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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.

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This is sequence A002816 at OEIS. You can find a recurrence and asymptotic expansion there.

There is a summation for it in this old Stanford research report (end of page 6).

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