W. Lang Sep 24 2008

A144357 tabf array: partition numbers  M31(-1).

Partitions of n listed in Abramowitz-Stegun order p. 831-2 (see the main page for an A-number with the reference).
 

   n\k     1       2      3      4       5      6       7       8       9      10     11     12     13     14    15    16    17    18   19   20   21  22 ...  
    
   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       1      0      0       0      0       0       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0 
   
   3       0       3      1      0       0      0       0       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0
    
   4       0       0      3      6       1      0       0       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0  
 
   5       0       0      0      0      15     10       1       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0 
        
   6       0       0      0      0       0      0      15       0      45      15      1      0      0      0     0     0     0     0    0    0    0   0  
 
   7       0       0      0      0       0      0       0       0       0       0    105      0    105     21     1     0     0     0    0    0    0   0  

   8       0       0      0      0       0      0       0       0       0       0      0      0      0      0   105     0     0   420    0  210   28   1
                                                                                                                     
   .
   .
   .

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

The next two rows, for n=9 and n=10, are:

n=9:  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 945, 0, 0, 1260, 0, 378, 36, 1], 


n=10: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 945, 0, 0, 0, 0,
 4725, 0, 0, 3150, 0, 630, 45, 1].


The row sums give, for n>=1: A000085 =  [1,2,4,10,26,76,232,764,2620,9496,...].
They coincide with the row sums of triangle A049403 = S1(-1).


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