%I #17 Oct 22 2018 11:05:06
%S 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,
%T 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,
%U 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
%N 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.
%H Alois P. Heinz, <a href="/A284249/b284249.txt">Rows n = 0..200, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Triangular_number">Triangular number</a>
%e Triangle T(n,k) begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 1, 1;
%e 1, 2, 2, 1, 1;
%e 1, 2, 3, 3, 1, 1;
%e 1, 3, 4, 5, 3, 1, 1;
%e 1, 3, 5, 8, 6, 4, 1, 1;
%e 1, 3, 7, 12, 11, 9, 4, 1, 1;
%e 1, 3, 9, 16, 20, 18, 11, 5, 1, 1;
%e 1, 4, 10, 22, 32, 35, 26, 14, 5, 1, 1;
%e 1, 4, 12, 29, 48, 61, 55, 36, 17, 6, 1, 1;
%e 1, 4, 14, 37, 70, 100, 106, 84, 48, 21, 6, 1, 1;
%p b:= proc(n, s) option remember; expand(`if`(n=0,
%p `if`(issqr(8*s+1), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
%p seq(T(n), n=0..16);
%t 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 T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
%t Table[T[n], {n, 0, 16}] // Flatten (*_Jean-François Alcover_, May 29 2018, from Maple *)
%Y Columns k=0-10 give: A000012, A003056, A320848, A320849, A320850, A320851, A320852, A320853, A320854, A320855, A320856.
%Y Second and third lower diagonals give: A008619(n+1), A008747(n+1).
%Y Row sums give A284250.
%Y T(2n,n) gives A284251.
%Y Cf. A000217, A281871.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Mar 23 2017