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A239550
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Number A(n,k) of compositions of n such that the first part is 1 and the second differences of the parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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13
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1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 4, 4, 3, 1, 1, 1, 2, 4, 7, 6, 2, 1, 1, 1, 2, 4, 7, 11, 9, 2, 1, 1, 1, 2, 4, 8, 13, 18, 13, 3, 1, 1, 1, 2, 4, 8, 15, 23, 32, 18, 3, 1, 1, 1, 2, 4, 8, 15, 28, 40, 53, 24, 2, 1, 1, 1, 2, 4, 8, 16, 29, 52, 73, 89, 34, 3
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OFFSET
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0,10
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LINKS
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EXAMPLE
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A(6,0) = 3: [1,1,1,1,1,1], [1,2,3], [1,5].
A(5,1) = 4: [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,4].
A(4,2) = 4: [1,1,1,1], [1,1,2], [1,2,1], [1,3].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, 2, 2, 2, ...
2, 3, 4, 4, 4, 4, 4, 4, 4, ...
2, 4, 7, 7, 8, 8, 8, 8, 8, ...
3, 6, 11, 13, 15, 15, 16, 16, 16, ...
2, 9, 18, 23, 28, 29, 31, 31, 32, ...
2, 13, 32, 40, 52, 56, 60, 61, 63, ...
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MAPLE
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b:= proc(n, i, j, k) option remember; `if`(n=0, 1,
`if`(i=0, add(b(n-h, j, h, k), h=1..n), add(
b(n-h, j, h, k), h=max(1, 2*j-i-k)..min(n, 2*j-i+k))))
end:
A:= (n, k)-> `if`(n=0, 1, b(n-1, 0, 1, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, 1, If[i == 0, Sum[b[n-h, j, h, k], {h, 1, n}], Sum[b[n-h, j, h, k], {h, Max[1, 2*j - i - k], Min[n, 2*j - i + k]}]]] ; A[n_, k_] := If[n == 0, 1, b[n-1, 0, 1, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 22 2015, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 gives: A129654, A239551, A239552, A239553, A239554, A239555, A239556, A239557, A239558, A239559, A239560.
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KEYWORD
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AUTHOR
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STATUS
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approved
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