A349449 - OEIS

%I #15 Feb 24 2022 08:57:16

%S 1,-2,-5,-10,-42,-112,-429,-1382,-4837,-16628,-58786,-207404,-742900,

%T -2672724,-9694425,-35351906,-129644790,-477618082,-1767263190,

%U -6564052564,-24466262730,-91482328496,-343059613650,-1289903299544,-4861946399688,-18367350100552,-69533550867509,-263747941045736,-1002242216651368

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

%H Antti Karttunen, <a href="/A349449/b349449.txt">Table of n, a(n) for n = 1..1001</a>

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

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

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

%o (PARI)

%o A000108(n) = binomial(2*n, n)/(n+1);

%o memoA349449 = Map();

%o A349449(n) = if(1==n,1,my(v); if(mapisdefined(memoA349449,n,&v), v, v = -sumdiv(n,d,if(d<n,A000108(n/d)*A349449(d),0)); mapput(memoA349449,n,v); (v)));

%Y Cf. A000108.

%Y Cf. also A349450.

%K sign

%O 1,2

%A _Antti Karttunen_, Nov 22 2021