A358334 Number of twice-partitions of n into odd-length partitions.

1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765

Offset

0,3

Comments

A twice-partition of n (A063834) 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 - A027193(k)*x^k). - Andrew Howroyd, Dec 30 2022

Example

The a(0) = 1 through a(5) = 13 twice-partitions:
  ()  ((1))  ((2))     ((3))        ((4))           ((5))
             ((1)(1))  ((111))      ((211))         ((221))
                       ((2)(1))     ((2)(2))        ((311))
                       ((1)(1)(1))  ((3)(1))        ((3)(2))
                                    ((111)(1))      ((4)(1))
                                    ((2)(1)(1))     ((11111))
                                    ((1)(1)(1)(1))  ((111)(2))
                                                    ((211)(1))
                                                    ((2)(2)(1))
                                                    ((3)(1)(1))
                                                    ((111)(1)(1))
                                                    ((2)(1)(1)(1))
                                                    ((1)(1)(1)(1)(1))

Mathematica

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

Prog

(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u, y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n, 1)-P(n, -1))/2); Vec(R(u, 1), -(n+1))} \\ Andrew Howroyd, Dec 30 2022

Crossrefs

For multiset partitions of integer partitions: A356932, ranked by A356935.

For odd length instead of lengths we have A358824.

For odd sums instead of lengths we have A358825.

For odd sums also we have A358827.

For odd length also we have A358834.

A000041 counts integer partitions.

A027193 counts odd-length partitions, ranked by A026424.

A055922 counts partitions with odd multiplicities, also odd parts A117958.

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

Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

Sequence in context: A102112 A102113 A034440 * A000074 A374517 A079958

Adjacent sequences: A358331 A358332 A358333 * A358335 A358336 A358337

Keyword

nonn

Author

Gus Wiseman, Dec 01 2022

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

Status

approved