Solving linear recurrence with binomials

Question

I need to solve linear recurrence \begin{align} a_n = \binom{9}{1}a_{n-1}-\binom{9}{2}a_{n-2}+\binom{9}{3}a_{n-3}-\dots+\binom{9}{9}a_{n-9} \end{align} with initial conditions \begin{align} a_0=a_1=a_2=\dots=a_7=0, a_8=1 \space and \space n\ge9 \end{align} I know method with generating functions, but i've stuck with binomials. Can anyone help?

Answer 1

The corresponding characteristic equation is $$x^9 - \binom{9}{1}x^8+x^7\binom{9}{2}-...+x\binom{9}{8} - 1 = (x-1)^9 = 0.$$

Therefore, the general formula is given by $$a_n = c_0+c_1n+c_2n^2+...+c_8n^8$$. Now the rest is simply to solve a $9\times 9$ matrix equation using the initial condition.