W. Lang, Sep 18 2008

A144341 tabf array: partition numbers  M32hat(-5)= 'M32(-5)/M3'. 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         5        1        0        0        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0
    
   4        935        55       25        5        1        0        0        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0  
 
   5      21505       935      275       55       25        5        1        0       0       0        0       0       0       0      0      0      0      0     0    0   0  0 
      
   6     623645     21505     4675     3025      935      275      125       55      25       5        1       0       0       0      0      0      0      0     0    0   0  0 

   7   21827575    623645   107525    51425    21505     4675     3025     1375     935     275      125      55      25       5      1      0      0      0     0    0   0  0  

   8  894930575  21827575  3118225  1182775   874225   623645   107525    51425   23375   15125    21505    4675    3025    1375    625    935    275    125    55   25   5  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 rows n=9 and n=10 are: 

n=9: [42061737025, 894930575, 109137875, 34300475, 20107175, 21827575, 3118225, 1182775, 874225, 537625, 257125, 
166375, 623645, 107525, 51425, 23375, 15125, 6875, 21505, 4675, 3025, 1375, 625, 935, 275, 125, 55, 25, 5, 1].

n=10: [2229272062325, 42061737025, 4474652875, 1200516625, 583108075, 462465025, 894930575, 109137875, 34300475, 
20107175, 15591125, 5913875, 4371125, 2828375, 21827575, 3118225, 1182775, 874225, 537625, 257125, 166375, 
116875, 75625, 623645, 107525, 51425, 23375, 15125, 6875, 3125, 21505, 4675, 3025, 1375, 625, 935, 275, 125, 
55, 25, 5, 1].

The first column gives A008543(n-1)=(6*n-7)(!^7),n>=2, (6-factorials) and 1 for n=1.

The row sums give for n>=1:  A144343= [1,6,61,1021,22801,654271,22642171,922787096,43149037646,2279170742696,...].
They coincide with the row sums of triangle  S2hat(-5)= A144342.


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