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Dirichlet inverse of right-shifted Catalan numbers [as when started from A000108(0): 1, 1, 2, 5, 14, 42, etc.]
6

%I #21 Feb 24 2022 08:57:23

%S 1,-1,-2,-4,-14,-38,-132,-420,-1426,-4834,-16796,-58688,-208012,

%T -742636,-2674384,-9693976,-35357670,-129641774,-477638700,

%U -1767253368,-6564119892,-24466233428,-91482563640,-343059494120,-1289904147128,-4861945985428,-18367353066440,-69533549429280,-263747951750360,-1002242211282032

%N Dirichlet inverse of right-shifted Catalan numbers [as when started from A000108(0): 1, 1, 2, 5, 14, 42, etc.]

%H Antti Karttunen, <a href="/A349450/b349450.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)-1) * a(d).

%F For n > 1, a(n) = -A035010(n) = A035102(n) - A000108(n-1).

%F G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Catalan(k-1) * A(x^k). - _Ilya Gutkovskiy_, Feb 23 2022

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

%o (PARI)

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

%o memoA349450 = Map();

%o 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)));

%Y Cf. A000108, A035010, A035102.

%Y Cf. also A349449, A349451, A349452, A349453, A349564.

%K sign

%O 1,3

%A _Antti Karttunen_, Nov 22 2021