A092413 - OEIS

%I #13 Nov 15 2025 09:54:17

%S 1,-1,2,-6,20,-70,256,-970,3772,-14960,60280,-246090,1015700,-4231216,

%T 17767456,-75126078,319588340,-1366846548,5873832384,-25350152100,

%U 109828012448,-477486940848,2082520454864,-9109146150050,39950535931956

%N Coefficient of x^n in solution of x = y + y^2 + y^4 + y^8 + ...

%C Reversion of Fredholm-Rueppel sequence (A036987) shifted right.

%F From _Paul D. Hanna_, Nov 14 2025: (Start)

%F G.f. A(x) satisfies the following formulas.

%F (1) x = Sum_{n>=0} A(x)^(2^n) by definition.

%F (2) x = A( Sum_{n>=0} x^(2^n) ).

%F (3) x^(2^n) = A( Sum_{k>=n} x^(2^k) ) for n >= 1.

%F (4) A(x)^2 = A(x - A(x)).

%F (5) A(x)^4 = A(x - A(x) - A(x)^2).

%F (6) A(x)^8 = A(x - A(x) - A(x)^2 - A(x)^4).

%F (7) A(x)^(2^n) = A( x - Sum_{k=0..n-1} A(x)^(2^k) ) for n >= 1.

%F (8) A(x)^(2^n) = A( Sum_{k>=n} A(x)^(2^k) ) for n >= 1. (End)

%p # Using function CompInv from A357588.

%p CompInv(25, n -> if 2^ilog2(n) = n then 1 else 0 fi); # _Peter Luschny_, Oct 05 2022

%o (PARI) serreverse(sum(k=0,8,x^(2^k))+O(x^257))

%Y Cf. A036987, A049140.

%K sign

%O 1,3

%A _Ralf Stephan_, Mar 22 2004