The first , 25, terms of the sequence enumerating the number of permutations \
    such that |pi[i+1]-pi[i]|<>1  are



[1, 0, 0, 2, 14, 90, 646, 5242, 47622, 479306, 5296790, 63779034, 831283558,

    11661506218, 175203184374, 2806878055610, 47767457130566, 860568917787402,

    16362838542699862, 327460573946510746, 6880329406055690790,

    151436547414562736234, 3484423186862152966838, 83655126041771262574458,

    2092014180086865279171334]





                          It satisfies the recurrence



(-n - 1) a(n) + (n - 1) a(n + 1) + (n + 2) a(n + 2) + (-n - 5) a(n + 3)

     + a(n + 4) = 0



                             and in Maple notation



(-n-1)*a(n)+(n-1)*a(n+1)+(n+2)*a(n+2)+(-n-5)*a(n+3)+a(n+4) = 0