A284251 -id:A284251 - cp's OEIS frontend

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

Alois P. Heinz, Mar 23 2017

			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;

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

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.

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);

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 *)

A281872 Number of sets of exactly n positive integers <= 2n having a square element sum.

Original entry on oeis.org

1, 1, 1, 3, 8, 24, 74, 238, 786, 2646, 9071, 31536, 110967, 394420, 1414053, 5107523, 18568628, 67894564, 249513890, 921134164, 3414439996, 12703212213, 47419589969, 177551521838, 666655929561, 2509526456940, 9469082269965, 35807434636137, 135681047696427

Offset: 0

Alois P. Heinz, Jan 31 2017

nonn

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 1: {1,3}.
a(3) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
a(4) = 8: {1,2,5,8}, {1,2,6,7}, {1,3,4,8}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}, {4,6,7,8}.

Alois P. Heinz, Table of n, a(n) for n = 0..130

Cf. A281871, A284251.

a(n) = A281871(2n,n).

cp's OEIS Frontend

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.

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