1,3

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.

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

a(n) = sum(binomial(n,k)*a(k)*a(n-k),k=1..n-1) with a(1)=1, a(2)=1. e.g.f.=-(z-1)/(2z^2+1-4z)^1/2.

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.

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:

Cf. A001147, A137051.

Adjacent sequences: A137588 A137589 A137590 this_sequence A137592 A137593 A137594

Sequence in context: A081132 A034001 A084062 this_sequence A072034 A138434 A049037

nonn

Thomas Wieder (thomas.wieder(AT)t-online.de), Jan 28 2008, Feb 07 2008