\begin{eqnarray*}&&\frac{4}{45}-A\left(1,q\right)\\& = & \frac{4}{45}-\frac{q}{2\left(q+1\right)\left(q+2\right)}\\ & \# & 8\left(q+1\right)\left(q+2\right)-45q\\ & = & 8q^{2}-21q+16 \end{eqnarray*}