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Nested factorial notation
Notation$$n!^m$$
TypeCombinatorial
Based onFactorial
Growth rate$$f_{3}(n)$$
AuthorAarex Tiaokhiao

Nested factorial notation is an extension of the factorial function, created by Aarex Tiaokhiao.[1] It is formally defined as follows:

n!m = ((...(((n!)!)!...)!)! w/m nested factorials (not to be confused with n! to the power of m).

For example, 3!3 = ((3!)!)! = (6!)! = 720! It's decimal expansion is:

2601218943565795100204903227081043611191521875016945785727541837850835631156947382240678577958130457082619920575892247259536641565162052015873791984587740832529105244690388811884123764341191951045505346658616243271940197113909845536727278537099345629855586719369774070003700430783758997420676784016967207846280629229032107161669867260548988445514257193985499448939594496064045132362140265986193073249369770477606067680670176491669403034819961881455625195592566918830825514942947596537274845624628824234526597789737740896466553992435928786212515967483220976029505696699927284670563747137533019248313587076125412683415860129447566011455420749589952563543068288634631084965650682771552996256790845235702552186222358130016700834523443236821935793184701956510729781804354173890560727428048583995919729021726612291298420516067579036232337699453964191475175567557695392233803056825308599977441675784352815913461340394604901269542028838347101363733824484506660093348484440711931292537694657354337375724772230181534032647177531984537341478674327048457983786618703257405938924215709695994630557521063203263493209220738320923356309923267504401701760572026010829288042335606643089888710297380797578013056049576342838683057190662205291174822510536697756603029574043387983471518552602805333866357139101046336419769097397432285994219837046979109956303389604675889865795711176566670039156748153115943980043625399399731203066490601325311304719028898491856203766669164468791125249193754425845895000311561682974304641142538074897281723375955380661719801404677935614793635266265683339509760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Extended Nested Factorial Notation

Extended Nested Factorial Notation
Notation$$n!^m[x]$$
TypeCombinatorial
Based onNested factorial notation
Growth rate$$f_{\omega}(n)$$
AuthorAarex Tiaokhiao

Aarex defined n![2], which is equal to n!n. Also he defined multorial of n, is same at n![2]. This can also be written as n% using warp notation.

Next is n!2[2], is equal to (n!n)!n!n, or (n![2])![2].

Then n!3[2] = (n!n)!2[2] = ((n![2])![2])![2], and so on.

He defined n!n[2], is same at n![3]. And also defined n![4], is equal to n!n[3].

In general, n![m] is equal to n!n[m-1]. For example, 7![3], is equal to 7!7[2].

The limit of this notation is n![n].

Multi-entry Factorial Notation

Multi-entry Factorial Notation
Notation$$n!^m[a,b,c,...]$$
TypeCombinatorial
Based onNested factorial notation
Growth rate$$f_{\omega^\omega}(n)$$
AuthorAarex Tiaokhiao

Aarex defined n![1,2], which is equal to n![n].

Then n![2,2] = n!n[1,2], n![3,2] = n!n[2,2], etc.

The same pattern continues for more entries, each corresponding to the next exponent on $$\omega$$ for $$f_{\omega^m}(n)$$ in the FGH.