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

1, 1, 2, 2, 2, -14, -68, -308, -1178, -4366, -15772, -56780, -203916, -734772, -2658088, -9662208, -35292134, -129514026, -477376556, -1766739436, -6563071972, -24464170892, -91478369336, -343051227304, -1289887370136, -4861912851116, -18367285963792, -69533416706328, -263747683314904, -1002241679797688

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• sign

Comments

Dirichlet convolution with A034731 gives A034729.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1001

Crossrefs

Cf. A011782, A349450.

Cf. A000108, A011782, A349452, A349563 (Dirichlet inverse).

Cf. also A034729, A034731, A349566, A349568.

Programs

Mathematica

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 *)

PARI

A000108(n) = (binomial(2*n, n)/(n+1));
memoA349450 = Map();
A349450(n) = if(1==n,1,my(v); if(mapisdefined(memoA349450,n,&v), v, v = -sumdiv(n,d,if(dA000108((n/d)-1)*A349450(d),0)); mapput(memoA349450,n,v); (v)));
A349564(n) = sumdiv(n,d,2^(d-1)*A349450(n/d));

Formula

a(n) = Sum_{d|n} 2^(d-1) * A349450(n/d).