central binomial coefficient
The th central binomial coefficient is defined to be
where is a binomial coefficient. These numbers have the generating function
They are closely related to the Catalan sequence, in that
Alternate definition
A less frequently-encountered definition for the th central binomial coefficient is .
Note that the set of these numbers meeting this alternate criterion is a superset of those meeting the first criterion, since for we have
By cancelling terms of one of the ’s against terms of the , one may rewrite the central binomial coefficient as follows:
Alternatively, one may cancel each term of the against twice itself, leaving ’s in the numerator:
Doubling the terms in the denominator, we obtain an expression for the central binomial coeficient in terms of a quotient of successive odd numbers by successive even numbers:
By means of these formulae, one may derive some important properties of the central binomial coeficients. By examining the first two formulae, one may deduce results about the prime factors of central binomial coefficients (for proofs, please see the attachments to this entry):
Theorem 1
If is an integer and is a prime number such that , then divides .
Theorem 2
If is an integer and is a prime number such that , then does not divide .
In conjunction with Wallis’ formula for , the third formula for the central binomial coefficient may be used to derive an asymptotic expression, as is done in an attachment to this entry:
Title | central binomial coefficient |
---|---|
Canonical name | CentralBinomialCoefficient |
Date of creation | 2013-03-22 14:25:40 |
Last modified on | 2013-03-22 14:25:40 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 8 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 05A10 |
Classification | msc 11B65 |
Related topic | BinomialCoefficient |
Related topic | CatalanNumbers |