%I #19 Apr 30 2022 12:19:08
%S 1,1,0,-4,-10,-18,112,520,-1188,-21700,459584,1186704,-33320120,
%T -917538776,433679040,203262002528,3173401795088,-28004721669360,
%U -854986923602432,-13072356448331840,-17371649304775584,4477993621700382176,159817807129635664640
%N Expansion of e.g.f. exp(Sum_{k>=1} mu(k) * x^k / k), where mu() is the Moebius function (A008683).
%F a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} mu(k) * a(n-k)/(n-k)!.
%t a[0] = 1; a[n_] := a[n] = (n - 1)! * Sum[MoebiusMu[k] * a[n - k]/(n - k)!, {k, 1, n}]; Array[a, 23, 0] (* _Amiram Eldar_, Apr 30 2022 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, moebius(k)*x^k/k))))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, moebius(j)*v[i-j+1]/(i-j)!)); v;
%Y Cf. A008683, A300673, A352868, A353191.
%K sign
%O 0,4
%A _Seiichi Manyama_, Apr 29 2022