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A284249
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Number T(n,k) of k-element subsets of [n] whose sum is a triangular number; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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13
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 3, 4, 5, 3, 1, 1, 1, 3, 5, 8, 6, 4, 1, 1, 1, 3, 7, 12, 11, 9, 4, 1, 1, 1, 3, 9, 16, 20, 18, 11, 5, 1, 1, 1, 4, 10, 22, 32, 35, 26, 14, 5, 1, 1, 1, 4, 12, 29, 48, 61, 55, 36, 17, 6, 1, 1, 1, 4, 14, 37, 70, 100, 106, 84, 48, 21, 6, 1, 1
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OFFSET
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0,8
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LINKS
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EXAMPLE
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Triangle T(n,k) begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 2, 2, 1, 1;
1, 2, 3, 3, 1, 1;
1, 3, 4, 5, 3, 1, 1;
1, 3, 5, 8, 6, 4, 1, 1;
1, 3, 7, 12, 11, 9, 4, 1, 1;
1, 3, 9, 16, 20, 18, 11, 5, 1, 1;
1, 4, 10, 22, 32, 35, 26, 14, 5, 1, 1;
1, 4, 12, 29, 48, 61, 55, 36, 17, 6, 1, 1;
1, 4, 14, 37, 70, 100, 106, 84, 48, 21, 6, 1, 1;
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MAPLE
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b:= proc(n, s) option remember; expand(`if`(n=0,
`if`(issqr(8*s+1), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..16);
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MATHEMATICA
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b[n_, s_] := b[n, s] = Expand[If[n == 0, If[IntegerQ @ Sqrt[8*s + 1], 1, 0], b[n - 1, s] + x*b[n - 1, s + n]]];
T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
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CROSSREFS
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Columns k=0-10 give: A000012, A003056, A320848, A320849, A320850, A320851, A320852, A320853, A320854, A320855, A320856.
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KEYWORD
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AUTHOR
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STATUS
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approved
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