%I #6 Dec 04 2022 08:33:36

%S 1,4,11,35,113,326,985,3124,8523,24519,71096,191940,530167,1442059,

%T 3833007,10243259,27151086,71032191,184492464,478339983,1227208513,

%U 3140958369,8016016201,20210235189,50962894061,127936646350,319022819270,794501931062,1969154638217

%N Number of ways to choose a sequence of partitions, one of each part of an odd-length partition of 2n+1 into odd parts.

%H Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>

%e The a(1) = 1 through a(5) = 11 twice-partitions:

%e (1) (3) (5)

%e (21) (32)

%e (111) (41)

%e (1)(1)(1) (221)

%e (311)

%e (2111)

%e (11111)

%e (3)(1)(1)

%e (21)(1)(1)

%e (111)(1)(1)

%e (1)(1)(1)(1)(1)

%t twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];

%t Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Total/@#]&]],{n,1,15,2}]

%Y For odd parts instead of length and sums we have A270995.

%Y Requiring odd lengths and odd parts gives A279374 aerated.

%Y This is the case of A358824 with odd sums.

%Y This is the odd-length case (hence odd bisection) of A358825.

%Y For odd lengths (instead of length) we have A358827.

%Y For odd lengths instead of sums we have A358834.

%Y A000009 counts partitions into odd parts.

%Y A027193 counts partitions of odd length.

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

%Y A078408 counts odd-length partitions into odd parts.

%Y A300301 aerated counts twice-partitions with odd sums and parts.

%Y Cf. A000041, A001970, A072233, A271619, A279785, A356932.

%K nonn

%O 0,2

%A _Gus Wiseman_, Dec 03 2022