A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 20, 20, 30, 31, 45, 46, 66, 68, 93, 97, 130, 136, 179, 188, 242, 256, 325, 344, 432, 459, 568, 606, 742, 793, 963, 1031, 1240, 1331, 1589, 1707, 2026, 2179, 2567, 2766, 3240, 3493, 4072, 4393, 5094, 5501, 6351

Offset: 0

Author

Gus Wiseman, Sep 20 2023

Keywords

• nonn

Comments

Also the number of strict integer partitions of n without two parts (allowing parts to be re-used) summing to n.

Examples

			The a(6) = 1 through a(12) = 7 strict partitions:
  (6)  (7)      (8)      (9)      (10)       (11)       (12)
       (4,2,1)  (5,2,1)  (4,3,2)  (6,3,1)    (5,4,2)    (5,4,3)
                         (5,3,1)  (7,2,1)    (6,3,2)    (7,3,2)
                         (6,2,1)  (4,3,2,1)  (6,4,1)    (7,4,1)
                                             (7,3,1)    (8,3,1)
                                             (8,2,1)    (9,2,1)
                                             (5,3,2,1)  (5,4,2,1)
		

Crossrefs

The second condition alone has bisections A078408 and A365828.

The complement is counted by A365659.

The non-strict version is A365825, complement A238628.

The first condition alone is A365827, complement A140106.

A000041 counts integer partitions, strict A000009.

A182616 counts partitions of 2n that do not contain n, strict A365828.

Cf. A004526, A005408, A008967, A035363, A058984, A086543, A100959, A344415, A365376, A365377, A365543.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Tuples[#,2],n]&]], {n,0,30}]