%I #5 Apr 07 2022 12:11:38
%S 1,1,1,2,3,4,5,7,12,20,31,48,78,131,217,354,581,971,1634,2739,4580,
%T 7699,13027,22092,37449,63551,108176,184637,315530,539625,924125,
%U 1585371,2723675,4683890,8062277,13892645,23966392,41384842,71522034,123706840,214148865
%N G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(4*n) + (-1)^n*A(x))^n.
%F G.f. A(x) satisfies:
%F (1) 1 - x = Sum_{n>=0} ( x^(4*n) + (-1)^n*A(x) )^n.
%F (2) 1 - x = Sum_{n>=0} x^(4*n^2) / (1 + (-1)^n*x^(4*n)*A(x))^(n+1).
%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 12*x^9 + 20*x^10 + 31*x^11 + 48*x^12 + 78*x^13 + 131*x^14 + ...
%e where
%e 1 - x = 1 + (x^4 - A(x)) + (x^8 + A(x))^2 + (x^12 - A(x))^3 + (x^16 + A(x))^4 + (x^20 - A(x))^5 + (x^24 + A(x))^6 + ...
%e Also,
%e 1 - x = 1/(1 + A(x)) + x^4/(1 - x^4*A(x))^2 + x^16/(1 + x^8*A(x))^3 + x^36/(1 - x^12*A(x))^4 + x^64/(1 + x^16*A(x))^5 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,#A, (x^(4*m) + (-1)^m*x*Ser(A))^m ),#A));A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,sqrtint(#A\4), x^(4*m^2)/(1 + (-x^4)^m*x*Ser(A))^(m+1) ),#A));A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A317997, A352818, A352819, A352821.
%K nonn
%O 1,4
%A _Paul D. Hanna_, Apr 05 2022
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