A327243 a(n) = n! * Sum_{d|n} (-1)^(n - d) / (n/d)!.

1, 1, 7, 35, 121, 479, 5041, 62159, 423361, 1844639, 39916801, 779042879, 6227020801, 43606442879, 1536517382401, 32256486662399, 355687428096001, 4259374594675199, 121645100408832001, 3568256949101644799, 59616236292028416001, 562000392047391897599

Offset: 1

Author

Ilya Gutkovskiy, Sep 14 2019

Keywords

• nonn

Crossrefs

Cf. A057625, A132958.

Programs

Magma

[Factorial(n)*(&+[(-1)^(n-d)/Factorial(n div d):d in Divisors(n)]):n in [1..22]]; // Marius A. Burtea, Sep 14 2019

Mathematica

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

PARI

a(n) = {n!*sumdiv(n, d, (-1)^(n - d) / (n/d)!)} \\ Andrew Howroyd, Sep 14 2019

Formula

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

E.g.f.: Sum_{k>=1} (-1)^k * (exp((-x)^k) - 1). [corrected by Ilya Gutkovskiy, May 14 2022]