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A381669 The function A(x) = x+(1/2)*x^2-(1/16)*x^4... = Sum_{k >= 0} x^k*a(k)/A381670(k) satisfies the functional equation: x*(A(x)+1) = A(A(x)).

%I A381669 #21 Mar 12 2025 08:35:22
%S A381669 0,1,1,0,-1,1,-1,-1,113,-19,-1049,849,10171,-67975,183735,143679,
%T A381669 -81627111,-135422127,3045667427,341639611,-225862086367,212228801943,
%U A381669 8911194501081,-5123304557653,-1496818714531027,6387545555294289,64005829810291411,-250179519280324047
%N A381669 The function A(x) = x+(1/2)*x^2-(1/16)*x^4... = Sum_{k >= 0} x^k*a(k)/A381670(k) satisfies the functional equation: x*(A(x)+1) = A(A(x)).
%o A381669 (PARI)
%o A381669 compose(v) = polcoeff(subst(Polrev(v),x,Polrev(v)),#v-1)
%o A381669 optimize(v) = { my(r=1,z = v[#v],t = compose(concat(v,r))); while(t<>z, r = r+(z-t)/2; t = compose(concat(v,r)));concat(v,r) }
%o A381669 listA(max_n) = { my(v=[0, 1], out=[0, 1]); while(#v<max_n, v=optimize(v); out=concat(out, numerator(v[#v]))); out }
%Y A381669 Cf. A381670 ( denominators ).
%Y A381669 Cf. A381666 ( A(x)+x = x*A(A(x)) ).
%Y A381669 Cf. A030266 ( A(x)-x = x*A(A(x)) ).
%Y A381669 Cf. A347080 ( A(x)-x = x*A(A(-x)) ).
%K A381669 sign,frac,eigen
%O A381669 0,9
%A A381669 _Thomas Scheuerle_, Mar 03 2025