A320782 - OEIS

%I #5 Oct 22 2018 22:54:55

%S 1,1,0,0,-1,1,0,0,-1,1,0,0,-2,3,0,-1,-3,6,-3,0,-6,12,-6,0,-9,23,-17,0,

%T -15,47,-40,8,-24,91,-101,34,-46,181,-230,109,-92,354,-534,323,-208,

%U 690,-1177,883,-520,1365,-2603,2297,-1377,2760,-5641,5789,-3721,5741

%N Inverse Euler transform of the unsigned Moebius function A008966.

%C 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 OEIS Wiki, <a href="https://oeis.org/wiki/Euler_transform">Euler transform</a>

%t 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 Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];

%t EulerInvTransform[Table[Abs[MoebiusMu[n]],{n,30}]]

%Y Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.

%Y Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.

%Y Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781.

%K sign

%O 0,13

%A _Gus Wiseman_, Oct 22 2018