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 A325406 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree. 9
 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 1, 3, 2, 0, 0, 1, 4, 2, 3, 1, 0, 0, 1, 1, 5, 5, 2, 1, 0, 0, 1, 3, 5, 6, 3, 3, 1, 0, 0, 1, 3, 4, 8, 7, 1, 4, 2, 0, 0, 1, 3, 6, 11, 7, 5, 2, 4, 2, 1, 0, 1, 1, 6, 13, 8, 9, 9, 0, 4, 3, 1, 0, 1, 6, 7, 11, 12, 9 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences. The distinct differences of any degree are the union of the k-th differences for all k >= 0. For example, the k-th differences of (1,1,2,4) for k = 0...3 are:   (1,1,2,4)   (0,1,2)   (1,1)   (0) so there are a total of 4 distinct differences of any degree, namely {0,1,2,4}. LINKS EXAMPLE Triangle begins:   1   0  1   0  1  1   0  1  2  0   0  1  2  2  0   0  1  1  3  2  0   0  1  4  2  3  1  0   0  1  1  5  5  2  1  0   0  1  3  5  6  3  3  1  0   0  1  3  4  8  7  1  4  2  0   0  1  3  6 11  7  5  2  4  2  1   0  1  1  6 13  8  9  9  0  4  3  1   0  1  6  7 11 12  9 10  8  4  3  2  2   0  1  1  7 18  9 14 19  5 10  3  5  4  1   0  1  3  9 17  9 22 20 15  9  7  6  5  4  1   0  1  4  8 22 11 16 24 22 19 10 11  2  8  7  2   0  1  4 10 23 15 24 23 27 27 12 14 11  8  8  5  5 Row n = 8 counts the following partitions:   (8)  (44)        (17)       (116)     (134)   (1133)   (111122)        (2222)      (26)       (125)     (233)   (11123)        (11111111)  (35)       (1115)    (1223)  (11222)                    (224)      (1124)                    (1111112)  (11114)                               (111113) MATHEMATICA Table[Length[Select[Reverse/@IntegerPartitions[n], Length[Union@@Table[Differences[#, i], {i, 0, Length[#]}]]==k&]], {n, 0, 16}, {k, 0, n}] CROSSREFS Row sums are A000041. Cf. A049597, A049988, A279945, A320348, A325324, A325325, A325349, A325404, A325466. Sequence in context: A144874 A333365 A303065 * A257900 A039971 A205593 Adjacent sequences:  A325403 A325404 A325405 * A325407 A325408 A325409 KEYWORD nonn,tabl AUTHOR Gus Wiseman, May 03 2019 STATUS approved

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Last modified March 31 19:16 EDT 2020. Contains 333151 sequences. (Running on oeis4.)