A320780 - OEIS

%I #12 Nov 21 2022 08:12:59

%S 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,

%T 11,-17,-25,198,-81,-312,89,326,325,-739,-275,572,-255,1287,-453,

%U -2062,-583,2155,5985,-6725,-6661,6968,3045,3876,-7205,-2773,-5447,-4902

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

%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).

%H OEIS Wiki, <a href="https://oeis.org/wiki/Euler_transform">Euler transform</a>

%p # The function EulerInvTransform is defined in A358451.

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

%p seq(a(n), n = 1..54); # _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[Table[DivisorSigma[1,n],{n,30}]]

%Y Cf. A000203.

%K sign

%O 1,2

%A _Gus Wiseman_, Oct 22 2018