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A301595
Number of thrice-partitions of n.
5
1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508
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.
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 *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 24 2018
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jan 25 2019
STATUS
approved