A284249 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.

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

Offset: 0

Author

Alois P. Heinz, Mar 23 2017

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Triangular number

Crossrefs

Columns k=0-10 give: A000012, A003056, A320848, A320849, A320850, A320851, A320852, A320853, A320854, A320855, A320856.

Second and third lower diagonals give: A008619(n+1), A008747(n+1).

Row sums give A284250.

T(2n,n) gives A284251.

Cf. A000217, A281871.

Programs

Maple

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);

Mathematica

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]];
Table[T[n], {n, 0, 16}] // Flatten (*Jean-François Alcover, May 29 2018, from Maple *)