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The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The n-th Catalan number can be expressed directly in terms of the central binomial coefficients by
The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ 1 or the aliquot sum function s in the following way: [1] = = = > = = = If the s n-1 = 0 condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these ...
6.2.1 Example: Generating function for the Catalan numbers. ... in that a series of terms can be said to be the generator of its sequence of term coefficients.
For the sequence as a whole, the number of open brackets must equal the number of closed brackets; ... one can generate all other Fuss–Catalan numbers if p is an ...
Catalan's conjecture; ... Halton sequences; Geometry of numbers. Minkowski's theorem; Pick's theorem; ... Pseudorandom number generator. Pseudorandomness;
The dimensions of Temperley-Lieb algebras are Catalan numbers: [2] ... The generator is the diagram in ... even, this (sequence A051255 in the OEIS) ...
The sequence begins: 1, 1, 2, 5, 14, 42, 132, ... The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n th Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves.
A modification of Lagged-Fibonacci generators. A SWB generator is the basis for the RANLUX generator, [19] widely used e.g. for particle physics simulations. Maximally periodic reciprocals: 1992 R. A. J. Matthews [20] A method with roots in number theory, although never used in practical applications. KISS: 1993 G. Marsaglia [21]