is named for Eugène Charles Catalan, but it was discovered in 1751 by Euler when he was trying to solve the problem of subdividing polygons into triangles. References [1] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley, 1998. Zbl 0836.00001. 2