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
Examples
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;
...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
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.
Programs
-
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 *)