Coprime Convolution Sum

[Leroy Quet]

I just submitted the following to the EIS:


%S A000001 1,1,2,4,12,24,88
%N A000001 a(1)=1; for n>1, a(n) = sum{1<=j<n,GCD(j,n)=1} a(j) 
a(n-j)
%e A000001 Since 1 and 5 are the positive integers < 6 and coprime to 
6,
a(6) = a(1)a(5) + a(5)a(1) = 1*12 +12*1 = 24.
%O A000001 1
%K A000001 ,more,nonn,
%A A000001 Leroy Quet (qq-quet at mindspring.com), Aug 08 2004

Maybe someone can extend this sequence.

Also a possibly interesting question, is there a non-recursive way to 
generate this sequence?

We also have the generating function identity:

sum{n=1 to oo} a(n) x^n =

x + sum{n=1 to oo; m=1 to oo; GCD(m,n)=1} a(m) a(n) x^(m+n)

thanks,
Leroy Quet