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A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

%I A300673 #12 Feb 27 2022 11:11:04
%S A300673 1,1,0,-3,-6,5,61,126,-308,-2772,-5669,25630,224730,486551,-3068155,
%T A300673 -29264219,-72173176,513535711,5625869262,16687752839,-113740116822,
%U A300673 -1496118902963,-5508392724427,31534346503605,523333047780288,2414704077547660,-10254467367668159
%N A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).
%C A300673 Exponential transform of A008683.
%H A300673 Seiichi Manyama, <a href="/A300673/b300673.txt">Table of n, a(n) for n = 0..592</a>
%H A300673 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A300673 E.g.f.: exp(Sum_{k>=1} A008683(k)*x^k/k!).
%F A300673 a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Feb 27 2022
%e A300673 E.g.f.: A(x) = 1 + x/1! - 3*x^3/3! - 6*x^4/4! + 5*x^5/5! + 61*x^6/6! + 126*x^7/7! - 308*x^8/8! - 2772*x^9/9! - 5669*x^10/10! + ...
%t A300673 nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A300673 a[n_] := a[n] = Sum[MoebiusMu[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
%o A300673 (PARI) a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n-1, k-1)*a(n-k))); \\ _Seiichi Manyama_, Feb 27 2022
%Y A300673 Cf. A008683, A050385, A050386, A068341, A073776, A104688, A117209, A117210, A195589, A204290, A300663.
%K A300673 sign
%O A300673 0,4
%A A300673 _Ilya Gutkovskiy_, Mar 11 2018