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%I #9 May 02 2014 18:15:00
%S 1,1,2,7,34,201,1357,10109,81397,698948,6341597,60391832,600661215,
%T 6215862360,66726103981,741259084280,8504902411004,100618874020119,
%U 1225724374602147,15356200178917791,197646961110310062,2610956607315266757,35370366025297098315
%N G.f. satisfies: A(x) = G(x*A(x)) where G(x) = -1+x + A(x) + 1/A(x).
%H Paul D. Hanna, <a href="/A237645/b237645.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. satisfies:
%F (1) A(x) = -1 + x*A(x) + A(x*A(x)) + 1/A(x*A(x)).
%F (2) A(x) = (1/x) * Series_Reversion( x / ( -1+x + A(x) + 1/A(x) ) ).
%F a(n) = [x^n] ( -1+x + A(x) + 1/A(x) )^(n+1) / (n+1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 201*x^5 + 1357*x^6 +...
%e Let G(x) = -1+x + A(x) + 1/A(x):
%e G(x) = 1 + x + x^2 + 3*x^3 + 13*x^4 + 70*x^5 + 436*x^6 + 3024*x^7 + 22828*x^8 + 184795*x^9 + 1587809*x^10 +...
%e then A(x) = G(x*A(x)) and G(x) = A(x/G(x)).
%e Related expansions.
%e A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 72*x^4 + 470*x^5 + 3449*x^6 + 27662*x^7 + 238209*x^8 + 2176591*x^9 + 20928935*x^10 +...
%e 1/A(x*A(x)) = 1 - x - 2*x^2 - 8*x^3 - 45*x^4 - 303*x^5 - 2293*x^6 - 18910*x^7 - 166921*x^8 - 1559040*x^9 - 15286286*x^10 +...
%e where A(x) = -1 + x*A(x) + A(x*A(x)) + 1/A(x*A(x)).
%o (PARI) {a(n)=local(A=[1,1]); for(m=2,n+1, A[m]=Vec((-1+x+ Ser(A) +1/Ser(A))^m)[m]/m;A=concat(A,0));A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 02 2014
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