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%I #30 Apr 11 2024 04:51:28
%S 0,0,0,1,2,5,8,14,22,34,50,75,106,151,210,291,394,535,712,949,1246,
%T 1634,2118,2745,3520,4508,5728,7266,9152,11512,14390,17959,22298,
%U 27634,34094,41993,51510,63075,76966,93752,113834,137992,166788,201269,242248,291102,348976,417727
%N a(n) is the sum over all partitions of n of the number of right angles that are not the largest right angle.
%C a(n) is also the sum of the sizes of the Durfee squares of all partitions of n, minus the number of partitions of n.
%C a(n) is also the sum of positive cranks of all partitions of n, minus the number of partitions of n.
%D G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143.
%F a(n) = A115995(n) - A000041(n), n >= 1.
%e For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:
%e .
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 1| |8 2| _|8 3| _ _|8 4| _ _ _|8 5| _ _ _ _|8
%e 1| | 1| | 1| | 1| | 1| |
%e 1| | 1| | 1| | 1| | 1| |
%e 1| | 1| | 1| | 1| | 1|_|
%e 1| | 1| | 1| | 1|_|
%e 1| | 1| | 1|_|
%e 1| | 1|_|
%e 1|_|
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 6| _ _ _ _ _|8 7| _ _ _ _ _ _|8 8|_ _ _ _ _ _ _ _|8
%e 1| | 1|_|
%e 1|_|
%e .
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 2| _|7 3| _ _|7 4| _ _ _|7 5| _ _ _ _|7 6| _ _ _ _ _|7
%e 2| |_|1 2| |_| 1 2| |_| 1 2| |_| 1 2|_|_| 1
%e 1| | 1| | 1| | 1|_|
%e 1| | 1| | 1|_|
%e 1| | 1|_|
%e 1|_|
%e .
%e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 2| _|6 3| _ _|6 3| _ _|6 4| _ _ _|6 4| _ _ _|6 5| _ _ _ _|6
%e 2| | |2 2| | | 2 3| |_ _|2 2| | | 2 3| |_ _| 2 3|_|_ _| 2
%e 2| |_| 2| |_| 1| | 2|_|_| 1|_|
%e 1| | 1|_| 1|_|
%e 1|_|
%e .
%e _ _ _ _ _ _ _ _ _
%e 2| _|5 3| _ _|5 4| _ _ _|5
%e 2| | |3 3| | _|3 4|_|_ _ _|3
%e 2| | | 2|_|_|
%e 2|_|_|
%e .
%e In total there are 14 right angles that are not the largest right angle of the partitions of 8, so a(8) = 14.
%Y Cf. A000041, A115995, A330369, A330375, A330379.
%K nonn
%O 1,5
%A _Omar E. Pol_, Jan 01 2020
%E a(17) and a(47) corrected by _Georg Fischer_, Apr 11 2024