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A352868 Expansion of e.g.f. exp(Sum_{k>=1} mu(k) * x^k), where mu() is the Moebius function (A008683).

%I A352868 #18 Mar 01 2024 06:29:11
%S A352868 1,1,-1,-11,-23,-19,991,4369,-11311,-356903,5389471,7875341,
%T A352868 -430708871,-16579950971,45417621887,3629980647721,93982540029601,
%U A352868 -1077931879771471,-29167938898699841,-486520057714400603,7973931691642326281,205214099791890382621
%N A352868 Expansion of e.g.f. exp(Sum_{k>=1} mu(k) * x^k), where mu() is the Moebius function (A008683).
%F A352868 a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * mu(k) * a(n-k)/(n-k)!.
%F A352868 E.g.f. A(x) satisfies Product_{n>=1} A(x^n) = exp(x). - _Paul D. Hanna_, Feb 29 2024
%e A352868 E.g.f.: A(x) = 1 + x - x^2/2! - 11*x^3/3! - 23*x^4/4! - 19*x^5/5! + 991*x^6/6! + 4369*x^7/7! - 11311*x^8/8! - 356903*x^9/9! + 5389471*x^10/10! + ...
%e A352868 where A(x) = exp(x - x^2 - x^3 - x^5 + x^6 - x^7 + ... + mu(n)*x^n +....);
%e A352868 thus, exp(x) = A(x) * A(x^2) * A(x^3) * ... * A(x^n) * ...
%t A352868 nmax = 20; A[_] = 1; Do[A[x_] = Exp[x]/Product[A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]*Range[0, nmax]! (* _Vaclav Kotesovec_, Mar 01 2024 *)
%o A352868 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, moebius(k)*x^k))))
%o A352868 (PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*moebius(k)*a(n-k)/(n-k)!));
%Y A352868 Cf. A008683, A300663, A300673.
%K A352868 sign
%O A352868 0,4
%A A352868 _Seiichi Manyama_, Apr 06 2022