%I #11 Mar 30 2012 18:37:25

%S 1,1,2,6,20,70,255,961,3717,14663,58758,238524,978844,4054152,

%T 16924986,71145392,300876074,1279225578,5464762332,23444755016,

%U 100968932096,436355661280,1891770895672,8225329914410,35858439130786,156708088823482,686392551655052

%N G.f. A(x) satisfies: x = Sum_{n>=0} -(-A(x))^A000069(n), where A000069 is the odious numbers.

%C The odious numbers (A000069) have an odd number of 1's in their binary expansion.

%F Equals the series reversion of the g.f. of the Thue-Morse sequence (A010060) evaluated at x = -x.

%e G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 255*x^7 +...

%e Series reversion of the g.f. yields:

%e G(x) = x - x^2 - x^4 + x^7 - x^8 + x^11 + x^13 - x^14 - x^16 + x^19 + x^21 - x^22 + x^25 +..+ -(-x)^A000069(n) +...

%o (PARI) {odious(n)=if(n==0, 1, if(n%2==0, odious(n/2)+n, -odious((n-1)/2)+3*n))}

%o {a(n)=polcoeff(serreverse(sum(k=1,n+1,-(-x)^odious(k-1)+x^2*O(x^n))),n)}

%Y Cf. A000069 (odious numbers), A010060 (Thue-Morse).

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jan 24 2012