W. Lang and M. Sjodahl, Mar 06 2009

a(n,m)  tabf head (staircase) for  A145574

    n\m    1    2    3    4    5    6    7   8   9 10 ....

    2      1    
    3      1    
    4      1    1    
    5      1    1   
    6      1    2    1    
    7      1    2    1    
    8      1    3    2    1    
    9      1    3    3    1    
   10      1    4    4    2    1    
   11      1    4    5    3    1    
   12      1    5    7    5    2    1
   13      1    5    8    6    3    1
   14      1    6   10    9    5    2    1
   15      1    6   12   11    7    3    1
   16      1    7   14   15   10    5    2   1
   17      1    7   16   18   13    7    3   1
   18      1    8   19   23   18   11    5   2   1
   19      1    8   21   27   23   14    7   3   1
   20      1    9   24   34   30   20   11   5   2  1
    .
    .
    .


The second column gives floor(n-2)/2), n>=4, which is A004526 (integers repeated)
a(n,2), n>=5, is also the number of partitions of n-4 into at most 2 parts (one could
include n=4 if the partition of 0 is defined as having no part). 
In general a(n,m), n>=2m+1, is also the number of partitions of n-2m into at most m parts 
(one could include n=2m if the partition of 0 is defined as having no part).
The proof can be given with the Ferrers diagram of a partition. 
From a partition of n without part 1 one deletes the first two columns of the diagram 
with m rows to obtain a partition of n-2m with at most m parts, i.e., rows. 
This works also the other way around.

The column sequences are therefore: A000012, A004526, A001399, A001400, A001401, A001402, A026813 for m=1..7.
Their o.g.f. is x^(2*m)/(product((1-x^j),j=1..m)), m>=1. 

The row sums give  [1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137], see A002865(n).


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