User blog comment:MilkyWay90/Finishing my very first notation - The Generalized Factorial/@comment-35470197-20190710234255/@comment-35470197-20190716221026

How about the following simplified one?

For an array \(s\) of natural numbers and exclamation marks, define \(F_!(s)\) in the following way:
 * 1) Suppose that all entries of \(s\) are natural numbers.
 * 2) Set \(F_!(s) = F(s)\).
 * 3) Otherwise, suppose that all entries of \(s\) are \(1\)'s or exclamation marks.
 * 4) Denote by \(t\) the array obtained by replacing all exclamation marks by \(1\).
 * 5) Set \(F_!(s) = F(t)\).
 * 6) Otherwise, suppose that some entry of \(s\) is a natural number greater than \(1\).
 * 7) Denote by \(s'\) the array obtained by decrementing the rightmost natural number greater than \(1\).
 * 8) Denote by \(t\) the array obtained by replacing the leftmost exclamation mark by \(F_!(s')\).
 * 9) Set \(F_!(s) = F(t)\).