OFFSET
0,4
COMMENTS
A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: ((1/Product_{k>=1} (1-A027193(k)*x^k)) - (1/Product_{k>=1} (1+A027193(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022
EXAMPLE
The a(1) = 1 through a(6) = 11 twice-partitions:
(1) (2) (3) (4) (5) (6)
(111) (211) (221) (222)
(1)(1)(1) (2)(1)(1) (311) (321)
(11111) (411)
(2)(2)(1) (21111)
(3)(1)(1) (2)(2)(2)
(111)(1)(1) (3)(2)(1)
(1)(1)(1)(1)(1) (4)(1)(1)
(111)(2)(1)
(211)(1)(1)
(2)(1)(1)(1)(1)
MATHEMATICA
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], OddQ[Length[#]]&&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) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022
CROSSREFS
The version for set partitions is A003712.
If the parts are also odd we get A279374.
The version for multiset partitions of integer partitions is the odd-length case of A356932, ranked by A026424 /\ A356935.
This is the odd-length case of A358334.
This is the odd-lengths case of A358824.
For odd sums instead of lengths we have A358826.
The case of odd sums also is the bisection of A358827.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 04 2022
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022
STATUS
approved