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A345907
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Triangle giving the main antidiagonals of the matrices counting integer compositions by length and alternating sum (A345197).
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5
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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, 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, 14, 8, 1, 1, 0, 0, 0, 0, 80, 100, 90, 63, 16, 9, 1, 1
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OFFSET
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0,12
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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. Here, the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
Problem: What are the column sums? They appear to match A239201, but it is not clear why.
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LINKS
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EXAMPLE
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Triangle begins:
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 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 14 8 1 1
0 0 0 0 80 100 90 63 16 9 1 1
0 0 0 0 35 200 200 126 84 18 10 1 1
0 0 0 0 70 175 400 350 168 108 20 11 1 1
0 0 0 0 0 350 525 700 560 216 135 22 12 1 1
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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, {n-k}], k==(n+ats[#])/2-1&]], {k, 0, n-1}], {n, 0, 15}]
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CROSSREFS
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Rows are the antidiagonals of the matrices given by A345197.
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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