A343942 Number of even-length strict integer partitions of 2n+1.

0, 1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 52, 71, 96, 128, 170, 224, 292, 380, 491, 630, 805, 1024, 1295, 1632, 2049, 2560, 3189, 3959, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29249, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937, 172928

Offset

0,3

Comments

By conjugation, also the number of integer partitions of 2n+1 covering an initial interval of positive integers with greatest part even.

Links

Table of n, a(n) for n=0..48.

Formula

The Heinz numbers are A005117 /\ A028260 /\ A300063.

Example

The a(1) = 1 through a(7) = 13 strict partitions:
  (2,1)  (3,2)  (4,3)  (5,4)  (6,5)      (7,6)      (8,7)
         (4,1)  (5,2)  (6,3)  (7,4)      (8,5)      (9,6)
                (6,1)  (7,2)  (8,3)      (9,4)      (10,5)
                       (8,1)  (9,2)      (10,3)     (11,4)
                              (10,1)     (11,2)     (12,3)
                              (5,3,2,1)  (12,1)     (13,2)
                                         (5,4,3,1)  (14,1)
                                         (6,4,2,1)  (6,4,3,2)
                                         (7,3,2,1)  (6,5,3,1)
                                                    (7,4,3,1)
                                                    (7,5,2,1)
                                                    (8,4,2,1)
                                                    (9,3,2,1)

Mathematica

Table[Length[Select[IntegerPartitions[2n+1], UnsameQ@@#&&EvenQ[Length[#]]&]], {n, 0, 15}]

Crossrefs

Ranked by A005117 (strict), A028260 (even length), and A300063 (odd sum).

Odd bisection of A067661 (non-strict: A027187).

The non-strict version is A236914.

The opposite type of strict partition (odd length and even sum) is A344650.

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A344610 counts partitions by sum and positive reverse-alternating sum.

Cf. A000070, A000097, A030229, A035294, A067659, A236559, A338907, A343941, A344649, A344654, A344739.

Sequence in context: A098889 A061481 A017824 * A094054 A001521 A003143

Adjacent sequences: A343939 A343940 A343941 * A343943 A343944 A343945

Keyword

nonn

Author

Gus Wiseman, Jun 09 2021

Extensions

More terms from Bert Dobbelaere, Jun 12 2021

Status

approved