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A168443
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Triangle, T(n,k) = number of compositions a(1),...,a(k) of n, such that a(i+1) <= a(i) + 1 for 1 <= i < k.
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2
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 4, 4, 1, 1, 3, 6, 7, 5, 1, 1, 4, 7, 11, 11, 6, 1, 1, 4, 9, 15, 19, 16, 7, 1, 1, 5, 11, 19, 29, 31, 22, 8, 1, 1, 5, 13, 25, 39, 52, 48, 29, 9, 1, 1, 6, 15, 30, 53, 76, 88, 71, 37, 10, 1, 1, 6, 18, 37, 67, 107, 140, 142, 101, 46, 11, 1, 1, 7, 20, 44, 84, 143, 207, 245, 220, 139, 56, 12, 1
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OFFSET
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1,5
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LINKS
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EXAMPLE
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Triangle T(n,k) begins:
1;
1, 1;
1, 2, 1;
1, 2, 3, 1;
1, 3, 4, 4, 1;
1, 3, 6, 7, 5, 1;
1, 4, 7, 11, 11, 6, 1;
1, 4, 9, 15, 19, 16, 7, 1;
...
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MAPLE
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b:= proc(n, k) option remember; expand(`if`(n=0, 1,
x*add(b(n-j, j), j=1..min(n, k+1))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
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MATHEMATICA
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b[n_, k_] := b[n, k] = Expand[If[n == 0, 1,
x*Sum[b[n - j, j], {j, 1, Min[n, k + 1]}]]];
T[n_] := Rest@CoefficientList[b[n, n], x];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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