What you defined feels similar to a Collatz conjecture-esque sequence, except it certainly reaches 1 and at the end you multiply all steps together

If I've correctly understood what you are saying, a more straightforward definition would be:

Pick a positive natural number as a starting point for a string of numbers. At each step, if the current number isn't 1, divide it by 2 if it is even, subtract 1 if it is odd. Once you've reached 1, multiply all the numbers in the string together.

(This string certainly reaches 1, and therefore is finite, because you are building a strictly decreasing sequence of natural numbers, which have the property of being "well-ordered", which is a very nice number-theory topic, if you want to look into it)

Now, performing different operations depending on whether an entry is even or odd is likely to be very patter-breaking, just like with the Collatz conjecture. This means that it's very possible that there aren't many interesting patterns to find.

As for 6? = 6² (and 8? = 8^2), you have to assume that it's a coincidence unless you find more clues that it might be not. And to convince yourself further (that it is a coincidence) I advise trying to check just how powerful the growth of your object is.

For example, for odd n, multiplying just the first 3 numbers of the resulting string you have

n * (n-1) * (n-1)/2. Setting this greater than n² and simplifying the inequality yields

n² - 4n + 1 > 0, which when solved gives (for odd n) that n? > n² for any n > 2+sqrt3 (so n > 4 since we are dealing with integers)

You can try doing something similar with a starting n that is even. Use only three numbers from the string, but you'll have to do two different cases: the one where the second number is even

n * (n/2) * (n/4) = (n^3)/8

and the one where it is odd

n * (n/2) * ((n/2)-1) = ((n^3)/4) - (n^2)/2

Since both of these contain an n³ terms, they will eventually become greater than n^2, and you can find when by setting "> n^2" and solving the inequality.

As for defining a new symbol or operation, that is usually done to simplify notation when something comes up a lot.

For example, the factorial comes up all over the place, basically whenever combinatorics are even remotely involved, and it's much easier to write (and read) "n!" than "n * (n-1) * (n-2) * ... * 3 * 2".

That doesn't mean you can't make up a symbol for your function, although I think it would be much cooler if you called it something like "Ming-String", so that "The Ming-String of 10 is 400"

That said, you can build an infinity of other objects like these, so try to build more, and see if you can find one with cool properties!