A330242 Sum of largest emergent parts of the partitions of n.

0, 0, 0, 2, 3, 9, 12, 24, 33, 54, 72, 112, 144, 210, 273, 379, 485, 661, 835, 1112, 1401, 1825, 2284, 2944, 3652, 4645, 5745, 7223, 8879, 11080, 13541, 16760, 20406, 25062, 30379, 37102, 44761, 54351, 65347, 78919, 94517, 113645, 135603, 162331, 193088, 230182, 272916, 324195, 383169, 453571

Offset

1,4

Comments

In other words: a(n) is the sum of the largest parts of all partitions of n that contain emergent parts.

The partitions of n that contain emergent parts are the partitions that contain neither 1 nor n as a part. All parts of these partitions are emergent parts except the last part of every partition.

For the definition of emergent part see A182699.

Links

Table of n, a(n) for n=1..50.

Formula

a(n) = A138137(n) - n.

a(n) = A207031(n,1) - n.

Example

For n = 9 the diagram of
the partitions of 9 that
do not contain 1 as a part
is as shown below:           Partitions
.
    |_ _ _|   |   |   |      [3, 2, 2, 2]
    |_ _ _ _ _|   |   |      [5, 2, 2]
    |_ _ _ _|     |   |      [4, 3, 2]
    |_ _ _ _ _ _ _|   |      [7, 2]
    |_ _ _|     |     |      [3, 3, 3]
    |_ _ _ _ _ _|     |      [6, 3]
    |_ _ _ _ _|       |      [5, 4]
    |_ _ _ _ _ _ _ _ _|      [9]
.
Note that the above diagram is also the "head" of the last section of the set of partitions of 9, where the "tail" is formed by A000041(9-1)= 22 1's.
The diagram of the
emergent parts is as
shown below:                 Emergent parts
.
    |_ _ _|   |   |          [3, 2, 2]
    |_ _ _ _ _|   |          [5, 2]
    |_ _ _ _|     |          [4, 3]
    |_ _ _ _ _ _ _|          [7]
    |_ _ _|     |            [3, 3]
    |_ _ _ _ _ _|            [6]
    |_ _ _ _ _|              [5]
.
The sum of the largest emergent parts is 3 + 5 + 4 + 7 + 3 + 6 + 5 = 33, so a(9) = 33.

Crossrefs

Cf. A000041, A002865, A006128, A135010, A138135, A138137, A141285, A182699, A182703, A182709, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A207031, A299474, A299475.

Sequence in context: A067719 A057017 A330298 * A239859 A047162 A155929

Adjacent sequences: A330239 A330240 A330241 * A330243 A330244 A330245

Keyword

nonn

Author

Omar E. Pol, Dec 06 2019

Status

approved