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

%I A381670 #20 Mar 12 2025 08:35:35
%S A381670 1,1,2,1,16,16,64,16,1024,1024,4096,2048,32768,32768,131072,16384,
%T A381670 4194304,4194304,16777216,8388608,134217728,134217728,536870912,
%U A381670 134217728,8589934592,8589934592,34359738368,17179869184,274877906944,274877906944,1099511627776
%N A381670 The function A(x) = x+(1/2)*x^2-(1/16)*x^4... = Sum_{k >= 0} x^k*A381669(k)/a(k) satisfies the functional equation: x*(A(x)+1) = A(A(x)).
%C A381670 Conjecture: All terms are powers of two.
%o A381670 (PARI)
%o A381670 compose(v) = polcoeff(subst(Polrev(v),x,Polrev(v)),#v-1)
%o A381670 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 A381670 listA(max_n) = { my(v=[0, 1], out=[1,1]); while(#v<max_n, v=optimize(v); out=concat(out, denominator(v[#v]))); out }
%Y A381670 Cf. A381669 ( numerator ).
%Y A381670 Cf. A381666 ( A(x)+x = x*A(A(x)) ).
%Y A381670 Cf. A030266 ( A(x)-x = x*A(A(x)) ).
%Y A381670 Cf. A347080 ( A(x)-x = x*A(A(-x)) ).
%K A381670 nonn,frac,eigen
%O A381670 0,3
%A A381670 _Thomas Scheuerle_, Mar 03 2025