I’m already aware of what factorial are used for, but the other versions are less intuitive. I understand how they work on paper, but what problems are they used to solve?
Also how high do they go in practical terms? Are there quadruple, quintuple, etc, versions?
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It’s hard to think of anything specific, and the examples on the “double factorial” Wikipedia page seem fairly obscure. You don’t actually *need* this notation as you can use, for example, product notation (the big pi, similar to the big sigma used for sums). But factorials show up all the time, and double factorials show up often enough that it’s useful to have a shorthand for them.
I don’t think I’ve ever seen a triple factorial, and I find it hard to imagine anyone writing four or more exclamation marks in a row, since at that point it starts getting hard to see how many there are at first glance. If someone needed to write a lot of expressions that involve, say, quadruple or quintuple factorials, they might well make up a new notation for it.
The number of possible bifurcating trees with labeled tips is, I think, “(3n-1)!!”. Going from memory, so I probably got that formula slightly wrong. That’s how many possible evolutionary history topologies there are for n species under some mild assumptions.
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