Central binomial coefficient

In mathematics, the middle -th binomial coefficient for a non-negative integer is given by

The name " middle binomial coefficient " comes from the fact that these binomial coefficients are exactly in the middle of line in Pascal's triangle:

The first central binomial coefficients are therefore ( A000984 in OEIS sequence ):

Representations

It is

The fracture is related to the Wallis product.

Assessments

Using the Stirling formula we obtain for the assessment:

So true ( for notation see Landau symbol):

More precisely:

Generating function

The generating function is

Number theoretic properties

By the theorem of Wolstenholme applies to primes

(for the symbols see congruence ( number theory ) ).

In addition, no odd numbers come out before.

Erdős and Graham also showed that the figures are never square-free.

Integral representation

An integral representation is as follows:

Rows of the reciprocals

The following applies:

The individual decimal form sequence A073016 in OEIS.

Some other series are:

See sequence A073010 in OEIS, sequence A086463 in OEIS, -, sequence A086464 in OEIS, -. This means the Digamma function Trigammafunktion and generally th Polygammafunktion; the Riemann zeta function and the circuit number.

Generally speaking the following beautiful formula:

For, where the hypergeometric function called; cf.

The corresponding alternating series converge, and indeed to the following limits:

See sequence A086465 in OEIS, sequence A086466 in OEIS, sequence A086467 in OEIS, sequence A086468 in OEIS.

Analogously, generally write:

Related terms

Closely related to the central binomial coefficients are the Catalan numbers. They are given by

Generalization

In Pascal's triangle, only the rows with an even-numbered index have a unique middle node, the rows with odd index, however, have two lying in the middle entries. Since these two items, however, are always the same, they are sometimes included in the definition of the middle binomial coefficients with, it is then:

The first definition is obtained if we consider here the even numbers.