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 $A(n,0) = 1$, $A(n,1) = n-2$.