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A338682 a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).

%I A338682 #24 May 28 2021 17:16:08
%S A338682 1,1,4,0,6,3,8,-8,20,0,12,-12,14,-7,72,-65,18,10,20,-61,142,-33,24,
%T A338682 -203,152,-52,248,-183,30,121,32,-617,398,-102,828,-619,38,-133,600,
%U A338682 -896,42,140,44,-870,2864,-207,48,-4438,1766,751,1192,-1587,54,-348,4424,-3011,1598,-348,60
%N A338682 a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).
%H A338682 Seiichi Manyama, <a href="/A338682/b338682.txt">Table of n, a(n) for n = 1..10000</a>
%F A338682 G.f.: Sum_{k >= 1} (1 - 1/(1 + x^k)^k).
%F A338682 G.f.: - Sum_{k >= 1} (-x)^k/(1 - x^k)^(k+1).
%F A338682 If p is prime, a(p) = (-1)^(p-1) + p.
%t A338682 a[n_] := DivisorSum[n, (-1)^(# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* _Amiram Eldar_, Apr 24 2021 *)
%o A338682 (PARI) a(n) = sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-1, d));
%o A338682 (PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+x^k)^k))
%o A338682 (PARI) N=66; x='x+O('x^N); Vec(-sum(k=1, N, (-x)^k/(1-x^k)^(k+1)))
%Y A338682 Cf. A081543, A217670, A318636, A338683, A338684.
%K A338682 sign
%O A338682 1,3
%A A338682 _Seiichi Manyama_, Apr 23 2021