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

Author

Gus Wiseman, Jun 09 2021

Keywords

• nonn

Comments

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

Examples

			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)
		

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.

Programs

Mathematica

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

Formula

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

Extensions

More terms from Bert Dobbelaere, Jun 12 2021