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a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).
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%I #12 Aug 24 2023 02:30:49

%S 2,1,2,4,10,25,70,200,600,1845,5830,18772,61542,204659,689410,2347920,

%T 8074762,28009524,97909318,344615860,1220539390,4347310451,

%U 15564141262,55985418344,202256970300,733607281875,2670698800548,9755982857964,35751803209918,131405090455065,484316704740126,1789672012052256

%N a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

%C 6*a(n) is divisible by n (cf. A268592).

%F a(n) = (1/n^2)* Sum_{d|n} A008683(n/d)*A000984(d).

%F a(n) = A007727(n)/n^2 = A045630(n)*2/n^2 = A060165(n)/n = A022553(n)*2/n.

%t a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[2*#, #] &] / n^2; Array[a, 35] (* _Amiram Eldar_, Aug 24 2023 *)

%o (PARI) { a(n) = sumdiv(n, d, moebius(n/d)*binomial(2*d, d))/n^2; }

%Y Cf. A000984, A007727, A008683, A022553, A045630, A060165, A268592, A268617.

%K nonn

%O 1,1

%A _Max Alekseyev_, Feb 09 2016