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
Andrew Howroyd, Table of n, a(n) for n = 0..1000
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 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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 01 2022
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022
STATUS
approved