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A358823 Number of odd-length twice-partitions of n into partitions with all odd parts.

%I A358823 #13 Dec 31 2022 14:51:33
%S A358823 0,1,1,3,3,7,10,20,29,58,83,150,230,399,605,1037,1545,2547,3879,6241,
%T A358823 9437,15085,22622,35493,53438,82943,124157,191267,284997,434634,
%U A358823 647437,979293,1452182,2185599,3228435,4826596,7112683,10575699,15530404,22990800,33651222
%N A358823 Number of odd-length twice-partitions of n into partitions with all odd parts.
%C A358823 A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
%C A358823 Also the number of odd-length twice-partitions of n into strict partitions.
%H A358823 Andrew Howroyd, <a href="/A358823/b358823.txt">Table of n, a(n) for n = 0..1000</a>
%H A358823 Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>
%F A358823 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 A358823 The a(1) = 1 through a(6) = 10 twice-partitions with all odd parts:
%e A358823   (1)  (11)  (3)        (31)        (5)              (33)
%e A358823              (111)      (1111)      (311)            (51)
%e A358823              (1)(1)(1)  (11)(1)(1)  (11111)          (3111)
%e A358823                                     (3)(1)(1)        (111111)
%e A358823                                     (11)(11)(1)      (3)(11)(1)
%e A358823                                     (111)(1)(1)      (31)(1)(1)
%e A358823                                     (1)(1)(1)(1)(1)  (11)(11)(11)
%e A358823                                                      (111)(11)(1)
%e A358823                                                      (1111)(1)(1)
%e A358823                                                      (11)(1)(1)(1)(1)
%e A358823 The a(1) = 1 through a(6) = 10 twice-partitions into strict partitions:
%e A358823   (1)  (2)  (3)        (4)        (5)              (6)
%e A358823             (21)       (31)       (32)             (42)
%e A358823             (1)(1)(1)  (2)(1)(1)  (41)             (51)
%e A358823                                   (2)(2)(1)        (321)
%e A358823                                   (3)(1)(1)        (2)(2)(2)
%e A358823                                   (21)(1)(1)       (3)(2)(1)
%e A358823                                   (1)(1)(1)(1)(1)  (4)(1)(1)
%e A358823                                                    (21)(2)(1)
%e A358823                                                    (31)(1)(1)
%e A358823                                                    (2)(1)(1)(1)(1)
%t A358823 twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t A358823 Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Flatten[#]]&]],{n,0,10}]
%o A358823 (PARI)
%o A358823 R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
%o A358823 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 A358823 This is the odd-length case of A270995.
%Y A358823 Requiring odd sums also gives A279374 aerated.
%Y A358823 This is the case of A358824 with all odd parts.
%Y A358823 A000009 counts partitions into odd parts.
%Y A358823 A027193 counts partitions of odd length.
%Y A358823 A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y A358823 A078408 counts odd-length partitions into odd parts.
%Y A358823 A300301 aerated counts twice-partitions with odd sums and parts.
%Y A358823 A358334 counts twice-partitions into odd-length partitions.
%Y A358823 Cf. A000041, A001970, A072233, A271619, A279785, A356932.
%K A358823 nonn
%O A358823 0,4
%A A358823 _Gus Wiseman_, Dec 03 2022
%E A358823 Terms a(26) and beyond from _Andrew Howroyd_, Dec 31 2022