Catalan Number
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the Belgian mathematician Eugène Charles Catalan (1814–1894).
The nth Catalan number is given directly in terms of binomial coefficients by
The first Catalan numbers for n = 0, 1, 2, 3, … are
- 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, … (sequence A000108 in OEIS)
Read more about Catalan Number: Properties, Applications in Combinatorics, Proof of The Formula, Hankel Matrix, Quadruple Factorial, History
Famous quotes containing the words catalan and/or number:
“It’s better that it should make you sick than that you don’t eat it at all.”
—Catalan proverb, quoted in Colman Andrews, Catalan Cuisine.
“Even in ordinary speech we call a person unreasonable whose outlook is narrow, who is conscious of one thing only at a time, and who is consequently the prey of his own caprice, whilst we describe a person as reasonable whose outlook is comprehensive, who is capable of looking at more than one side of a question and of grasping a number of details as parts of a whole.”
—G. Dawes Hicks (1862–1941)