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A281871
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Number T(n,k) of k-element subsets of [n] having a square element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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24
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1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 2, 0, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 4, 5, 5, 2, 1, 0, 1, 2, 5, 8, 8, 6, 3, 0, 1, 1, 3, 6, 11, 14, 13, 7, 4, 1, 0, 1, 3, 7, 15, 23, 24, 19, 10, 3, 1, 0, 1, 3, 8, 20, 34, 43, 39, 25, 13, 3, 1, 0, 1, 3, 9, 26, 49, 71, 74, 60, 34, 14, 5, 0, 0
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OFFSET
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0,12
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LINKS
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FORMULA
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T(n,n) = 1 for n in { A001108 }, T(n,n) = 0 otherwise.
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EXAMPLE
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T(7,0) = 1: {}.
T(7,1) = 2: {1}, {4}.
T(7,2) = 4: {1,3}, {2,7}, {3,6}, {4,5}.
T(7,3) = 5: {1,2,6}, {1,3,5}, {2,3,4}, {3,6,7}, {4,5,7}.
T(7,4) = 5: {1,2,6,7}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}.
T(7,5) = 2: {1,2,3,4,6}, {3,4,5,6,7}.
T(7,6) = 1: {1,2,4,5,6,7}.
T(7,7) = 0.
T(8,8) = 1: {1,2,3,4,5,6,7,8}.
Triangle T(n,k) begins:
1;
1, 1;
1, 1, 0;
1, 1, 1, 0;
1, 2, 1, 1, 0;
1, 2, 2, 2, 0, 0;
1, 2, 3, 3, 2, 1, 0;
1, 2, 4, 5, 5, 2, 1, 0;
1, 2, 5, 8, 8, 6, 3, 0, 1;
1, 3, 6, 11, 14, 13, 7, 4, 1, 0;
1, 3, 7, 15, 23, 24, 19, 10, 3, 1, 0;
1, 3, 8, 20, 34, 43, 39, 25, 13, 3, 1, 0;
1, 3, 9, 26, 49, 71, 74, 60, 34, 14, 5, 0, 0;
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MAPLE
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b:= proc(n, s) option remember; expand(`if`(n=0,
`if`(issqr(s), 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[s], 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, A000196, A176615, A281706, A281864, A281865, A281866, A281867, A281868, A281869, A281870.
Main diagonal is characteristic function of A001108.
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KEYWORD
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AUTHOR
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STATUS
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approved
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