A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731

Offset: 0

Author

Gus Wiseman, Sep 19 2023

Keywords

• nonn

Comments

Also the number of integer partitions of n with no two possibly equal parts summing to n.

Examples

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (111)  (1111)  (221)    (222)     (322)      (332)
                           (311)    (411)     (331)      (521)
                           (2111)   (2211)    (421)      (611)
                           (11111)  (21111)   (511)      (2222)
                                    (111111)  (2221)     (3221)
                                              (3211)     (3311)
                                              (4111)     (5111)
                                              (22111)    (22211)
                                              (31111)    (32111)
                                              (211111)   (221111)
                                              (1111111)  (311111)
                                                         (2111111)
                                                         (11111111)
		

Crossrefs

First condition alone is A058984, complement A004526, ranks A100959.

Second condition alone is A086543, complement A035363, ranks !A344415.

The complement is counted by A238628.

The strict case is A365826, complement A365659.

A000041 counts integer partitions, strict A000009.

A046663 counts partitions with no submultiset summing to k, strict A365663.

A140106 counts strict partitions of length 2, complement A365827.

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

Cf. A005408, A008967, A068911, A365377, A365544, A365543.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]

Python

from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023

Formula

Heinz numbers are A100959 /\ !A344415.

a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - Chai Wah Wu, Sep 23 2023

Extensions

a(31)-a(47) from Chai Wah Wu, Sep 23 2023