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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. 3
1, 1, 6, 54, 660, 10260, 194040, 4326840, 111177360, 3234848400, 105135861600, 3775206204000, 148426878600000, 6341634955656000, 292576856395824000, 14496220038251952000, 767691210706291872000, 43274547687106768032000, 2587028200730649643968000, 163484729048197101504960000 (list; graph; refs; listen; history; text; internal format)



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.


Vincenzo Librandi, Table of n, a(n) for n = 1..200

Dimitar L. Vandev, Random Dendrograms. Statistical Data Analysis, Proceedings SDA-95, SDA-96, pp. 186-196. [Cached copy from citeseerx.ist.psu.edu]


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-x)/sqrt(1-4*x+2*x^2). - Thomas Wieder, May 02 2009

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


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).


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:


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

(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


Cf. A001147, A137051.

Sequence in context: A034001 A084062 A292633 * A292716 A072034 A167571

Adjacent sequences:  A137588 A137589 A137590 * A137592 A137593 A137594




Thomas Wieder, Jan 28 2008, Feb 07 2008


Added more terms, Joerg Arndt, Oct 08 2013

Name corrected by Andrey Zabolotskiy, Mar 06 2018



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