W. Lang Oct 10 2008

A145366 tabf array: partition numbers  M31hat(-3).

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       3       1      0      0       0      0       0       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0 
  
   3       6       3      1      0       0      0       0       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0
    
   4       6       6      9      3       1      0       0       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0  
 
   5       0       6     18      6       9      3       1       0       0       0      0      0      0      0     0     0     0     0    0    0    0   0 
        
   6       0       0     18     36       6     18      27       6       9       3      1      0      0      0     0     0     0     0    0    0    0   0  
 
   7       0       0      0     36       0     18      36      54       6      18     27      6      9      3     1     0     0     0    0    0    0   0  
  
   8       0       0      0      0      36      0       0      36      54     108      0     18     36     54    81     6    18    27    6    9    3   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, 36, 0, 108, 216, 0, 0, 36, 54, 108, 162, 0, 18, 36, 54, 81, 6, 18, 27, 6,
 9, 3, 1],

n=10:  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 108, 216, 0, 0, 0, 36, 0, 108, 216, 162, 324, 0, 0, 36, 54, 
108, 162, 243, 0, 18, 36, 54, 81, 6, 18, 27, 6, 9, 3, 1].
      

The row sums give, for n>=1, A145368: [1,4,10,25,43,124,214,493,979,2032,...].
They coincide with the row sums of triangle A145367 = S1hat(-3).


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