A349563 Dirichlet convolution of right-shifted Catalan numbers with A349452 (Dirichlet inverse of A011782, 2^(n-1)).

1, -1, -2, -1, -2, 18, 68, 311, 1182, 4370, 15772, 56754, 203916, 734636, 2658096, 9661591, 35292134, 129511602, 477376556, 1766730706, 6563071700, 24464139348, 91478369336, 343051112482, 1289887370140, 4861912443284, 18367285959072, 69533415236716, 263747683314904, 1002241674463968, 3814985428350480, 14544633872450487

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• sign

Comments

Dirichlet convolution with A034729 gives A034731.

Links

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

Crossrefs

Cf. A000108, A011782, A349452, A349564 (Dirichlet inverse).

Cf. also A034729, A034731, A349565, A349567, A349569.

Programs

Mathematica

s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * 2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, CatalanNumber[# - 1] * s[n/#] &]; Array[a, 32] (* Amiram Eldar, Nov 22 2021 *)

PARI

A000108(n) = (binomial(2*n, n)/(n+1));
A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349563(n) = sumdiv(n,d,A000108(d-1)*A349452(n/d));

Formula

a(n) = Sum_{d|n} A000108(d-1) * A349452(n/d).