Has this number triangle been studied? Let's make a number triangle like this.
$\sum 1 = x$
$ \sum \sum 1 = \sum x = 1/2 x + 1/2 x^2 $
$\sum \sum \sum 1 = \sum \sum x = \sum 1/2 x + 1/2 x^2 = 1/6 x^3 + 1/2 x^2 + 1/3 x $
...
So the triangle starts
1
1/2 1/2
1/6 1/2 1/3
...
So we have polynomials of degree n and we use the coëfficiënts for the triangle.
Has this number triangle been studied ?
Does it have a name ?
 A: *

*The matrix is likely this one:            


$\qquad \qquad $  
which is the matrix of unsigned Stirling numbers first kind, rowwise scaled by the reciprocal factorials.
Note, if the columns are also rescaled by the (original = non-recipocal) factorials (and the entries are alternating signed) then this becomes -if has infinite size- the Carleman-matrix fS1F for the function $\log(1+x)$ where the coefficients of the function's (and of their powers) formal power series are read column-wise, such that with a type of infinite vector $V(x)=[1,x,x^2,x^3,...]$ we have
$$ V(x) \cdot \text{fS1F} = V(\log(1+x)) \qquad \qquad \text{convergent for } |x|<1 $$
(This relation to the generating function $\log(1+x)$ is also in the "Handbook of mathematical functions" of Abramowitz/Stegun, however without this explicite matrix-display) 


*The matrix can also be this one:            


But without having decoded the summation-noation it can also be the matrix of Faulhaber-polynomials (which I call "Gp" in my matrix-toolbox). It starts whith the same values in the first three rows:     
$\qquad \qquad $ 
After seeing the many sum-signs in the definition I rather believe it is this one meant and the matrix can be used to compute the "sums-of-like-powers" like in this geometric matrix-multiplication-scheme:            
$\qquad \qquad $ 
A: It turns out what i want is the stirling Numbers 
http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html
, divided by the n th factorial.
