%I #19 Aug 11 2021 05:49:14
%S 1,1,1,0,1,1,0,1,1,1,0,2,2,1,1,0,0,4,3,1,1,0,0,3,6,4,1,1,0,0,6,9,8,5,
%T 1,1,0,0,0,18,18,10,6,1,1,0,0,0,10,36,30,12,7,1,1,0,0,0,20,40,60,45,
%U 14,8,1,1,0,0,0,0,80,100,90,63,16,9,1,1
%N Triangle giving the main antidiagonals 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. Here, the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C Problem: What are the column sums? They appear to match A239201, but it is not clear why.
%e Triangle begins:
%e 1
%e 1 1
%e 0 1 1
%e 0 1 1 1
%e 0 2 2 1 1
%e 0 0 4 3 1 1
%e 0 0 3 6 4 1 1
%e 0 0 6 9 8 5 1 1
%e 0 0 0 18 18 10 6 1 1
%e 0 0 0 10 36 30 12 7 1 1
%e 0 0 0 20 40 60 45 14 8 1 1
%e 0 0 0 0 80 100 90 63 16 9 1 1
%e 0 0 0 0 35 200 200 126 84 18 10 1 1
%e 0 0 0 0 70 175 400 350 168 108 20 11 1 1
%e 0 0 0 0 0 350 525 700 560 216 135 22 12 1 1
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t Table[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{n-k}],k==(n+ats[#])/2-1&]],{k,0,n-1}],{n,0,15}]
%Y Row sums are A163493.
%Y Rows are the antidiagonals of the matrices given by A345197.
%Y The main diagonals of A345197 are A346632, with sums 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, A114121, A344610.
%K nonn,tabl
%O 0,12
%A _Gus Wiseman_, Jul 26 2021