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A345907 Triangle giving the main antidiagonals of the matrices counting integer compositions by length and alternating sum (A345197).

%I A345907 #19 Aug 11 2021 05:49:14
%S A345907 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 A345907 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 A345907 14,8,1,1,0,0,0,0,80,100,90,63,16,9,1,1
%N A345907 Triangle giving the main antidiagonals of the matrices counting integer compositions by length and alternating sum (A345197).
%C A345907 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 A345907 Problem: What are the column sums? They appear to match A239201, but it is not clear why.
%e A345907 Triangle begins:
%e A345907    1
%e A345907    1   1
%e A345907    0   1   1
%e A345907    0   1   1   1
%e A345907    0   2   2   1   1
%e A345907    0   0   4   3   1   1
%e A345907    0   0   3   6   4   1   1
%e A345907    0   0   6   9   8   5   1   1
%e A345907    0   0   0  18  18  10   6   1   1
%e A345907    0   0   0  10  36  30  12   7   1   1
%e A345907    0   0   0  20  40  60  45  14   8   1   1
%e A345907    0   0   0   0  80 100  90  63  16   9   1   1
%e A345907    0   0   0   0  35 200 200 126  84  18  10   1   1
%e A345907    0   0   0   0  70 175 400 350 168 108  20  11   1   1
%e A345907    0   0   0   0   0 350 525 700 560 216 135  22  12   1   1
%t A345907 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345907 Table[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{n-k}],k==(n+ats[#])/2-1&]],{k,0,n-1}],{n,0,15}]
%Y A345907 Row sums are A163493.
%Y A345907 Rows are the antidiagonals of the matrices given by A345197.
%Y A345907 The main diagonals of A345197 are A346632, with sums A345908.
%Y A345907 A011782 counts compositions.
%Y A345907 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345907 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345907 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345907 Other diagonals are A008277 of A318393 and A055884 of A320808.
%Y A345907 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345907 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345907 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345907 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345907 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345907 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345907 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345907 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345907 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345907 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345907 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345907 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345907 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345907 Cf. A000070, A000346, A007318, A008549, A025047, A114121, A344610.
%K A345907 nonn,tabl
%O A345907 0,12
%A A345907 _Gus Wiseman_, Jul 26 2021