Central binomial coefficient
Encyclopedia
In mathematics
the nth central binomial coefficient is defined in terms of the binomial coefficient
by
They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle
. The first few central binomial coefficients starting at n = 0 are:
By Stirling's formula we have
Some useful bounds are
and, if more accuracy is required,
for all 
The closely related Catalan numbers Cn are given by:
A slight generalization of central binomial coefficients is to take them as
and so the former definition is a particular case when m = 2n, that is, when m is even.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
the nth central binomial coefficient is defined in terms of the binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
by
They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
. The first few central binomial coefficients starting at n = 0 are:
- , , , , , 252, 924, 3432, 12870, 48620, …
Properties
These numbers have the generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
By Stirling's formula we have
Some useful bounds are
and, if more accuracy is required,
for all The closely related Catalan numbers Cn are given by:
A slight generalization of central binomial coefficients is to take them as
and so the former definition is a particular case when m = 2n, that is, when m is even.
See also
- Erdős squarefree conjecture
- Central multinomial coefficients