Originally developed sometime in the last year or two during a personal quest for a probability distribution function, the hyperfactorial family may — or may not — have a direct bearing on the problem I am considering.

I haven't quite been able to figure out how to use them to express the probabilities I want to calculate, but they are interesting in themselves, as operations that produce rapidly-growing series of outputs (faster than conventional factorial operations, which are themselves faster than exponential operations).

Here's a handwritten note:


And the explanation:

The hyperfactorial operation is like rolling up several layers of factorial operation into one function. Imagined this way, the usual factorial function is then just the foundational hyperfactorial function with k=1, and we can define N!k where k=0 as just the plain old whole number N.

As in the figure above, 3!1 = 3x2x1 = 6, where k = 1.

Larger values of k beget much more complicated calculations that lead to huge numbers (though not as huge as in the last entry's "Star Wars" numbers).

I have found a few "shortcut" calculations — formulas — for getting values for some of the lower-level hyperfactorial functions; if memory serves, I have formulas for k=2 and k=3 figured out. I may have a formula for k=4, but cannot recall and my math workbook is not handy.

I cannot yet derive a general formula for calculating a value, given only N and k, which would be neat if I could!

Trivia:

1!k always = 1, for any level of k. This is hardly surprising, but it took me a moment or two to realize that
2!k always = 2, for any level of k.

Things get more interesting after that, though. And we can close by here showing the calculations for 4!3. You will notice that the same factors keep popping up, which makes deriving formulas for various levels of "k" attractive and no doubt possible:

4!3 = 4!! x 3!! x 2!! x 1!!

= (4! x 3! x 2! x 1!) x (3! x 2! x 1!) x 2 x 1

= (24 x 6 x 2 x 1) x (6 x 2 x 1) x 2 x 1

= 288 x 12 x 2 x 1

= 6,912

Again, a nice, lovely, rather large number from an economical package: 4!3

***

I don't know if this beats singing in the shower, overall, but I do find this sort of thing very arresting. Don't know why.