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

%I A320780 #12 Nov 21 2022 08:12:59
%S A320780 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 A320780 11,-17,-25,198,-81,-312,89,326,325,-739,-275,572,-255,1287,-453,
%U A320780 -2062,-583,2155,5985,-6725,-6661,6968,3045,3876,-7205,-2773,-5447,-4902
%N A320780 Inverse Euler transform of the sum-of-divisors or sigma function A000203.
%C A320780 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 A320780 OEIS Wiki, <a href="https://oeis.org/wiki/Euler_transform">Euler transform</a>
%p A320780 # The function EulerInvTransform is defined in A358451.
%p A320780 a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-SumOfDivisors(n, 1))):
%p A320780 seq(a(n), n = 1..54); # _Peter Luschny_, Nov 21 2022
%t A320780 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 A320780 Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
%t A320780 EulerInvTransform[Table[DivisorSigma[1,n],{n,30}]]
%Y A320780 Cf. A000203.
%K A320780 sign
%O A320780 1,2
%A A320780 _Gus Wiseman_, Oct 22 2018