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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf7f73a4-246e-4c35-8f96-56bf14bc4c1c%2F152ff490-b9cd-40eb-b57a-aaf9417f4da7%2Fqgvs9t_processed.png&w=3840&q=75)
Transcribed Image Text: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
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