%I #11 Dec 30 2022 21:38:48
%S 0,1,2,4,7,15,32,61,121,260,498,967,1890,3603,6839,12972,23883,44636,
%T 82705,150904,275635,501737,905498,1628293,2922580,5224991,9296414,
%U 16482995,29125140,51287098,90171414,157704275,275419984,479683837,833154673,1442550486,2493570655
%N Number of twice-partitions of n of odd length.
%C A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
%H Andrew Howroyd, <a href="/A358824/b358824.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: ((1/Product_{k>=1} (1-A000041(k)*x^k)) - (1/Product_{k>=1} (1+A000041(k)*x^k)))/2. - _Andrew Howroyd_, Dec 30 2022
%e The a(1) = 1 through a(5) = 15 twice-partitions:
%e (1) (2) (3) (4) (5)
%e (11) (21) (22) (32)
%e (111) (31) (41)
%e (1)(1)(1) (211) (221)
%e (1111) (311)
%e (2)(1)(1) (2111)
%e (11)(1)(1) (11111)
%e (2)(2)(1)
%e (3)(1)(1)
%e (11)(2)(1)
%e (2)(11)(1)
%e (21)(1)(1)
%e (11)(11)(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[#]]&]],{n,0,10}]
%o (PARI)
%o R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
%o seq(n) = {my(u=vector(n,k,numbpart(k))); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ _Andrew Howroyd_, Dec 30 2022
%Y The version for set partitions is A024429.
%Y For odd lengths (instead of length) we have A358334.
%Y The case of odd parts also is A358823.
%Y The case of odd sums also is A358826.
%Y The case of odd lengths also is A358834.
%Y For multiset partitions of integer partitions: A358837, ranked by A026424.
%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, A270995, A271619, A279374, A279785, A306319, A336342, A356932.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 03 2022
%E Terms a(26) and beyond from _Andrew Howroyd_, Dec 30 2022