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A196018 G.f. satisfies: A(x) = 1/(1-x) - 1/(1-x*A(x)) + 1/(1-x*A(x)^2).

%I A196018 #19 Nov 18 2017 09:25:45
%S A196018 1,1,2,6,23,98,440,2044,9742,47384,234289,1174214,5951877,30459550,
%T A196018 157168265,816777857,4271248777,22459464722,118678530165,629867928597,
%U A196018 3356148860975,17946684482409,96280344449069,518058601390577,2795121781871727,15118502434518352
%N A196018 G.f. satisfies: A(x) = 1/(1-x) - 1/(1-x*A(x)) + 1/(1-x*A(x)^2).
%H A196018 Vaclav Kotesovec, <a href="/A196018/b196018.txt">Table of n, a(n) for n = 0..400</a>
%F A196018 G.f. satisfies: A(x) = 1 - x*(A(x) - A(x)^2 - A(x)^3) + x^2*(A(x) - 2*A(x)^2 - A(x)^4) + x^3*A(x)^4.
%F A196018 a(n) ~ sqrt((1/(-1 + r)^2 - s/(-1 + r*s)^2 + s^2/(-1 + r*s^2)^2) / (Pi*(1/(-1 + r*s^2)^2 + r*(1/(-1 + r*s)^3 - (4*s^2) / (-1 + r*s^2)^3)))) / (2*n^(3/2)*r^n), where r = 0.1741099691155951761402154753241071226265020289369... and s = 1.469614426933947254586622522985062658500679266649... are roots of the system of equations 1/(1-r) + 1/(-1+r*s) + 1/(1-r*s^2) = s, 2*r*s / (-1+r*s^2)^2 = 1 + r/(-1+r*s)^2. - _Vaclav Kotesovec_, Nov 18 2017
%e A196018 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 98*x^5 + 440*x^6 +...
%e A196018 Related series begin:
%e A196018 1/(1-x*A(x)) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 53*x^5 + 211*x^6 +...
%e A196018 1/(1-x*A(x)^2) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 150*x^5 + 650*x^6 +...
%o A196018 (PARI) {a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1/(1-X)-1/(1-X*A)+1/(1-X*A^2));polcoeff(A,n)}
%K A196018 nonn
%O A196018 0,3
%A A196018 _Paul D. Hanna_, Sep 26 2011