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%I #16 Aug 04 2022 05:07:56
%S 1,0,0,0,1,0,0,1,2,0,0,1,2,0,0,0,1,2,3,0,0,0,1,2,6,6,0,0,0,1,2,9,12,0,
%T 0,0,0,1,2,12,18,10,0,0,0,0,1,2,15,24,30,20,0,0,0,0,1,2,18,30,60,60,0,
%U 0,0,0,0,1,2,21,36,100,120,35,0,0,0,0
%N Triangle read by rows giving the main diagonals of the matrices counting integer compositions by length and alternating sum (A345197).
%C The matrices (A345197) count the integer compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2. The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e Triangle begins:
%e 1
%e 0 0
%e 0 1 0
%e 0 1 2 0
%e 0 1 2 0 0
%e 0 1 2 3 0 0
%e 0 1 2 6 6 0 0
%e 0 1 2 9 12 0 0 0
%e 0 1 2 12 18 10 0 0 0
%e 0 1 2 15 24 30 20 0 0 0
%e 0 1 2 18 30 60 60 0 0 0 0
%e 0 1 2 21 36 100 120 35 0 0 0 0
%e 0 1 2 24 42 150 200 140 70 0 0 0 0
%e 0 1 2 27 48 210 300 350 280 0 0 0 0 0
%e 0 1 2 30 54 280 420 700 700 126 0 0 0 0 0
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t Table[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],k==(n+ats[#])/2&]],{k,n}],{n,0,15}]
%Y The first nonzero element in each column appears to be A001405.
%Y These are the diagonals of the matrices given by A345197.
%Y Antidiagonals of the same matrices are A345907.
%Y Row sums are A345908.
%Y A011782 counts compositions.
%Y A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y Other diagonals are A008277 of A318393 and A055884 of A320808.
%Y Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y - k = 0: counted by A088218, ranked by A344619/A344619.
%Y - k = 1: counted by A000984, ranked by A345909/A345911.
%Y - k = -1: counted by A001791, ranked by A345910/A345912.
%Y - k = 2: counted by A088218, ranked by A345925/A345922.
%Y - k = -2: counted by A002054, ranked by A345924/A345923.
%Y - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y - k > 0: counted by A027306, ranked by A345917/A345918.
%Y - k < 0: counted by A294175, ranked by A345919/A345920.
%Y - k != 0: counted by A058622, ranked by A345921/A345921.
%Y - k even: counted by A081294, ranked by A053754/A053754.
%Y - k odd: counted by A000302, ranked by A053738/A053738.
%Y Cf. A000070, A000346, A007318, A008549, A025047, A163493, A344610.
%K nonn,tabl
%O 0,9
%A _Gus Wiseman_, Jul 26 2021
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