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%I #6 Feb 19 2021 11:54:20
%S 1,1,2,2,4,7,16,38,97,257,704,1985,5742,17013,51532,159356,502470,
%T 1613880,5275917,17543426,59307258,203759160,711246902,2521876015,
%U 9081377033,33207738613,123289411854,464675856111,1777656126662,6901581699899,27187917004378
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n/(1 - x^2*A(x)^n).
%C The g.f. A(x) of this sequence is motivated by the following identity:
%C Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1 - q*r^n)) ;
%C here, p = x, q = x^2, and r = A(x).
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} x^n / (1 - x^2*A(x)^n).
%F (2) A(x) = Sum_{n>=0} x^(2*n) / (1 - x*A(x)^n).
%F (3) A(x) = Sum_{n>=0} x^(3*n) * A(x)^(n^2) * (1 - x^3*A(x)^(2*n)) / ((1 - x*A(x)^n)*(1 - x^2*A(x)^n)).
%e G.f.: A(x) = = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 16*x^6 + 38*x^7 + 97*x^8 + 257*x^9 + 704*x^10 + 1985*x^11 + 5742*x^12 + ...
%e where
%e A(x) = 1/(1 - x^2) + x/(1 - x^2*A(x)) + x^2/(1 - x^2*A(x)^2) + x^3/(1 - x^2*A(x)^3) + x^4/(1 - x^2*A(x)^4) + ...
%e also
%e A(x) = 1/(1 - x) + x^2/(1 - x*A(x)) + x^4/(1 - x*A(x)^2) + x^6/(1 - x*A(x)^3) + x^8/(1 - x*A(x)^4) + ...
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m /(1 - x^2*A^m +x*O(x^n)) )); polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^(2*m) /(1 - x*A^m +x*O(x^n)) )); polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A340356.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 11 2021
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