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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