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Wikipedia / Central binomial coefficient

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. , , , , , , 924, 3432, 12870, 48620, ...;

Combinatorial interpretations and other properties

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.

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 than A, 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.

is equal to the number of 1s in the binary representation of n. As a consequence, 1 is the only odd central binomial coefficient.

Generating function

The ordinary generating function This can be proved using the binomial series is a generalized binomial coefficient.

The central binomial coefficients have exponential generating function where I0 is a modified Bessel function of the first kind.

The generating function of the squares of the central binomial coefficients can be written in terms of the complete elliptic integral of the first kind:

Asymptotic growth

The asymptotic behavior can be described quite accurately:

Related sequences

The closely related Catalan numbers Cn are given by:

, with appropriate real numbers n is the gamma function 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.

.

The number lattice paths of length 2n .

Properties

) is seen in Wolstenholme's theorem.

By the Erdős squarefree conjecture, proved in 1996, no central binomial coefficient with n > 4 is squarefree.

is the sum of the squares of the n-th row of Pascal's Triangle:

.

Erdős uses central binomial coefficients extensively in his proof of Bertrand's postulate.

is exactly .

See also

References

  • .