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A211854 G.f. satisfies: A(x) = (1+x*A(x)^2)*(1+x^2*A(x)^2)*(1+x^3*A(x)^2).

%I A211854 #7 Nov 22 2017 07:42:07
%S A211854 1,1,3,11,42,173,746,3321,15155,70516,333282,1595620,7722036,37715028,
%T A211854 185661034,920244770,4588778327,23003827327,115867080623,586089365947,
%U A211854 2975978506450,15163583668774,77507719810688,397320926569995,2042152353063874
%N A211854 G.f. satisfies: A(x) = (1+x*A(x)^2)*(1+x^2*A(x)^2)*(1+x^3*A(x)^2).
%H A211854 Vaclav Kotesovec, <a href="/A211854/b211854.txt">Table of n, a(n) for n = 0..500</a>
%F A211854 a(n) ~ s * sqrt((1 + 2*r + 4*r^3*s^2 + 5*r^4*s^2 + 6*r^5*s^4 + 3*r^2*(1 + s^2)) / (Pi*(1 + r + 6*r^3*s^2 + 6*r^4*s^2 + 15*r^5*s^4 + r^2*(1 + 6*s^2)))) / (2*n^(3/2)*r^n), where r = 0.1829152018931276962733907918487144062831105492965... and s = 1.828118673659452305128580127483211657533668751760... are real roots of the system of equations (1 + r*s^2)*(1 + r^2*s^2)*(1 + r^3*s^2) = s, 2*r*s*(1 + r + 2*r^3*s^2 + 2*r^4*s^2 + 3*r^5*s^4 + r^2*(1 + 2*s^2)) = 1. - _Vaclav Kotesovec_, Nov 22 2017
%e A211854 G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 42*x^4 + 173*x^5 + 746*x^6 +...
%e A211854 Related expansions:
%e A211854 A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 115*x^4 + 496*x^5 + 2211*x^6 +...
%e A211854 A(x)^4 = 1 + 4*x + 18*x^2 + 84*x^3 + 391*x^4 + 1844*x^5 + 8800*x^6 +...
%e A211854 A(x)^6 = 1 + 6*x + 33*x^2 + 176*x^3 + 912*x^4 + 4674*x^5 + 23842*x^6 +...
%e A211854 where A(x) = 1 + x*(1+x+x^2)*A(x)^2 + x^3*(1+x+x^2)*A(x)^4 + x^6*A(x)^6.
%o A211854 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A^2)*(1+x^2*A^2)*(1+x^3*A^2)+x*O(x^n)); polcoeff(A, n)}
%o A211854 for(n=0, 40, print1(a(n), ", "))
%Y A211854 Cf. A182053, A211855.
%K A211854 nonn
%O A211854 0,3
%A A211854 _Paul D. Hanna_, Apr 22 2012