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COMMENTS
| See the double factorials A001147 for the case when the product is commutative and nonassociative.
Another interpretation of A137591 is possible in terms of dendrograms. A001147 gives the number labeled, non-ranked, binary dendrograms, so called L-NR dendrograms. A137591 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: Dimitar L. Vandev, Random Dendrograms. Statistical Data Analysis, Proceedings SDA-95, SDA-96, pp.186-196 (https://www.fmi.uni-sofia.bg/fmi/statist/Personal/Vandev/papers/dendro.pdf)
Vandev's formula (1) is our recurrence for A137591, but it seems that Vandev meant a(n) = sum(binomial(n-1,k)*a(k)*a(n-k),k=1..n-1) with a(1)=1, a(2)=1. This recurrence gives the double factorials.
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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)), w((yz),x), (x(yz))w, ((yz),y)w,
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(w(xz)), (w(xz))y, ((xz)w)y,
y(x(wz)), y(x(wz)), (x(wz))y, ((wz)x)y,
y(z(wx)), y(z(wx)), (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.
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