A281994 -id:A281994 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Jan 31 2017

			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;
  ...

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

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, A377572.

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);

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 *)

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 }.
Sum_{k=0..n} k * T(n,k) = A377572(n).

A278339 Number of set partitions of [n] into subsets whose element sums are distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 23, 9, 41, 248, 277, 1556, 2854, 5233, 20701, 145137, 1626890, 4118910, 9963276, 9260756, 64027363, 365237571, 1002679107, 21594036300, 24465529531, 144914973347, 1921444799766

Offset: 0

Alois P. Heinz, Nov 18 2016

nonn

			a(0) = 1: {}.
a(1) = 1: 1.
a(4) = 1: 1|234.
a(6) = 1: 1|2356|4.
a(8) = 1: 12345678.
a(9) = 23: 12345678|9, 123568|4|79, 1236789|45, 1245789|36, 126|345789, 12589|367|4, 1258|3679|4, 12679|358|4, 1267|3589|4, 1345689|27, 135|246789, 13|24568|79, 13579|268|4, 1357|2689|4, 13678|259|4, 13|259|4678, 13|2689|457, 13|268|4579, 156789|234, 18|2345679, 169|23578|4, 1789|2356|4, 178|23569|4.
a(10) = 9: 1|2356|4|78(10)|9, 1|23578|4|6(10)|9, 1|258(10)|367|4|9, 1|258(10)|36|4|79, 1|259|36|4|78(10), 1|267(10)|358|4|9, 1|268|357(10)|4|9, 1|27|3589|4|6(10), 1|27|358|4|69(10).

Cf. A252897, A275780, A278329, A278340, A281994.

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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.

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