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%I #5 May 14 2019 22:08:32
%S 1,1,1,1,2,1,1,3,0,1,1,4,4,0,1,1,5,5,0,0,1,1,6,12,4,0,0,1,1,7,12,5,0,
%T 0,0,1,1,8,25,8,4,0,0,0,1,1,9,24,12,3,0,0,0,0,1,1,10,40,32,8,4,0,0,0,
%U 0,1,1,11,41,41,6,3,0,0,0,0,0,1
%N Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.
%C A composition of n is a finite sequence of positive integers summing to n.
%e The distinct consecutive subsequences of (1,1,3,3) are (1), (1,1), (3), (1,3), (1,1,3), (3,3), (1,3,3), (1,1,3,3), all of which have different sums, so (1,1,3,3) is counted under a(8).
%e Triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 3 0 1
%e 1 4 4 0 1
%e 1 5 5 0 0 1
%e 1 6 12 4 0 0 1
%e 1 7 12 5 0 0 0 1
%e 1 8 25 8 4 0 0 0 1
%e 1 9 24 12 3 0 0 0 0 1
%e 1 10 40 32 8 4 0 0 0 0 1
%e 1 11 41 41 6 3 0 0 0 0 0 1
%e 1 12 60 76 14 4 4 0 0 0 0 0 1
%e 1 13 60 88 16 6 3 0 0 0 0 0 0 1
%e Row n = 8 counts the following compositions:
%e (8) (17) (116) (1115) (11111111)
%e (26) (125) (1133)
%e (35) (143) (2222)
%e (44) (152) (3311)
%e (53) (215) (5111)
%e (62) (233)
%e (71) (251)
%e (332)
%e (341)
%e (512)
%e (521)
%e (611)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]&]],{n,15},{k,n}]
%Y Row sums are A325676.
%Y Column k = 2 is A000027.
%Y Column k = 3 is A325688.
%Y Cf. A000079, A007318, A048004, A108917, A143823, A169942, A266223, A325592, A325680, A325685.
%K nonn,tabl
%O 1,5
%A _Gus Wiseman_, May 13 2019
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