%I #19 Jul 31 2023 02:25:39
%S 1,3,7,31,121,851,5041,43261,369601,3748249,39916801,490801081,
%T 6227020801,87861842641,1310800947457,21018206008801,355687428096001,
%U 6419518510204801,121645100408832001,2435836129700029057,51102829650622464001,1124549558817839481601
%N Expansion of e.g.f. Sum_{k>0} x^k / (k! * (1 - x^k/k)).
%F a(n) = n! * Sum_{d|n} 1 / (d^(n/d-1) * d!).
%F If p is prime, a(p) = 1 + p! = A038507(p).
%t a[n_] := n! * DivisorSum[n, 1/(#^(n/#-1) * #!) &]; Array[a, 20] (* _Amiram Eldar_, Jul 31 2023 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k/k)))))
%o (PARI) a(n) = n!*sumdiv(n, d, 1/(d^(n/d-1)*d!));
%Y Cf. A038507, A057625, A327578, A354891.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Feb 23 2023
|