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A303927 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^2)^n - A(x) )^n.

%I A303927 #9 Dec 11 2018 22:48:55
%S A303927 1,1,3,19,199,2863,51280,1087107,26492959,728234294,22273547313,
%T A303927 750180870861,27591387247199,1100527782602563,47324815446060104,
%U A303927 2182852921566858499,107515416285928793865,5632697086212688424650,312779421789041421062682,18351511395587408908636348,1134459736825581425674735933
%N A303927 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^2)^n - A(x) )^n.
%F A303927 G.f. A(x) satisfies:
%F A303927 (1) 1 = Sum_{n>=0} ( (1 + x*A(x)^2)^n - A(x) )^n.
%F A303927 (2) 1 = Sum_{n>=0} (1 + x*A(x)^2)^(n^2) / (1 + A(x)*(1 + x*A(x)^2)^n)^(n+1). - _Paul D. Hanna_, Dec 11 2018
%F A303927 G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^n - F(x))^n, where F(x) is the g.f. of A303926.
%F A303927 G.f.: x/Series_Reversion( x*G(x) ) such that 1 = Sum_{n>=0} ((1 + x*G(x)^3)^n - G(x))^n, where G(x) is the g.f. of A303928.
%e A303927 G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 199*x^4 + 2863*x^5 + 51280*x^6 + 1087107*x^7 + 26492959*x^8 + 728234294*x^9 + 22273547313*x^10 + ...
%e A303927 such that
%e A303927 1 = 1 + ((1 + x*A(x)^2) - A(x)) + ((1 + x*A(x)^2)^2 - A(x))^2 + ((1 + x*A(x)^2)^3 - A(x))^3 + ((1 + x*A(x)^2)^4 - A(x))^4 + ((1 + x*A(x)^2)^5 - A(x))^5 + ...
%o A303927 (PARI) {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( (1 + x*Ser(A)^2)^m - Ser(A))^m ) )[#A] ); A[n+1]}
%o A303927 for(n=0,30, print1(a(n),", "))
%Y A303927 Cf. A303926, A303928.
%K A303927 nonn
%O A303927 0,3
%A A303927 _Paul D. Hanna_, May 03 2018