%I #23 Mar 13 2015 22:54:38

%S 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,

%T 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,

%U 63,31,31,15,15,15,15,7,7,7,7,7,7,7,7,3,3

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

%C 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.

%C Row lengths is A000079.

%C Row sums give A001787, n >= 1.

%e For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:

%e ------------------------------------------------------

%e . A038712 as

%e . a tree of sum Diagram

%e Region of all parts of regions Composition

%e ------------------------------------------------------

%e . _ _ _ _ _

%e 1 | 1 | |_| | | | | 1, 1, 1, 1, 1

%e 2 | 3 | |_ _| | | | 2, 1, 1, 1

%e 3 | 1 | |_| | | | 1, 2, 1, 1

%e 4 | 7 | |_ _ _| | | 3, 1, 1

%e 5 | 1 | |_| | | | 1, 1, 2, 1

%e 6 | 3 | |_ _| | | 2, 2, 1

%e 7 | 1 | |_| | | 1, 3, 1

%e 8 | 15 | |_ _ _ _| | 4, 1

%e 9 | 1 | |_| | | | 1, 1, 1, 2

%e 10 | 3 | |_ _| | | 2, 1, 2

%e 11 | 1 | |_| | | 1, 2, 2

%e 12 | 7 | |_ _ _| | 3, 2

%e 13 | 1 | |_| | | 1, 1, 3

%e 14 | 3 | |_ _| | 2, 3

%e 15 | 1 | |_| | 1, 4

%e 16 | 31 | |_ _ _ _ _| 5

%e .

%e 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.

%e Triangle begins:

%e 1;

%e 3,1;

%e 7,3,1,1;

%e 15,7,3,3,1,1,1,1;

%e 31,15,7,7,3,3,3,3,1,1,1,1,1,1,1,1;

%e 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;

%e ...

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

%K nonn,tabf,easy

%O 1,2

%A _Omar E. Pol_, Sep 10 2013