A358824 Number of twice-partitions of n of odd length.

0, 1, 2, 4, 7, 15, 32, 61, 121, 260, 498, 967, 1890, 3603, 6839, 12972, 23883, 44636, 82705, 150904, 275635, 501737, 905498, 1628293, 2922580, 5224991, 9296414, 16482995, 29125140, 51287098, 90171414, 157704275, 275419984, 479683837, 833154673, 1442550486, 2493570655

Offset

0,3

Comments

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

Links

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

Formula

G.f.: ((1/Product_{k>=1} (1-A000041(k)*x^k)) - (1/Product_{k>=1} (1+A000041(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Example

The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             (1)(1)(1)  (211)       (221)
                        (1111)      (311)
                        (2)(1)(1)   (2111)
                        (11)(1)(1)  (11111)
                                    (2)(2)(1)
                                    (3)(1)(1)
                                    (11)(2)(1)
                                    (2)(11)(1)
                                    (21)(1)(1)
                                    (11)(11)(1)
                                    (111)(1)(1)
                                    (1)(1)(1)(1)(1)

Mathematica

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], OddQ[Length[#]]&]], {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=vector(n, k, numbpart(k))); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

Crossrefs

The version for set partitions is A024429.

For odd lengths (instead of length) we have A358334.

The case of odd parts also is A358823.

The case of odd sums also is A358826.

The case of odd lengths also is A358834.

For multiset partitions of integer partitions: A358837, ranked by A026424.

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.

Cf. A000041, A001970, A072233, A270995, A271619, A279374, A279785, A306319, A336342, A356932.

Sequence in context: A049885 A129682 A129981 * A072964 A247291 A030033

Adjacent sequences: A358821 A358822 A358823 * A358825 A358826 A358827

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Extensions

Terms a(26) and beyond from Andrew Howroyd, Dec 30 2022

Status

approved