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A320781 Inverse Euler transform of the Moebius function A008683.

%I A320781 #17 Jul 16 2024 16:14:59
%S A320781 1,-2,0,0,-1,2,-4,5,-7,9,-10,7,-5,-2,19,-44,70,-103,138,-166,154,-83,
%T A320781 -70,346,-797,1413,-2160,2931,-3479,3380,-2080,-1259,7593,-17743,
%U A320781 32014,-49818,68683,-82985,82807,-53462,-24942,176139,-422887,777357,-1226688
%N A320781 Inverse Euler transform of the Moebius function A008683.
%C A320781 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 A320781 OEIS Wiki, <a href="https://oeis.org/wiki/Euler_transform">Euler transform</a>
%p A320781 # The function EulerInvTransform is defined in A358451.
%p A320781 a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Moebius(n))):
%p A320781 seq(a(n), n = 1..45); # _Peter Luschny_, Nov 21 2022
%t A320781 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 A320781 Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
%t A320781 EulerInvTransform[Table[MoebiusMu[n],{n,30}]]
%o A320781 (Python)
%o A320781 from functools import lru_cache
%o A320781 from sympy import mobius, divisors
%o A320781 def A320781(n):
%o A320781     @lru_cache(maxsize=None)
%o A320781     def b(n): return mobius(n)
%o A320781     @lru_cache(maxsize=None)
%o A320781     def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
%o A320781     return sum(b(d)*c(n//d) for d in divisors(n,generator=True))//n # _Chai Wah Wu_, Jul 15 2024
%Y A320781 Cf. A008683,
%K A320781 sign
%O A320781 1,2
%A A320781 _Gus Wiseman_, Oct 22 2018