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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.

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