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%I #13 Dec 31 2022 14:51:33
%S 0,1,1,3,3,7,10,20,29,58,83,150,230,399,605,1037,1545,2547,3879,6241,
%T 9437,15085,22622,35493,53438,82943,124157,191267,284997,434634,
%U 647437,979293,1452182,2185599,3228435,4826596,7112683,10575699,15530404,22990800,33651222
%N Number of odd-length twice-partitions of n into partitions with all odd parts.
%C A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
%C Also the number of odd-length twice-partitions of n into strict partitions.
%H Andrew Howroyd, <a href="/A358823/b358823.txt">Table of n, a(n) for n = 0..1000</a>
%H Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>
%F G.f.: ((1/Product_{k>=1} (1-A000009(k)*x^k)) - (1/Product_{k>=1} (1+A000009(k)*x^k)))/2. - _Andrew Howroyd_, Dec 31 2022
%e The a(1) = 1 through a(6) = 10 twice-partitions with all odd parts:
%e (1) (11) (3) (31) (5) (33)
%e (111) (1111) (311) (51)
%e (1)(1)(1) (11)(1)(1) (11111) (3111)
%e (3)(1)(1) (111111)
%e (11)(11)(1) (3)(11)(1)
%e (111)(1)(1) (31)(1)(1)
%e (1)(1)(1)(1)(1) (11)(11)(11)
%e (111)(11)(1)
%e (1111)(1)(1)
%e (11)(1)(1)(1)(1)
%e The a(1) = 1 through a(6) = 10 twice-partitions into strict partitions:
%e (1) (2) (3) (4) (5) (6)
%e (21) (31) (32) (42)
%e (1)(1)(1) (2)(1)(1) (41) (51)
%e (2)(2)(1) (321)
%e (3)(1)(1) (2)(2)(2)
%e (21)(1)(1) (3)(2)(1)
%e (1)(1)(1)(1)(1) (4)(1)(1)
%e (21)(2)(1)
%e (31)(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@@Flatten[#]]&]],{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=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ _Andrew Howroyd_, Dec 31 2022
%Y This is the odd-length case of A270995.
%Y Requiring odd sums also gives A279374 aerated.
%Y This is the case of A358824 with all odd parts.
%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 A358334 counts twice-partitions into odd-length partitions.
%Y Cf. A000041, A001970, A072233, A271619, A279785, A356932.
%K nonn
%O 0,4
%A _Gus Wiseman_, Dec 03 2022
%E Terms a(26) and beyond from _Andrew Howroyd_, Dec 31 2022