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

0,4

Comments

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

Links

Table of n, a(n) for n=0..47.

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

Example

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)

Mathematica

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

Prog

(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

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.

Sequence in context: A238517 A335240 A356367 * A099926 A355021 A098600

Adjacent sequences: A365822 A365823 A365824 * A365826 A365827 A365828

Keyword

nonn

Author

Gus Wiseman, Sep 19 2023

Extensions

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

Status

approved