A301595 Number of thrice-partitions of n.

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.

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

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.

Column k=3 of A323718.

Sequence in context: A261577 A224217 A066454 * A022445 A091003 A140725

Adjacent sequences: A301592 A301593 A301594 * A301596 A301597 A301598

Keyword

nonn

Author

Gus Wiseman, Mar 24 2018

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

Status

approved