Meaning of wavelet and scaling coefficients
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The fast wavelet transform (FWT) is a mathematical transform which results in a series of orthogonal coefficients of varying resolution.   

Assume an input signal or time series $S$ of length $n=2^{j}$.  The length of $S$ must be equal to some power of 2, that is, $n$ equal to 8, 16, 32, 64, 128,  256, 512, 1024, etc.  To transform a signal into wavelets, at each resolution the signal is convolved using a series of scaling coefficients, $a_0$, $a_1$, $a_2$, $\ldots$ representing a low pass filter and a sequence of $d_0$, $d_1$, $d_2$, $\ldots$ of a high-pass filter.  

Scaling function coefficients, $\phi$.      At the $j-1$th resolution, the convolution is taken over the signal's original length, $2^j$, in the form 
\begin{equation}
\phi_k =  a_{-1}s_{2k+1} + a_0 s_{2k},
\end{equation}
where $k=0,1,2,\ldots,2^{j-1}-1$ are the indices of each scaling function, $s_0, s_1, \ldots, s_{2^{j}-1}$ are the original input signal elements, and $a_{-1}$ and $a_0$ are constants from the Haar scaling function.  An example of expanding the notation for the $j-1$ resolution is
\begin{equation}
\begin{split}
\phi_0 &amp;= a_{-1}s_1 + a_0 s_0\\
\phi_1 &amp;= a_{-1}s_3 + a_0 s_2\\
\vdots\\
\phi_{2^{j-1}-1} &amp;= a_{-1}s_{(2^j-1)} + a_0 s_{(2^j-2)}.\\
\end{split}
\end{equation}
Consider a signal $S$ of length $n=1024=2^{10}$ $(j=10)$, and assume Haar scaling function values of $a_{-1}=a_0=0.5$.  Although the length of $S$ is 1024, the notation used to represent the series of values is $s_0$, $s_1$, $s_2$, $\ldots$, $s_{1023}$.   The first convolution of $S$ will result in 512 ($2^{10-1}=2^9$) scaling function coefficients, which are obtained using the relationship
\begin{equation}
\phi_k = 0.5 s_{2k+1} + 0.5 s_{2k}
\end{equation}
where $k=1,2,\ldots,511$.  This translates to 
\begin{equation}
\begin{split}
\phi_0 &amp;= 0.5 s_{1} + 0.5 s_{0}\\
\phi_1 &amp;= 0.5 s_{3} + 0.5 s_{2}\\
\vdots\\
\phi_{511} &amp;= 0.5 s_{1023} + 0.5 s_{1022}\\
\end{split}
\end{equation}
It is easily noticed that the scaling function coefficients are the average of each neighboring pair of signal elements. 

Wavelet coefficients, $\psi$.  The wavelet coefficient is essentially based on the difference between each neighboring pair of signal elements.  At the $j-1$ resolution, these are 
\begin{equation}
\begin{split}
\psi_0 &amp;= -d_{-1}s_1 + d_0 s_0\\
\psi_1 &amp;= -d_{-1}s_3 + d_0 s_2\\
\vdots\\
\psi_{2^{j-1}-1} &amp;= -d_{-1}s_{(2^{j-1}-1)} + d_0 s_{(2^{j-1}-2)}\\
\end{split}
\end{equation}
where $d_{-1}$ and $d_0$ are both 0.5 based on the Haar wavelet.  

For our first convolution of signal $S$ of length 1024 at resolution $j-1$, the 512 wavelet coefficients are
\begin{equation}
\begin{split}
\psi_0 &amp;= -0.5 s_{1} + 0.5 s_{0}\\
\psi_1 &amp;= -0.5 s_{3} + 0.5 s_{2}\\
\vdots\\
\psi_{511} &amp;= -0.5 s_{1023} + 0.5 s_{1022}\\
\end{split}
\end{equation}