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%I #9 Apr 15 2021 13:36:57
%S 1,1,1,2,3,3,4,6,7,8,10,12,14,16,20,23,26,30,36,41,47,55,64,73,83,96,
%T 111,125,144,165,187,211,241,272,306,346,391,439,493,553,622,695,779,
%U 871,974,1086,1211,1348,1502,1671,1857,2061,2288,2533,2808,3107,3439,3800,4199,4634,5113
%N The sequence e, 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="/A041003/b041003.txt">Table of n, a(n) for n = 0..1000</a>
%F From _Andrew Howroyd_, Apr 15 2021: (Start)
%F Euler transform of A041004.
%F G.f.: A(x) = 1/Product_{k>=1} (1 - x^k)^(a(k+1) mod 2). (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); concat([1], EulerT(v[2..n+1]))} \\ _Andrew Howroyd_, Apr 14 2021
%Y Cf. A041002, A041004.
%K nonn,easy,eigen
%O 0,4
%A _N. J. A. Sloane_
%E Terms a(50) and beyond from _Andrew Howroyd_, Apr 14 2021
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