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A182267 G.f. satisfies: A(x) = (1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)).

%I A182267 #15 Nov 18 2017 09:03:34
%S A182267 1,1,2,6,16,46,140,435,1382,4474,14687,48787,163703,554009,1888794,
%T A182267 6481220,22366415,77575617,270277602,945480612,3319582632,11693824752,
%U A182267 41318554495,146399071577,520042511448,1851657641932,6607352892709,23624965371264
%N A182267 G.f. satisfies: A(x) = (1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)).
%H A182267 Vaclav Kotesovec, <a href="/A182267/b182267.txt">Table of n, a(n) for n = 0..400</a>
%F A182267 a(n) ~ sqrt(s*(1 + 2*r*s + 4*r^3*s + 5*r^4*s^2 + 6*r^5*s^3 + 3*r^2*(1 + s^2)) / (Pi*(1 + r^2 + 3*r*s + 3*r^3*s + 6*r^4*s^2))) / (2 * n^(3/2) * r^(n + 1/2)), where r = 0.2649675733882333627400730579639429790476557486165... and s = 2.383929237709193665917448862090331200952809331679... are roots of the system of equations (1 + r*s)*(1 + r^3*s)*(1 + r^2*s^2) = s, r*(1 + r^2 + 2*r*s + 2*r^3*s + 3*r^2*s^2 + 3*r^4*s^2 + 4*r^5*s^3) = 1. - _Vaclav Kotesovec_, Nov 18 2017
%e A182267 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 46*x^5 + 140*x^6 + 435*x^7 +...
%e A182267 Related expansions:
%e A182267 A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 148*x^5 + 472*x^6 +...
%e A182267 A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 102*x^4 + 336*x^5 + 1124*x^6 +...
%e A182267 A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 185*x^4 + 648*x^5 + 2272*x^6 +...
%e A182267 where A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*(A(x) + A(x)^3) + x^4*A(x)^2 + x^5*A(x)^3 + x^6*A(x)^4.
%o A182267 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x^2*A^2)*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
%o A182267 for(n=0, 40, print1(a(n), ", "))
%Y A182267 Cf. A182053, A211854, A211855.
%K A182267 nonn
%O A182267 0,3
%A A182267 _Paul D. Hanna_, Apr 22 2012