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a(n) = n! * Sum_{d|n} d^(n - d) / (n/d)!.
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%I #16 Jun 11 2022 07:52:15

%S 1,3,7,73,121,12361,5041,5308801,44452801,5681370241,39916801,

%T 16800125569921,6227020801,35897693762810881,2134168822456070401,

%U 190139202281277849601,355687428096001,3563095308471181273190401,121645100408832001

%N a(n) = n! * Sum_{d|n} d^(n - d) / (n/d)!.

%H Seiichi Manyama, <a href="/A354893/b354893.txt">Table of n, a(n) for n = 1..291</a>

%F E.g.f.: Sum_{k>0} (exp((k * x)^k) - 1)/k^k.

%F If p is prime, a(p) = 1 + p! = A038507(p).

%t a[n_] := n! * DivisorSum[n, #^(n - #)/(n/#)! &]; Array[a, 19] (* _Amiram Eldar_, Jun 10 2022 *)

%o (PARI) a(n) = n!*sumdiv(n, d, d^(n-d)/(n/d)!);

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

%Y Cf. A038507, A342628, A354845, A354891, A354892.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jun 10 2022