A137732 Repeated set splitting, unlabeled elements. Repeated integer partitioning into two parts.

1, 1, 2, 5, 16, 55, 224, 935, 4400, 21262, 111624, 596805, 3457354, 20147882, 125455512, 792576243, 5277532388, 35519373064, 252120178596, 1800810613940, 13492153025558, 102095379031327, 804122472505530, 6395239610004277

Offset

1,3

Comments

Consider a set of n unlabeled elements. Form all splittings into two subsets. Consider the resulting sets and perform the splittings on all their subsets and so on. In order to understand this structure, imagine that each of the two parts can be put either 'to the left or to the right.

E.g., (4) gives (3,1) and (1,3). That is, the order of parts counts. H(n+1) = number of splittings of the n-set {*,*,...,*} composed of n elements '*'. E.g., H(4)=5 because we have (***), (**,*), (*,**), ((*,*),*), (*,(*,*)).

Equivalently, we have (3), (2,1), (1,2), ((1,1),1), (1,(1,1)). The case for labeled elements is described by A137731. This structure is related to the Double Factorials A000142 for which the recurrence is a(n) = Sum_{k=1..n-1} binomial(n-1,k)*a(k)*a(n-k), with a(1)=1, a(2)=1.

See also A137591 = Number of parenthesizings of products formed by n factors assuming noncommutativity and nonassociativity. See also the Catalan numbers A000108.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..250

Formula

a(n) = Sum_{k=1..n-1} p(n-1,k)*a(k)*a(n-k), with a(1)=1 and where p(n,k) denotes the number of integer partitions of n into k parts.

Example

(1)
(2), (1,1).
(3), (2,1), (1,2), ((1,1),1), (1,(1,1)).
(4), (3,1), (1,3), ((2,1),1), (1,(2,1)), ((1,2),1), (1,(1,2)),
(((1,1),1),1), (1,((1,1),1)), ((1,(1,1)),1), (1,(1,(1,1))),
(2,2), ((1,1),2), (2,(1,1)), ((1,1),(1,1)), ((1,1),(1,1)).
Observe that for (4) we obtain ((1,1),(1,1)), ((1,1),(1,1)) twice.

Maple

A008284 := proc(n, k) combinat[numbpart](n, k)-combinat[numbpart](n, k-1) ; end: A137732 := proc(n) option remember ; local i ; if n =1 then 1; else add(A008284(n-1, k)*procname(k)*procname(n-k), k=1..n-1) ; fi ; end: for n from 1 to 40 do printf("%d, ", A137732(n)) ; od: # R. J. Mathar, Aug 25 2008

Mathematica

p[_, 1] = 1; p[n_, k_] /; 1 <= k <= n := p[n, k] = Sum[p[n-i, k-1], {i, 1, n-1}] - Sum[p[n-i, k], {i, 1, k-1}]; p[_, _] = 0; a[1] = 1; a[n_] := a[n] = Sum[p[n-1, k]*a[k]*a[n-k], {k, 1, n-1}]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 03 2017 *)

Prog

Sub A137714() ' This is a VBA program.
Dim n As Long, nstart As Long, nend As Long
Dim k As Long, HSumme As Long, H(100) As Long
nstart = 2
nend = 15
H(1) = 1
For n = nstart To nend
HSumme = 0
For k = 1 To n - 1
HSumme = HSumme + ZahlPartitionen(n - 1, k) * H(k) * H(n - k)
Next k
H(n) = HSumme
Next n
Debug.Print H(1), H(2), H(3), H(4), H(5), H(6), H(7), H(8), H(9), H(10), H(11), H(12), H(13), H(14), H(15)
End Sub
Public Function ZahlPartitionen(n As Long, k As Long)
Dim imsgbox As Integer
If n > 2147483648# Or k > 2147483648# Then
imsgbox = MsgBox("n and k need to be smaller than 2147483648 !", vbOKOnly, "ZahlPartitionen")
End
End If
If (n < 0 Or k < 0) Then
imsgbox = MsgBox("n and k need to be greater than 0 !", vbOKOnly, "ZahlPartitionen")
End
End If
If k = 1 Then
ZahlPartitionen = 1
Exit Function
ElseIf k = n Then
ZahlPartitionen = 1
Exit Function
ElseIf k > n Then
ZahlPartitionen = 0
Exit Function
End If
ZahlPartitionen = ZahlPartitionen(n - 1, k - 1) + ZahlPartitionen(n - k, k)
End Function

Crossrefs

Cf. A000108, A000142, A137591, A137731.

Sequence in context: A066642 A333233 A019988 * A119611 A057973 A102461

Adjacent sequences: A137729 A137730 A137731 * A137733 A137734 A137735

Keyword

nonn

Author

Thomas Wieder, Feb 09 2008

Extensions

Extended by R. J. Mathar, Aug 25 2008

Status

approved