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  "body": "\n## Statement\n\nFor 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|0SJNCJCCQG]] of $r$ of length $k$ divided by $k!$.\n\nLet $k \\ge 1$ be an integer. Then\n$$\\binom{1/2}{k} = \\frac{(-1)^{k-1}}{k} \\cdot \\frac{\\binom{2k-2}{k-1}}{2^{2k-1}}.$$\n\n## Remarks\n\nThe number $\\binom{2k-2}{k-1}$ is the $(k-1)$-th [[central binomial coefficient|01WHD9T0V6]].\n\nExample: 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.\n",
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