W. Lang, Sep 16 2008

A144267 tabf array: partition numbers  M32(-4). 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          4         1        0        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
      
   3         36        12        1        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0
    
   4        504       144       48       24        1        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0  
 
   5       9576      2520     1440      360      240       40        1        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
       
   6     229824     57456    30240    12960     7560     8640      960      720     720      60        1       0       0       0      0      0      0      0     0    0   0  0   

   7    6664896   1608768   804384   635040   201096   211680    90720    60480   17640   30240     6720    1260    1680      84      1      0      0      0     0    0   0  0

   8  226606464  53319168 25740288 19305216  8890560  6435072  6435072  5080320 1693440 1451520   536256  846720  362880  483840  26880  35280  80640  26880  2016 3360 112  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: [8837652096, 2039458176, 959745024, 694987776, 608114304, 239936256, 231662592, 173746944, 
80015040, 57915648, 91445760, 13063680, 19305216, 28957824, 22861440, 15240960, 13063680, 2903040, 
1206576, 2540160, 1088640, 2177280, 241920, 63504, 181440, 80640, 3024, 6048, 144, 1],

n=10: [388856692224, 88376520960, 40789163520, 28792350720, 24324572160, 11554171776, 10197290880, 
9597450240, 6949877760, 6081143040, 2316625920, 3474938880, 1600300800, 1371686400, 799787520, 
1158312960, 868734720, 400075200, 579156480, 914457600, 130636800, 101606400, 130636800, 
48263040, 96526080, 76204800, 76204800, 65318400, 29030400, 967680, 2413152, 6350400, 2721600, 
7257600, 1209600, 105840, 362880, 201600, 4320, 10080, 180, 1].


The first column gives A008546(n-1)=(5*n-6)(!^4),n>=2, (4-factorials) and 1 for n=1: 
[1,4,36,504,9576,229824,6664896,226606464,8837652096,...].

The row sums give, for n>=1: A028575 = [1,5,49,721,14177,349141,10334689,357361985,14137664833,629779342213,...].
They coincide with the row sums of triangle A011801.



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