A128628 An irregular triangular array read by rows, with shape sequence A000041(n) related to sequence A060850.

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, 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, 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

Offset

1,3

Comments

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.

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.

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]

Links

T. D. Noe, Rows n = 1..25 of irregular triangle, flattened

Example

The values at level three are 1, 2, and 3.
The 1 generates 1 and 2; the 2 generates 2 and 3; the 3 only generates 4.
The array begins
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

Mathematica

Flatten[Table[Length /@ IntegerPartitions[n], {n, 9}]] (* T. D. Noe, Feb 27 2014 *)

Crossrefs

Cf. A000041, A060850, A128629.

Cf. A006128 (row sums), A036043.

Cf. A177740.

Cf. A308355 (limiting row sequence).

Sequence in context: A252230 A036043 A333486 * A238966 A353510 A334230

Adjacent sequences: A128625 A128626 A128627 * A128629 A128630 A128631

Keyword

nonn,tabf

Author

Alford Arnold, Mar 27 2007, Aug 01 2007

Status

approved