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%I #6 May 23 2022 05:37:56
%S 0,1,2,3,24,50,720,4095,35840,267624,3628800,35724150,479001600,
%T 5240149200,82614884352,1188272460375,20922789888000,320893244672000,
%U 6402373705728000,113803149223980216,2379913632645120000,46396417566975840000,1124000727777607680000
%N Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = exp(-x) / (1 - x).
%F a(1) = 0; a(n) = (n-1)! * (1 - Sum_{d|n, 1 < d < n} d * (a(d)/d!)^(n/d)).
%t a[1] = 0; a[n_] := a[n] = (n - 1)! (1 - Sum[d (a[d]/d!)^(n/d), {d, Divisors[n]~Complement~{1, n}}]); Table[a[n], {n, 1, 23}]
%Y Cf. A000166, A006973, A137852, A353822, A354277.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, May 22 2022