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A345908
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Traces of the matrices (A345197) counting integer compositions by length and alternating sum.
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13
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1, 1, 0, 1, 3, 3, 6, 15, 24, 43, 92, 171, 315, 629, 1218, 2313, 4523, 8835, 17076, 33299, 65169
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OFFSET
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0,5
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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. So a(n) is the number of compositions of n of length (n + s)/2, where s is the alternating sum of the composition.
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LINKS
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EXAMPLE
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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):
() (1) . (2,1) (2,2) (2,3) (2,4) (2,5)
(1,1,2) (1,2,2) (1,3,2) (1,4,2)
(2,1,1) (2,2,1) (2,3,1) (2,4,1)
(1,1,3,1) (1,1,3,2)
(2,1,2,1) (1,2,3,1)
(3,1,1,1) (2,1,2,2)
(2,2,2,1)
(3,1,1,2)
(3,2,1,1)
(1,1,1,1,3)
(1,1,2,1,2)
(1,1,3,1,1)
(2,1,1,1,2)
(2,1,2,1,1)
(3,1,1,1,1)
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MATHEMATICA
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ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[#]==(n+ats[#])/2&]], {n, 0, 15}]
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CROSSREFS
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Traces of the matrices given by A345197.
Diagonals and antidiagonals of the same matrices are A346632 and 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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nonn,more
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AUTHOR
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STATUS
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approved
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