%I #66 Aug 06 2024 10:06:41
%S 1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600,
%T 3775206204000,148426878600000,6341634955656000,292576856395824000,
%U 14496220038251952000,767691210706291872000,43274547687106768032000,2587028200730649643968000,163484729048197101504960000
%N 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.
%C See the double factorials A001147 for the case when the product is commutative and nonassociative.
%C 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.
%C See the Murtagh paper cited in A001147 for more on dendrograms. See also Vandev.
%C 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.
%H Vincenzo Librandi, <a href="/A137591/b137591.txt">Table of n, a(n) for n = 1..200</a>
%H May Cai, Kisun Lee, and Josephine Yu, <a href="https://arxiv.org/abs/2404.08121">Symmetric Tropical Rank 2 Matrices</a>, arXiv:2404.08121 [math.CO], 2024. See p. 12.
%H Dimitar L. Vandev, <a href="https://citeseerx.ist.psu.edu/pdf/0cf8e97e5531a639afd0aff20387da6d12b49335">Random Dendrograms. Statistical Data Analysis</a>, Proceedings SDA-95, SDA-96, pp. 186-196. [Cached copy from citeseerx.ist.psu.edu]
%F a(n) = Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k), with a(1)=1, a(2)=1.
%F 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
%F a(n) ~ sqrt(2+2*sqrt(2))/2 * n^n * (2+sqrt(2))^n / exp(n). - _Vaclav Kotesovec_, Oct 07 2013
%e a(4)=54 because we have
%e w(x(yz)), w((yz)x), (x(yz))w, ((yz)x)w,
%e w(y(xz)), w((xz)y), (y(xz))w, ((xz)y)w,
%e w(z(xy)), w((xy)z), (z(xy))w, ((xy)z)w,
%e x(w(yz)), x((yz)w), (w(yz))x, ((yz)w)x,
%e x(y(wz)), x((wz)y), (y(wz))x, ((wz)y)x,
%e x(z(wy)), x((wy)z), (z(wy))x, ((wy)z)x,
%e y(w(xz)), y((xz)w), (w(xz))y, ((xz)w)y,
%e y(x(wz)), y((wz)x), (x(wz))y, ((wz)x)y,
%e y(z(wx)), y((wx)z), (z(wx))y, ((wx)z)y,
%e z(w(xy)), z((xy)w), (w(xy))z, ((xy)w)z,
%e z(x(wy)), z((wy)x), (x(wy))z, ((wy)x)z,
%e z(y(wx)), z((wx)y), (y(wx))z, ((wx)y)z,
%e (wx)(yz), (yz)(wx)
%e (wy)(xz), (xz)(wy)
%e (wz)(xy), (xy)(wz)
%e and 12*4 + 3*2 = 48 + 6 = 54.
%e Note that:
%e w(x(yz)) is equivalent to w(x(zy)) but not to (x(yz))w or w((yz)x);
%e (wx)(yz) is equivalent to (xw)(yz) or (wx)(zy) but not to (yz)(wx).
%p 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:
%t CoefficientList[Series[(1-x)/Sqrt[1-4*x+2*x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 07 2013 *)
%o (PARI) x='x+O('x^66); Vec( serlaplace((1-x)/sqrt(1-4*x+2*x^2)) ) \\ _Joerg Arndt_, Oct 08 2013
%o (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
%Y Cf. A001147, A137051.
%K nonn
%O 1,3
%A _Thomas Wieder_, Jan 28 2008, Feb 07 2008
%E Added more terms, _Joerg Arndt_, Oct 08 2013
%E Name corrected by _Andrey Zabolotskiy_, Mar 06 2018