A320767 - OEIS

%I #9 Oct 24 2018 13:44:02

%S 1,1,-2,1,-1,2,-3,4,-5,8,-13,18,-25,40,-62,90,-135,210,-324,492,-750,

%T 1164,-1809,2786,-4305,6710,-10460,16264,-25350,39650,-62057,97108,

%U -152145,238818,-375165,589520,-927200,1459960,-2300346,3626200,-5720274,9030450

%N Inverse Euler transform applied once to {1,-1,0,0,0,...}, twice to {1,0,0,0,0,...}, or three times to {1,1,1,1,1,...}.

%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.

%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 Nest[EulerInvTransform,Array[DiscreteDelta,50,0],2]

%Y Cf. A000081, A001970, A007562, A007294, A034691, A059966, A061255, A061256, A061257, A065490, A073576, A117209.

%K sign

%O 0,3

%A _Gus Wiseman_, Oct 20 2018