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a(n) = Sum_{d|n} (-1)^(n/d+1)*d!.
2

%I #9 Nov 14 2018 14:02:20

%S 1,1,7,21,121,715,5041,40293,362887,3628681,39916801,479000895,

%T 6227020801,87178286161,1307674368127,20922789847653,355687428096001,

%U 6402373705365835,121645100408832001,2432902008173011101,51090942171709445047,1124000727777567763201

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

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

%F a(n) ~ n!. - _Vaclav Kotesovec_, Nov 13 2018

%t Table[Sum[(-1)^(n/d + 1) d!, {d, Divisors[n]}], {n, 22}]

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

%o (PARI) a(n) = sumdiv(n, d, (-1)^(n/d+1)*d!); \\ _Michel Marcus_, Nov 12 2018

%Y Cf. A000142, A062363, A321522.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Nov 12 2018