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%I #18 Jun 03 2019 19:47:19
%S 1,1,2,1,2,3,1,2,2,3,4,1,2,2,3,3,4,5,1,2,2,3,2,3,4,3,4,5,6,1,2,2,3,2,
%T 3,4,3,3,4,5,4,5,6,7,1,2,2,3,2,3,4,2,3,3,4,5,3,4,4,5,6,4,5,6,7,8,1,2,
%U 2,3,2,3,4,2,3,3,4,5,3,3,4,4,5,6,3,4,5,4,5,6,7,5,6,7,8,9
%N An irregular triangular array read by rows, with shape sequence A000041(n) related to sequence A060850.
%C The next level gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent.
%C Sequence A036043 counts the parts of numeric partitions and contains the same values on each row as the current sequence. When a node generates two branches the first branch can be mapped to cyclic partitions; all other branches map to matching partitions.
%C Appears to be the triangle in which the n-th row contains the number of parts of each partition of n, where the partitions are ordered as in A080577. - _Jason Kimberley_, May 12 2010]
%H T. D. Noe, <a href="/A128628/b128628.txt">Rows n = 1..25 of irregular triangle, flattened</a>
%e The values at level three are 1, 2, and 3.
%e The 1 generates 1 and 2; the 2 generates 2 and 3; the 3 only generates 4.
%e The array begins
%e 1
%e 1 2
%e 1 2 3
%e 1 2 2 3 4
%e 1 2 2 3 3 4 5
%e 1 2 2 3 2 3 4 3 4 5 6
%t Flatten[Table[Length /@ IntegerPartitions[n], {n, 9}]] (* _T. D. Noe_, Feb 27 2014 *)
%Y Cf. A000041, A060850, A128629.
%Y Cf. A006128 (row sums), A036043.
%Y Cf. A177740.
%Y Cf. A308355 (limiting row sequence).
%K nonn,tabf
%O 1,3
%A _Alford Arnold_, Mar 27 2007, Aug 01 2007
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