A320785 Inverse Euler transform of the number of factorizations function A001055.

1, 1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 1, -1, 0, 0, 1, 1, -3, 3, -3, 0, 4, -6, 6, -5, 5, -1, -7, 13, -16, 15, -8, -3, 12, -25, 41, -40, 21, 10, -51, 83, -93, 81, -38, -44, 148, -234, 258, -190, 35, 184, -429, 616, -660, 480, -18, -640, 1289, -1714, 1693, -1039, -268

Offset: 0

Author

Gus Wiseman, Oct 22 2018

Keywords

• sign

Comments

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.

Links

• OEIS Wiki, Euler transform

Crossrefs

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

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

Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.

Programs

Mathematica

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}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
EulerInvTransform[Table[Length[facs[n]],{n,100}]]