10. The Catalan numbers, defined by Cn = 1 n + 1 ( 2n n = (2n)! n = 0, 1, 2, . . . n!(n + 1)! form the sequence 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, .... They first appeared in 1838 when Eugène Catalan (1814-1894) showed that there are C, ways of parenthesizing a nonassociative product of n + 1 factors. [For instance, when n = 3 there are five ways: ((ab)c)d, (a(bc))d, a((bc)d), a(b(cd)), (ab)(ac).] For n ≥ 1, prove that С can be given inductively by 2(2n - Сп = 1) ·Сn−1 n+1

10. The Catalan numbers, defined by Cn = 1 n + 1 ( 2n n = (2n)! n = 0, 1, 2, . . . n!(n + 1)! form the sequence 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, .... They first appeared in 1838 when Eugène Catalan (1814-1894) showed that there are C, ways of parenthesizing a nonassociative product of n + 1 factors. [For instance, when n = 3 there are five ways: ((ab)c)d, (a(bc))d, a((bc)d), a(b(cd)), (ab)(ac).] For n ≥ 1, prove that С can be given inductively by 2(2n - Сп = 1) ·Сn−1 n+1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

See similar textbooks