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A228528 Triangle read by rows in which row n lists the compositions (ordered partitions) of n (see Comments lines for definition).

%I A228528 #13 Sep 17 2013 15:52:51
%S A228528 1,1,1,2,2,1,1,1,1,1,2,3,3,1,1,2,1,2,1,1,1,1,1,1,1,1,2,2,2,1,3,4,4,1,
%T A228528 1,3,1,2,2,1,1,1,2,1,3,1,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,2,1,2,1,
%U A228528 2,2,3,2,1,1,3,2,3,1,4,5,5,1,1,4,1
%N A228528 Triangle read by rows in which row n lists the compositions (ordered partitions) of n (see Comments lines for definition).
%C A228528 In order to construct this sequence we use the following rules: T(1,1) = 1. For n >= 2, row n lists the last 2^(n-2) compositions from the n-th row of triangle A228351 and then the last 2^(n-2) compositions from the n-th row of triangle A228525. In both cases these compositions are listed in the same order as they are listed in the mentioned triangles.
%C A228528 Row n has length A001792(n-1).
%C A228528 Row sums give A001787, n >= 1.
%C A228528 First differs from A227736 at a(18).
%e A228528 Illustration of initial terms:
%e A228528 ---------------------------------------
%e A228528 n   j     Diagram        Composition j
%e A228528 ---------------------------------------
%e A228528 .          _
%e A228528 1   1     |_|_           1;
%e A228528 2   1     |_| |          1, 1,
%e A228528 2   2     |_ _|_         2;
%e A228528 3   1     |_  | |        2, 1,
%e A228528 3   2     |_|_| |        1, 1, 1,
%e A228528 3   3     |_|   |        1, 2,
%e A228528 3   4     |_ _ _|_       3;
%e A228528 4   1     |_    | |      3, 1,
%e A228528 4   2     |_|_  | |      1, 2, 1,
%e A228528 4   3     |_  | | |      2, 1, 1,
%e A228528 4   4     |_|_|_| |      1, 1, 1, 1,
%e A228528 4   5     |_| |   |      1, 1, 2,
%e A228528 4   6     |_ _|   |      2, 2,
%e A228528 4   7     |_|     |      1, 3,
%e A228528 4   8     |_ _ _ _|_     4;
%e A228528 5   1     |_      | |    4, 1,
%e A228528 5   2     |_|_    | |    1, 3, 1,
%e A228528 5   3     |_  |   | |    2, 2, 1,
%e A228528 5   4     |_|_|_  | |    1, 1, 2, 1,
%e A228528 5   5     |_    | | |    3, 1, 1,
%e A228528 5   6     |_|_  | | |    1, 2, 1, 1,
%e A228528 5   7     |_  | | | |    2, 1, 1, 1,
%e A228528 5   8     |_|_|_|_| |    1, 1, 1, 1, 1,
%e A228528 5   9     |_| | |   |    1, 1, 1, 2,
%e A228528 5  10     |_ _| |   |    2, 1, 2,
%e A228528 5  11     |_|   |   |    1, 2, 2,
%e A228528 5  12     |_ _ _|   |    3, 2,
%e A228528 5  13     |_| |     |    1, 1, 3,
%e A228528 5  14     |_ _|     |    2, 3,
%e A228528 5  15     |_|       |    1, 4,
%e A228528 5  16     |_ _ _ _ _|    5;
%e A228528 .
%e A228528 Triangle begins:
%e A228528 [1];
%e A228528 [1,1], [2];
%e A228528 [2,1], [1,1,1], [1,2], [3];
%e A228528 [3,1], [1,2,1], [2,1,1], [1,1,1,1], [1,1,2], [2,2], [1,3], [4];
%e A228528 [4,1], [1,3,1], [2,2,1], [1,1,2,1], [3,1,1], [1,2,1,1], [2,1,1,1], [1,1,1,1,1], [1,1,1,2], [2,1,2], [1,2,2], [3,2], [1,1,3], [2,3], [1,4], [5];
%Y A228528 Cf. A001792, A001787, A011782, A066099, A227736, A228525, A228351, A228369.
%K A228528 nonn,tabf
%O A228528 1,4
%A A228528 _Omar E. Pol_, Sep 07 2013