Iz Wikipedije, proste enciklopedije
n -ti središčni binomski koeficient je v matematiki določen z binomskim koeficientom kot:
(
2
n
n
)
=
(
2
n
)
!
(
n
!
)
2
=
2
n
(
2
n
−
1
)
!
!
n
!
,
(
n
≥
0
)
.
{\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 :
1
_
{\displaystyle {\underline {1}}}
1
{\displaystyle 1}
1
{\displaystyle 1}
1
{\displaystyle 1}
2
_
{\displaystyle {\underline {2}}}
1
{\displaystyle 1}
1
{\displaystyle 1}
3
{\displaystyle 3}
3
{\displaystyle 3}
1
{\displaystyle 1}
1
{\displaystyle 1}
4
{\displaystyle 4}
6
_
{\displaystyle {\underline {6}}}
4
{\displaystyle 4}
1
{\displaystyle 1}
1
{\displaystyle 1}
5
{\displaystyle 5}
10
{\displaystyle 10}
10
{\displaystyle 10}
5
{\displaystyle 5}
1
{\displaystyle 1}
1
{\displaystyle 1}
6
{\displaystyle 6}
15
{\displaystyle 15}
20
_
{\displaystyle {\underline {20}}}
15
{\displaystyle 15}
6
{\displaystyle 6}
1
{\displaystyle 1}
1
{\displaystyle 1}
7
{\displaystyle 7}
21
{\displaystyle 21}
35
{\displaystyle 35}
35
{\displaystyle 35}
21
{\displaystyle 21}
7
{\displaystyle 7}
1
{\displaystyle 1}
1
{\displaystyle 1}
8
{\displaystyle 8}
28
{\displaystyle 28}
56
{\displaystyle 56}
70
_
{\displaystyle {\underline {70}}}
56
{\displaystyle 56}
28
{\displaystyle 28}
8
{\displaystyle 8}
1
{\displaystyle 1}
1
{\displaystyle 1}
9
{\displaystyle 9}
36
{\displaystyle 36}
84
{\displaystyle 84}
126
{\displaystyle 126}
126
{\displaystyle 126}
84
{\displaystyle 84}
36
{\displaystyle 36}
9
{\displaystyle 9}
1
{\displaystyle 1}
1
{\displaystyle 1}
10
{\displaystyle 10}
45
{\displaystyle 45}
120
{\displaystyle 120}
210
{\displaystyle 210}
252
_
{\displaystyle {\underline {252}}}
210
{\displaystyle 210}
120
{\displaystyle 120}
45
{\displaystyle 45}
10
{\displaystyle 10}
1
{\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:
A
10
,
10
=
[
1
_
1
1
1
1
1
1
1
1
1
1
2
_
3
4
5
6
7
8
9
10
1
3
6
_
10
15
21
28
36
45
55
1
4
10
20
_
35
56
84
120
165
220
1
5
15
35
70
_
126
210
330
495
715
1
6
21
56
126
252
_
462
792
1287
2002
1
7
28
84
210
462
924
_
1716
3003
5005
1
8
36
120
330
792
1716
3432
_
6435
11440
1
9
45
165
495
1287
3003
6435
12870
_
24310
1
10
55
220
715
2002
5005
11440
24310
48620
_
]
,
{\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}}\;,}
Za središčne binomske koeficiente velja rodovna funkcija :
1
1
−
4
x
=
1
+
2
x
+
6
x
2
+
20
x
3
+
70
x
4
+
252
x
5
+
⋯
.
{\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:
(
2
n
n
)
=
2
2
n
⋅
1
⋅
3
⋅
5
⋯
(
2
n
−
1
)
2
⋅
4
⋅
6
⋯
(
2
n
)
∼
4
n
π
n
,
ko gre
n
→
∞
.
{\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
2
π
{\displaystyle {\sqrt {2\pi }}}
Enostavni meji sta dani z:
4
n
2
n
+
1
≤
(
2
n
n
)
≤
4
n
,
(
n
≥
1
)
.
{\displaystyle {\frac {4^{n}}{2n+1}}\leq {2n \choose n}\leq 4^{n},\qquad (n\geq 1)\!\,.}
Boljši meji sta:
4
n
4
n
≤
(
2
n
n
)
≤
4
n
3
n
+
1
,
(
n
≥
1
)
,
{\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:
(
2
n
n
)
=
4
n
π
n
(
1
−
c
n
n
)
,
{\displaystyle {2n \choose n}={\frac {4^{n}}{\sqrt {\pi n}}}\left(1-{\frac {c_{n}}{n}}\right)\!\,,}
kjer je:
1
9
<
c
n
<
1
8
,
(
n
≥
1
)
.
{\displaystyle {\frac {1}{9}}<c_{n}<{\frac {1}{8}},\qquad (n\geq 1)\!\,.}
Edini lihi središčni binomski koeficient je 1.[1]
Sorodna Catalanova števila C n
C
n
=
1
n
+
1
(
2
n
n
)
=
(
2
n
n
)
−
(
2
n
n
+
1
)
=
(
2
n
)
!
n
!
(
n
+
1
)
!
,
(
n
≥
0
)
.
{\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:
Γ
(
2
n
+
1
)
Γ
(
n
+
1
)
2
=
1
n
B
(
n
+
1
,
n
)
,
{\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
Γ
(
x
)
{\displaystyle \Gamma (x)\,}
funkcija gama in
B
(
x
,
y
)
{\displaystyle \operatorname {\mathrm {B} } (x,y)\,}
funkcija beta .
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
