OFFSET
0,3
COMMENTS
A twice-partition of n 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-A000041(k)*x^k)) - (1/Product_{k>=1} (1+A000041(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022
EXAMPLE
The a(1) = 1 through a(5) = 15 twice-partitions:
(1) (2) (3) (4) (5)
(11) (21) (22) (32)
(111) (31) (41)
(1)(1)(1) (211) (221)
(1111) (311)
(2)(1)(1) (2111)
(11)(1)(1) (11111)
(2)(2)(1)
(3)(1)(1)
(11)(2)(1)
(2)(11)(1)
(21)(1)(1)
(11)(11)(1)
(111)(1)(1)
(1)(1)(1)(1)(1)
MATHEMATICA
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], OddQ[Length[#]]&]], {n, 0, 10}]
PROG
(PARI)
R(u, y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=vector(n, k, numbpart(k))); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022
CROSSREFS
The version for set partitions is A024429.
For odd lengths (instead of length) we have A358334.
The case of odd parts also is A358823.
The case of odd sums also is A358826.
The case of odd lengths also is A358834.
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 03 2022
EXTENSIONS
Terms a(26) and beyond from Andrew Howroyd, Dec 30 2022
STATUS
approved