# CatalanNumber

gives the n Catalan number .

# Details

• is generically defined as .
• Catalan numbers are integers for integer arguments, and appear in various tree enumeration problems.
• CatalanNumber can be used with Interval and CenteredInterval objects: »

# Examples

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## Basic Examples(1)

The first 10 Catalan numbers:

## Scope(9)

Evaluate for large arguments:

Evaluate for half-integer arguments:

Evaluate numerically:

Evaluate for complex arguments:

Plot the Catalan number as a function of its index:

Compute sums involving CatalanNumber:

CatalanNumber can be used with Interval and CenteredInterval objects:

## Applications(3)

Compute the number of different ways to parenthesize an expression:

Distribute over lists in CirclePlus:

Use the pattern matcher to repeatedly split the list into two parts in all possible ways:

The number of ways to parenthesize the expression abcd:

Check:

The Catalan numbers can be characterized as the unique set of numbers such that two Hankel determinants are both equal to one. Verify for the first few cases:

Verify an expression for the Catalan numbers in terms of double factorials:

## Properties & Relations(6)

The generating function for Catalan numbers:

Catalan numbers can be represented as a difference of binomial coefficients:

Catalan numbers can be represented in terms of the generalized Bell polynomial:

CatalanNumber can be represented as a DifferenceRoot:

FindSequenceFunction can recognize the CatalanNumber sequence:

The exponential generating function for CatalanNumber:

## Possible Issues(1)

The Catalan number is, by convention, defined using its representation in terms of binomials:

This value is different from the limiting value of the analytic function:

## Neat Examples(2)

The only odd Catalan numbers are those of the form CatalanNumber[2k-1]:

Determinants of Hankel matrices made out of sums of Catalan numbers:

Compare with an expression in terms of the Fibonacci numbers:

Wolfram Research (2007), CatalanNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/CatalanNumber.html (updated 2014).

#### Text

Wolfram Research (2007), CatalanNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/CatalanNumber.html (updated 2014).

#### CMS

Wolfram Language. 2007. "CatalanNumber." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/CatalanNumber.html.

#### APA

Wolfram Language. (2007). CatalanNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CatalanNumber.html

#### BibTeX

@misc{reference.wolfram_2024_catalannumber, author="Wolfram Research", title="{CatalanNumber}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/CatalanNumber.html}", note=[Accessed: 25-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_catalannumber, organization={Wolfram Research}, title={CatalanNumber}, year={2014}, url={https://reference.wolfram.com/language/ref/CatalanNumber.html}, note=[Accessed: 25-April-2024 ]}