User blog comment:Googleaarex/Extended Factorial/@comment-25418284-20130328202935

My extension is

\[n!_0 = n\]

\[n!_1 = n!\]

\[n!_2 = n! \cdot (n - 1)! \cdots 2! \cdot 1!\]

\[n!_3 = n!_2 \cdot (n - 1)!_2 \cdots 2!_2 \cdot 1!_2\]

\[n!_\omega = n!_n\]

\[n!_{\omega + 1} = n!_n \cdot (n - 1)!_{n - 1} \cdots 2!_2 \cdot 1!_1\]

\[n!_{\omega + 2} = n!_{\omega + 1} \cdot (n - 1)!_{\omega + 1} \cdots 2!_{\omega + 1} \cdot 1!_{\omega + 1}\]

etc.