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%I #14 Aug 04 2022 05:07:42
%S 1,1,0,1,3,3,6,15,24,43,92,171,315,629,1218,2313,4523,8835,17076,
%T 33299,65169
%N Traces of the matrices (A345197) counting integer compositions by length and alternating sum.
%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. So a(n) is the number of compositions of n of length (n + s)/2, where s is the alternating sum of the composition.
%e The a(0) = 1 through a(7) = 15 compositions of n = 0..7 of length (n + s)/2 where s = alternating sum (empty column indicated by dot):
%e () (1) . (2,1) (2,2) (2,3) (2,4) (2,5)
%e (1,1,2) (1,2,2) (1,3,2) (1,4,2)
%e (2,1,1) (2,2,1) (2,3,1) (2,4,1)
%e (1,1,3,1) (1,1,3,2)
%e (2,1,2,1) (1,2,3,1)
%e (3,1,1,1) (2,1,2,2)
%e (2,2,2,1)
%e (3,1,1,2)
%e (3,2,1,1)
%e (1,1,1,1,3)
%e (1,1,2,1,2)
%e (1,1,3,1,1)
%e (2,1,1,1,2)
%e (2,1,2,1,1)
%e (3,1,1,1,1)
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==(n+ats[#])/2&]],{n,0,15}]
%Y Traces of the matrices given by A345197.
%Y Diagonals and antidiagonals of the same matrices are A346632 and A345907.
%Y Row sums of A346632.
%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, A001405, A007318, A008549, A025047, A114121, A163493.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Jul 26 2021
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