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A228347 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

%I A228347 #13 Oct 22 2013 12:37:36
%S A228347 1,1,2,0,0,1,1,1,2,3,0,0,0,0,1,0,0,0,0,1,2,0,0,0,0,0,0,1,1,1,1,1,2,2,
%T A228347 3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U A228347 0,0,0,0,0,0,1,1,2,3,0,0,0,0,0,0,0,0
%N A228347 Triangle of regions and compositions of the positive integers (see Comments lines for definition).
%C A228347 Triangle read by rows in which row n lists A129760(n) zeros followed by the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.
%C A228347 The equivalent sequence for partitions is A186114.
%e A228347 ----------------------------------------------------------
%e A228347 .             Diagram                Triangle
%e A228347 Compositions    of            of compositions (rows)
%e A228347 of 5          regions          and regions (columns)
%e A228347 ----------------------------------------------------------
%e A228347 .            _ _ _ _ _
%e A228347 5           |_        |                                 5
%e A228347 1+4         |_|_      |                               1 4
%e A228347 2+3         |_  |     |                             2 0 3
%e A228347 1+1+3       |_|_|_    |                           1 1 0 3
%e A228347 3+2         |_    |   |                         3 0 0 0 2
%e A228347 1+2+2       |_|_  |   |                       1 2 0 0 0 2
%e A228347 2+1+2       |_  | |   |                     2 0 1 0 0 0 2
%e A228347 1+1+1+2     |_|_|_|_  |                   1 1 0 1 0 0 0 2
%e A228347 4+1         |_      | |                 4 0 0 0 0 0 0 0 1
%e A228347 1+3+1       |_|_    | |               1 3 0 0 0 0 0 0 0 1
%e A228347 2+2+1       |_  |   | |             2 0 2 0 0 0 0 0 0 0 1
%e A228347 1+1+2+1     |_|_|_  | |           1 1 0 2 0 0 0 0 0 0 0 1
%e A228347 3+1+1       |_    | | |         3 0 0 0 1 0 0 0 0 0 0 0 1
%e A228347 1+2+1+1     |_|_  | | |       1 2 0 0 0 1 0 0 0 0 0 0 0 1
%e A228347 2+1+1+1     |_  | | | |     2 0 1 0 0 0 1 0 0 0 0 0 0 0 1
%e A228347 1+1+1+1+1   |_|_|_|_|_|   1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1
%e A228347 .
%e A228347 For the positive integer k consider the first 2^(k-1) rows of triangle, as shown below. The positive terms of the n-th row are the parts of the n-th region of the diagram of regions of the set of compositions of k. The positive terms of the n-th column are the parts of the n-th composition of k, with compositions in colexicographic order.
%e A228347 Triangle begins:
%e A228347 1;
%e A228347 1,2;
%e A228347 0,0,1;
%e A228347 1,1,2,3;
%e A228347 0,0,0,0,1;
%e A228347 0,0,0,0,1,2;
%e A228347 0,0,0,0,0,0,1;
%e A228347 1,1,1,1,2,2,3,4;
%e A228347 0,0,0,0,0,0,0,0,1;
%e A228347 0,0,0,0,0,0,0,0,1,2;
%e A228347 0,0,0,0,0,0,0,0,0,0,1;
%e A228347 0,0,0,0,0,0,0,0,1,1,2,3;
%e A228347 0,0,0,0,0,0,0,0,0,0,0,0,1;
%e A228347 0,0,0,0,0,0,0,0,0,0,0,0,1,2;
%e A228347 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e A228347 1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
%e A228347 ...
%Y A228347 Mirror of A228348. Column 1 is A036987. Also column 1 gives A209229, n >= 1. Right border gives A001511. Positive terms give A228349.
%Y A228347 Cf. A001792, A001787, A006519, A011782, A065120, A096996, A129760, A186114, A187816, A187818, A206437, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.
%K A228347 nonn,tabl
%O A228347 1,3
%A A228347 _Omar E. Pol_, Aug 26 2013