Solving recurrence relation using generating function's

Question

Solve the recurrence relation $$h_n=5h_{n-1}-6h_{n-2}-4h_{n-3}+8h_{n-4}$$ With initial conditions $$h_0=0$$$$h_1=1$$$$h_2=1$$$$h_3=2$$ EDIT:When I worked it out I got a solution of $$h_n=\frac{-13}{18}2^n+\frac{4}{3}(n+1)2^n-\frac{1}{4}(n+1)(n+2)2^n-\frac{1}{9}(-1)^n$$ which works for $h_0,h_1,h_2$ however not for $h_3$