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A258314 G.f. B(x) satisfies: B(x) = 1 + x*A(x)*C(x) where A(x) = B(x)*C(x) and C(x) = 1 + 2*x*A(x)*B(x).

%I A258314 #7 May 25 2015 23:25:49
%S A258314 1,1,5,29,193,1389,10525,82729,668321,5514873,46285861,393889605,
%T A258314 3390819777,29475696197,258371636989,2281190100625,20268424498753,
%U A258314 181090741905393,1625999443354501,14664436054418477,132781726001339713,1206629736349162909,11000943167309740701,100596316305081808185
%N A258314 G.f. B(x) satisfies: B(x) = 1 + x*A(x)*C(x) where A(x) = B(x)*C(x) and C(x) = 1 + 2*x*A(x)*B(x).
%F A258314 G.f. B(x) satisfies:
%F A258314 (1) B(x) = 1 + x*B(x)*(1 - 2*B(x))^2 + 4*x^2*B(x)^4*(1-B(x)).
%F A258314 (2) B(x) = sqrt( (1/x)*Series_Reversion( x*(1-2*x)^2 / (1-x + x^2*G(-x^2))^2 ) ), where G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers.
%F A258314 (3) x = ( sqrt(1 - 8*B(x) + 8*B(x)^2) - (1 - 2*B(x))^2 ) / (8*B(x)^3*(1-B(x))).
%F A258314 Other relations involving A=A(x), B=B(x), and C=C(x) are:
%F A258314 (a) B = (1 + x*A) / (1 - 2*x^2*A^2).
%F A258314 (b) C = (1 + 2*x*A) / (1 - 2*x^2*A^2).
%F A258314 (c) B = 1/(1 - x*C^2).
%F A258314 (d) C = 1/(1 - 2*x*B^2).
%e A258314 G.f.: B(x) = 1 + x + 5*x^2 + 29*x^3 + 193*x^4 + 1389*x^5 + 10525*x^6 +...
%e A258314 where B(x) = 1 + x*A(x)*C(x):
%e A258314 A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 641*x^4 + 4719*x^5 + 36335*x^6 +...
%e A258314 C(x) = 1 + 2*x + 8*x^2 + 46*x^3 + 304*x^4 + 2178*x^5 + 16456*x^6 +...
%e A258314 Related series:
%e A258314 A(x)*B(x) = 1 + 4*x + 23*x^2 + 152*x^3 + 1089*x^4 + 8228*x^5 +...
%e A258314 A(x)*C(x) = 1 + 5*x + 29*x^2 + 193*x^3 + 1389*x^4 + 10525*x^5 +...
%o A258314 (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(B,n)}
%o A258314 for(n=0,30,print1(a(n),", "))
%o A258314 (PARI) {a(n)=local(B=1); B = sqrt( (1/x)*serreverse( x*(1-2*x)^2 / (1-x + x*serreverse(x/(1-x^2 +x*O(x^n))))^2 ) ); polcoeff(B,n)}
%o A258314 for(n=0,30,print1(a(n),", "))
%Y A258314 Cf. A258313 (A(x)), A258315 (C(x)).
%K A258314 nonn
%O A258314 0,3
%A A258314 _Paul D. Hanna_, May 25 2015