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%I #11 Nov 08 2021 15:42:08
%S 1,1,2,3,7,22,92,440,2281,12493,71765,430651,2693870,17539117,
%T 118715136,834501419,6086215582,46008381159,360118132277,
%U 2915376645033,24382943109124,210428145134993,1871657093564422,17137046392867008,161334811357004021,1559974013680482615,15475431343550867360,157351220236056642597,1638298847460905532116,17451514257450951873260,190036890804241449007404,2113879533614295483104495
%N G.f. A(x) = Sum_{n>=0} x^n*(A(x)^n + sqrt(2))^n/(1 + sqrt(2)*x*A(x)^n)^(n+1).
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} x^n*(A(x)^n + sqrt(2))^n / (1 + sqrt(2)*x*A(x)^n)^(n+1).
%F (2) A(x) = Sum_{n>=0} x^n*(A(x)^n - sqrt(2))^n / (1 - sqrt(2)*x*A(x)^n)^(n+1).
%F From _Paul D. Hanna_, Nov 05 2021: (Start)
%F (3) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} sqrt(2)^k * binomial(n,k) * (A(x)^n - sqrt(2)*A(x)^k)^(n-k).
%F (4) A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} (-sqrt(2))^k * binomial(n,k) * (A(x)^n + sqrt(2)*A(x)^k)^(n-k). (End)
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 22*x^5 + 92*x^6 + 440*x^7 + 2281*x^8 + 12493*x^9 + 71765*x^10 + 430651*x^11 + 2693870*x^12 + ...
%e Let q = sqrt(2), then
%e A(x) = 1/(1 + q*x) + x*(A(x) + q)/(1 + q*x*A(x))^2 + x^2*(A(x)^2 + q)^2 / (1 + q*x*A(x)^2)^3 + x^3*(A(x)^3 + q)^3/(1 + q*x*A(x)^3)^4 + x^4*(A(x)^4 + q)^4 / (1 + q*x*A(x)^4)^5 + x^5*(A(x)^5 + q)^5/(1 + q*x*A(x)^5)^6 + ...
%e also
%e A(x) = 1/(1 - q*x) + x*(A(x) - q)/(1 - q*x*A(x))^2 + x^2*(A(x)^2 - q)^2 / (1 - q*x*A(x)^2)^3 + x^3*(A(x)^3 - q)^3/(1 - q*x*A(x)^3)^4 + x^4*(A(x)^4 - q)^4 / (1 - q*x*A(x)^4)^5 + x^5*(A(x)^5 - q)^5/(1 - q*x*A(x)^5)^6 + ...
%o (PARI) {a(n) = my(A=[1,1]); for(i=0,n, A = concat(A,0);
%o A[#A] = polcoeff( sum(n=0,#A+1, x^n*(Ser(A)^n + sqrt(2))^n/(1 + sqrt(2)*x*Ser(A)^n)^(n+1) ),#A-1)); round( polcoeff(Ser(A),n) )}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A324618, A324619.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 12 2019
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