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

1,5

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.

References

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

Links

Table of n, a(n) for n=1..48.

Formula

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

Example

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.

Crossrefs

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

Sequence in context: A000094 A182377 A327380 * A058578 A261526 A295392

Adjacent sequences: A330375 A330376 A330377 * A330379 A330380 A330381

Keyword

nonn

Author

Omar E. Pol, Jan 01 2020

Extensions

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

Status

approved