A003319

[Paul D. Hanna]

Hello Gottfried,
        As is common with deep thinkers, you may be looking deeper into
the question than is needed.
The matrix you gave is constructed by making all columns equal to the
factorial sequence. 
When all columns of a matrix are equal, certain operations applied upon
that matrix such as matrix multiplication, inverse, logarithm, etc., will
result in a matrix that likewise has all columns equal; 
the g.f. of the column vector of the resultant matrix will equal the
operation applied upon the g.f. of the column vector of the original
matrix.  I believe that you are already aware of this fact.
  
The appearance of A003319 arises due to the relation: 
log(1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 120*x^5 +...+ n!*x^n +...) =
x + 3/2*x^2 + 13/3*x^3 + 71/4*x^4 + 461/5*x^5 + ... + A003319(n)/n*x^n
+... (n>0)
 
If your question is directed more to the combinatorial interpretation of
the above, 
than it is beyond me. 
 
Regards,
Paul