W. Lang, Sep 16 2008

A143173 tabf array: partition numbers  M32(-3). Row n is filled with zeros for k>p(n), the partition number.

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         21         9        1        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0
    
   4        231        84       27       18        1        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0  
 
   5       3465      1155      630      210      135       30        1        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
       
   6      65835     20790    10395     4410     3465     3780      405      420     405      45        1       0       0       0      0      0      0      0     0    0   0  0   

   7    1514205    460845   218295   169785    72765    72765    30870    19845    8085   13230     2835     735     945      63      1      0      0      0     0    0   0  0

   8   40883535  12113640  5530140  4074840  1867635  1843380  1746360  1358280  436590  370440   194040  291060  123480  158760   8505  16170  35280  11340  1176 1890  84  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: [1267389585, 367951815, 163534140, 116132940, 100852290, 54511380, 49771260, 36673560, 16808715, 
11787930, 18336780, 2593080, 5530140, 7858620, 6112260, 3929310, 3333960, 714420, 436590, 873180, 
370440, 714420, 76545, 29106, 79380, 34020, 1764, 3402, 108, 1], 

n=10:[44358635475, 12673895850, 5519277225, 3815796600, 3193655850, 1512784350, 1839759075, 1635341400, 
1161329400, 1008522900, 373284450, 550103400, 252130725, 213929100, 181704600, 248856300, 183367800, 
84043575, 117879300, 183367800, 25930800, 19646550, 25004700, 13825350, 26195400, 20374200, 19646550, 
16669800, 7144200, 229635, 873180, 2182950, 926100, 2381400, 382725, 48510, 158760, 85050, 2520, 5670, 
135, 1]. 


The first column gives A008545(n-1)=(4*n-5)(!^4),n>=2, (4-factorials) and 1 for n=1: 
[1, 3, 21, 231, 3465, 65835, 1514205, 40883535, 1267389585, 44358635475,...].

The row sums give, for n>=1: A016036 = [1, 4, 31, 361, 5626, 109951, 2585269, 71066626, 2236441141, 79289379361,...].
They coincide with the row sums of triangle A000369.



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