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A249933 G.f. A(x) satisfies: 3*A(x) - 2*(1+x) = Series_Reversion( x/(2*A(x) - (1+x)) )/x.

%I A249933 #17 Nov 30 2014 03:01:37
%S A249933 1,1,1,7,77,1111,19389,391959,8942541,226359559,6281444909,
%T A249933 189409746967,6163853164333,215283653614823,8033015072981677,
%U A249933 318963167792376087,13430607648848415309,597871639955925563463,28059143187405536116653,1384849551267166030082199,71712434188939188553335789
%N A249933 G.f. A(x) satisfies: 3*A(x) - 2*(1+x) = Series_Reversion( x/(2*A(x) - (1+x)) )/x.
%H A249933 Vaclav Kotesovec, <a href="/A249933/b249933.txt">Table of n, a(n) for n = 0..230</a>
%F A249933 G.f. A(x) satisfies: 2*A(x) - (1+x) = x/Series_Reversion( x*(3*A(x) - 2*(1+x)) ).
%F A249933 G.f. A(x) satisfies: 2*A(3*x*A(x) - 2*x*(1+x)) = 3*(1+x)*A(x) - 2*(1+x)^2 + 1.
%F A249933 Define B(x) = 2*A(x) - (1+x) and C(x) = 3*A(x) - 2*(1+x), then B(x) = C(x/B(x)) and C(x) = B(x*C(x)).
%F A249933 a(n) ~ c * n^(n + 1/2 + 2*log(3/2)) / (exp(n) * (log(3/2))^n), where c = 0.104713270868379... . - _Vaclav Kotesovec_, Nov 30 2014
%e A249933 G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 77*x^4 + 1111*x^5 + 19389*x^6 +...
%e A249933 If we define
%e A249933 B(x) = 2*A(x) - (1+x) = 1 + x + 2*x^2 + 14*x^3 + 154*x^4 + 2222*x^5 +...
%e A249933 C(x) = 3*A(x) - 2*(1+x) = 1 + x + 3*x^2 + 21*x^3 + 231*x^4 + 3333*x^5 +...
%e A249933 then
%e A249933 B(x) = C(x/B(x)) and
%e A249933 C(x) = B(x*C(x)).
%e A249933 Further, A(x), B(x), and C(x) satisfy
%e A249933 B(x) = 3*A(x/B(x)) - 2 - 2*x/B(x) and
%e A249933 C(x) = (2*A(x*C(x)) - 1)/(1+x).
%o A249933 (PARI) /* 3*A(x) - 2*(1+x) = Series_Reversion(x/(2*A(x) - (1+x)))/x */
%o A249933 {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=Ser(concat(Vec(A),0));
%o A249933 A = serreverse(x/(2*A - (1+x)))/x + 2*(1+x) - 2*A); polcoeff(A,n)}
%o A249933 for(n=0, 25, print1(a(n), ", "))
%o A249933 (PARI) /* 2*A(x) - (1+x) = x/Series_Reversion(x*(3*A(x) - 2*(1+x))) */
%o A249933 {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=Ser(concat(Vec(A),0));
%o A249933 A = 3*A - x/serreverse(x*(3*A - 2*(1+x))) - (1+x)); polcoeff(A,n)}
%o A249933 for(n=0, 25, print1(a(n), ", "))
%Y A249933 Cf. A120955.
%K A249933 nonn
%O A249933 0,4
%A A249933 _Paul D. Hanna_, Nov 26 2014