User blog:ArtismScrub/Generalized Factorial Notation

n! = Γ(n+1) ≈ f2(n)

a!b = a ↑b (a-1)!b ≈ fω(n)

a!c0 = a

a!!b = a!(a!!b-1) ≈ fω+1(n)

a!cb = a!c-1(a!cb-1) ≈ fω2(n)

fact(a) = a!

fact(a,b) = a!b

fact(a,b,c) = a!cb

fact(a,b,...,x,y,0) = a

fact(a,b,...,x,y,z) = fact(a,b,...,x,fact(a,b,...,x,y-1,z),z-1) ≈ fω2(n)

Simple and not very strong, but it iterates the b in a!b, which imo is more elegant than iterating f(n) = n!n.

If I think of ways to improve or extend this, I'll revise.