Help with this hard recurrence relation question

Question

Please help with this.

Suppose $\{a_n\}$ satisfies $$a_n=(n+1)a_{n-1}-(n-2)a_{n-2}-(n-5)a_{n-3}+(n-3)a_{n-4},$$ and $a_0=a_1=1,a_2=a_3=0$.

Please sort out the general form of $a_n$.

I guess $a_n$ is some linear combination of binomial coeffs but cannot sort it out.

Thanks.

Answer 1

Hint: work out the next 4 or 5 entries and search on The Online Encyclopedia of Integer Sequences