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Inverse Euler transform of the Euler totient function phi = A000010.
10

%I #6 Nov 21 2022 07:58:03

%S 1,1,0,1,0,2,-3,4,-4,4,-9,14,-19,30,-42,50,-76,128,-194,286,-412,598,

%T -909,1386,-2100,3178,-4763,7122,-10758,16414,-25061,38056,-57643,

%U 87568,-133436,203618,-311128,475536,-726355,1109718,-1697766,2601166,-3987903,6114666

%N Inverse Euler transform of the Euler totient function phi = A000010.

%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.

%p # The function EulerInvTransform is defined in A358451.

%p a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Totient(n))):

%p seq(a(n), n = 0..43); # _Peter Luschny_, Nov 21 2022

%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[EulerPhi,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, A320779, A320780, A320781, A320782.

%K sign

%O 0,6

%A _Gus Wiseman_, Oct 22 2018