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). ###################################### e.o.f. ###############################################################