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%I #5 Apr 07 2022 12:09:33
%S 1,1,2,3,4,8,15,27,54,107,211,429,870,1776,3669,7600,15832,33183,
%T 69766,147267,312000,662801,1412103,3016367,6457814,13856491,29792374,
%U 64174954,138483001,299324140,647967785,1404737352,3049490131,6628499431,14425441436
%N G.f. A(x) satisfies: 1 - x = Sum_{n>=0} x^n * (x^(2*n) + (-1)^n*A(x))^n.
%F G.f. A(x) satisfies:
%F (1) 1 - x = Sum_{n>=0} x^n * ( x^(2*n) + (-1)^n*A(x) )^n.
%F (2) 1 - x = Sum_{n>=0} x^(n*(2*n+1)) / (1 + (-1)^n*x^(2*n+1)*A(x))^(n+1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 8*x^5 + 15*x^6 + 27*x^7 + 54*x^8 + 107*x^9 + 211*x^10 + 429*x^11 + 870*x^12 + ...
%e where
%e 1 - x = 1 + x*(x^2 - A(x)) + x^2*(x^4 + A(x))^2 + x^3*(x^6 - A(x))^3 + x^4*(x^8 + A(x))^4 + x^5*(x^10 - A(x))^5 + x^6*(x^12 + A(x))^6 + ...
%e Also,
%e 1 - x = 1/(1 + x*A(x)) + x^3/(1 - x^3*A(x))^2 + x^10/(1 + x^5*A(x))^3 + x^21/(1 - x^7*A(x))^4 + x^36/(1 + x^9*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^m * (x^(2*m) + (-1)^m*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\2), x^((2*m+1)*m)/(1 + (-1)^m*x^(2*m+1)*Ser(A))^(m+1) ),#A));A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A352816, A352818.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 05 2022
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