a(n,k)  tabf head (staircase) for  A111785 
   
  partitions of n  listed in Abramowitz-Stegun order p. 831-2 (see the main page for the ref.)

   n\k   1  2   3   4   5   6    7    8   9   10  11   12   13   14   15   16    17    18   19   20    21   22 ...  
     

   0     1  0   0   0   0   0    0    0   0    0   0    0    0    0    0    0     0     0    0    0     0    0 

   1    -1  0   0   0   0   0    0    0   0    0   0    0    0    0    0    0     0     0    0    0     0    0 
     
   2    -1  2   0   0   0   0    0    0   0    0   0    0    0    0    0    0     0     0    0    0     0    0       
  
   3    -1  5  -5   0   0   0    0    0   0    0   0    0    0    0    0    0     0     0    0    0     0    0      

   4    -1  6   3 -21  14   0    0    0   0    0   0    0    0    0    0    0     0     0    0    0     0    0  
 
   5    -1  7   7 -28 -28  84  -42    0   0    0   0    0    0    0    0    0     0     0    0    0     0    0 
  
   6    -1  8   8   4 -36 -72  -12  120 180 -330 132    0    0    0    0    0     0     0    0    0     0    0

   7    -1  9   9   9 -45 -90   -4  -45 165  495 165 -495 -990 1287 -429    0     0     0    0    0     0    0   
  
   8    -1 10  10  10   5 -55 -110 -110 -55  -55 220  660  330  660   55 -715 -2860 -1430 2002 5005 -5005 1430 
   .
   .
   .

   n\k   1  2   3   4   5   6    7    8   9   10  11   12   13   14   15   16    17    18   19   20    21   22 ...        
 


##################################################################################################################
 
 As row sequences for n=0..9:

 0. [1] 

 1. [-1]

 2. [-1,2] 

 3. [-1,5,-5] 

 4. [-1,6,3,-21,14]

 5. [-1,7,7,-28,-28,84,-42]

 6. [-1,8,8,4,-36,-72,-12,120,180,-330,132] 

 7. [-1,9,9,9,-45,-90,-45,-45,165,495,165,-495,-990,1287,-429] 
 
 8. [-1,10,10,10,5,-55,-110,-110,-55,-55,220,660,330,660,55,-715,-2860,-1430,2002,5005,-5005,1430]

 9. [-1,11,11,11,11,-66,-132,-132,-66,-66,-132,-22,286,858,858,858,858,286,-1001,-4004,-2002,-6006,-1001,3003,15015,10010,-8008,-24024,19448,-4862] 
 
 .
 .

#######################################################################################################################################

The row sums are : (-1)^n, n=0..9.

The unsigned row sums are : A001003(n+1) , n=0..9 ('little' Schroeder numbers).

#######################################################################################################################################

Note added (Jan 30 2007):

See the MathWorld entry "Series reversion", formula (11) (Morse-Feshbach). This translates into the 

following formula for the entries of row n pertaining to the partitions of n with m parts 

(in Abramowitz-Stegun order): 

List(n,m):= [seq(((-1)^m)*risefac(n+1,m)/((n+1)*product(e(m,k,j)!,j=1..n)),k=1..p(n,m))]

with p(n,m) the number of m part partitions of n (see A008284(n,m)), risefac(n,m):= n*(n+1)*...*(n+(m-1)) 

(rising factorials), e(m,k,j) the exponent of j in the k-th partition of n with m part 

(in Abramowitz-Stegun order).

The list of list of all row n entries belonging to partitions of n with m parts is then:

List(n):=[seq(List(n,m),m=1..n)].

For example: 

n=5, m=3, produces List(5,3)= [-28, -28] corresponding to the partitions [(1^2,3),(1,2^2)].

List(5)= [[-1], [7, 7], [-28, -28], [84], [-42]].


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