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a(n) = Sum_{d|n} mu(d) * binomial(d+n/d-1, d).
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%I #20 Apr 23 2021 06:38:58

%S 1,1,2,1,4,-3,6,-2,-1,-10,10,-22,12,-21,-40,-20,16,-55,18,-80,-98,-55,

%T 22,-90,-101,-78,-138,-182,28,-271,30,-104,-330,-136,-756,-37,36,-171,

%U -520,-676,40,-476,42,-550,-1786,-253,46,648,-1667,-1276,-1088,-832,52,1539,-4312

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

%H Seiichi Manyama, <a href="/A338657/b338657.txt">Table of n, a(n) for n = 1..10000</a>

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

%F If p is prime, a(p) = p - 1.

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

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

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

%Y Cf. A008683, A338656, A343565, A338658.

%K sign

%O 1,3

%A _Seiichi Manyama_, Apr 22 2021