(Sum from i = 1 to n ((-1)^(i+1) * (n choose 2i) * (2i)! * (n-1)^(n-2i)) / (2^i * i!)) / (n-1)^n - Wolfram|Alpha

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Result:

link to /input/?i=%28%28-1%2Bn%29%5E%28-n%29+%28sqrt%282%29+%28-1%2Bn%29%5En-2%5E%28n%2F2%29+%28-1%2Bn%29+U%28%281-n%29%2F2%2C+3%2F2%2C+1%2F2+%28-1%2Bn%29%5E2%29%29%29%2Fsqrt%282%29&lk=1&a=ClashPrefs_*Math-
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Values:

link to /input/?i=3%2F4&lk=1&a=ClashPrefs_*Math-link to /input/?i=17%2F27&lk=1&a=ClashPrefs_*Math-link to /input/?i=145%2F256&lk=1&a=ClashPrefs_*Math-link to /input/?i=1653%2F3125&lk=1&a=ClashPrefs_*Math-link to /input/?i=0.75&lk=1&a=ClashPrefs_*Math-link to /input/?i=0.62963&lk=1&a=ClashPrefs_*Math-link to /input/?i=0.566406&lk=1&a=ClashPrefs_*Math-link to /input/?i=0.52896&lk=1&a=ClashPrefs_*Math-
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