In mathematics the nth central binomial coefficient is the particular binomial coefficient
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:
1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...; (sequence A000984 in the OEIS)
Combinatorial interpretations and other properties
The central binomial coefficient is the number of arrangements where there are an equal number of two types of objects. For example, when , the binomial coefficient is equal to 6, and there are six arrangements of two copies of A and two copies of B: AABB, ABAB, ABBA, BAAB, BABA, BBAA.
The same central binomial coefficient is also the number of words of length 2n made up of A and B within which, as one reads from left to right, there are never more B's than A's at any point. For example, when , there are six words of length 4 in which each prefix has at least as many copies of A as of B: AAAA, AAAB, AABA, AABB, ABAA, ABAB.
The number of factors of 2 in is equal to the number of 1s in the binary representation of n.[1] As a consequence, 1 is the only odd central binomial coefficient.
Generating function
The ordinary generating function for the central binomial coefficients is
This can be proved using the binomial series and the relation
Simple bounds that immediately follow from are[citation needed]
The asymptotic behavior can be described more precisely:[4]
Related sequences
The closely related Catalan numbers Cn are given by:
A slight generalization of central binomial coefficients is to take them as , with appropriate real numbers n, where is the gamma function and is the beta function.
The powers of two that divide the central binomial coefficients are given by Gould's sequence, whose nth element is the number of odd integers in row n of Pascal's triangle.