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A179862 An unrestricted partition statistic: sum of A179864 over row n.

%I A179862 #26 Dec 21 2015 03:03:31
%S A179862 1,4,9,19,33,59,93,150,226,342,494,721,1011,1425,1960,2695,3633,4903,
%T A179862 6506,8633,11312,14796,19157,24773,31744,40608,51578,65372,82341,
%U A179862 103522,129428,161505,200589,248614,306869,378051,463987,568387,693989,845754,1027625
%N A179862 An unrestricted partition statistic: sum of A179864 over row n.
%C A179862 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 A179862 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 A179862 a(n) = Sum_{k=1..A000041(n)} A179864(n,k).
%F A179862 a(n) = A211978(n) - A000041(n). - _Omar E. Pol_, Jul 15 2013
%F A179862 a(n) = A225600(A139582(n)-1), n>= 1. - _Omar E. Pol_, Jul 25 2013
%e A179862 From _Omar E. Pol_, Jul 15 2013: (Start)
%e A179862 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 A179862 .
%e A179862 7..................................
%e A179862 .                                 /\
%e A179862 5....................            /  \                /\
%e A179862 .                   /\          /    \          /\  /
%e A179862 3..........        /  \        /      \        /  \/
%e A179862 2.....    /\      /    \    /\/        \      /
%e A179862 1..  /\  /  \  /\/      \  /            \  /\/
%e A179862 0 /\/  \/    \/          \/              \/
%e A179862 . 0,2,  6,   12,         24,             40... = A211978
%e A179862 .  1, 4,   9,       19,           33... = this sequence (End)
%Y A179862 Cf. A179864.
%K A179862 nonn
%O A179862 1,2
%A A179862 _Alford Arnold_, Aug 02 2010
%E A179862 More terms from _Omar E. Pol_, Jul 15 2013