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G.f.: A(x) = Sum_{n>=0} x^n*A(x)^A001969(n+1), where A001969 lists numbers with an even number of 1's in their binary expansion.
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%I #6 Mar 30 2012 18:37:29

%S 1,1,4,21,125,805,5459,38403,277667,2050771,15405655,117344350,

%T 904175038,7035182178,55197856415,436221495843,3469249248383,

%U 27744896161177,222987118478532,1800106801933350,14589674016207940,118674224290447850,968474133792224994

%N G.f.: A(x) = Sum_{n>=0} x^n*A(x)^A001969(n+1), where A001969 lists numbers with an even number of 1's in their binary expansion.

%e G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 125*x^4 + 805*x^5 + 5459*x^6 +...

%e where

%e A(x) = 1 + x*A(x)^3 + x^2*A(x)^5 + x^3*A(x)^6 + x^4*A(x)^9 + x^5*A(x)^10 + x^6*A(x)^12 + x^7*A(x)^15 + x^8*A(x)^17 +...

%e and exponents A001969(n) begin:

%e [0,3,5,6,9,10,12,15,17,18,20,23,24,27,29,30,33,34,36,39,40,...].

%o (PARI) {A000120(n)=n-sum(k=1,#binary(n),floor(n/2^k))}

%o {A001969(n) = (1/2)*(4*n+1-(-1)^A000120(n))}

%o {a(n)=local(A=1+x+x*O(x^n)); for(k=1, n, A=1+sum(j=1, n, x^j*A^A001969(j))); polcoeff(A, n)}

%Y Cf. A001969 (evil numbers), A195261.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 13 2011