A358823 - OEIS

%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