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)).

1, 1, 2, 1, 16, 16, 64, 16, 1024, 1024, 4096, 2048, 32768, 32768, 131072, 16384, 4194304, 4194304, 16777216, 8388608, 134217728, 134217728, 536870912, 134217728, 8589934592, 8589934592, 34359738368, 17179869184, 274877906944, 274877906944, 1099511627776

Offset: 0

Author

Thomas Scheuerle, Mar 03 2025

Keywords

• nonn
• frac
• eigen

Comments

Conjecture: All terms are powers of two.

Crossrefs

Cf. A381669 ( numerator ).

Cf. A381666 ( A(x)+x = x*A(A(x)) ).

Cf. A030266 ( A(x)-x = x*A(A(x)) ).

Cf. A347080 ( A(x)-x = x*A(A(-x)) ).

Programs

PARI

compose(v) = polcoeff(subst(Polrev(v),x,Polrev(v)),#v-1)
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) }
listA(max_n) = { my(v=[0, 1], out=[1,1]); while(#v