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A171199 G.f. satisfies: A(x) = exp( Sum_{n>=1} [A(x)^n + A(x)^-n]*x^n/n ).

%I A171199 #5 Jul 20 2021 17:20:10
%S A171199 1,2,3,8,25,83,289,1041,3847,14504,55569,215727,846761,3354858,
%T A171199 13398965,53888063,218053915,887107888,3626373205,14887942624,
%U A171199 61358959587,253771944529,1052917272543,4381374717994,18280470530047
%N A171199 G.f. satisfies: A(x) = exp( Sum_{n>=1} [A(x)^n + A(x)^-n]*x^n/n ).
%C A171199 Same as A143330 after initial terms.
%F A171199 G.f.: A(x) = (1+x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).
%F A171199 G.f. satisfies: 1 = (A(x) - x)*(1 - x*A(x)).
%F A171199 a(0) = 1, a(1) = 2; a(n) = a(n-1) + a(n-2) + Sum_{k=2..n-1} a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jul 20 2021
%e A171199 G.f.: A(x) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 +...
%e A171199 log(A(x)) = [A(x)+1/A(x)]*x + [A(x)^2+1/A(x)^2]*x^2/2 + [A(x)^3+1/A(x)^3]*x^3/3 +...
%o A171199 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,(A^m+A^-m+x*O(x^n))*x^m/m)));polcoeff(A,n)}
%o A171199 (PARI) {a(n)=polcoeff((1+x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x), n)}
%Y A171199 Cf. A171190, A171191, A143330.
%K A171199 nonn
%O A171199 0,2
%A A171199 _Paul D. Hanna_, Dec 05 2009