Limit looks like

                                   11 cosh(1)

                          4      2        3      2
                      (9 n  + 3 n  + 2) (n  - 2 n  + 11 n - 3)
  Hyper(---------------------------------------------------------------------)
                        3    2                 4       3       2
        2 n (2 n - 1) (n  + n  + 10 n + 7) (9 n  - 36 n  + 57 n  - 42 n + 14)

                                   4      2
                                9 n  + 3 n  + 2
                          ---------------------------
                            3    2
                          (n  + n  + 10 n + 7) (2 n)!