# [Pari/GP], 26 bytes

 n->n!*sum(k=0,n,(-1)^k/k!)

[Try it online!][TIO-kyu0787z]

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# [Pari/GP], 28 bytes

 f(n)=if(n,n*f(n-1)+(-1)^n,1)

[Try it online!][TIO-kyu0f1z1]

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# [Pari/GP], 15 bytes (invalid because of finite precision)

 n->n!\/exp(!!n)

[Try it online!][TIO-kypaaqe2]

[Pari/GP]: http://pari.math.u-bordeaux.fr/
[TIO-kyu0787z]: https://tio.run/##K0gsytRNL/ifpmD7P0/XLk9Rq7g0VyPb1kAnT0dD11AzLls/W1Hzf1p@kUaegq2CgY6CERAXFGXmlQAFlBR07YBEmkaepqbmfwA "Pari/GP – Try It Online"
[TIO-kyu0f1z1]: https://tio.run/##K0gsytRNL/j/P00jT9M2E0jq5GkBSV1DTW0NIBGXp2Oo@T8tv0gjT8FWwUBHwQiIC4oy80qAAkoKunZAAqRVU/M/AA "Pari/GP – Try It Online"
[TIO-kypaaqe2]: https://tio.run/##K0gsytRNL/ifpmD7P0/XLk8xRj@1okBDUTFP839afpFGnoKtgoGOghEQFxRl5pUABZQUdO2ARJpGnqam5n8A "Pari/GP – Try It Online"