As functions of $k$, it appears that both sides satisfy the recurrence $$4\, \left( -n+2\,k+1 \right) \left( -n+2\,k \right) A \left( n,k \right) + \left( 8\,{k}^{2}-8\,kn+{n}^{2}+10\,k-9\,n \right) A \left( n,k+1 \right) + \left( k+2 \right) \left( -n+k \right) A \left( n,k+2 \right)=0 $$ with $A(n,0) = 1$, $A(n,1) = n-2$.

As functions of $k$, it appears that both sides satisfy the recurrence \begin{align} & 4(-n+2\,k+1) (-n+2k) A(n,k) \\[6pt] & {} + (8k^2-8\,kn+n^2+10k-9n) A(n,k+1)\\[6pt] & {} + (k+2)(-n+k) A(n,k+2)=0 \end{align} with $A(n,0) = 1$, $A(n,1) = n-2$.