%I
%S 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,
%T 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,
%U 1,6,13,8,9,9,0,4,3,1,0,1,6,7,11,12,9
%N Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.
%C 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).
%C The zeroth differences of a sequence are the sequence itself, while the kth differences for k > 0 are the differences of the (k1)th differences. The distinct differences of any degree are the union of the kth differences for all k >= 0. For example, the kth differences of (1,1,2,4) for k = 0...3 are:
%C (1,1,2,4)
%C (0,1,2)
%C (1,1)
%C (0)
%C so there are a total of 4 distinct differences of any degree, namely {0,1,2,4}.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 2 0
%e 0 1 2 2 0
%e 0 1 1 3 2 0
%e 0 1 4 2 3 1 0
%e 0 1 1 5 5 2 1 0
%e 0 1 3 5 6 3 3 1 0
%e 0 1 3 4 8 7 1 4 2 0
%e 0 1 3 6 11 7 5 2 4 2 1
%e 0 1 1 6 13 8 9 9 0 4 3 1
%e 0 1 6 7 11 12 9 10 8 4 3 2 2
%e 0 1 1 7 18 9 14 19 5 10 3 5 4 1
%e 0 1 3 9 17 9 22 20 15 9 7 6 5 4 1
%e 0 1 4 8 22 11 16 24 22 19 10 11 2 8 7 2
%e 0 1 4 10 23 15 24 23 27 27 12 14 11 8 8 5 5
%e Row n = 8 counts the following partitions:
%e (8) (44) (17) (116) (134) (1133) (111122)
%e (2222) (26) (125) (233) (11123)
%e (11111111) (35) (1115) (1223) (11222)
%e (224) (1124)
%e (1111112) (11114)
%e (111113)
%t Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union@@Table[Differences[#,i],{i,0,Length[#]}]]==k&]],{n,0,16},{k,0,n}]
%Y Row sums are A000041.
%Y Cf. A049597, A049988, A279945, A320348, A325324, A325325, A325349, A325404, A325466.
%K nonn,tabl
%O 0,9
%A _Gus Wiseman_, May 03 2019
