A099740 -id:A099740 - cp's OEIS frontend

A332466 a(n) = n! * Sum_{d|n} mu(d) / d!.

1, 1, 5, 12, 119, 241, 5039, 20160, 302400, 1784161, 39916799, 160332480, 6227020799, 43571848321, 1078831353601, 10461394944000, 355687428095999, 2143016754278400, 121645100408831999, 1196177491129420800, 42565648051390464001, 562000335730215782401

Offset: 1

Ilya Gutkovskiy, Feb 13 2020

nonn

Cf. A008683, A057625, A068107, A099740, A132958, A332467.

with(numtheory):
a:= n-> n! * add(mobius(d)/d!, d=divisors(n)):
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 13 2020

Table[n! DivisorSum[n, MoebiusMu[#]/#! &], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] x^k/(k! (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n)={sumdiv(n, d, moebius(d)*n!/d!)} \\ Andrew Howroyd, Feb 13 2020

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

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

A354022 a(n) = n! * Sum_{d|n} mu(n/d) / d!.

Original entry on oeis.org

1, -1, -5, -11, -119, 241, -5039, -1679, -60479, 1784161, -39916799, 218877121, -6227020799, 43571848321, 1078831353601, -518918399, -355687428095999, 1058152455360001, -121645100408831999, 1115079416638387201, 42565648051390464001, 562000335730215782401

Offset: 1

Ilya Gutkovskiy, May 14 2022

sign

Winston de Greef, Table of n, a(n) for n = 1..448

Cf. A008683, A057625, A062794, A068107, A099740, A132958, A327243, A332466.

Table[n! Sum[MoebiusMu[n/d]/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] (Exp[x^k] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n)=n! * sumdiv(n, d, moebius(n/d) / d!) \\ Winston de Greef, Sep 19 2023

E.g.f.: Sum_{k>=1} mu(k) * (exp(x^k) - 1).

Sum_{n>=1} a(n) * x^n / (n! * (1 - x^n)) = exp(x) - 1.

cp's OEIS Frontend

A332466 a(n) = n! * Sum_{d|n} mu(d) / d!.

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