I need help reindexing the sum I know this is probably exceedingly simple, but I'm just stuck and keep making some mistake.  Here, $t_n$ represents the n-th Tribonacci number.  That is, $t_0 = 0, t_1 = 0, t_2 = 1$ and $t_n = t_{n-1}+t_{n-2}+t_{n-3}$
Also, $T(z) = \sum_{k \geq 0} t_nz^n$
I have the task of cleaning up the following sum: 
$T(z) - zT(z) - z^2T(z) - z^3T(z) =  $
$\sum_{k \geq 0} t_nz^n -z\sum_{k \geq 0} t_nz^n -z^2\sum_{k \geq 0} t_nz^n - z^3\sum_{k \geq 0} t_nz^n = $
$\sum_{k \geq 0} t_nz^n -\sum_{k \geq 0} t_nz^{n+1} -\sum_{k \geq 0} t_nz^{n+2} - \sum_{k \geq 0} t_nz^{n+3} = $
I need to make it so that the each sum has $z^n$ and the index all start at $k = 0$, but I keep messing up.  Can somebody please help me with this?  
 A: You can’t have each sum with $z^n$ in the general term and the index starting at $0$ unless you define $t_{-1}=t_{-2}=t_{-3}=0$. If you do that, you have
$$\begin{align*}
(1-z-z^2-z^3)T(z)&=(1-z-z^2-z^3)\sum_{n\ge 0}t_nz^n\\
&=\sum_{n\ge 0}t_nz^n-\sum_{n\ge 0}t_nz^{n+1}-\sum_{n\ge 0}t_nz^{n+2}-\sum_{n\ge 0}t_nz^{n+3}\\
&=\color{red}{\sum_{n\ge 0}t_nz^n}-\color{blue}{\sum_{n\ge 1}t_{n-1}z^n}-\color{brown}{\sum_{n\ge 2}t_{n-2}z^n}-\sum_{n\ge 3}t_{n-3}z^n\\
&=\color{red}{(t_0+t_1z+t_2z^2)}-\color{blue}{(t_0z+t_1z^2)}-\color{brown}{t_0z^2}\\
&\qquad+\sum_{n\ge 3}(\color{red}{t_n}-\color{blue}{t_{n-1}}-\color{brown}{t_{n-2}}-t_{n-3})z^n\\
&=\sum_{n\ge 3}(t_n-t_{n-1}-t_{n-2}-t_{n-3})z^n+(t_2-t_1-t_0)z^2+(t_1-t_0)z+t_0\\
&=\sum_{n\ge 0}(t_n-t_{n-1}-t_{n-2}-t_{n-3})z^n\;,
\end{align*}$$
but you really need that in order to make the last step. Keeping the summations separate doesn’t help: you still need negative subscripts if you want the index to start at $0$ in every summation.
A: $\begin{align}&T(z) - zT(z) - z^2T(z) - z^3T(z) \\&=\sum_{n \geq 0} t_nz^n -z\sum_{n \geq 0} t_nz^n -z^2\sum_{n \geq 0} t_nz^n - z^3\sum_{n \geq 0} t_nz^n \\&=t_0+t_1z+t_2z^2+\sum_{n \geq 3} t_nz^n -z\sum_{n \geq 0} t_nz^n -z^2\sum_{n \geq 0} t_nz^n - z^3\sum_{n \geq 0} t_nz^n \\&=t_0+t_1z+t_2z^2+\sum_{n \geq 3} (t_{n-1}+t_{n-2}+t_{n-3})z^n -z\sum_{n \geq 0} t_nz^n -z^2\sum_{n \geq 0} t_nz^n - z^3\sum_{n \geq 0} t_nz^n \\&=t_0+t_1z+t_2z^2+z\sum_{n \geq 2} t_nz^n +z^2\sum_{n \geq 1} t_nz^n +z^3\sum_{n \geq 0} t_nz^n-z\sum_{n \geq 0} t_nz^n -z^2\sum_{n \geq 0} t_nz^n - z^3\sum_{n \geq 0} t_nz^n \\&=t_0+t_1z+t_2z^2-zt_0-z^2t_1-z^2t_0 \\&=t_0+z(t_1-t_0)+z^2(t_2-t_1-t_0)\\&=z^2\\&=\sum_{n\ge 0}u_nz^n \end{align}$
Where $u_0=0,u_1=0,u_2=1,\forall n\geq3 \space u_n=0.$
