Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #4 Oct 22 2018 22:55:03
%S 1,1,-2,2,-3,6,-11,18,-30,56,-105,186,-335,630,-1179,2182,-4080,7710,
%T -14588,27594,-52377,99858,-190743,364722,-698870,1342176,-2581425,
%U 4971008,-9586395,18512790,-35792449,69273666,-134215680,260300986,-505294125,981706806
%N Inverse Euler transform of (-1)^(n - 1).
%C After a(1) and a(2), same as A038063.
%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[Array[(-1)^(#-1)&,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, A320782.
%K sign
%O 0,3
%A _Gus Wiseman_, Oct 22 2018