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%I #19 Jun 11 2022 07:52:07
%S 1,3,7,73,121,9721,5041,1760641,44452801,562615201,39916801,
%T 3156125575681,6227020801,192873372531841,222245415808416001,
%U 14806216643368550401,355687428096001,34884164976924636172801,121645100408832001
%N a(n) = n! * Sum_{d|n} d^(n - d) / d!.
%H Seiichi Manyama, <a href="/A354891/b354891.txt">Table of n, a(n) for n = 1..305</a>
%F E.g.f.: Sum_{k>0} x^k/(k! * (1 - (k * x)^k)).
%F If p is prime, a(p) = 1 + p! = A038507(p).
%t a[n_] := n! * DivisorSum[n, #^(n - #)/#! &]; Array[a, 19] (* _Amiram Eldar_, Jun 10 2022 *)
%o (PARI) a(n) = n!*sumdiv(n, d, d^(n-d)/d!);
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-(k*x)^k)))))
%Y Cf. A038507, A342628, A354888, A354890, A354893.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Jun 10 2022