A349449 Dirichlet inverse of Catalan numbers, when started from A000108(1): 1, 2, 5, 14, 42, ...

1, -2, -5, -10, -42, -112, -429, -1382, -4837, -16628, -58786, -207404, -742900, -2672724, -9694425, -35351906, -129644790, -477618082, -1767263190, -6564052564, -24466262730, -91482328496, -343059613650, -1289903299544, -4861946399688, -18367350100552, -69533550867509, -263747941045736, -1002242216651368

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• sign

Links

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

Crossrefs

Cf. A000108.

Cf. also A349450.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * CatalanNumber[n/#] &, # < n &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)

PARI

A000108(n) = binomial(2*n, n)/(n+1);
memoA349449 = Map();
A349449(n) = if(1==n,1,my(v); if(mapisdefined(memoA349449,n,&v), v, v = -sumdiv(n,d,if(dA000108(n/d)*A349449(d),0)); mapput(memoA349449,n,v); (v)));

Formula

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

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Catalan(k) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022