Hmm... is there a notation called 7!!! where it's 7 x 4 x 1?

Prof. Loh said here that 7!! = 7 x 5 x 3 x 1.
Following that logic, is there a notation called 7!!! where it's 7 x 4 x 1?

Would (7!)! be (7 x 6 x 5 x 4 x 3 x 2 x 1)! ? Also, I remember in a math contest it asked something like "What is 4!?" and I was really confused because I thought it was just an exciting question and that the answer was 4 because 4 is 4.

Lol these notations are weird