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# Catalan numbers

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The Catalan numbers are also called Segner numbers.

## Contents

## Definitions

(...)

## Formulae

where are central binomial coefficients.

## Recurrence relation

## Generating function

The ordinary generating function for the Catalan numbers is

which may be represented by the continued fraction

since is one of the two solutions of the quadratic functional equation

## Order of basis

(...)

## Forward differences

## Partial sums

## Partial sums of reciprocals

## Sum of reciprocals

## Sequences

Catalan numbers: gives the count of balanced parenthesizations of "(" and ")" (represented by "1" and "0" respectively) (Cf. A000108)

- {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, ...} =

- {#{ { } }, #{ {10} }, #{ {1010}, {1100} }, #{ {101010}, {101100}, {110010}, {110100}, {111000} }, #{ {10101010}, {10101100}, {10110010}, {10110100}, {10111000}, {11001010}, {11001100}, {11010010}, {11010100}, {11011000}, {11100010}, {11100100}, {11101000}, {11110000} }, ...}