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A179862 An unrestricted partition statistic: sum of A179864 over row n. 5
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
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.
Sequence in context: A038403 A009856 A009920 * A008036 A009863 A301249
KEYWORD
nonn
AUTHOR
Alford Arnold, Aug 02 2010
EXTENSIONS
More terms from Omar E. Pol, Jul 15 2013
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)