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%I #16 Aug 22 2022 10:06:07
%S 1,1,4,12,72,240,2520,10080,127680,816480,11037600,79833600,
%T 1514177280,12454041600,261655954560,2699348652000,62869385779200,
%U 711374856192000,19407798693803520,243290200817664000,7300765959334848000,102980278869910041600
%N Expansion of e.g.f. Sum_{k>=0} x^k / (1 - k*x^k/k!).
%F a(n) = n! * Sum_{d|n} 1/((d-1)!^(n/d-1)) for n > 0.
%F a(p) = 2 * p! for prime p.
%t a[n_]:= n! * DivisorSum[n, 1/(# - 1)!^(n/# - 1) &]; a[0] = 1; Array[a, 22, 0] (* _Amiram Eldar_, Aug 22 2022 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(1-k*x^k/k!))))
%o (PARI) a(n) = if(n==0, 1, n!*sumdiv(n, d, 1/(d-1)!^(n/d-1)));
%Y Cf. A356632, A356633, A356634.
%Y Cf. A356668.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 22 2022
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