A179862 - OEIS

A179862

An unrestricted partition statistic: sum of A179864 over row n.

1, 4, 9, 19, 33, 59, 93, 150, 226, 342, 494, 721, 1011, 1425, 1960, 2695, 3633, 4903, 6506, 8633, 11312, 14796, 19157, 24773, 31744, 40608, 51578, 65372, 82341, 103522, 129428, 161505, 200589, 248614, 306869, 378051, 463987, 568387, 693989, 845754, 1027625

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n minus the number of partitions of n. - Omar E. Pol, Jul 15 2013

Sum of the hook-lengths of the (1,1)-cells of the Ferrers diagrams over all partitions of n. Example: a(3) = 9 because in each of the partitions 3, 21, and 111 the (1,1)-cell has hook-length 3. Comment follows at once from the previous comment. - Emeric Deutsch, Dec 20 2015

LINKS

Table of n, a(n) for n=1..41.

FORMULA

a(n) = Sum_{k=1..A000041(n)} A179864(n,k).

a(n) = A211978(n) - A000041(n). - Omar E. Pol, Jul 15 2013

a(n) = A225600(A139582(n)-1), n>= 1. - Omar E. Pol, Jul 25 2013

EXAMPLE

From Omar E. Pol, Jul 15 2013: (Start)

Illustration of initial terms using a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). a(n) is the x-coordinate of the mentioned largest peak. Note that this Dyck path is infinite.

7..................................

. /\

5.................... / \ /\

. /\ / \ /\ /

3.......... / \ / \ / \/

2..... /\ / \ /\/ \ /

1.. /\ / \ /\/ \ / \ /\/

0 /\/ \/ \/ \/ \/

. 0,2, 6, 12, 24, 40... = A211978

. 1, 4, 9, 19, 33... = this sequence (End)

CROSSREFS

Cf. A179864.