A177740 -id:A177740 - cp's OEIS frontend

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

Alford Arnold, Mar 27 2007, Aug 01 2007

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

			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

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

Cf. A000041, A060850, A128629.
Cf. A006128 (row sums), A036043.
Cf. A177740.
Cf. A308355 (limiting row sequence).

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

A177739 In those partitions of n with every part >=3, the total number of parts (counted with multiplicity).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 5, 8, 10, 13, 22, 25, 34, 49, 62, 77, 108, 132, 172, 221, 276, 345, 448, 544, 680, 851, 1050, 1280, 1596, 1931, 2366, 2884, 3496, 4220, 5135, 6144, 7403, 8890, 10644, 12679, 15177, 18007, 21419, 25399, 30066, 35488, 41971, 49344, 58088

Offset: 0

Jason Kimberley, May 13 2010

Also the number of components (counted with multiplicity) of the 2-regular simple graphs of order n.

The number of such partitions is given by A008483.
Lengths of the rows of triangle A176210.
Row sums of triangle A177740.
Cf. A006128, A138135.

[ #&cat RestrictedPartitions(n,{3..n}):n in [0..50]];

Table[Length[Flatten[Select[IntegerPartitions[n],Min[#]>2&]]],{n,0,50}] (* Harvey P. Dale, May 12 2020 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica