OFFSET
0,3
COMMENTS
A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..3244
Gus Wiseman, The a(4) = 34 thrice-partitions of 4.
FORMULA
O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).
EXAMPLE
The a(3) = 10 thrice-partitions:
((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
((2))((1)), ((11))((1)), ((1)(1))((1)),
((1))((1))((1)).
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..35); # Alois P. Heinz, Jan 25 2019
MATHEMATICA
twie[n_]:=Sum[Times@@PartitionsP/@ptn, {ptn, IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn, {ptn, IntegerPartitions[n]}];
Array[thrie, 30]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
a[n_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 24 2018
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jan 25 2019
STATUS
approved