W. Lang, Sep 16 2008

A144268 tabf array: partition numbers  M32(-5). 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          5         1        0        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
      
   3         55        15        1        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0
    
   4        935       220       75       30        1        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0  
 
   5      21505      4675     2750      550      375       50        1        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
       
   6     623645    129030    70125    30250    14025    16500     1875     1100    1125      75        1       0       0       0      0      0      0      0      0   0   0  0 

   7   21827575   4365515  2258025  1799875   451605   490875   211750   144375   32725   57750    13125    1925    2625     105      1      0      0      0     0    0   0  0

   8  894930575 174620600 87310300 66235400 30597875 17462060 18064200 14399000 4908750 4235000  1204280 1963500  847000 1155000  65625  65450 154000  52500  3080 5250  140 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: [42061737025, 8054375175, 3928963500, 2881239900, 2533504050, 785792700, 785792700, 596118600,
 275380875, 203222250, 323977500, 46585000, 52386180, 81288900, 64795500, 44178750, 38115000, 
8662500, 2709630, 5890500, 2541000, 5197500, 590625, 117810, 346500, 157500, 4620, 9450, 180, 1],

n=10: [2229272062325, 420617370250, 201359379375, 144061995000, 122452695750, 58270593150, 
40271875875, 39289635000, 28812399000, 25335040500, 9822408750, 14902965000, 6884521875, 5939587500,
 2619309000, 3928963500, 2980593000, 1376904375, 2032222500, 3239775000, 465850000, 368156250, 
476437500, 130965450, 270963000, 215985000, 220893750, 190575000, 86625000, 2953125, 5419260, 
14726250, 6352500, 17325000, 2953125, 196350, 693000, 393750, 6600, 15750, 225, 1].


The first column gives A008543(n-1)=(6*n-7)(!^6),n>=2, (6-factorials) and 1 for n=1: 
[1, 5, 55, 935, 21505, 623645, 21827575, 894930575, 42061737025, 2229272062325,...].

The row sums give, for n>=1: A028844 = [1, 6, 71, 1261, 29906, 887751, 31657851, 1318279586, 62783681421, 
3365947782611,...].
They coincide with the row sums of triangle A013988.



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