A096137 Table read by rows: row n contains the sum of each nonempty subset of {1, 2, ..., n}. In each row, the sums are arranged in ascending order.

1, 1, 2, 3, 1, 2, 3, 3, 4, 5, 6, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 14, 15, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12

Offset: 1

Author

Amarnath Murthy, Jul 06 2004

Keywords

• nonn
• easy
• look
• tabf

Comments

The n-th row has 2^n-1 members. A001788 gives the row sums. The sums of the k-element subsets of {1, 2, ..., n} add up to A094305(n-1, k-1).

Examples

			The nonempty subsets of {1, 2, 3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}, which have sums 1, 2, 3, 3, 4, 5 and 6 respectively, so these are the terms of row 3.
Triangle T(n,k) begins:
  1;
  1, 2, 3;
  1, 2, 3, 3, 4, 5, 6;
  1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..15, flattened

Crossrefs

Cf. A001788, A094305.

Programs

Maple

T:= proc(n) option remember; `if`(n=0, [][], subsop(1=[][],
      sort(map(x-> (x, x+n), [0, T(n-1)])))[])
    end:
seq(T(n), n=1..7);  # Alois P. Heinz, Jul 24 2019

Mathematica

T[n_] := T[n] = Total /@ Subsets[Range[n], {1, n}] // Sort;
Array[T, 7] // Flatten (* Jean-François Alcover, Feb 14 2021 *)

Extensions

Edited and extended by David Wasserman, Oct 04 2007