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Catalan numbers

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The Catalan numbers are also called Segner numbers.

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} }, ...}

See also