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%I #9 Dec 30 2022 21:38:54
%S 0,1,1,3,3,8,11,24,35,74,109,213,336,624,986,1812,2832,5002,7996,
%T 13783,21936,37528,59313,99598,158356,262547,415590,684372,1079576,
%U 1759984,2779452,4491596,7069572,11370357,17841534,28509802,44668402,70975399,110907748
%N Number of odd-length twice-partitions of n into odd-length partitions.
%C A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.
%H Andrew Howroyd, <a href="/A358834/b358834.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: ((1/Product_{k>=1} (1-A027193(k)*x^k)) - (1/Product_{k>=1} (1+A027193(k)*x^k)))/2. - _Andrew Howroyd_, Dec 30 2022
%e The a(1) = 1 through a(6) = 11 twice-partitions:
%e (1) (2) (3) (4) (5) (6)
%e (111) (211) (221) (222)
%e (1)(1)(1) (2)(1)(1) (311) (321)
%e (11111) (411)
%e (2)(2)(1) (21111)
%e (3)(1)(1) (2)(2)(2)
%e (111)(1)(1) (3)(2)(1)
%e (1)(1)(1)(1)(1) (4)(1)(1)
%e (111)(2)(1)
%e (211)(1)(1)
%e (2)(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@@Length/@#]&]],{n,0,10}]
%o (PARI)
%o P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
%o seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ _Andrew Howroyd_, Dec 30 2022
%Y The version for set partitions is A003712.
%Y If the parts are also odd we get A279374.
%Y The version for multiset partitions of integer partitions is the odd-length case of A356932, ranked by A026424 /\ A356935.
%Y This is the odd-length case of A358334.
%Y This is the odd-lengths case of A358824.
%Y For odd sums instead of lengths we have A358826.
%Y The case of odd sums also is the bisection of A358827.
%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, A270995, A271619, A279785, A358823, A358837.
%K nonn
%O 0,4
%A _Gus Wiseman_, Dec 04 2022
%E Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2022
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