I would like to know information regarding the function $h(n)$ where $h(n)$ is the number of hamiltonian cycles the graph $K\_n$ has after removing the edges that make up a hamiltonian cycle of $K\_n$. Thank you very much in advance.


Regards.



Determining the number of hamiltonian paths of $K_n-C_n$
This is sequence [A002816](http://oeis.org/A002816) at OEIS. You can find a recurrence and asymptotic expansion there.


There is a summation for it in [this old Stanford research report](http://i.stanford.edu/pub/cstr/reports/cs/tr/80/829/CS-TR-80-829.pdf) (end of page 6).


