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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} }, ...}