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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
OFFSET
0,8
LINKS
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
Second and third lower diagonals give: A008619(n+1), A008747(n+1).
Row sums give A284250.
T(2n,n) gives A284251.
Sequence in context: A239550 A058398 A091499 * A137350 A334607 A166240
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 23 2017
STATUS
approved