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a(n) = Sum_{d|n} mu(d) * binomial(d + n/d - 2, d-1).
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%I #24 Apr 23 2021 11:58:44

%S 1,0,0,-1,0,-4,0,-3,-5,-8,0,-9,0,-12,-28,-7,0,-8,0,-34,-54,-20,0,9,

%T -69,-24,-44,-83,0,0,0,-15,-130,-32,-418,157,0,-36,-180,-129,0,0,0,

%U -285,-494,-44,0,633,-923,-24,-304,-454,0,1090,-2000,-1183,-378,-56,0,3050,0,-60,-3002,-31,-3638,0

%N a(n) = Sum_{d|n} mu(d) * binomial(d + n/d - 2, d-1).

%H Seiichi Manyama, <a href="/A338656/b338656.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k >= 1} mu(k) * (x/(1 - x^k))^k.

%F If p is prime, a(p) = 0.

%t a[n_] := DivisorSum[n, MoebiusMu[#] * Binomial[# + n/# - 2, # - 1] &]; Array[a, 100] (* _Amiram Eldar_, Apr 22 2021 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(d)*binomial(d+n/d-2, d-1));

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*(x/(1-x^k))^k))

%Y Cf. A008683, A157020, A332470, A338657.

%K sign,look

%O 1,6

%A _Seiichi Manyama_, Apr 22 2021