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%I #21 Apr 08 2019 12:09:50
%S 1,1,5,21,109,573,3209,18409,108569,651897,3977677,24578237,153522117,
%T 967698037,6147885585,39325620689,253063546417,1637141058033,
%U 10641294559637,69461189473381,455143665718301,2992662818159277,19739469132843673,130576261281640377,866046528206946377,5758054698392120425,38369626772586994013,256215306174255250701,1714211196437154993013,11489680147035224255717,77140806921880072224801
%N G.f.: A(x) = Sum_{n>=0} x^n * (1 + (-1)^n * A(x))^n / (1 - (-1)^n * x*A(x))^(n+1).
%C Compare the g.f. to the following related identities.
%C (1.a) F(x) = Sum_{n>=0} x^n * (1 + F(x))^n / (1 + x*F(x))^(n+1) holds when G(x) = 1/(1-x).
%C (1.b) F(x) = Sum_{n>=0} x^n * (1 - F(x))^n / (1 - x*F(x))^(n+1) holds when G(x) = 1/(1-x).
%C (2.a) G(x) = Sum_{n>=0} x^n * (1 + (-1)^n/G(x))^n / (1 - (-1)^n*x/G(x))^(n+1) holds when G(x) = (1 + sqrt(1 + 16*x^2))/(2*(1 - x)).
%C (2.b) G(x) = Sum_{n>=0} x^n * (1 - (-1)^n/G(x))^n / (1 + (-1)^n*x/G(x))^(n+1) holds when G(x) = (1 + sqrt(1 + 16*x^2))/(2*(1 - x)).
%H Paul D. Hanna, <a href="/A325157/b325157.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} x^n * (1 + (-1)^n * A(x))^n / (1 - (-1)^n * x*A(x))^(n+1).
%F (2) A(x) = Sum_{n>=0} x^n * (1 - (-1)^n * A(x))^n / (1 + (-1)^n * x*A(x))^(n+1).
%F (3) 0 = 4*x^2*A(x)^3 - (1-x)^2*A(x) + (1-x).
%F FORMULAS INVOLVING TERMS.
%F a(n) = 1 (mod 4) for n >= 0.
%e G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 109*x^4 + 573*x^5 + 3209*x^6 + 18409*x^7 + 108569*x^8 + 651897*x^9 + 3977677*x^10 + 24578237*x^11 + 153522117*x^12 + ...
%e such that
%e A(x) = 1/(1 + x*A(x)) + x*(1 + A(x))/(1 - x*A(x))^2 + x^2*(1 - A(x))^2/(1 + x*A(x))^3 + x^3*(1 + A(x))^3/(1 - x*A(x))^4 + x^4*(1 - A(x))^4/(1 + x*A(x))^5 + x^5*(1 + A(x))^5/(1 - x*A(x))^6 + x^6*(1 - A(x))^6/(1 + x*A(x))^7 + ...
%e also,
%e A(x) = 1/(1 - x*A(x)) + x*(1 - A(x))/(1 + x*A(x))^2 + x^2*(1 + A(x))^2/(1 - x*A(x))^3 + x^3*(1 - A(x))^3/(1 + x*A(x))^4 + x^4*(1 + A(x))^4/(1 - x*A(x))^5 + x^5*(1 - A(x))^5/(1 + x*A(x))^6 + x^6*(1 + A(x))^6/(1 - x*A(x))^7 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=Vec(sum(n=0, #A, x^n*(1 + (-1)^n*Ser(A))^n/(1 - (-1)^n*x*Ser(A))^(n+1)))); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=Vec(sum(n=0, #A, x^n*(1 - (-1)^n*Ser(A))^n/(1 + (-1)^n*x*Ser(A))^(n+1)))); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A325156, A307123.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 06 2019