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This is an extension series to the factorial I just thought of:

n!(1) = n!

n!(2) = n!(1)*n-1!(1)*...*2!(!)*1!(!) (this is the superfactorial)

n!(3) is the product of 1!(2) through n!(2). This is the product of the first n superfactorials.

Continue with n!(4), n!(5), ...

Then n!(1,2) = the product o f 1!(!), 2!(2), ... n-1!(n-1). n!(n). Then n!(2,2) makes the product of n!(1,2)'s, n(3,2) makes a product of !(2,2), ...

n!(1,3) makes a product of 1!(1,2), 2!(2,2), 3!(3,2) ... This continues up to n!(n,n). Then we can have n!(1,1,2), n!(1,1,1,2), etc.

The rules are like this:

n!(1) = n! n!(#,1) = n!(#) 1!(#) = 1 n!(m+1#) = n!(m#)*(n-1!(m+1#)) n!(0,0,...,0,0,m#) = n!(n,n,...,n,n,m-1#)*(n-1!(0,0,...,0,m#))

I don't know if this has been done before, but I'm just putting a random thought out there.

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