OFFSET
1,3
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.
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]
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
EXAMPLE
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).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Jan 28 2008, Feb 07 2008
EXTENSIONS
Added more terms, Joerg Arndt, Oct 08 2013
Name corrected by Andrey Zabolotskiy, Mar 06 2018
STATUS
approved