Convolution Semi-Square

[Christian G.Bower]

I usually call that transform E_2.  Although a quick search of the EIS shows
me that is probably not a good name to avoid confustion.  I gave an example of
it in the formula line for A003080.

I got the name from the Bergeron/Labelle/Leroux book (on the references line
of A003080) where "E" is the species of sets (E for the exponential nature of
such compositions) and 2 for limiting the number to exactly 2 of them.

B/L/L at one point uses the term symmetric square for this operation (as a
species operation, thus equally applicable to the unlabeled version here and
the labeled version A(x)^2/2 applied to the e.g.f.)

I would lean to names SET_2, SET_UNLABELED_2, EULER_2 (to show its relation to
the EULER transform.)

I have a library of transforms that need some EIS expression some day.

Christian
------ Original Message ------
From: franktaw at netscape.net
To: seqfan at ext.jussieu.fr
Subject: Convolution Semi-Square

> I have recently encountered a transformation several times that I have never
seen described.  It basically takes a sequence a, where a(n) is the number of
objects (of some type) of size n, and produces the sequence of the number of
unordered pairs of such objects.  The formula is:
>  
> b(2n) = C(b(n)+1,2) + sum_{k=0}^{n-1} a(k)*a(2n-k), b(2n+1)=sum_{k=0}^n
a(k)*a(2n+1-k).
>  
> I'm calling this the convolution semi-square, because it is (in some sense)
half the summation for the convolution square.
>  
> Has anyone identified and named this transformation before?  Should it be
included in the transformations page?
>  
> Franklin T. Adams-Watters
> 16 W. Michigan Ave.
> Palatine, IL 60067
> 847-776-7645