a(n,m) asociated Sheffer triangle A111596 (assciated to Sheffer triangle A111595) 


  n\m    0        1          2         3          4          5           6           7           8           9 ...
 
  
  0      1        0          0         0          0          0           0           0           0           0

  1      0        1          0         0          0          0           0           0           0           0

  2      0       -2          1         0          0          0           0           0           0           0

  3      0        6         -6         1          0          0           0           0           0           0

  4      0      -24         36       -12          1          0           0           0           0           0

  5      0      120       -240       120        -20          1           0           0           0           0

  6      0     -720       1800     -1200        300        -30           1           0           0           0

  7      0     5040     -15120     12600      -4200        630         -42           1           0           0

  8      0   -40320     141120   -141120      58800     -11760        1176         -56           1           0

  9      0   362880   -1451520   1693440    -846720     211680      -28224        2016         -72           1
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Added April 12, 2007:
  
The unsigned triangle |a(n,m)| is the matrix product of the lower triangular Stirling number matrices |S1|*S2.
(remember that S1*S2=S2*S1=1(matrix)).
 
|a(n,m)|=sum(|S1(n,k)|*S2(k,m),k=m..n), n>=0. S2(n,m):=A048993. S1(n,m):=A048994.

This follows from the Jabotinsky type of the Stirling number matrices. See the D. E. Knuth reference 
given in A039692 for Jabotinsky type matrices.
|S1| has the e.g.f. of the m=1 column -ln(1-x) and S2 has the e.g.f. of the m=1 column exp(x)-1. Therefore,
the matrix product |S1|*S2, which is again of the Jabotinsky type, has corresponding e.g.f.  
exp(-ln(1-x))-1 = x/(1-x).
This means that the e.g.f. for the row polynomials of P(n,x):=sum(|a(n,m)|*x^m,m=0..n) is exp(x*z/(1-z)).  
As a Jobotinsky matrix |a(n,m)| has the e.g.f. for column nr. m:  (f(x)^m)/m! with f(x):= x/(1-x).

########################################### e.o.f. ###########################################################