Half-integer binomial coefficient formula

Builds on Central binomial coefficient Falling factorial

Statement

For a real number $r$ and a natural number $k$ , let $\binom{r}{k}$ be the generalized binomial coefficient $\frac{r(r-1)\cdots(r-k+1)}{k!}$ , that is, the falling factorial of $r$ of length $k$ divided by $k!$ .

Let $k \ge 1$ be an integer. Then $\binom{1/2}{k} = \frac{(-1)^{k-1}}{k} \cdot \frac{\binom{2k-2}{k-1}}{2^{2k-1}}.$

Remarks

The number $\binom{2k-2}{k-1}$ is the $(k-1)$ -th central binomial coefficient.

Example: for $k = 1$ the right side is $\frac{1}{1} \cdot \frac{1}{2} = \frac{1}{2}$ . For $k = 2$ it is $-\frac{1}{2} \cdot \frac{2}{8} = -\frac{1}{8}$ . Both agree with the left side.

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Central binomial coefficient definition
Falling factorial concept
Half-integer binomial coefficient formula statement

Tag	`2WXZAJSBKV`
URI	`m2://2WXZAJSBKV`
Kind	statement, lemma
Writing	W0 stub
MSC	05A10
Status	This copy of the site has no build data. Run `m2 status 2WXZAJSBKV`.
Uses	Central binomial coefficient, Falling factorial
Changed	4 November 2026

Half-integer binomial coefficient formula

Statement

Remarks

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