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%I #26 Dec 21 2015 03:03:31
%S 1,4,9,19,33,59,93,150,226,342,494,721,1011,1425,1960,2695,3633,4903,
%T 6506,8633,11312,14796,19157,24773,31744,40608,51578,65372,82341,
%U 103522,129428,161505,200589,248614,306869,378051,463987,568387,693989,845754,1027625
%N An unrestricted partition statistic: sum of A179864 over row n.
%C 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
%C 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
%F a(n) = Sum_{k=1..A000041(n)} A179864(n,k).
%F a(n) = A211978(n) - A000041(n). - _Omar E. Pol_, Jul 15 2013
%F a(n) = A225600(A139582(n)-1), n>= 1. - _Omar E. Pol_, Jul 25 2013
%e From _Omar E. Pol_, Jul 15 2013: (Start)
%e 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.
%e .
%e 7..................................
%e . /\
%e 5.................... / \ /\
%e . /\ / \ /\ /
%e 3.......... / \ / \ / \/
%e 2..... /\ / \ /\/ \ /
%e 1.. /\ / \ /\/ \ / \ /\/
%e 0 /\/ \/ \/ \/ \/
%e . 0,2, 6, 12, 24, 40... = A211978
%e . 1, 4, 9, 19, 33... = this sequence (End)
%Y Cf. A179864.
%K nonn
%O 1,2
%A _Alford Arnold_, Aug 02 2010
%E More terms from _Omar E. Pol_, Jul 15 2013
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