中心二項係数

数学における中心二項係数（ちゅうしんにこうけいすう、英: Central binomial coefficient）は、n番目の中心二項係数を

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

とする。パスカルの三角形の奇数行の真ん中にあるため、中心二項係数と呼ばれる。

${\displaystyle {\underline {1}}}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle {\underline {2}}}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 3}$

${\displaystyle 3}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 4}$

${\displaystyle {\underline {6}}}$

${\displaystyle 4}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 5}$

${\displaystyle 10}$

${\displaystyle 10}$

${\displaystyle 5}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 6}$

${\displaystyle 15}$

${\displaystyle {\underline {20}}}$

${\displaystyle 15}$

${\displaystyle 6}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 7}$

${\displaystyle 21}$

${\displaystyle 35}$

${\displaystyle 35}$

${\displaystyle 21}$

${\displaystyle 7}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 8}$

${\displaystyle 28}$

${\displaystyle 56}$

${\displaystyle {\underline {70}}}$

${\displaystyle 56}$

${\displaystyle 28}$

${\displaystyle 8}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 9}$

${\displaystyle 36}$

${\displaystyle 84}$

${\displaystyle 126}$

${\displaystyle 126}$

${\displaystyle 84}$

${\displaystyle 36}$

${\displaystyle 9}$

${\displaystyle 1}$

${\displaystyle 1}$

${\displaystyle 10}$

${\displaystyle 45}$

${\displaystyle 120}$

${\displaystyle 210}$

${\displaystyle {\underline {252}}}$

${\displaystyle 210}$

${\displaystyle 120}$

${\displaystyle 45}$

${\displaystyle 10}$

${\displaystyle 1}$

中心二項係数の n ≧ 0 の値は

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100, 126410606437752, 495918532948104, 1946939425648112, …（オンライン整数列大辞典の数列 A000984）

パスカル行列（英語版）では、対角成分に表示される。

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