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

1, 1, 0, -3, -6, 5, 61, 126, -308, -2772, -5669, 25630, 224730, 486551, -3068155, -29264219, -72173176, 513535711, 5625869262, 16687752839, -113740116822, -1496118902963, -5508392724427, 31534346503605, 523333047780288, 2414704077547660, -10254467367668159

Offset: 0

Author

Ilya Gutkovskiy, Mar 11 2018

Keywords

• sign

Comments

Exponential transform of A008683.

Examples

			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! + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..592
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A008683, A050385, A050386, A068341, A073776, A104688, A117209, A117210, A195589, A204290, A300663.

Programs

Mathematica

nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
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}]

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

Formula

E.g.f.: exp(Sum_{k>=1} A008683(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022