Central binomial coefficient

Central binomial coefficient

In mathematics the "n"th central binomial coefficient is defined in terms of the binomial coefficient by

: {2n choose n} = frac{(2n)!}{(n!)^2}.

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 OEIS|id=A000984:

:num|1, num|2, num|6, num|20, num|70, 252, 924, 3432, 12870, 48620, …

These numbers have the generating function

:frac{1}{sqrt{1-4x = 1 + 2x + 6x^2 + 20x^3 + 70x^4 + 252x^5 + cdots.

By Stirling's formula we have

: {2n choose n} sim frac{4^n}{sqrt{pi n as n ightarrowinfty.

Some useful bounds are

:frac{4^n}{sqrt{4n leq {2n choose n} leq frac{4^n}{sqrt{3n+1 for all n geq 1

and, if more accuracy is required,

:frac{4^n}{sqrt22 over 7}n + {6 over 7} leq {2n choose n} leq frac{4^n}{sqrt28 over 9}n+{8 over 9} for all n geq 1.

The closely related Catalan numbers "C""n" are given by:

:C_n = frac{1}{n+1} {2n choose n}.

A slight generalization of central binomial coefficients is to take them as { m choose {lfloor frac{m}{2} floor} }and so the former definition is a particular case when "m" = 2"n", that is, when "m" is even.

ee also

* Erdős Squarefree Conjecture

External links

*
*
*
*


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Binomial coefficient — The binomial coefficients can be arranged to form Pascal s triangle. 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… …   Wikipedia

  • Binomial distribution — Probability distribution name =Binomial type =mass pdf cdf Colors match the image above parameters =n geq 0 number of trials (integer) 0leq p leq 1 success probability (real) support =k in {0,dots,n}! pdf ={nchoose k} p^k (1 p)^{n k} ! cdf =I {1… …   Wikipedia

  • List of factorial and binomial topics — This is a list of factorial and binomial topics in mathematics, by Wikipedia page. See also binomial (disambiguation).*Alternating factorial *Antichain *Beta function *Binomial coefficient *Binomial distribution *Binomial proportion confidence… …   Wikipedia

  • Central limit theorem — This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 1 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result… …   Wikipedia

  • Negative binomial distribution — Probability mass function The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation. notation: parameters: r > 0 number of failures until the experiment is stopped (integer,… …   Wikipedia

  • Multinomial theorem — In mathematics, the multinomial theorem says how to write a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem to polynomials. Contents 1 Theorem 1.1 Number of multinomial coefficients 1.2 …   Wikipedia

  • Proof of Bertrand's postulate — In mathematics, Bertrand s postulate (actually a theorem) states that for each n ≥ 2 there is a prime p such that n < p < 2 n . It was first proven by Pafnuty Chebyshev, and a short but advanced proof was given by Srinivasa Ramanujan. The gist of …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Square-free integer — In mathematics, a square free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square free but 18 is not, as it is divisible by 9 = 32. The smallest square free numbers are :1, 2, 3, 5, 6, 7, 10, 11, 13 …   Wikipedia

  • 250 (number) — 250 is the natural number following 249 and preceding 251.As a Mandarin slang term, 250 (二百五 pinyin èrbǎiwǔ) is an insult meaning stupid person or simpleton . The expression is based on bàndiàozi (半弔子 or 半吊子). In ancient China, copper coins were… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”