Изменения

<tex dpi = "180150">zB'(z)=z(\sum_{n=0}^\infty b_n z^n)'=z\sum_{n=1}^\infty nb_n z^{n-1}=\sum_{n=0}^\infty nb_n z^n</tex>

<tex dpi = "180150">\sum_{n=2}^\infty n z^n=z \sum_{n=2}^\infty n z^{n-1}= z (\sum_{n=2}^\infty z^n)'</tex>

<tex dpi = "180150">\sum_{n=2}^\infty z^n=\sum_{n=0}^\infty z^n-1-z=\frac{1}{1-z}-1-z=\frac{z^2}{1-z}</tex>

<tex dpi = "180150">z (\frac{z^2}{1-z})'=\frac{z^2(2-z)}{(1-z)^2}</tex>

<tex dpi = "180150">G(z)=1-4z+6zG(z) - 8z^2G(z)+\frac{z^2(2-z)}{(1-z)^2}</tex>

<tex dpi = "180">G(z)=\frac{1-6z+11z^2-5z^3}{(1-6z+8z^2)(1-z)^2}=\frac{1-6z+11z^2-5z^3}{(1-2z)(1-4z))(1-z)^2}=\frac{1/3}{(1-z)^2}+\frac{7/9}{1-z}-\frac{1/2}{1-2z}+\frac{7/18}{1-4z}</tex>

<tex dpi = "180160">\frac{1}{(1-z)^2}=(1-z)^{-2}=\sum_{n=0}^{\infty} {-2\choose n}(-z)^n=</tex>

<tex dpi = "180160">=\sum_{n=0}^{\infty} (-1)^n{n+1\choose 1}(-z)^n=\sum_{n=0}^{\infty}(n+1)z^n</tex>

<tex dpi = "180160">G(z)=\frac{1/3}{(1-z)^2}+\frac{7/9}{1-z}-\frac{1/2}{1-2z}+\frac{7/18}{1-4z}=</tex>

<tex dpi = "180160">=\frac{1}{3}\sum_{n=0}^{\infty} (n+1)z^n +\frac{7}{9}\sum_{n=0}^{\infty} z^n - \frac{1}{2}\sum_{n=0}^{\infty} 2^n z^n + \frac{7}{18}\sum_{n=0}^{\infty} 4^n z^n</tex>

<tex dpi = "180160">a_n=\frac{n+1}{3}+\frac{7}{9}-\frac{2^n}{2}+\frac{7 \cdot 4^n}{18}=\frac{7 \cdot 4^n+6n+20}{18}-2^{n-1}</tex>