1,3

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(binomial(n-1,k)*a(k)*a(n-k),k=1..n-1) 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.

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

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

(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.

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

Cf. A000108, A000142, A137591, A137731.

Sequence in context: A026106 A066642 A019988 this_sequence A057973 A052708 A141201

Adjacent sequences: A137729 A137730 A137731 this_sequence A137733 A137734 A137735

nonn

Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 09 2008