Convolution Semi-Square

[franktaw at netscape.net]

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
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