Properties
An alternative expression for Cn is
which is equivalent to the expression given above because . This shows that Cn is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula.
The Catalan numbers satisfy the recurrence relation
moreover,
This is due to the fact that since choosing n numbers from a 2n set of numbers can be uniquely divided into 2 parts: choosing i numbers out of the first n numbers and then choosing n-i numbers from the remaining n numbers.
They also satisfy:
which can be a more efficient way to calculate them.
Asymptotically, the Catalan numbers grow as
in the sense that the quotient of the nth Catalan number and the expression on the right tends towards 1 as n → +∞. (This can be proved by using Stirling's approximation for n!.)
The only Catalan numbers Cn that are odd are those for which n = 2k − 1. All others are even.
The Catalan numbers have an integral representation
where This means that the Catalan numbers are a solution of the Hausdorff moment problem on the interval instead of . The Orthogonal polynomials having the weight function on are
Read more about this topic: Catalan Number
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)