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A320786 Inverse Euler transform of {1,0,1,0,0,0,...}.

%I A320786 #6 Jun 02 2025 15:22:19
%S A320786 1,1,-1,1,-1,1,-2,2,-2,3,-5,6,-7,11,-16,20,-27,39,-55,75,-102,145,
%T A320786 -207,286,-397,565,-802,1123,-1581,2248,-3193,4517,-6399,9112,-12984,
%U A320786 18457,-26270,37502,-53553,76416,-109146,156135,-223446,319764,-457884,656288,-941081
%N A320786 Inverse Euler transform of {1,0,1,0,0,0,...}.
%C A320786 The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.
%H A320786 OEIS Wiki, <a href="https://oeis.org/wiki/Euler_transform">Euler transform</a>
%t A320786 EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
%t A320786 Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
%t A320786 EulerInvTransform[PadRight[{1,0,1},50]]
%Y A320786 Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
%Y A320786 Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
%Y A320786 Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.
%K A320786 sign
%O A320786 0,7
%A A320786 _Gus Wiseman_, Oct 22 2018