A225596 Sum of largest parts of all partitions of n plus n. Also, total number of parts in all partitions of n plus n.

0, 2, 5, 9, 16, 25, 41, 61, 94, 137, 202, 286, 411, 569, 794, 1083, 1479, 1982, 2662, 3517, 4650, 6073, 7921, 10229, 13198, 16876, 21548, 27321, 34573, 43482, 54593, 68166, 84959, 105399, 130496, 160911, 198050, 242849, 297239, 362626, 441586, 536145

Offset

0,2

Comments

a(n) is also the number of horizontal toothpicks (or the total length of all horizontal boundary segments) in the diagram of regions of the set of partitions of n, see example. A093694(n) is the number of vertical toothpicks in the diagram. See also A225610. For a minimalist version of the diagram see A211978. For the definition of "region" see A206437.

Links

Table of n, a(n) for n=0..41.

Formula

a(n) = A006128(n) + n = A225610(n) - A093694(n).

a(n) = n + sum_{k=1..A000041(n)} A141285(k), n >= 1.

Example

For n = 7 the sum of largest parts of all partitions of 7 plus 7 is (7+4+5+3+6+3+4+2+5+3+4+2+3+2+1) + 7 = 54 + 7 = 61. On the other hand the number of toothpicks in horizontal direction in the diagram of regions of the set of partitions of 7 is equal to 61, so a(7) = 61.
.
.                  Diagram of regions       Horizontal
Partitions         and partitions of 7      toothpicks
of 7
.                     _ _ _ _ _ _ _
7                    |_ _ _ _      |             7
4+3                  |_ _ _ _|_    |             4
5+2                  |_ _ _    |   |             5
3+2+2                |_ _ _|_ _|_  |             3
6+1                  |_ _ _      | |             6
3+3+1                |_ _ _|_    | |             3
4+2+1                |_ _    |   | |             4
2+2+2+1              |_ _|_ _|_  | |             2
5+1+1                |_ _ _    | | |             5
3+2+1+1              |_ _ _|_  | | |             3
4+1+1+1              |_ _    | | | |             4
2+2+1+1+1            |_ _|_  | | | |             2
3+1+1+1+1            |_ _  | | | | |             3
2+1+1+1+1+1          |_  | | | | | |             2
1+1+1+1+1+1+1        |_|_|_|_|_|_|_|             1
.                                                7
.                                              _____
.                                       Total   61
.

Crossrefs

Cf. A000041, A006128, A093694, A135010, A139250, A141285, A186114, A194446, A194447, A206437, A211978, A220517, A225600, A225610.

Sequence in context: A007979 A097701 A362548 * A211881 A056870 A326507

Adjacent sequences: A225593 A225594 A225595 * A225597 A225598 A225599

Keyword

nonn

Author

Omar E. Pol, Aug 01 2013

Status

approved