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%I #9 Apr 15 2021 13:37:35
%S 1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0,0,0,1,1,1,0,1,1,0,1,1,0,1,1,1,1,0,0,
%T 0,1,1,1,1,0,1,1,1,0,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0,0,1,
%U 0,0,1,1,0,1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0,0,1
%N The sequence b, given that c is a left shift by one place of b.
%C Map a binary sequence b=[ b_1,... ] to a binary sequence c=[ c_1,... ] so that C=1/Product (1-x^i)^c_i == 1+Sum b_i*x^i mod 2.
%C This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C=1+Sum e_i*x^i.
%H Andrew Howroyd, <a href="/A041004/b041004.txt">Table of n, a(n) for n = 0..1000</a>
%F From _Andrew Howroyd_, Apr 15 2021: (Start)
%F a(n) = A041003(n+1) mod 2.
%F G.f.: A(x) = 1 + Sum_{k>=0} x^A041002(k). (End)
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o seq(n)={my(v=vector(n+1)); v[1]=1; for(n=2, #v, v[n]=(v[n-1] + EulerT(v[2..n])[n-1])%2); v} \\ _Andrew Howroyd_, Apr 14 2021
%Y Cf. A041002, A041003.
%K nonn,easy,eigen
%O 0,1
%A _N. J. A. Sloane_
%E Terms a(53) and beyond from _Andrew Howroyd_, Apr 14 2021
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