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

0, 0, 0, 1, 2, 5, 8, 14, 22, 34, 50, 75, 106, 151, 210, 291, 394, 535, 712, 949, 1246, 1634, 2118, 2745, 3520, 4508, 5728, 7266, 9152, 11512, 14390, 17959, 22298, 27634, 34094, 41993, 51510, 63075, 76966, 93752, 113834, 137992, 166788, 201269, 242248, 291102, 348976, 417727

Offset: 1

Author

Omar E. Pol, Jan 01 2020

Keywords

• nonn

Comments

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.

a(n) is also the sum of positive cranks of all partitions of n, minus the number of partitions of n.

Examples

			For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:
.
    _       _ _       _ _ _       _ _ _ _       _ _ _ _ _
  1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1|_|
  1| |    1| |      1| |        1|_|
  1| |    1| |      1|_|
  1| |    1|_|
  1|_|
    _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _
  6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8
  1| |              1|_|
  1|_|
.
    _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _
  2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7
  2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1
  1| |      1| |        1| |          1|_|
  1| |      1| |        1|_|
  1| |      1|_|
  1|_|
.
    _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _
  2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6
  2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2
  2| |_|    2| |_|      1| |        2|_|_|        1|_|
  1| |      1|_|        1|_|
  1|_|
.
    _ _       _ _ _        _ _ _ _
  2|  _|5   3|  _ _|5    4|  _ _ _|5
  2| | |3   3| |  _|3    4|_|_ _ _|3
  2| | |    2|_|_|
  2|_|_|
.
In total there are 14 right angles that are not the largest right angle of the partitions of 8, so a(8) = 14.
		

References

• G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143.

Crossrefs

Cf. A000041, A115995, A330369, A330375, A330379.

Formula

a(n) = A115995(n) - A000041(n), n >= 1.

Extensions

a(17) and a(47) corrected by Georg Fischer, Apr 11 2024