I don't know a straightforward way to do that; coeff
seems to handle only a single variable at a time. But here's a way to get the list you want. The basic idea is to extract the terms of Z[20] as a list, and then evaluate each term with z[1] = 1, z[2] = 1, ..., z[20] = 1.
(%i1) display2d : false $
(%i2) Z[0] : 1 $
(%i3) Z[n] := expand (sum ((n - 1)!/i!*z[n - i]*Z[i], i, 0, n-1)) $
(%i4) z1 : makelist (z[i] = 1, i, 1, 20);
(%o4) [z[1] = 1,z[2] = 1,z[3] = 1,z[4] = 1,z[5] = 1,z[6] = 1,z[7] = 1, ...]
(%i5) a : args (Z[20]);
(%o5) [121645100408832000*z[20],128047474114560000*z[1]*z[19],
67580611338240000*z[2]*z[18],67580611338240000*z[1]^2*z[18],
47703960944640000*z[3]*z[17],71555941416960000*z[1]*z[2]*z[17], ...]
(%i6) a1 : ev (a, z1);
(%o6) [121645100408832000,128047474114560000,67580611338240000, ...]
(%i7) apply ("+", a1);
(%o7) 2432902008176640000
(%i8) 20!;
(%o8) 2432902008176640000