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a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-2, d-1).
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%I #25 Apr 24 2021 09:57:47

%S 1,3,28,223,3126,44660,823544,16514047,387538588,9951176994,

%T 285311670612,8903202187413,302875106592254,11107259264162760,

%U 437894348359764856,18444492187995996159,827240261886336764178,39345059356329821149097

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

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

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

%t a[n_] := DivisorSum[n, (-1)^(# - 1) * (n/#)^n * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* _Amiram Eldar_, Apr 24 2021 *)

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

%o (PARI) N=20; x='x+O('x^N); Vec(sum(k=1, N, (k*x/(1+(k*x)^k))^k))

%Y Cf. A217670, A338661, A338684, A338688.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Apr 24 2021