Convolution Semi-Square

[franktaw at netscape.net]

If a(n) has (ordinary) generating function A(x), then the generating function for b(n) is B(x) = (A(x)^2 + A(x^2))/2.
 
If we take a(n) to be the Wedderburn-Etherington numbers (A001190), b(n) is the same sequence except that b(1) = 0.  This immediately gives rise to the generating function equation for A001190: A(x) = x + (A(x)^2 + A(x^2))/2.
 
Taking a(n)=n+1, b(n) is A005993, Alkane numbers l(6,n).  From triangular numbers (n+1)*(n+2)/2, we get A005995, Alkane numbers l(8,n).
 
Taking a(n)=P(n-1)-1=A000065(n-1), the resulting sequence is A000147, trees of diameter 5.  Similarly, starting with rooted trees of height 3 (A000235) will produce trees of diameter 7 (A000550), etc.
 
Finally, I just submitted:
%I A000001
%S A000001 1,1,2,3,7,14,35,85,226,600,1658,4622,13141,37699,109419,320017,943329,2797788,
8346030,25019401,75340824,227777899,691146578,2104028507,6424449318,19670277332,
60378290912,185763773723,572764664975,1769533823103,5477077872168,16982128750507
%N A000001 Number of binary trees of weight n where leaves have positive integer 
weights, where the order of subtrees is insignificant. Commutative 
non-associative version of partitions of n.
%F A000001 a(2n) = 1 + C(a(n)+1,2) + sum_{k=1}^{n-1} a(k)*a(2n-k).  a(2n+1) = 
1 + sum_{k=1}^n a(k)*a(2n+1-k), with a(0)=0.
%e A000001 For a(4)=7, we have the following 7 sums: 4, 3+1, 2+2, (2+1)+1, 
(1+1)+2, ((1+1)+1)+1, (1+1)+(1+1).
%Y A000001 Cf A007317, A000041.
%O A000001 0
%K A000001 ,easy,nonn,
%A A000001 Frank Adams-Watters (FrankTAW at Netscape.net), Jan 23 2006

To which I should add
 
%F A000001 G.f. A(x) = ((A(x)-1)^2 + A(x^2) - 1)/2 + 1/(1-x).
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
 
 
-----Original Message-----

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.  ...
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