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