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A346632
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Triangle read by rows giving the main diagonals of the matrices counting integer compositions by length and alternating sum (A345197).
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3
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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, 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, 0, 0, 0, 0, 1, 2, 21, 36, 100, 120, 35, 0, 0, 0, 0
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OFFSET
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0,9
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COMMENTS
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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.
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LINKS
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EXAMPLE
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Triangle begins:
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 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 0 0 0
0 1 2 21 36 100 120 35 0 0 0 0
0 1 2 24 42 150 200 140 70 0 0 0 0
0 1 2 27 48 210 300 350 280 0 0 0 0 0
0 1 2 30 54 280 420 700 700 126 0 0 0 0 0
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MATHEMATICA
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ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Table[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], k==(n+ats[#])/2&]], {k, n}], {n, 0, 15}]
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CROSSREFS
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The first nonzero element in each column appears to be A001405.
These are the diagonals of the matrices given by A345197.
Antidiagonals of the same matrices are A345907.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
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KEYWORD
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AUTHOR
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STATUS
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approved
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