[seqfan] Number of ways to decompose K_2n into n spanning trees (unlabeled nodes)

[Andrew Weimholt]

So far, I have

1 way to decompose k0 (null graph) into 0 trees.
1 way to decompose k2 into 1 tree.
1 way to decompose k4 into 2 trees.
62 ways to decompose k6 into 3 trees.
2796 ways to decompose k8 into 4 trees.

If my terms are correct, then this is not yet in the OEIS.

Can anyone confirm these terms and/or extend the sequence?

Andrew