A358825 Number of ways to choose a sequence of integer partitions, one of each part of an integer partition of n into odd parts.

1, 1, 1, 4, 4, 11, 20, 35, 56, 113, 207, 326, 602, 985, 1777, 3124, 5115, 8523, 15011, 24519, 41571, 71096, 115650, 191940, 320651, 530167, 865781, 1442059, 2358158, 3833007, 6325067, 10243259, 16603455, 27151086, 43734197, 71032191, 115091799, 184492464

Offset

0,4

Links

Table of n, a(n) for n=0..37.

Formula

G.f.: Product_{k odd} 1/(1-A000041(k)*x^k).

Example

The a(1) = 1 through a(5) = 11 twice-partitions:
  (1)  (1)(1)  (3)        (3)(1)        (5)
               (21)       (21)(1)       (32)
               (111)      (111)(1)      (41)
               (1)(1)(1)  (1)(1)(1)(1)  (221)
                                        (311)
                                        (2111)
                                        (11111)
                                        (3)(1)(1)
                                        (21)(1)(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[Times@@Total/@#]&]], {n, 0, 10}]

Crossrefs

For odd parts instead of sums we have A270995.

For distinct instead of odd sums we have A271619.

Requiring odd length, odd lengths, and odd parts gives A279374 aerated.

For odd lengths instead of sums we have A358334.

The odd-length case is A358826.

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, A279785, A356932, A358824.

Sequence in context: A302516 A254205 A048223 * A214112 A211950 A014012

Adjacent sequences: A358822 A358823 A358824 * A358826 A358827 A358828

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Status

approved