A187818 Triangle read by rows in which row n lists the first 2^(n-1) terms of A038712 in nonincreasing order, n >= 1.

1, 3, 1, 7, 3, 1, 1, 15, 7, 3, 3, 1, 1, 1, 1, 31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 63, 31, 15, 15, 7, 7, 7, 7, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 63, 31, 31, 15, 15, 15, 15, 7, 7, 7, 7, 7, 7, 7, 7, 3, 3

Offset: 1

Author

Omar E. Pol, Sep 10 2013

Keywords

• nonn
• tabf
• easy

Comments

T(n,k) is also the sum of all parts of the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.

Row lengths is A000079.

Row sums give A001787, n >= 1.

Examples

			For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
------------------------------------------------------
.         A038712 as
.       a tree of sum      Diagram
Region   of all parts    of regions     Composition
------------------------------------------------------
.                         _ _ _ _ _
1      | 1          |    |_| | | | |    1, 1, 1, 1, 1
2      |   3        |    |_ _| | | |    2, 1, 1, 1
3      | 1          |    |_|   | | |    1, 2, 1, 1
4      |      7     |    |_ _ _| | |    3, 1, 1
5      | 1          |    |_| |   | |    1, 1, 2, 1
6      |   3        |    |_ _|   | |    2, 2, 1
7      | 1          |    |_|     | |    1, 3, 1
8      |       15   |    |_ _ _ _| |    4, 1
9      | 1          |    |_| | |   |    1, 1, 1, 2
10     |   3        |    |_ _| |   |    2, 1, 2
11     | 1          |    |_|   |   |    1, 2, 2
12     |      7     |    |_ _ _|   |    3, 2
13     | 1          |    |_| |     |    1, 1, 3
14     |   3        |    |_ _|     |    2, 3
15     | 1          |    |_|       |    1, 4
16     |         31 |    |_ _ _ _ _|    5
.
The first largest region in the diagram is the 16th region which contains 16 parts and the sum of parts is 31, so T(5,1) = 31. The second largest region is the 8th region which contains 8 parts and the sum of parts is 15, so T(5,2) = 15. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts and the sum of parts is 7, so T(5,3) = 7 and T(5,4) = 7. And so on. The sequence of the sum of all parts of the k-th largest region of the diagram is [31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
Triangle begins:
1;
3,1;
7,3,1,1;
15,7,3,3,1,1,1,1;
31,15,7,7,3,3,3,3,1,1,1,1,1,1,1,1;
63,31,15,15,7,7,7,7,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...
		

Crossrefs

Cf. A000079, A000225, A001511, A001787, A001792, A006519, A011782, A038712, A065120, A187816, A228525, A228369.