Multifactorials

Lately I found, to my great surprise, that ATON Exponential Set is in fact already defined in the number domain as the second multifactorial, also called Barnes G-function;

G2(z+1)=Gamma(z)*G2(z)
G2(1)=1
Gamma(z+1)=z*Gamma(z)

Higher multifactorials follow. To see how it works, expand 3 to depth 6;

G0(3) = 3 = 3*2^0
G1(3)=3.2.1.1 = 6 = 3*2^1
G2(3)=(3.2.1.1).(2.1.1).(1.1).(1) = 12 = 3*2^2
G3(3)=[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)] = 24 = 3*2^3

G4(3)={[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]} = 48 = 3*2^4

G5(3)={{[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]}}.

.{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}} = 96 = 3*2^5

G6(3)={{{[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]}}.
.{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(1)]}}} = 192 = 3*2^6

Therefore Gn(3)=3*2^n.

Instead of multiplying, one could add the expanded elements (with no apparent benefits);

GS0(3)=3
GS1(3)=7
GS2(3)=14
GS3(3)=25
GS4(3)=41
GS5(3)=63
GS6(3)=92