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

Author

Thomas Wieder, Feb 09 2008

Keywords

• nonn

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.

Examples

			(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.
		

Links

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

Crossrefs

Cf. A000108, A000142, A137591, A137731.

Programs

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 *)

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.

Extensions

Extended by R. J. Mathar, Aug 25 2008