A330378 - OEIS

%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