Exercise · ZW-CATALAN-NUMBER-006 · ★★☆ Intermediate
Compute \(C_4\) using the recurrence \(C_{n+1}=\sum_{k=0}^{n} C_k C_{n-k}\) with \(C_0=1\), \(C_1=1\), \(C_2=2\), \(C_3=5\).
Choose one option.
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- A. 14
Explanation
For \(n=3\) the sum is \(C_0C_3+C_1C_2+C_2C_1+C_3C_0=5+2+2+5=14\). The value 42 is \(C_5\), a common index shift.
Review the concept: Catalan number
The Catalan number \(C_n\) counts, among many other structures, the triangulations of a convex polygon with \(n+2\) sides and the Dyck paths of length \(2n\).
\[C_n=\frac{1}{n+1}\binom{2n}{n}=\frac{(2n)!}{n!\,(n+1)!}\]
More practice problems: catalan number
- ZW-CATALAN-NUMBER-001 What is the value of C_2?
- ZW-CATALAN-NUMBER-002 What is the value of C_0?
- ZW-CATALAN-NUMBER-003 How many distinct triangulations does a convex hexagon have, using diagonals that do not cross in i…
- ZW-CATALAN-NUMBER-004 In how many different ways can the product x_1x_2x_3 be parenthesized?
- ZW-CATALAN-NUMBER-005 Which of the following expressions equals C_n?
- ZW-CATALAN-NUMBER-007 Compute binom105 and use it to find C_5 through C_n=(1)/(n+1)binom2nn.
- ZW-CATALAN-NUMBER-008 Compute (C_n+1)/(C_n) for n=3 from C_3=5 and C_4=14.
- ZW-CATALAN-NUMBER-009 How many Dyck paths of length 6 exist?