A358823 Number of odd-length twice-partitions of n into partitions with all odd parts.

0, 1, 1, 3, 3, 7, 10, 20, 29, 58, 83, 150, 230, 399, 605, 1037, 1545, 2547, 3879, 6241, 9437, 15085, 22622, 35493, 53438, 82943, 124157, 191267, 284997, 434634, 647437, 979293, 1452182, 2185599, 3228435, 4826596, 7112683, 10575699, 15530404, 22990800, 33651222

Offset

0,4

Comments

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

Also the number of odd-length twice-partitions of n into strict partitions.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Gus Wiseman, Sequences enumerating triangles of integer partitions

Formula

G.f.: ((1/Product_{k>=1} (1-A000009(k)*x^k)) - (1/Product_{k>=1} (1+A000009(k)*x^k)))/2. - Andrew Howroyd, Dec 31 2022

Example

The a(1) = 1 through a(6) = 10 twice-partitions with all odd parts:
  (1)  (11)  (3)        (31)        (5)              (33)
             (111)      (1111)      (311)            (51)
             (1)(1)(1)  (11)(1)(1)  (11111)          (3111)
                                    (3)(1)(1)        (111111)
                                    (11)(11)(1)      (3)(11)(1)
                                    (111)(1)(1)      (31)(1)(1)
                                    (1)(1)(1)(1)(1)  (11)(11)(11)
                                                     (111)(11)(1)
                                                     (1111)(1)(1)
                                                     (11)(1)(1)(1)(1)
The a(1) = 1 through a(6) = 10 twice-partitions into strict partitions:
  (1)  (2)  (3)        (4)        (5)              (6)
            (21)       (31)       (32)             (42)
            (1)(1)(1)  (2)(1)(1)  (41)             (51)
                                  (2)(2)(1)        (321)
                                  (3)(1)(1)        (2)(2)(2)
                                  (21)(1)(1)       (3)(2)(1)
                                  (1)(1)(1)(1)(1)  (4)(1)(1)
                                                   (21)(2)(1)
                                                   (31)(1)(1)
                                                   (2)(1)(1)(1)(1)

Mathematica

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], OddQ[Length[#]]&&OddQ[Times@@Flatten[#]]&]], {n, 0, 10}]

Prog

(PARI)
R(u, y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 31 2022

Crossrefs

This is the odd-length case of A270995.

Requiring odd sums also gives A279374 aerated.

This is the case of A358824 with all odd parts.

A000009 counts partitions into odd parts.

A027193 counts partitions of odd length.

A063834 counts twice-partitions, strict A296122, row-sums of A321449.

A078408 counts odd-length partitions into odd parts.

A300301 aerated counts twice-partitions with odd sums and parts.

A358334 counts twice-partitions into odd-length partitions.

Cf. A000041, A001970, A072233, A271619, A279785, A356932.

Sequence in context: A136445 A326269 A052989 * A252750 A287274 A305099

Adjacent sequences: A358820 A358821 A358822 * A358824 A358825 A358826

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Extensions

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

Status

approved