# Središčni binomski koeficient

n-ti središčni binomski koeficient je v matematiki določen z binomskim koeficientom kot:

${\displaystyle {2n \choose n}={\frac {(2n)!}{(n!)^{2}}}={\frac {2^{n}(2n-1)!!}{n!}},\qquad (n\geq 0)\!\,.}$

Tu je n! funkcija fakulteta in n!! dvojna fakulteta. Binomski koeficienti se imenujejo središčni (centralni), ker se pojavljajo točno na sredi sodih vrstic v Pascalovem trikotniku:

 ${\displaystyle {\underline {1}}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle {\underline {2}}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 4}$ ${\displaystyle {\underline {6}}}$ ${\displaystyle 4}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 5}$ ${\displaystyle 10}$ ${\displaystyle 10}$ ${\displaystyle 5}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 6}$ ${\displaystyle 15}$ ${\displaystyle {\underline {20}}}$ ${\displaystyle 15}$ ${\displaystyle 6}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 7}$ ${\displaystyle 21}$ ${\displaystyle 35}$ ${\displaystyle 35}$ ${\displaystyle 21}$ ${\displaystyle 7}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 8}$ ${\displaystyle 28}$ ${\displaystyle 56}$ ${\displaystyle {\underline {70}}}$ ${\displaystyle 56}$ ${\displaystyle 28}$ ${\displaystyle 8}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 9}$ ${\displaystyle 36}$ ${\displaystyle 84}$ ${\displaystyle 126}$ ${\displaystyle 126}$ ${\displaystyle 84}$ ${\displaystyle 36}$ ${\displaystyle 9}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 10}$ ${\displaystyle 45}$ ${\displaystyle 120}$ ${\displaystyle 210}$ ${\displaystyle {\underline {252}}}$ ${\displaystyle 210}$ ${\displaystyle 120}$ ${\displaystyle 45}$ ${\displaystyle 10}$ ${\displaystyle 1}$

Prve vrednosti središčnih binomskih koeficientov za n ≥ 0 so (OEIS A000984):

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, ... .

V Pascalovi matriki se pojavljajo po njeni diagonali:

${\displaystyle A_{10,10}={\begin{bmatrix}{\underline {1}}&1&1&1&1&1&1&1&1&1\\1&{\underline {2}}&3&4&5&6&7&8&9&10\\1&3&{\underline {6}}&10&15&21&28&36&45&55\\1&4&10&{\underline {20}}&35&56&84&120&165&220\\1&5&15&35&{\underline {70}}&126&210&330&495&715\\1&6&21&56&126&{\underline {252}}&462&792&1287&2002\\1&7&28&84&210&462&{\underline {924}}&1716&3003&5005\\1&8&36&120&330&792&1716&{\underline {3432}}&6435&11440\\1&9&45&165&495&1287&3003&6435&{\underline {12870}}&24310\\1&10&55&220&715&2002&5005&11440&24310&{\underline {48620}}\end{bmatrix}}\;,}$

## Značilnosti

Za središčne binomske koeficiente velja rodovna funkcija:

${\displaystyle {\frac {1}{\sqrt {1-4x}}}=1+2x+6x^{2}+20x^{3}+70x^{4}+252x^{5}+\cdots \!\,.}$,

Wallisov produkt se lahko zapiše v asimptotični obliki za središčni binomski koeficient:

${\displaystyle {2n \choose n}=2^{2n}\cdot {\frac {1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots (2n)}}\sim {\frac {4^{n}}{\sqrt {\pi n}}},{\text{ ko gre }}n\rightarrow \infty \!\,.}$

Zadnji izraz se lahko preprosto izpelje s pomočjo Stirlingove formule. Lahko se na drugi strani uporabi za določitev konstante ${\displaystyle {\sqrt {2\pi }}}$ pred Stirlingovo formulo s primerjavo.

Enostavni meji sta dani z:

${\displaystyle {\frac {4^{n}}{2n+1}}\leq {2n \choose n}\leq 4^{n},\qquad (n\geq 1)\!\,.}$

Boljši meji sta:

${\displaystyle {\frac {4^{n}}{\sqrt {4n}}}\leq {2n \choose n}\leq {\frac {4^{n}}{\sqrt {3n+1}}},\qquad (n\geq 1)\!\,,}$

in, če je potrebna še večja točnost:

${\displaystyle {2n \choose n}={\frac {4^{n}}{\sqrt {\pi n}}}\left(1-{\frac {c_{n}}{n}}\right)\!\,,}$

kjer je:

${\displaystyle {\frac {1}{9}}

Edini lihi središčni binomski koeficient je 1.[1]

## Sorodna zaporedja

Sorodna Catalanova števila Cn so dana z:

${\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}={2n \choose n}-{2n \choose n+1}={\frac {(2n)!}{n!\;(n+1)!}},\qquad (n\geq 0)\!\,.}$

Preprosta posplošitev središčnih binomskih koeficientov je dana kot:

${\displaystyle {\frac {\Gamma (2n+1)}{\Gamma (n+1)^{2}}}={\frac {1}{n\operatorname {\mathrm {B} } (n+1,n)}}\!\,,}$

z odgovarjajočimi realnimi števili n, kjer je ${\displaystyle \Gamma (x)\,}$ funkcija gama in ${\displaystyle \operatorname {\mathrm {B} } (x,y)\,}$ funkcija beta.

## Viri

• Banakh, Iryna; Banakh, Taras; Trisch, Pavel; Vovk, Myroslava (2012), Toehold Purchase Problem: A comparative analysis of two strategies, arXiv:1204.2065
• Koshy, Thomas (2008), Catalan Numbers with Applications, Oxford University Press, COBISS 62943745, ISBN 978-0-19533-454-8