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A358334
Number of twice-partitions of n into odd-length partitions.
10
1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765
OFFSET
0,3
COMMENTS
A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.
LINKS
FORMULA
G.f.: 1/Product_{k>=1} (1 - A027193(k)*x^k). - Andrew Howroyd, Dec 30 2022
EXAMPLE
The a(0) = 1 through a(5) = 13 twice-partitions:
() ((1)) ((2)) ((3)) ((4)) ((5))
((1)(1)) ((111)) ((211)) ((221))
((2)(1)) ((2)(2)) ((311))
((1)(1)(1)) ((3)(1)) ((3)(2))
((111)(1)) ((4)(1))
((2)(1)(1)) ((11111))
((1)(1)(1)(1)) ((111)(2))
((211)(1))
((2)(2)(1))
((3)(1)(1))
((111)(1)(1))
((2)(1)(1)(1))
((1)(1)(1)(1)(1))
MATHEMATICA
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], OddQ[Times@@Length/@#]&]], {n, 0, 10}]
PROG
(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u, y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n, 1)-P(n, -1))/2); Vec(R(u, 1), -(n+1))} \\ Andrew Howroyd, Dec 30 2022
CROSSREFS
For multiset partitions of integer partitions: A356932, ranked by A356935.
For odd length instead of lengths we have A358824.
For odd sums instead of lengths we have A358825.
For odd sums also we have A358827.
For odd length also we have A358834.
A000041 counts integer partitions.
A027193 counts odd-length partitions, ranked by A026424.
A055922 counts partitions with odd multiplicities, also odd parts A117958.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Sequence in context: A102112 A102113 A034440 * A000074 A374517 A079958
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 01 2022
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022
STATUS
approved