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%I #6 Aug 30 2017 19:46:44
%S 1,3,21,213,2649,37683,591345,10018881,180728985,3438000843,
%T 68495707461,1421921135541,30636666967857,683015673332955,
%U 15717420795468609,372590181990474273,9083851303291071633,227458241320778287731,5842792897560424283157,153812115285865111498869,4146016380695723021134761,114341728157990237568733731,3224107003868451067670682993
%N G.f. A(x) satisfies: A(x - 2*x*A(x)) = x + x*A(x).
%H Paul D. Hanna, <a href="/A291743/b291743.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = 3/2 * Series_Reversion( x - 2*x*A(x) ) - x/2.
%F (2) A(x) = x * (1 + A(B(x))) / (1 - 2*A(B(x))), where B(x) = (x + 2*A(x))/3.
%F (3) A( (x + 2*A(x))/3 ) = (A(x) - x) / (2*A(x) + x).
%e G.f.: A(x) = x + 3*x^2 + 21*x^3 + 213*x^4 + 2649*x^5 + 37683*x^6 + 591345*x^7 + 10018881*x^8 + 180728985*x^9 + 3438000843*x^10 + 68495707461*x^11 + 1421921135541*x^12 +...
%e such that A(x - 2*x*A(x)) = x + x*A(x).
%e RELATED SERIES.
%e A(x - 2*x*A(x)) = x + x^2 + 3*x^3 + 21*x^4 + 213*x^5 + 2649*x^6 + 37683*x^7 + 591345*x^8 +...
%e which equals x + x*A(x).
%e Series_Reversion( x - x*A(x) ) = x + 2*x^2 + 14*x^3 + 142*x^4 + 1766*x^5 + 25122*x^6 + 394230*x^7 + 6679254*x^8 + 120485990*x^9 + 2292000562*x^10 + 45663804974*x^11 + 947947423694*x^12 +...
%e which equals (x + 2*A(x))/3.
%e A( (x + 2*A(x))/3 ) = x + 5*x^2 + 47*x^3 + 577*x^4 + 8273*x^5 + 131865*x^6 + 2275299*x^7 + 41821401*x^8 + 810261881*x^9 + 16426252597*x^10 + 346605967151*x^11 + 7582215510713*x^12 +...
%e which equals (A(x) - x) / (2*A(x) + x).
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = 3/2*serreverse( x - 2*x*A +x*O(x^n) ) - x/2 ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=x, B); for(i=1, n, B = (x + 2*A)/3 +x*O(x^n); A = x*(1 + subst(A, x, B))/(1 - 2*subst(A, x, B)) ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A291744, A276358.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 30 2017
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