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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. 13
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified May 27 18:56 EDT 2020. Contains 334664 sequences. (Running on oeis4.)