A301706
Number of rooted thrice-partitions of n.
Original entry on oeis.org
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
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))()(), (()())()(), (())(())(),
(())()()(),
()()()()().
Cf.
A000041,
A001383,
A002865,
A063834,
A093637,
A119442,
A196545,
A281113,
A289501,
A300383,
A301422,
A301462,
A301467,
A301480,
A301595,
A301598.
-
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]
A301595
Number of thrice-partitions of n.
Original entry on oeis.org
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
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)).
Cf.
A000041,
A001383,
A001970,
A061260,
A063834,
A119442,
A196545,
A281113,
A289501,
A300383,
A301422,
A301462,
A301480,
A301595,
A301598,
A301706.
-
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
-
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 *)
Showing 1-2 of 2 results.
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