Barnes G at positive integer

Theorem: The following formula holds: $$G(n) = \left\{ \begin{array}{ll} 0&\quad n=-1,-2,\ldots \\ \displaystyle\prod_{i=0}^{n-2} i!&\quad n=0,1,2,\ldots, \end{array} \right.$$ where $G$ denotes the Barnes G function and $i!$ denotes the factorial.

Proof: █