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

%I A187818 #23 Mar 13 2015 22:54:38
%S A187818 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 A187818 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 A187818 63,31,31,15,15,15,15,7,7,7,7,7,7,7,7,3,3
%N A187818 Triangle read by rows in which row n lists the first 2^(n-1) terms of A038712 in nonincreasing order, n >= 1.
%C A187818 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 A187818 Row lengths is A000079.
%C A187818 Row sums give A001787, n >= 1.
%e A187818 For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
%e A187818 ------------------------------------------------------
%e A187818 .         A038712 as
%e A187818 .       a tree of sum      Diagram
%e A187818 Region   of all parts    of regions     Composition
%e A187818 ------------------------------------------------------
%e A187818 .                         _ _ _ _ _
%e A187818 1      | 1          |    |_| | | | |    1, 1, 1, 1, 1
%e A187818 2      |   3        |    |_ _| | | |    2, 1, 1, 1
%e A187818 3      | 1          |    |_|   | | |    1, 2, 1, 1
%e A187818 4      |      7     |    |_ _ _| | |    3, 1, 1
%e A187818 5      | 1          |    |_| |   | |    1, 1, 2, 1
%e A187818 6      |   3        |    |_ _|   | |    2, 2, 1
%e A187818 7      | 1          |    |_|     | |    1, 3, 1
%e A187818 8      |       15   |    |_ _ _ _| |    4, 1
%e A187818 9      | 1          |    |_| | |   |    1, 1, 1, 2
%e A187818 10     |   3        |    |_ _| |   |    2, 1, 2
%e A187818 11     | 1          |    |_|   |   |    1, 2, 2
%e A187818 12     |      7     |    |_ _ _|   |    3, 2
%e A187818 13     | 1          |    |_| |     |    1, 1, 3
%e A187818 14     |   3        |    |_ _|     |    2, 3
%e A187818 15     | 1          |    |_|       |    1, 4
%e A187818 16     |         31 |    |_ _ _ _ _|    5
%e A187818 .
%e A187818 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 A187818 Triangle begins:
%e A187818 1;
%e A187818 3,1;
%e A187818 7,3,1,1;
%e A187818 15,7,3,3,1,1,1,1;
%e A187818 31,15,7,7,3,3,3,3,1,1,1,1,1,1,1,1;
%e A187818 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 A187818 ...
%Y A187818 Cf. A000079, A000225, A001511, A001787, A001792, A006519, A011782, A038712, A065120, A187816, A228525, A228369.
%K A187818 nonn,tabf,easy
%O A187818 1,2
%A A187818 _Omar E. Pol_, Sep 10 2013