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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 (list; graph; refs; listen; history; text; internal format)
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]

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

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Last modified June 24 21:16 EDT 2019. Contains 324337 sequences. (Running on oeis4.)