fst[5](x) is defined to be repeated fst[4](x)  example fst[5](3)=fst[4](3)^fst[4](3)^fst[4](3)^fst[4](2)^fst[4](2)^fst[4](1) if you would like to know more about fst[4](x) go check out my blog post entitled factorial summation tree function.fst[6](x) is defined to be repeated fst[5](x) for example fst[6](3) is equal to fst[5](3)^fst[5](3)^fst[5](3)^fst[5](2)^fst[5](2)^fst[5](1). As fst[6](x) is repeated fst[5](x) then it is safe to assume that fst[7](x) is repeated fst[6](x). The greater the input  into the [ ] the greater the output will be for example fst[500](5)>fst[200](5). This first level of the fst(x) function is called the zeroth level as it is nothing compared to whats coming next. Beyond this the 2nd level these functions are denoted with a # and then brackets with an input x. for example fst[500](5)#(5) this the defined to the fst[500](5)th term of 5 in the fst(x) hiearchy another example is fst[6](6)#(6) this is equal the fst[6](6)th term of 6 in the fst(x) hierarchy. Now what is this fst(x) hierarchy well its the way of organising all the fst(x) functions from smallest to largest although technically there is no largest input as you could continue to input larger and larger numbers into the function although this would eventually get tedious and dull. The first level of the zeroth level of the fst(x) hierarchy are the fst[1](x) numbers the second are the fst[2](x) and so and so forth. Beyond even the ungodly huge numbers in the 2nd level of the fst(x) hierarchy we come to the 3rd level denoted with ## this works the same as the fst[x](n)#(p) or the 2nd level but instead of using fst[x](n) it uses fst[x](n)#(p) for example fst[4](4)##(4) is equal to the fst[4](4)th term of 4 in the fst(x) hierarchy. Using a bit of common sense we can deduce that the 4th level of of the fst(x) hiearchy is like the 2nd and 3rd level it is denoted with a ###.

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