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A349566 Dirichlet convolution of A011782 (2^(n-1)) with A349451 (Dirichlet inverse of Fibonacci numbers).

%I A349566 #10 Nov 25 2021 12:15:40
%S A349566 1,1,2,4,11,20,51,100,218,441,935,1862,3863,7751,15742,31648,63939,
%T A349566 128180,257963,516974,1037502,2078417,4165647,8339900,16702136,
%U A349566 33428943,66911942,133891584,267921227,536021340,1072395555,2145272320,4291440670,8584166169,17170641321,34344672290,68695318919,137399603159,274814652766
%N A349566 Dirichlet convolution of A011782 (2^(n-1)) with A349451 (Dirichlet inverse of Fibonacci numbers).
%C A349566 Dirichlet convolution of this sequence with A034748 produces A034738.
%H A349566 Antti Karttunen, <a href="/A349566/b349566.txt">Table of n, a(n) for n = 1..1001</a>
%F A349566 a(n) = Sum_{d|n} 2^(d-1) * A349451(n/d).
%t A349566 s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * Fibonacci[n/#] &, # < n &]; a[n_] := DivisorSum[n, 2^(# - 1) * s[n/#] &]; Array[a, 40] (* _Amiram Eldar_, Nov 22 2021 *)
%o A349566 (PARI)
%o A349566 memoA349451 = Map();
%o A349566 A349451(n) = if(1==n,1,my(v); if(mapisdefined(memoA349451,n,&v), v, v = -sumdiv(n,d,if(d<n,fibonacci(n/d)*A349451(d),0)); mapput(memoA349451,n,v); (v)));
%o A349566 A349566(n) = sumdiv(n,d,(2^(d-1)) * A349451(n/d));
%Y A349566 Cf. A000045, A011782, A349451, A349565 (Dirichlet inverse).
%K A349566 nonn
%O A349566 1,3
%A A349566 _Antti Karttunen_, Nov 22 2021