OFFSET
0,4
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.
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