A344650 Number of strict odd-length integer partitions of 2n.

0, 1, 1, 2, 3, 5, 8, 11, 16, 23, 32, 44, 61, 82, 111, 148, 195, 256, 334, 432, 557, 713, 908, 1152, 1455, 1829, 2291, 2859, 3554, 4404, 5440, 6697, 8222, 10066, 12288, 14964, 18176, 22023, 26625, 32117, 38656, 46432, 55661, 66592, 79523, 94793, 112792, 133984

Offset

0,4

Comments

Also the number of strict integer partitions of 2n with reverse-alternating sum >= 0.

Also the number of reversed strict integer partitions of 2n with alternating sum >= 0.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Formula

Sum of odd-indexed terms in row 2n of A008289.

a(n) = A067659(2n).

Example

The a(1) = 1 through a(8) = 16 partitions:
  (2)  (4)  (6)      (8)      (10)     (12)     (14)      (16)
            (3,2,1)  (4,3,1)  (5,3,2)  (5,4,3)  (6,5,3)   (7,5,4)
                     (5,2,1)  (5,4,1)  (6,4,2)  (7,4,3)   (7,6,3)
                              (6,3,1)  (6,5,1)  (7,5,2)   (8,5,3)
                              (7,2,1)  (7,3,2)  (7,6,1)   (8,6,2)
                                       (7,4,1)  (8,4,2)   (8,7,1)
                                       (8,3,1)  (8,5,1)   (9,4,3)
                                       (9,2,1)  (9,3,2)   (9,5,2)
                                                (9,4,1)   (9,6,1)
                                                (10,3,1)  (10,4,2)
                                                (11,2,1)  (10,5,1)
                                                          (11,3,2)
                                                          (11,4,1)
                                                          (12,3,1)
                                                          (13,2,1)
                                                          (6,4,3,2,1)

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&]], {n, 0, 30, 2}]

Crossrefs

The Heinz numbers are the intersection of A030059 and A300061.

Allowing even length gives A035294 (non-strict: A058696).

Even bisection of A067659.

The opposite type of strict partition (even length and odd sum) is A343942.

The non-strict version is A236559 or A344611.

Row sums of A344649.

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

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

A120452 counts partitions of 2n with reverse-alternating sum 2.

A124754 gives alternating sums of standard compositions (reverse: A344618).

A152146 interleaved with A152157 counts strict partitions by sum and alternating sum.

A316524 is the alternating sum of the prime indices of n (reverse: A344616).

A343941 counts strict partitions of 2n with reverse-alternating sum 4.

A344604 counts wiggly compositions with twins.

A344739 counts strict partitions by sum and reverse-alternating sum.

A344741 counts partitions of 2n with reverse-alternating sum -2.

Cf. A000070, A000097, A027187, A114121, A239829, A239830, A344607, A344609, A344610, A344651, A344654.

Sequence in context: A070228 A173599 A006304 * A238591 A039847 A046938

Adjacent sequences: A344647 A344648 A344649 * A344651 A344652 A344653

Keyword

nonn

Author

Gus Wiseman, Jun 05 2021

Status

approved