A320781 - OEIS

%I #17 Jul 16 2024 16:14:59

%S 1,-2,0,0,-1,2,-4,5,-7,9,-10,7,-5,-2,19,-44,70,-103,138,-166,154,-83,

%T -70,346,-797,1413,-2160,2931,-3479,3380,-2080,-1259,7593,-17743,

%U 32014,-49818,68683,-82985,82807,-53462,-24942,176139,-422887,777357,-1226688

%N Inverse Euler transform of the Moebius function A008683.

%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:-Moebius(n))):

%p seq(a(n), n = 1..45); # _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[MoebiusMu[n],{n,30}]]

%o (Python)

%o from functools import lru_cache

%o from sympy import mobius, divisors

%o def A320781(n):

%o     @lru_cache(maxsize=None)

%o     def b(n): return mobius(n)

%o     @lru_cache(maxsize=None)

%o     def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))

%o     return sum(b(d)*c(n//d) for d in divisors(n,generator=True))//n # _Chai Wah Wu_, Jul 15 2024

%Y Cf. A008683,

%K sign

%O 1,2

%A _Gus Wiseman_, Oct 22 2018