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%I #14 Sep 05 2016 21:56:33
%S 1,2,8,46,324,2608,23136,221370,2252872,24153284,270922880,3163154736,
%T 38291322000,479133266432,6181998751808,82084129578414,
%U 1119798740473788,15674024566862424,224843628257016920,3302256609111585300,49613275311027132672,761926428688868584400,11952618573953745931536,191418290850831848697272,3127755564602007721663352,52118116918762815035493760,885205781290692080951844800,15318116453244882343710519680,269953482313408263924956600000
%N G.f. A(x) satisfies: A(x - x*A(x)) = x + x*A(x).
%H Paul D. Hanna, <a href="/A276358/b276358.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = 2 * Series_Reversion( x - x*A(x) ) - x.
%F (2) A(x) = x * (1 + A(B(x))) / (1 - A(B(x))), where B(x) = (x + A(x))/2.
%F (3) A( (x + A(x))/2 ) = (A(x) - x) / (A(x) + x).
%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 46*x^4 + 324*x^5 + 2608*x^6 + 23136*x^7 + 221370*x^8 + 2252872*x^9 + 24153284*x^10 + 270922880*x^11 + 3163154736*x^12 +...
%e such that A(x - x*A(x)) = x + x*A(x).
%e RELATED SERIES.
%e A(x - x*A(x)) = x + x^2 + 2*x^3 + 8*x^4 + 46*x^5 + 324*x^6 + 2608*x^7 +...
%e which equals x + x*A(x).
%e Series_Reversion( x - x*A(x) ) = x + x^2 + 4*x^3 + 23*x^4 + 162*x^5 + 1304*x^6 + 11568*x^7 + 110685*x^8 + 1126436*x^9 + 12076642*x^10 + 135461440*x^11 + 1581577368*x^12 +...
%e which equals (x + A(x))/2.
%e A( (x + A(x))/2 ) = x + 3*x^2 + 16*x^3 + 111*x^4 + 898*x^5 + 8068*x^6 + 78400*x^7 + 810875*x^8 + 8832804*x^9 + 100592970*x^10 + 1191393144*x^11 + 14616198024*x^12 +...
%e which equals (A(x) - x)/(A(x) + x).
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = 2*serreverse( x - x*A +x*O(x^n) ) - 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 = (x + A)/2 +x*O(x^n); A = x*(1 + subst(A,x,B))/(1 - subst(A,x,B)) ); polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%Y f. A275765.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 04 2016
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