A301706 Number of rooted thrice-partitions of n.

1, 1, 2, 4, 9, 19, 43, 91, 201, 422, 918, 1896, 4089, 8376, 17793, 36445, 76446, 155209, 324481, 655426, 1355220, 2741092, 5617505, 11291037, 23086423, 46227338, 93753196, 187754647, 378675055, 754695631, 1518414812, 3016719277, 6037006608, 11984729983

Offset

1,3

Comments

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.

Links

Table of n, a(n) for n=1..34.

Example

The a(5) = 9 rooted thrice-partitions:
((2)), ((11)), ((1)()), (()()()),
((1))(), (()())(), (())(()),
(())()(),
()()()().
The a(6) = 19 rooted thrice-partitions:
((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()),
((2))(), ((11))(), ((1)())(), (()()())(), ((1))(()), (()())(()),
((1))()(), (()())()(), (())(())(),
(())()()(),
()()()()().

Mathematica

twire[n_]:=twire[n]=Sum[Times@@PartitionsP/@(ptn-1), {ptn, IntegerPartitions[n-1]}];
thrire[n_]:=Sum[Times@@twire/@ptn, {ptn, IntegerPartitions[n-1]}];
Array[thrire, 30]

Crossrefs

Cf. A000041, A001383, A002865, A063834, A093637, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301467, A301480, A301595, A301598.

Sequence in context: A192673 A193021 A112569 * A289845 A101463 A319379

Adjacent sequences: A301703 A301704 A301705 * A301707 A301708 A301709

Keyword

nonn

Author

Gus Wiseman, Mar 25 2018

Status

approved