central binomial coefficient
The th central binomial coefficient is defined to be
They are closely related to the Catalan sequence, in that
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:
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):
If is an integer and is a prime number such that , then does not divide .
|Title||central binomial coefficient|
|Date of creation||2013-03-22 14:25:40|
|Last modified on||2013-03-22 14:25:40|
|Last modified by||rspuzio (6075)|