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A349564 Dirichlet convolution of A011782 [2^(n-1)] with A349450 [Dirichlet inverse of right-shifted Catalan numbers].

%I A349564 #14 Jan 09 2024 13:17:56
%S A349564 1,1,2,2,2,-14,-68,-308,-1178,-4366,-15772,-56780,-203916,-734772,
%T A349564 -2658088,-9662208,-35292134,-129514026,-477376556,-1766739436,
%U A349564 -6563071972,-24464170892,-91478369336,-343051227304,-1289887370136,-4861912851116,-18367285963792,-69533416706328,-263747683314904,-1002241679797688
%N A349564 Dirichlet convolution of A011782 [2^(n-1)] with A349450 [Dirichlet inverse of right-shifted Catalan numbers].
%C A349564 Dirichlet convolution with A034731 gives A034729.
%H A349564 Antti Karttunen, <a href="/A349564/b349564.txt">Table of n, a(n) for n = 1..1001</a>
%F A349564 a(n) = Sum_{d|n} 2^(d-1) * A349450(n/d).
%t A349564 s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * CatalanNumber[n/# - 1] &, # < n &]; a[n_] := DivisorSum[n, 2^(# - 1) * s[n/#] &]; Array[a, 30] (* _Amiram Eldar_, Nov 22 2021 *)
%o A349564 (PARI)
%o A349564 A000108(n) = (binomial(2*n, n)/(n+1));
%o A349564 memoA349450 = Map();
%o A349564 A349450(n) = if(1==n,1,my(v); if(mapisdefined(memoA349450,n,&v), v, v = -sumdiv(n,d,if(d<n,A000108((n/d)-1)*A349450(d),0)); mapput(memoA349450,n,v); (v)));
%o A349564 A349564(n) = sumdiv(n,d,2^(d-1)*A349450(n/d));
%Y A349564 Cf. A011782, A349450.
%Y A349564 Cf. A000108, A011782, A349452, A349563 (Dirichlet inverse).
%Y A349564 Cf. also A034729, A034731, A349566, A349568.
%K A349564 sign
%O A349564 1,3
%A A349564 _Antti Karttunen_, Nov 22 2021