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.
LINKS
FORMULA
G.f.: Product_{k odd} 1/(1-A027193(k)*x^k).
EXAMPLE
The a(1) = 1 through a(6) = 11 twice-partitions:
(1) (1)(1) (3) (3)(1) (5) (3)(3)
(111) (111)(1) (221) (5)(1)
(1)(1)(1) (1)(1)(1)(1) (311) (111)(3)
(11111) (221)(1)
(3)(1)(1) (3)(111)
(111)(1)(1) (311)(1)
(1)(1)(1)(1)(1) (111)(111)
(11111)(1)
(3)(1)(1)(1)
(111)(1)(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/@#]&&OddQ[Times@@Total/@#]&]], {n, 0, 10}]
CROSSREFS
This is the case of A358334 with odd sums.
This is the case of A358825 with odd lengths.
The case of odd length is the odd bisection.
For odd parts instead of lengths and sums we have A270995.
Requiring odd parts also gives A279374 aerated.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 03 2022
STATUS
approved