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%I #5 Aug 30 2017 19:46:24
%S 1,3,15,105,897,8739,93663,1080909,13246017,170728251,2298619851,
%T 32162768805,465875706873,6964550221215,107193366978651,
%U 1695277029466917,27504875620268325,457183442035485927,7776605660061178251,135234473290510961097,2402252449086179775861,43557766261735276367055,805650777590230815177879,15191845940176304945626737,291896599103455803872483709,5712079123789080942126760083
%N G.f. A(x) satisfies: A(x - x*A(x)) = x + 2*x*A(x).
%H Paul D. Hanna, <a href="/A291744/b291744.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = 3 * Series_Reversion( x - x*A(x) ) - 2*x.
%F (2) A(x) = x * (1 + 2*A(B(x))) / (1 - A(B(x))), where B(x) = (2*x + A(x))/3.
%F (3) A( (2*x + A(x))/3 ) = (A(x) - x) / (A(x) + 2*x).
%e G.f.: A(x) = x + 3*x^2 + 15*x^3 + 105*x^4 + 897*x^5 + 8739*x^6 + 93663*x^7 + 1080909*x^8 + 13246017*x^9 + 170728251*x^10 + 2298619851*x^11 + 32162768805*x^12 +...
%e such that A(x - x*A(x)) = x + 2*x*A(x).
%e RELATED SERIES.
%e A(x - x*A(x)) = x + 2*x^2 + 6*x^3 + 30*x^4 + 210*x^5 + 1794*x^6 + 17478*x^7 + 187326*x^8 + 2161818*x^9 + 26492034*x^10 + 341456502*x^11 + 4597239702*x^12 +...
%e which equals x + 2*x*A(x).
%e Series_Reversion( x - x*A(x) ) = x + x^2 + 5*x^3 + 35*x^4 + 299*x^5 + 2913*x^6 + 31221*x^7 + 360303*x^8 + 4415339*x^9 + 56909417*x^10 + 766206617*x^11 + 10720922935*x^12 +...
%e which equals (2*x + A(x))/3.
%e A( (2*x + A(x))/3 ) = x + 4*x^2 + 26*x^3 + 218*x^4 + 2126*x^5 + 22986*x^6 + 268410*x^7 + 3331482*x^8 + 43492370*x^9 + 592851806*x^10 + 8393229602*x^11 + 122922601030*x^12 +...
%e which equals (A(x) - x) / (A(x) + 2*x).
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = 3*serreverse( x - x*A +x*O(x^n) ) - 2*x ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=x, B); for(i=1, n, B = (2*x + A)/3 +x*O(x^n); A = x*(1 + 2*subst(A, x, B))/(1 - subst(A, x, B)) ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A291743, A276358.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 30 2017
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