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Why does the Gamma function interpolate $(n-1)!$ and not $n!$ instead? What is the historical reason?

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$\Gamma(s+1)=s\Gamma(s)$ for all $s$, and $\Gamma(1)=1$. –  i707107 Apr 4 '13 at 15:39
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Very related. –  J. M. Apr 4 '13 at 15:44
    
@i707107 You should write that as answer. Just, for all $s>0$ –  Cortizol Apr 4 '13 at 15:44

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Historically, it is Legendre's "fault". Before that, several analytical expressions to interpolate $n!$ where introduced by Bernoulli and Euler and Gauß introduced $\Pi(x)$ as notation such that $\Pi(x)=\Gamma(x+1)$, but this has not become dominant, cf. J.M.'s comment above.

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