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A330378 a(n) is the sum over all partitions of n of the number of right angles that are not the largest right angle. 3

%I

%S 0,0,0,1,2,5,8,14,22,34,50,75,106,151,210,291,294,535,712,949,1246,

%T 1634,2118,2745,3520,4508,5728,7266,9152,11512,14390,17959,22298,

%U 27634,34094,41993,51510,63075,76966,93752,113834,137992,166788,201269,242248,291102,312976,417727

%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 fllows:

%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

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Last modified October 25 17:22 EDT 2021. Contains 348255 sequences. (Running on oeis4.)