A352013
a(n) = Sum_{d|n} (-1)^(n/d+1) * (n-1)!/(d-1)!.
Original entry on oeis.org
1, 0, 3, -11, 25, -59, 721, -10919, 60481, -15119, 3628801, -93471839, 479001601, -8648639, 134399865601, -2833553923199, 20922789888001, -174888473759999, 6402373705728001, -228084898487846399, 3652732042831872001, -14079294028799
Offset: 1
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restart;
f:= proc(n) local d;
add((-1)^(n/d + 1) * (n-1)!/(d-1)!, d = numtheory:-divisors(n))
end proc:
map(f, [$1..30]); # Robert Israel, Nov 14 2024
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a[n_] := DivisorSum[n, (-1)^(n/#+1) * (n-1)!/(#-1)! &]; Array[a, 22] (* Amiram Eldar, Aug 30 2023 *)
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a(n) = sumdiv(n, d, (-1)^(n/d+1)*(n-1)!/(d-1)!);
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my(N=40, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k!)))
A327587
a(n) = n! * Sum_{d|n} (-1)^(n - d) / (n/d)!^d.
Original entry on oeis.org
1, 1, 7, 29, 121, 649, 5041, 42909, 364561, 3515651, 39916801, 486821873, 6227020801, 86497214231, 1307843292757, 21004582611869, 355687428096001, 6390006277567483, 121645100408832001, 2435277595236694779, 51091124681475552961, 1123451899297248225431
Offset: 1
-
a[n_] := n! Sum[(-1)^(n - d)/(n/d)!^d, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[-(-x)^k/(k! + (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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a(n) = n!*sumdiv(n, d, (-1)^(n-d)/(n/d)!^d); \\ Michel Marcus, Sep 19 2019
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
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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
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a(n)=n! * sumdiv(n, d, moebius(n/d) / d!) \\ Winston de Greef, Sep 19 2023
Showing 1-3 of 3 results.