A320780 Inverse Euler transform of the sum-of-divisors or sigma function A000203.

1, 2, 1, 0, -3, 1, -1, 1, 3, -5, -1, 4, 3, -3, -7, 8, 1, -9, 7, 8, -13, -12, 27, 7, -19, -14, 11, -17, -25, 198, -81, -312, 89, 326, 325, -739, -275, 572, -255, 1287, -453, -2062, -583, 2155, 5985, -6725, -6661, 6968, 3045, 3876, -7205, -2773, -5447, -4902

Offset: 1

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).

Links

• OEIS Wiki, Euler transform

Crossrefs

Cf. A000203.

Programs

Maple

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-SumOfDivisors(n, 1))):
seq(a(n), n = 1..54); # Peter Luschny, Nov 21 2022

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]}]];
EulerInvTransform[Table[DivisorSigma[1,n],{n,30}]]