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 A137732 Repeated set splitting, unlabeled elements. Repeated integer partitioning into two parts. 4
 1, 1, 2, 5, 16, 55, 224, 935, 4400, 21262, 111624, 596805, 3457354, 20147882, 125455512, 792576243, 5277532388, 35519373064, 252120178596, 1800810613940, 13492153025558, 102095379031327, 804122472505530, 6395239610004277 (list; graph; refs; listen; history; text; internal format)
 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; 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

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Last modified September 16 12:25 EDT 2021. Contains 347472 sequences. (Running on oeis4.)