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A358823
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Number of odd-length twice-partitions of n into partitions with all odd parts.
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4
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0, 1, 1, 3, 3, 7, 10, 20, 29, 58, 83, 150, 230, 399, 605, 1037, 1545, 2547, 3879, 6241, 9437, 15085, 22622, 35493, 53438, 82943, 124157, 191267, 284997, 434634, 647437, 979293, 1452182, 2185599, 3228435, 4826596, 7112683, 10575699, 15530404, 22990800, 33651222
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OFFSET
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0,4
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COMMENTS
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A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
Also the number of odd-length twice-partitions of n into strict partitions.
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LINKS
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FORMULA
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EXAMPLE
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The a(1) = 1 through a(6) = 10 twice-partitions with all odd parts:
(1) (11) (3) (31) (5) (33)
(111) (1111) (311) (51)
(1)(1)(1) (11)(1)(1) (11111) (3111)
(3)(1)(1) (111111)
(11)(11)(1) (3)(11)(1)
(111)(1)(1) (31)(1)(1)
(1)(1)(1)(1)(1) (11)(11)(11)
(111)(11)(1)
(1111)(1)(1)
(11)(1)(1)(1)(1)
The a(1) = 1 through a(6) = 10 twice-partitions into strict partitions:
(1) (2) (3) (4) (5) (6)
(21) (31) (32) (42)
(1)(1)(1) (2)(1)(1) (41) (51)
(2)(2)(1) (321)
(3)(1)(1) (2)(2)(2)
(21)(1)(1) (3)(2)(1)
(1)(1)(1)(1)(1) (4)(1)(1)
(21)(2)(1)
(31)(1)(1)
(2)(1)(1)(1)(1)
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MATHEMATICA
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twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], OddQ[Length[#]]&&OddQ[Times@@Flatten[#]]&]], {n, 0, 10}]
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PROG
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(PARI)
R(u, y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
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
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CROSSREFS
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This is the odd-length case of A270995.
Requiring odd sums also gives A279374 aerated.
This is the case of A358824 with all odd parts.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
A358334 counts twice-partitions into odd-length partitions.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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