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

0,8

Links

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

Wikipedia, Triangular number

Example

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;

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 *)

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.

Sequence in context: A239550 A058398 A091499 * A137350 A334607 A166240

Adjacent sequences: A284246 A284247 A284248 * A284250 A284251 A284252

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 23 2017

Status

approved