%I #19 May 24 2017 02:39:54
%S 1,1,2,1,1,1,2,3,1,1,2,1,1,1,1,1,2,2,3,4,1,1,2,1,1,1,2,3,1,1,2,1,1,1,
%T 1,1,1,1,1,1,2,2,2,2,3,3,4,5,1,1,2,1,1,1,2,3,1,1,2,1,1,1,1,1,2,2,3,4,
%U 1,1,2,1,1,1,2,3,1,1,2,1,1,1,1,1,1,1
%N Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).
%C Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.
%C The equivalent sequence for partitions is A220482.
%H Michael De Vlieger, <a href="/A228349/b228349.txt">Table of n, a(n) for n = 1..13312</a> (rows 1 <= n <= 2^11 = 2048).
%e ----------------------------------------------------------
%e . Diagram Triangle
%e Compositions of of compositions (rows)
%e of 5 regions and regions (columns)
%e ----------------------------------------------------------
%e . _ _ _ _ _
%e 5 |_ | 5
%e 1+4 |_|_ | 1 4
%e 2+3 |_ | | 2 3
%e 1+1+3 |_|_|_ | 1 1 3
%e 3+2 |_ | | 3 2
%e 1+2+2 |_|_ | | 1 2 2
%e 2+1+2 |_ | | | 2 1 2
%e 1+1+1+2 |_|_|_|_ | 1 1 1 2
%e 4+1 |_ | | 4 1
%e 1+3+1 |_|_ | | 1 3 1
%e 2+2+1 |_ | | | 2 2 1
%e 1+1+2+1 |_|_|_ | | 1 1 2 1
%e 3+1+1 |_ | | | 3 1 1
%e 1+2+1+1 |_|_ | | | 1 2 1 1
%e 2+1+1+1 |_ | | | | 2 1 1 1
%e 1+1+1+1+1 |_|_|_|_|_| 1 1 1 1 1
%e .
%e Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
%e 1;
%e 1,2;
%e 1;
%e 1,1,2,3;
%e 1;
%e 1,2;
%e 1;
%e 1,1,1,1,2,2,3,4;
%e 1;
%e 1,2;
%e 1;
%e 1,1,2,3;
%e 1;
%e 1,2;
%e 1;
%e 1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
%e ...
%e Alternative interpretation of this sequence:
%e Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
%e [1];
%e [1,2];
%e [1],[1,1,2,3];
%e [1],[1,2],[1],[1,1,1,1,2,2,3,4];
%e [1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];
%t Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* _Michael De Vlieger_, May 23 2017 *)
%Y Main triangle: Right border gives A001511. Row j has length A006519(j). Row sums give A038712.
%Y Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A090996, A186114, A187816, A187818, A206437, A220482, A228347, A228348, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Aug 26 2013