a(n,m) tabl head (triangle) for  A039683 (W. Lang, Mar 5, 2004)
 
   n\m       1         2         3         4         5         6         7         8         9
 

   1         1         0         0         0         0         0         0         0         0
    
   2        -2         1         0         0         0         0         0         0         0
   
   3         8        -6         1         0         0         0         0         0         0
   
   4       -48        44       -12         1         0         0         0         0         0
   
   5       384      -400       140       -20         1         0         0         0         0
   
   6     -3840      4384     -1800       340       -30         1         0         0         0
   
   7     46080    -56448     25984     -5880       700       -42         1         0         0
   
   8   -645120    836352   -420224    108304    -15680      1288       -56         1         0
   
   9  10321920 -14026752   7559936  -2153088    359184    -36288      2184       -72         1
  
   etc.

   a(n,m)= (2^(n-m))*S1(n,m), with the (signed) Stirling1 triangle A008275. 
   
   I.e. Stirling1 diagonal k scaled with power 2^k, k=0,1,2,.... 
  
   This unsigned triangle, when multiplied (read as infinite dimensional matrix) from the right with 

   S2(n,m)=A008277(n,m) (Stirling2 numbers), produces triangle S2(3;n,m)=A046089(n,m) (genealized Stirling2).
   
   This observation is due to E. Neuwirth (priv. comm. 2001; see also the 2001 ref. given in A046089).