2-derangements

A derangement is a well known kind of permutation with no fixed point.

Define a 2-derangement as a pair P1, P2 of derangements where:
P1[i] != P2[i] for all i

Here are 2 examples of 2-derangements for 5 elements:

 2 4 5 3 1
 4 5 1 2 3

 4 5 1 2 3
 2 4 5 3 1

How many 2-derangements are there for 208 elements?

As the answer is a BIG number, use the following condensed representation:
First 10 digits[number of remaining digits]Last 10 digits

For instance, for 2^127, the representation is:
1701411834[19]5884105728

[My timing: 40 sec]