Here, i will compare various notations.

Up to w^w in FGH

FGH SGH Hardy BEAF BAN ExE HAN My EUAN Aarex's EUAN NFN Hyphen Notation Dollar Function R Function
0 w+1 1 {n+1} {n+1} nE0 1! n+1 n+1 1! n-1 n$1 0R0
1 w*2 w {n*2} {n*2} nE1 2! n*2 n*2 2! n-1 n$[0] 1R0
2 w^2 w^2 {n,2} {n,2} nE2 n! n^2 n^2 n! n-2 n$[1] 2R0
3 w^w w^3 {n,n} {n,n} nEn n![2] n^n n^n n![2] n-n n$[2] nR0
4 e(0) w^4 {n,n,2} {n,n,2} nEn#n n![3] n^^n n^n n![3] n-n-2 n$[3] nR1


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