A327243 - OEIS

%I #24 Sep 08 2022 08:46:24

%S 1,1,7,35,121,479,5041,62159,423361,1844639,39916801,779042879,

%T 6227020801,43606442879,1536517382401,32256486662399,355687428096001,

%U 4259374594675199,121645100408832001,3568256949101644799,59616236292028416001,562000392047391897599

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

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

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

%t a[n_] := n! Sum[(-1)^(n - d)/(n/d)!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]

%t nmax = 22; CoefficientList[Series[Sum[-(-x)^k/(k! (1 + (-x)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%o (PARI) a(n) = {n!*sumdiv(n, d, (-1)^(n - d) / (n/d)!)} \\ _Andrew Howroyd_, Sep 14 2019

%o (Magma) [Factorial(n)*(&+[(-1)^(n-d)/Factorial(n div d):d in Divisors(n)]):n in [1..22]]; // _Marius A. Burtea_, Sep 14 2019

%Y Cf. A057625, A132958.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Sep 14 2019