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

1, 1, -1, -11, -23, -19, 991, 4369, -11311, -356903, 5389471, 7875341, -430708871, -16579950971, 45417621887, 3629980647721, 93982540029601, -1077931879771471, -29167938898699841, -486520057714400603, 7973931691642326281, 205214099791890382621

Offset

0,4

Links

Table of n, a(n) for n=0..21.

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * mu(k) * a(n-k)/(n-k)!.

E.g.f. A(x) satisfies Product_{n>=1} A(x^n) = exp(x). - Paul D. Hanna, Feb 29 2024

Example

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! + ...
where A(x) = exp(x - x^2 - x^3 - x^5 + x^6 - x^7 + ... + mu(n)*x^n +....);
thus, exp(x) = A(x) * A(x^2) * A(x^3) * ... * A(x^n) * ...

Mathematica

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 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, moebius(k)*x^k))))
(PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*moebius(k)*a(n-k)/(n-k)!));

Crossrefs

Cf. A008683, A300663, A300673.

Sequence in context: A180481 A110044 A032663 * A119815 A046624 A225587

Adjacent sequences: A352865 A352866 A352867 * A352869 A352870 A352871

Keyword

sign

Author

Seiichi Manyama, Apr 06 2022

Status

approved