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A258315 G.f. C(x) satisfies: C(x) = 1 + 2*x*A(x)*B(x) where A(x) = B(x)*C(x) and B(x) = 1 + x*A(x)*C(x).

%I A258315 #9 May 25 2015 23:24:56
%S A258315 1,2,8,46,304,2178,16456,129086,1041248,8582274,71964232,611954286,
%T A258315 5264786448,45741886786,400776143752,3537136653566,31417018218688,
%U A258315 280616550025218,2518975669228936,22712641808517166,205612543320237808,1868112977079278594,17028815533533595080
%N A258315 G.f. C(x) satisfies: C(x) = 1 + 2*x*A(x)*B(x) where A(x) = B(x)*C(x) and  B(x) = 1 + x*A(x)*C(x).
%F A258315 G.f. C(x) satisfies:
%F A258315 (1) C(x) = 1 + 2*x*C(x)*(1 - C(x) + C(x)^2) + x^2*C(x)^4*(1 - C(x)).
%F A258315 (2) C(x) = (1/x)*Series_Reversion( x^2/(x + 2*Series_Reversion( x*(1-2*x^2)/(1+x) )^2) ).
%F A258315 (3) x = (sqrt(1 - 2*C(x) + 2*C(x)^2) - (1 - C(x) + C(x)^2)) / (C(x)^3*(1 - C(x))).
%F A258315 Other relations involving A=A(x), B=B(x), and C=C(x) are:
%F A258315 (a) B = (1 + x*A) / (1 - 2*x^2*A^2).
%F A258315 (b) C = (1 + 2*x*A) / (1 - 2*x^2*A^2).
%F A258315 (c) B = 1/(1 - x*C^2).
%F A258315 (d) C = 1/(1 - 2*x*B^2).
%e A258315 G.f.: C(x) = 1 + 2*x + 8*x^2 + 46*x^3 + 304*x^4 + 2178*x^5 + 16456*x^6 +...
%e A258315 where C(x) = 1 + 2*x*A(x)*B(x):
%e A258315 A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 641*x^4 + 4719*x^5 + 36335*x^6 +...
%e A258315 B(x) = 1 + x + 5*x^2 + 29*x^3 + 193*x^4 + 1389*x^5 + 10525*x^6 +...
%e A258315 Related series:
%e A258315 A(x)*B(x) = 1 + 4*x + 23*x^2 + 152*x^3 + 1089*x^4 + 8228*x^5 +...
%e A258315 A(x)*C(x) = 1 + 5*x + 29*x^2 + 193*x^3 + 1389*x^4 + 10525*x^5 +...
%o A258315 (PARI) {a(n)=local(A=1+x,B=1+x,C=1+2*x);for(i=1,n, A = B*C +x*O(x^n); B = 1 + x*A*C + x*O(x^n); C = 1 + 2*x*A*B + x*O(x^n)); polcoeff(C,n)}
%o A258315 for(n=0,30,print1(a(n),", "))
%o A258315 (PARI) {a(n)=local(C=1); C = (1/x)*serreverse( x^2/(x + 2*serreverse( x*(1-2*x^2)/(1+x +x*O(x^n)) )^2) ); polcoeff(C,n)}
%o A258315 for(n=0,30,print1(a(n),", "))
%Y A258315 Cf. A258313 (A(x)), A258314 (B(x)).
%K A258315 nonn
%O A258315 0,2
%A A258315 _Paul D. Hanna_, May 25 2015