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A192318 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^n * A(x)^A038722(n), where A038722(n) = floor(sqrt(2*n)+1/2)^2 - n + 1.

%I A192318 #10 Mar 30 2012 18:37:27
%S A192318 1,1,2,6,18,61,218,804,3052,11831,46646,186487,754177,3079767,
%T A192318 12681568,52595999,219515014,921264092,3885468897,16459470468,
%U A192318 70001813240,298785285316,1279450906737,5495145204550,23665623371950,102175095587827
%N A192318 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^n * A(x)^A038722(n), where A038722(n) = floor(sqrt(2*n)+1/2)^2 - n + 1.
%C A192318 A038722 is a self-inverse permutation of the natural numbers. Thus, the function defined by g(x,y) = Sum_{n>=0} x^n*y^A038722(n) is symmetric: g(x,y) = g(y,x). What are the properties of a function A(x) that satisfies: A(x) = g(x,A(x)) = g(A(x),x)?
%F A192318 G.f. satisfies: A(x) = 1 + Sum_{n>=1} (x*A(x))^(n*(n-1)/2+1) * (A(x)^n - x^n)/(A(x)-x).
%F A192318 G.f. satisfies: A(x) = Sum_{n>=0} x^A038722(n) * A(x)^n.
%e A192318 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 61*x^5 + 218*x^6 + 804*x^7 +...
%e A192318 which satisfies:
%e A192318 A(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^2 + x^4*A(x)^6 + x^5*A(x)^5 + x^6*A(x)^4 +...
%e A192318 A(x) = 1 + x*A(x) + x^2*A(x)^2*(A(x)^2-x^2)/(A(x)-x) + x^4*A(x)^4*(A(x)^3-x^3)/(A(x)-x) + x^7*A(x)^7*(A(x)^4-x^4)/(A(x)-x) + x^11*A(x)^11*(A(x)^5-x^5)/(A(x)-x) +...
%e A192318 Sequence A038722 begins:
%e A192318 [1, 3,2, 6,5,4, 10,9,8,7, 15,14,13,12,11, 21,20,19,18,17,16, 28,27,...].
%o A192318 (PARI) {b(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,sqrtint(2*n)+2,(x*A+x*O(x^n))^(m*(m-1)/2+1)*(A^m-x^m)/(A-x)));polcoeff(A,n)}
%o A192318 (PARI) {A038722(n)=local(t=floor(1/2+sqrt(2*n))); if(n<1, 0, t^2-n+1)}
%o A192318 {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(A+x*O(x^n))^A038722(m)));polcoeff(A,n)}
%Y A192318 Cf. A038722.
%K A192318 nonn
%O A192318 0,3
%A A192318 _Paul D. Hanna_, Jun 27 2011