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A281871 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. 24
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

T(n,n) = 1 for n in { A001108 }, T(n,n) = 0 otherwise.

T(n,n-1) = 1 for n in { A214857 }, T(n,n-1) = 0 for n in { A214858 }.

EXAMPLE

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;

MAPLE

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);

MATHEMATICA

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]];

Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

CROSSREFS

Columns k=0-10 give: A000012, A000196, A176615, A281706, A281864, A281865, A281866, A281867, A281868, A281869, A281870.

Main diagonal is characteristic function of A001108.

Diagonals T(n+k,n) for k=2-10 give: A281965, A281966, A281967, A281968, A281969, A281970, A281971, A281972, A281973.

Row sums give A126024.

T(2n,n) gives A281872.

Cf. A000217, A000290, A007318, A214857, A214858, A278339, A281994, A284249.

Sequence in context: A259538 A099314 A214341 * A131334 A004602 A247418

Adjacent sequences: A281868 A281869 A281870 * A281872 A281873 A281874

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jan 31 2017

STATUS

approved

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Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)