A137591 Number of parenthesizings of products formed by n factors assuming nonassociativity and partial commutativity: individual factors commute, but bracketed expressions don't commute with anything.

1, 1, 6, 54, 660, 10260, 194040, 4326840, 111177360, 3234848400, 105135861600, 3775206204000, 148426878600000, 6341634955656000, 292576856395824000, 14496220038251952000, 767691210706291872000, 43274547687106768032000, 2587028200730649643968000, 163484729048197101504960000

Offset: 1

Author

Thomas Wieder, Jan 28 2008, Feb 07 2008

Keywords

• nonn

Comments

See the double factorials A001147 for the case when the product is commutative and nonassociative.

Another interpretation is possible in terms of dendrograms. A001147 gives the number labeled, non-ranked, binary dendrograms, so-called L-NR dendrograms. This sequence gives the number of L-NR dendrograms if the order of objects counts within a dendrogram class.

See the Murtagh paper cited in A001147 for more on dendrograms. See also Vandev.

Vandev's formula (1) is our recurrence for this sequence, but it seems that Vandev meant a(n) = Sum_{k=1..n-1} binomial(n-1, k)*a(k)*a(n-k) with a(1)=1, a(2)=1. This recurrence gives the double factorials.

Examples

			a(4)=54 because we have
w(x(yz)), w((yz)x), (x(yz))w, ((yz)x)w,
w(y(xz)), w((xz)y), (y(xz))w, ((xz)y)w,
w(z(xy)), w((xy)z), (z(xy))w, ((xy)z)w,
x(w(yz)), x((yz)w), (w(yz))x, ((yz)w)x,
x(y(wz)), x((wz)y), (y(wz))x, ((wz)y)x,
x(z(wy)), x((wy)z), (z(wy))x, ((wy)z)x,
y(w(xz)), y((xz)w), (w(xz))y, ((xz)w)y,
y(x(wz)), y((wz)x), (x(wz))y, ((wz)x)y,
y(z(wx)), y((wx)z), (z(wx))y, ((wx)z)y,
z(w(xy)), z((xy)w), (w(xy))z, ((xy)w)z,
z(x(wy)), z((wy)x), (x(wy))z, ((wy)x)z,
z(y(wx)), z((wx)y), (y(wx))z, ((wx)y)z,
(wx)(yz), (yz)(wx)
(wy)(xz), (xz)(wy)
(wz)(xy), (xy)(wz)
and 12*4 + 3*2 = 48 + 6 = 54.
Note that:
w(x(yz)) is equivalent to w(x(zy)) but not to (x(yz))w or w((yz)x);
(wx)(yz) is equivalent to (xw)(yz) or (wx)(zy) but not to (yz)(wx).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• May Cai, Kisun Lee, and Josephine Yu, Symmetric Tropical Rank 2 Matrices, arXiv:2404.08121 [math.CO], 2024. See p. 12.
• Dimitar L. Vandev, Random Dendrograms. Statistical Data Analysis, Proceedings SDA-95, SDA-96, pp. 186-196. [Cached copy from citeseerx.ist.psu.edu]

Crossrefs

Cf. A001147, A137051.

Programs

GAP

a := [1,1];; for n in [3..10^2] do a[n] := Sum([1..n-1],k->Binomial(n,k)*a[k]*a[n-k]); od; a; # Muniru A Asiru, Jan 30 2018

Maple

H(1):=1; H(2):=1; for n from 3 to 12 do H(n):=0: for k from 1 to n-1 do H(n):= H(n)+binomial(n,k)*H(k)*H(n-k) od: print(H(n)); od:

Mathematica

CoefficientList[Series[(1-x)/Sqrt[1-4*x+2*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)

PARI

x='x+O('x^66); Vec( serlaplace((1-x)/sqrt(1-4*x+2*x^2)) ) \\ Joerg Arndt, Oct 08 2013

Formula

a(n) = Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k), with a(1)=1, a(2)=1.

E.g.f.: (1/2)*(1 - sqrt(1 - 4*x + 2*x^2)). - Thomas Wieder, May 02 2009, edited by May Cai, Feb 13 2024

a(n) ~ sqrt(2+2*sqrt(2))/2 * n^n * (2+sqrt(2))^n / exp(n). - Vaclav Kotesovec, Oct 07 2013

Extensions

Added more terms, Joerg Arndt, Oct 08 2013

Name corrected by Andrey Zabolotskiy, Mar 06 2018