HMMT 2011 T10

Once it has been demonstrated that $\lim _{n \rightarrow \infty} p_{n}(0)$ exists and is greater than 0 , it follows that $\lim _{n \rightarrow \infty} p_{n}(k)$ exists and is greater than 0 for all even positive integers $k$ and that $\sum_{k=0}^{\infty} A_{2 k}=1$. It also follows that $A_{0}=a_{1} A_{0}+a_{2} A_{2}+a_{3} A_{4}, A_{2}=b_{1} A_{0}+b_{2} A_{2}+b_{3} A_{4}+b_{4} A_{6}$, and $A_{2 j}=c_{1} A_{2 j-2}+c_{2} A_{2 j}+$ $c_{3} A_{2 j+2}+c_{4} A_{2 j+4}$ for all positive integers $j \geq 2$, where $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, b_{4}$, and $c_{1}, c_{2}, c_{3}, c_{4}$ are the constants you found in problem 8. Assuming these results, determine, with proof, the value of $A_{0}$.