W. Lang, Sep 16 2008

A143171 tabf array: partition numbers  M32(-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          3         3        1        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0
    
   4         15        12        3        6        1        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0  
 
   5        105        75       30       30       15       10        1        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
       
   6        945       630      225       90      225      180       15       60      45      15        1       0       0       0      0      0      0      0     0    0   0  0   

   7      10395      6615     2205     1575     2205     1575      630      315     525     630      105     105     105      21      1      0      0      0     0    0   0  0

   8     135135     83160    26460    17640     7875    26460    17640    12600    3150    2520     5880    6300    2520    2520    105   1050   1680    420   168  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: [2027025, 1216215, 374220, 238140, 198450, 374220, 238140, 158760, 70875, 39690, 56700, 7560, 79380, 
79380, 56700, 28350, 22680, 3780, 13230, 18900, 7560, 11340, 945, 1890, 3780, 1260, 252, 378, 36, 1],

n=10: [34459425, 20270250, 6081075, 3742200, 2976750, 1389150, 6081075, 3742200, 2381400, 1984500, 595350, 
793800, 354375, 283500, 1247400, 1190700, 793800, 354375, 396900, 567000, 75600, 47250, 56700, 198450, 264600, 
189000, 141750, 113400, 37800, 945, 26460, 47250, 18900, 37800, 4725, 3150, 7560, 3150, 360, 630, 45, 1].


The first column gives A001147(n-1)=(2*n-3)(!^2),n>=2, (2-factorials) and 1 for n=1.

The row sums give, for n>=1: A001515(n)=[1,2,7,37,266,2431,27007,353522,5329837,90960751,...].
They coincide with the row sums of triangle A001497.



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