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YCor
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Formula for volume of n$n$-ball for negative n$n$

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James Propp
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Formula for volume of n-ball for negative n

Does the expression $$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n,$$ which gives the volume of an $n$-dimensional ball of radius $R$ when $n$ is a nonnegative integer, have any known significance when $n$ is an odd negative integer?