a(n) = Sum_{k=0..floor(n/4)} (n-4*k)!.

Terms

a(0) =1a(1) =1a(2) =2a(3) =6a(4) =25a(5) =121a(6) =722a(7) =5046a(8) =40345a(9) =363001a(10) =3629522a(11) =39921846a(12) =479041945a(13) =6227383801a(14) =87181920722