Limit looks like

                                 - 3/2 + exp(1)

                                       n
                                   4 (2  + n + 2)
                      Hyper(----------------------------)
                                      (n + 1)
                            (n + 1) (2        + 4 n + 4)


        (n - 1)              2      (n - 1)
1/2 (2 2        n! n + 2 n! n  - 2 2        n! - 3 (n + 1)! n! - 4 n!

                             (n + 1)              /
     + 2 (n + 1)!) (n + 1) (2        + 4 n + 4)  /  ((n + 1)! n!
                                                /

        (n + 1)      2    (n + 1)      n
    (n 2        + 4 n  + 2        - 4 2  + 4 n - 4))