W. Lang, Sep 16 2008

A143172 tabf array: partition numbers  M32(-2). 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          2         1        0        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
      
   3         10         6        1        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0
    
   4         80        40       12       12        1        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0  
 
   5        880       400      200      100       60       20        1        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
       
   6      12320      5280     2400     1000     1200     1200      120      200     180      30        1       0       0       0      0      0      0      0     0    0   0  0   

   7     209440     86240    36960    28000    18480    16800     7000     4200    2800    4200      840     350     420      42      1      0      0      0     0    0   0  0

   8    4188800   1675520   689920   492800   224000   344960   295680   224000   67200   56000    49280   67200   28000   33600   1680   5600  11200   3360   560  840  56  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: [96342400, 37699200, 15079680, 10348800, 8870400, 7539840, 6209280, 4435200, 2016000, 
1330560, 2016000, 280000, 1034880, 1330560, 1008000, 604800, 504000, 100800, 110880, 201600, 
84000, 151200, 15120, 10080, 25200, 10080, 840, 1512, 72, 1],

n=10: [2504902400, 963424000, 376992000, 251328000, 206976000, 97574400, 188496000, 150796800, 
103488000, 88704000, 31046400, 44352000, 20160000, 16800000, 25132800, 31046400, 22176000, 
10080000, 13305600, 20160000, 2800000, 2016000, 2520000, 2587200, 4435200, 3360000, 3024000, 
2520000, 1008000, 30240, 221760, 504000, 210000, 504000, 75600, 16800, 50400, 25200, 1200, 
2520, 90, 1].


The first column gives A008544(n-1)=(3*n-4)(!^3),n>=2, (3-factorials) and 1 for n=1: 
[1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400,...].

The row sums give, for n>=1: A015735 = [1,3,17,145,1661,23931,415773,8460257,197360985,5192853011,...].
They coincide with the row sums of triangle A004747.



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