% <html><head><title>Weights for fitting polynomial trends</title>

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{center}
{\Large\textbf{Weights for fitting polynomial trends}}
\end{center}

\noindent
The following tables are independently computed on the lines of those in
Dudley J. Cowden,  \textit{Weights for fitting polynomial secular
trends}, Technical Paper No. 4, School of Business Administration,
University of North Carolina, 1962.  Cowden gives values up to
$N = 25$, and also values for even $N$.  Similar tables can be found in
M~G~Kendall, \textit{Time-Series}, London: Griffin 1973 and 1976,
Appendix A (see also his Section 3.9 and 3.10, pp.\,37--38).

Except in one or two early tables, the tables give the weights required
to fit at one end of the series, those for the other end being given by
symmetry.  For example, for fitting a straight line (a simple moving
average) to nine points the weights for the first point are 
$\frac{1}{45}[ 17, 14, 11, 8, 5, 2, - 1, - 4, - 7]$. Those for the
second point are $\frac{1}{180}[56, 47, 38, 29, 20, 11, 2, - 7, - 16]$ and 
so on.  Similarly for the last but three point we use the column headed
4 reading the values \textit{up} the tables, so obtaining the values
$\frac{1}{180}[8, 11, 14, 17, 20, 23, 26, 29, 321]$, and so on. 

The columns headed 0 give the weights required to extrapolate the
fitting one unit beyond the end of the observed series. 

The sums in the last row but one are the sums of the integral weights
given in the table. 

The final row in the table is the square root of the error-reducing
factor, i.e.\ $\sqrt{\sum x_i^2/(\sum x_i)^2}$ where the values in that 
column are denoted $x_i$.

\newpage

{\small
\[
\begin{array}{rrr}
\multicolumn{3}{c}{\textit{LINEAR}} \\
\multicolumn{3}{c}{N=3} \\
\ \\
0&1&2 \\
\ \\
4&5&1 \\
1&2&1 \\
-2&-1&1 \\
3&6&3 \\
1.528&0.913&0.577
\end{array}
\]
\[
\begin{array}{rrrr}
\multicolumn{4}{c}{N=5} \\
\ \\
0&1&2&3 \\
\ \\
8&3&4&1 \\
5&2&3&1 \\
2&1&2&1 \\
-1&0&1&1 \\
-4&-1&0&1 \\
\ \\
10&5&10&5 \\
1.049&0.775&0.548&0.447
\end{array}
\]
\[
\begin{array}{rrrrr}
\multicolumn{5}{c}{N=7} \\
0&1&2&3&4 \\
\ \\
4&13&5&7&1 \\
3&10&4&6&1 \\
2&7&3&5&1 \\
1&4&2&4&1 \\
0&1&1&3&1 \\
-1&-2&0&2&1 \\
-2&-5&-1&1&1 \\
\ \\
7&28&14&28&7 \\
0.845&0.681&0..535&0.423&0.378 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrr}
\multicolumn{6}{c}{\text{LINEAR}} \\
\multicolumn{6}{c}{N=9} \\
\ \\
0&1&2&3&4&5 \\
\ \\
16&17&56&22&32&1 \\
13&14&47&19&29&1 \\
10&11&38&16&26&1 \\
7&8&29&13&23&1 \\
4&5&20&10&20&1 \\
1&2&11&7&17&1 \\
-2&-1&2&4&14&1 \\
-5&-4&-7&1&11&1 \\
-8&-7&-16&-2&8&1 \\
\ \\
36&45&180&90&180&9 \\
0.726&0.615&0.511&0.422&0.357&0.333 
\end{array}
\]
\[
\begin{array}{rrrrrrr}
\multicolumn{7}{c}{N=11} \\
\ \\
0&1&2&3&4&5&6 \\
\ \\
20&7&15&25&10&15&1 \\
17&6&13&22&9&14&1 \\
14&5&11&19&8&13&1 \\
11&4&9&16&7&12&1 \\
8&3&7&13&6&11&1 \\
5&2&5&10&5&10&1 \\
2&1&3&7&4&9&1 \\
-1&0&1&4&3&8&1 \\
-4&-1&-1&1&2&7&1 \\
-7&-2&-3&-2&1&6&1 \\
-10&-3&-5&-5&0&5&1 \\
\ \\
55&22&55&110&55&110&11 \\
0.647&0.564&0.486&0.416&0.357&0.316&0.302 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrr}
\multicolumn{8}{c}{\text{LINEAR}} \\
\multicolumn{8}{c}{N=13} \\
\ \\
0&1&2&3&4&5&6&7 \\
\ \\
8&25&44&19&32&13&20&1 \\
7&22&39&17&29&12&19&1 \\
6&19&34&15&26&11&18&1 \\
5&16&29&13&23&10&17&1 \\
4&13&24&11&20&9&16&1 \\
3&10&19&9&17&8&15&1 \\
2&7&14&7&14&7&14&1 \\
1&4&9&5&11&6&13&1 \\
0&1&4&3&8&5&12&1 \\
-1&-2&-1&1&5&4&11&1 \\
-2&-5&-6&-1&2&3&10&1 \\
-3&-8&-11&-3&-1&2&9&1 \\
-4&-11&-16&-5&-4&1&8&1 \\
\ \\
26&91&182&91&182&91&182&13 \\
0.588&0.524&0.463&0.406&0.355&0.314&0.287&0.277 
\end{array}
\]
\[
\begin{array}{rrrrrrrrr}
\multicolumn{9}{c}{N=15} \\
\ \\
0&1&2&3&4&5&6&7&8 \\
\ \\
28&29&91&161&35&119&49&77&1 \\
25&26&82&146&32&110&46&74&1 \\
22&23&73&131&29&101&43&71&1 \\
19&20&64&116&26&92&40&68&1 \\
16&17&55&101&23&83&37&65&1 \\
13&14&46&86&20&74&34&62&1 \\
10&11&37&71&17&65&31&59&1 \\
7&8&28&56&14&56&28&56&1 \\
4&5&19&41&11&47&25&53&1 \\
1&2&10&26&a&38&22&50&1 \\
-2&-1&1&11&5&29&19&47&1 \\
-5&-4&-8&-4&2&20&16&44&1 \\
-8&-7&-17&-19&-1&11&13&41&1 \\
-11&-10&-26&-34&-4&2&10&38&1 \\
-14&-13&-35&-49&-7&-7&7&35&1 \\
\ \\
105&120&420&840&210&840&420&840&15 \\
0.543&0.492&0.442&0.395&0.352&0.314&0.285&0.265&0.258 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrrr}
\multicolumn{10}{c}{\text{LINEAR}} \\
\multicolumn{10}{c}{N=17} \\
\ \\
0&1&2&3&4&5&6&7&8&9 \\
\ \\
32&11&80&12&64&14&16&20&32&1 \\
29&10&73&11&59&13&15&19&31&1 \\
26&9&66&10&54&12&14&18&30&1 \\
23&8&59&9&49&11&13&17&29&1 \\
20&7&52&8&44&10&12&16&28&1 \\
17&6&45&7&39&9&11&15&27&1 \\
14&5&38&6&34&8&10&14&26&1 \\
11&4&31&5&29&7&9&13&25&1 \\
8&3&24&4&24&6&8&12&24&1 \\
5&2&17&3&19&5&7&11&23&1 \\
2&1&10&2&14&4&6&10&22&1 \\
-1&0&3&1&9&3&5&9&21&1 \\
-4&-1&-4&0&4&2&4&8&20&1 \\
-7&-2&-11&-1&-1&1&3&7&19&1 \\
-10&-3&-18&-2&-6&0&2&6&18&1 \\
-13&-4&-25&-3&-11&-1&1&5&17&1 \\
-16&-5&-32&-4&-16&-2&0&4&16&1 \\
\ \\
136&51&408&68&408&102&136&204&408&17 \\
0.507&0.464&0.423&0.383&0.347&0.313&0.284&0.262&0.248&0.243 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrrrr}
\multicolumn{11}{c}{\text{LINEAR}} \\
\multicolumn{11}{c}{N=19} \\
\ \\
0&1&2&3&4&5&6&7&8&9&10 \\
\ \\
12&37&51&93&14&15&33&19&24&39&1 \\
11&34&47&86&13&14&31&18&23&38&1 \\
10&31&43&79&12&13&29&17&22&37&1 \\
9&28&39&72&11&12&27&16&21&36&1 \\
8&25&35&65&10&11&25&15&20&35&1 \\
7&22&31&58&9&10&23&14&19&34&1 \\
6&19&27&51&8&9&21&13&18&33&1 \\
5&16&23&44&7&8&19&12&17&32&1 \\
4&13&19&37&6&7&17&11&16&31&1 \\
3&10&15&30&5&6&15&10&15&30&1 \\
2&7&11&23&4&5&13&9&14&29&1 \\
1&4&7&16&3&4&11&8&13&28&1 \\
0&1&3&9&2&3&9&7&12&27&1 \\
-1&-2&-1&2&1&2&7&6&11&26&1 \\
-2&-5&-5&-5&0&1&5&5&10&25&1 \\
-3&-8&-9&-12&-1&0&3&4&9&24&1 \\
-4&-11&-13&-19&-2&-1&1&3&8&23&1 \\
-5&-14&-17&-26&-3&-2&-1&2&7&22&1 \\
-6&-17&-21&-33&-4&-3&-3&1&6&21 \\
\ \\
57&190&285&570&95&114&285&190&285&570&19 \\
0.478&0.441&0.406&0.372&0.340&0.311&0.284&0.262&0.244&0.233&0.229 
\end{array}
\]

\newpage

\[
\begin{array}{rrrr}
\multicolumn{4}{c}{\text{QUADRATIC}} \\
\multicolumn{4}{c}{N=5} \\
\ \\
0&1&2&3 \\
\ \\
9&31&9 &-3 \\
0&9&13&12 \\
-4&-3&12&17 \\
-3&-5&6&12 \\
3&3&-5&-3 \\
\ \\
5&35&35&35 \\
2.145&0.941&0.609&0.697 
\end{array}
\]
\[
\begin{array}{rrrrr}
\multicolumn{5}{c}{N=7} \\
\ \\
0&1&2&3&4 \\
\ \\
9&32&5&1&-2 \\
3&15&4&3&3 \\
-1&3&3&4&6 \\
-3&-4&2&4&7 \\
-3&-6&1&3&6 \\
-1&-3&0&1&3 \\
3&5&-1&-2&-2 \\
\ \\
7&42&14&14&21 \\
1.558&0.873&0.535&0.535&0.577 
\end{array}
\]
\[
\begin{array}{rrrrrr}
\multicolumn{6}{c}{N=9} \\
\ \\
0&1&2&3&4&5 \\
\ \\
42&109&126&378&14&-21 \\
21&63&92&441&273&14 \\
5&27&63&464&447&39 \\
-6&1&39&447&536&54 \\
-12&-15&20&390&540&59 \\
-13&-21&6&293&459&54 \\
-9&-17&-3&156&293&39 \\
0&-3&-7&-21&42&14 \\
21&-6&-238&-294&-21&14 \\
\ \\
42&165&330&2310&2310&231 \\
1.272&0.813&0.528&0.448&0.482&0.505 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrr}
\multicolumn{7}{c}{\text{QUADRATIC}} \\
\multicolumn{7}{c}{N=11} \\
\ \\
0&1&2&3&4&5&6 \\
\ \\
135&83&270&450&55&-15&-36 \\
81&54&199&414&87&46&9 \\
37&30&138&373&109&92&44 \\
3&11&87&327&121&123&69 \\
-21&-3&46&276&123&139&84 \\
-35&-12&15&220&115&140&89 \\
-39&-16&-6&159&97&126&84 \\
-33&-15&-17&93&69&97&69 \\
-17&-9&-18&22&31&53&44 \\
9&2&-9&-54&-17&-6&9 \\
45&18&10&-135&-75&-80&-36 \\
\ \\
165&143&715&2145&715&715&429 \\
1.098&0.762&0.528&0.417&0.411&0.441&0.455 
\end{array}
\]
\[
\begin{array}{rrrrrrrr}
\multicolumn{8}{c}{N=13} \\
\ \\
01&1&2&3&4&5&6&7 \\
99&47&33&231&121&33&-33&-11 \\
66&33&25&198&132&77&33&0 \\
38&21&18&167&138&ill&86&9 \\
15&11&12&138&139&135&126&16 \\
-3&3&7&ill&135&149&153&21 \\
-16&-3&3&86&126&153&167&24 \\
-24&-7&0&63&112&147&168&25 \\
-27&-9&-2&42&93&131&156&24 \\
-25&-9&-3&23&69&105&131&21 \\
-18&-7&-3&6&40&69&93&16 \\
-6&-3&-2&-9&6&23&42&9 \\
11&3&0&-22&-33&-33&-22&0 \\
33&11&3&-33&-77&-99&-99&-11 \\
\ \\
143&91&91&1001&1001&1001&1001&143 \\
0.979&0.719&0.524&0.408&0.373&0.386&0.408&0.418 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrr}
\multicolumn{9}{c}{\text{QUADRATIC}} \\
\multicolumn{9}{c}{N=15} \\
\ \\
0&1&2&3&4&5&6&7&8 \\
\ \\
273&158&819&7371&2275&2184&273&-1183&-78 \\
195&117&638&6201&2184&2795&1482&429&-13 \\
127&81&477&5126&2073&3261&2471&1776&42 \\
69&50&336&4146&1942&3582&3240&2858&87 \\
21&24&215&3261&1791&3758&3789&3675&122 \\
-17&3&114&2471&1620&3789&4118&4227&147 \\
-45&-13&33&1776&1429&3675&4227&4514&162 \\
-63&-24&-28&1176&1218&3416&4116&4536&167 \\
-71&-30&-69&671&987&3012&3785&4293&162 \\
-69&-31&-90&261&736&2463&3234&3785&147 \\
-57&-27&-91&-54&465&1769&2463&3012&122 \\
-35&-18&-72&-274&174&930&1472&1974&87 \\
-3&-4&-33&-399&-137&-54&261&671&42 \\
39&15&26&-429&-468&-1183&-1170&-897&-13 \\
91&39&105&-364&-819&-2457&-2821&-2730&-78 \\
\ \\
455&340&2380&30940&15470&30940&30940&30940&1105 \\
0.891&0.682&0.518&0.407&0.354&0.349&0.3P5&0.382&0.389 
\end{array}
\]
\[
\begin{array}{rrrrrrrrrr}
\multicolumn{10}{c}{N=17} \\
\ \\
0&1&2&3&4&5&6&7&8&9 \\
\ \\
36&409&1260&308&628&186&52&-20&-156&-21 \\
27&315&1006&259&573&197&80&ill&7&-6 \\
19&231&777&214&519&204&103&222&147&7 \\
12&157&573&173&466&207&121&313&264&18 \\
6&93&394&136&414&206&134&384&358&27 \\
1&39&240&103&363&201&142&435&429&34 \\
-3&-5&ill&74&313&192&145&466&477&39 \\
-6&-39&7&49&264&179&143&477&502&42 \\
-8&-63&-72&28&216&162&136&468&504&43 \\
-9&-77&-126&11&169&141&124&439&483&42 \\
-9&-81&-155&-2&123&116&107&390&439&39 \\
-8&-75&-159&-11&78&87&85&321&372&34 \\
-6&-59&-138&-16&34&54&58&232&282&27 \\
-3&-33&-92&-17&-9&17&26&123&169&18 \\
1&3&-21&-14&-51&-24&-11&-6&33&7 \\
6&49&75&-7&-92&-69&-53&-155&-126&-6 \\
12&105&196&4&-132&-118&-100&-324&-308&-21 \\
\ \\
68&969&3876&1292&3876&1938&1292&3876&3876&323 \\
0.822&0.650&0.509&0.407&0.347&0.326&0.332&0.347&0.360&0.365 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrrrr}
\multicolumn{11}{c}{\text{QUADRATIC}} \\
\multicolumn{11}{c}{N=19} \\
\ \\
0&1&2&3&4&5&6&7&8&9&10 \\
\ \\
459&257&612&7956&1921&765&306&34&-459&-1377&-136 \\
357&204&499&6732&1717&748&357&68&527&-204&-51 \\
265&156&396&5603&1523&726&398&97&1383&824&24 \\
183&113&303&4569&1339&699&429&121&2109&1707&89 \\
111&75&220&3630&1165&667&450&140&2705&2445&144 \\
49&42&147&2786&1001&630&461&154&3171&3038&189 \\
-3&14&84&2037&847&588&462&163&3507&3486&224 \\
-45&-9&31&1383&703&541&453&167&3713&3789&249 \\
-77&-27&-12&824&569&489&434&166&3789&3947&264 \\
-99&-40&-45&360&445&432&405&160&3735&3960&269 \\
-111&-48&-68&-9&331&370&366&149&3551&3828&264 \\
-113&-51&-81&-283&227&303&317&133&3237&3551&249 \\
-105&-49&-84&-462&133&231&258&112&2793&3129&224 \\
-87&-42&-77&-546&49&154&189&86&2219&2562&189 \\
-59&-30&-60&-535&-25&72&110&55&1515&1850&144 \\
-21&-13&-33&-429&-89&-15&21&19&681&993&89 \\
27&9&4&-228&-143&-107&-78&-22&-283&-9&24 \\
85&36&51&68&-187&-204&-17&-68&-1377&-1156&-51 \\
153&68&108&459&-221&-306&-306&-119&-2601&-2448&-136 \\
\ \\
969&665&1995&33915&11305&6783&4845&1615&33915&33915&2261 \\
0.768&0.622&0.500&0.406&0.344&0.314&0.308&0.318&0.331&0.341&0.345 
\end{array}
\]

\newpage

\[
\begin{array}{rrrr}
\multicolumn{4}{c}{\text{CUBIC}} \\
\multicolumn{4}{c}{N=5} \\
\ \\
0&1&2&3 \\
\ \\
16&69&2&-3 \\
-14&4&27&12 \\
-4&-6&12&17 \\
11&4&-8&2 \\
-4&-1&2&-3 \\
\ \\
5&70&35&35 \\
4.919&0.993&0.878&0.697 
\end{array}
\]
\[
\begin{array}{rrrrr}
\multicolumn{5}{c}{N=7} \\
\ \\
0&1&2&3&4 \\
\ \\
16&39&8&-4&-2 \\
-4&8&19&16&3 \\
-8&-4&16&19&6 \\
-3&-4&6&12&7 \\
4&1&-4&2&6 \\
6&4&-7&-4&3 \\
-4&-2&4&1&-2 \\
\ \\
7&42&42&42&21 \\
2.903&0.964&0.673&0.673&0.577 
\end{array}
\]
\[
\begin{array}{rrrrrr}
\multicolumn{6}{c}{N=9} \\
\ \\
0&1&2&3&4&5 \\
\ \\
224&85&56&-28&-56&-21 \\
14&28&65&392&84&14 \\
-76&-2&56&515&144&39 \\
-81&-12&36&432&145&54 \\
-36&-9&12&234&108&59 \\
24&0&-9&12&54&54 \\
64&8&-20&-143&4&39 \\
49&8&-14&-140&-21&14 \\
-56&-7&16&112&0&-21 \\
\ \\
126&99&198&1386&462&231 \\
2.162&0.927&0.573&0.610&0.560&0.505 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrr}
\multicolumn{7}{c}{\text{CUBIC}} \\
\multicolumn{7}{c}{N=11} \\
\ \\
0&1&2&3&4&5&6 \\
\ \\
96&113&48&24&-72&-51&-36 \\
24&48&41&96&132&36&9 \\
-16&8&32&123&232&86&44 \\
-31&-12&22&116&251&106&69 \\
-28&-17&12&86&212&103&84 \\
-14&-12&3&44&138&84&89 \\
4&-2&-4&1&52&56&84 \\
19&8&-8&-32&-23&26&69 \\
24&13&-8&-44&-64&1&44 \\
12&8&-3&-24&-48&-12&9 \\
-24&-12&8&39&48&-6&-36 \\
\ \\
66&143&143&429&858&429&429 \\
1.771&0.889&0.535&0.535&0.541&0.490&0.455 
\end{array}
\]
\[
\begin{array}{rrrrrrrr}
\multicolumn{8}{c}{N=13} \\
\ \\
0&1&2&3&4&5&6&7 \\
\ \\
176&265&33&231&-33&-407&-110&-11 \\
66&132&25&396&132&308&33&0 \\
-4&42&18&460&222&738&128&9 \\
-41&-12&12&444&251&932&182&16 \\
-52&-37&7&369&233&939&202&21 \\
-44&-40&3&256&182&808&195&24 \\
-24&-28&0&126&112&588&168&25 \\
1&-8&-2&0&37&328&128&24 \\
24&13&-3&-101&-29&77&82&21 \\
38&28&-3&-156&-72&-116&37&16 \\
36&30&-2&-144&-78&-202&0&9 \\
11&12&0&-44&-33&-132&-22&0 \\
-44&-33&3&165&77&143&-22&-11 \\
\ \\
143&364&91&2002&1001&4004&1001&143 \\
1.526&0.853&0.524&0.479&0.501&0.484&0.441&0.418 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrr}
\multicolumn{9}{c}{\text{CUBIC}} \\
\multicolumn{9}{c}{N=15} \\
\ \\
0&1&2&3&4&5&6&7&8 \\
\ \\
1456&2059&8008&44044&2002&-19201&-28756&-3094&-78 \\
676&1144&5833&52624&17017&19604&8879&156&-13 \\
136&484&4048&54709&25762&44294&34244&2511&42 \\
-199&44&2618&51524&29252&57004&49054&4076&87 \\
-364&-211&1508&44294&28502&59869&55024&4956&122 \\
-394&-316&683&34244&24527&55024&53869&5256&147 \\
-324&-306&108&22599&18342&44604&47304&5081&162 \\
-189&-216&-252&10584&10962&30744&37044&4536&167 \\
-24&-81&-432&-576&3402&15579&24804&3726&162 \\
136&64&-467&-9656&-3323&1244&12299&2756&147 \\
256&184&-392&-15431&-8198&-10126&1244&1731&122 \\
301&244&-242&-16676&-10208&-16396&-6646&756&87 \\
236&209&-52&-12166&-8338&-15431&-9656&-64&42 \\
26&44&143&-676&-1573&-5096&-6071&-624&-13 \\
-364&-286&308&19019&11102&16744&5824&-819&-78 \\
\ \\
1365&3060&21420&278460&139230&278460&278460&30940&1105 \\
1.356&0.820&0.522&0.443&0.458&0.464&0.410&0.405&0.389 
\end{array}
\]
\[
\begin{array}{rrrrrrrrrr}
\multicolumn{10}{c}{N=17} \\
\ \\
0&1&2&3&4&5&6&7&8&9 \\
\ \\
64&605&1456&723&208&-66&-320&-384&-352&-21 \\
34&364&1055&728&468&134&121&20&-42&-6 \\
12&182&728&691&624&267&428&313&196&7 \\
-3&52&468&624&691&342&618&508&369&18 \\
-12&-33&268&534&684&368&708&618&484&27 \\
-16&-80&121&428&618&354&715&656&548&34 \\
-16&-96&20&313&508&309&656&635&568&39 \\
-13&-88&-42&196&369&242&548&568&551&42 \\
-8&-63&-72&84&216&162&408&468&504&43 \\
-2&-28&-77&-16&64&78&253&348&434&42 \\
4&10&-64&-97&-72&-1&100&221&348&39 \\
9&44&-40&-152&-177&-66&-34&100&253&34 \\
12&67&-12&-174&-236&-108&-132&-2&156&27 \\
12&72&13&-156&-234&-118&-177&-72&64&18 \\
8&52&28&-91&-156&-87&-152&-97&-16&7 \\
-1&0&26&28&13&-6&-40&-64&-77&-6 \\
-16&-91&0&208&288&134&176&40&-112&-21 \\
\ \\
68&969&3876&3876&3876&1938&3876&3876&3876&323 \\
1.231&0.790&0.522&0.422&0.422&0.436&0.429&0.405&0.377&0.365 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrrrr}
\multicolumn{11}{c}{\text{CUBIC}} \\
\multicolumn{11}{c}{N=19} \\
\ \\
0&1&2&3&4&5&6&7&8&9&10 \\
\ \\
3264&851&1632&15504&25296&-153&-1224&-918&-27744&-561&-136 \\
1904&544&1193&14144&36856,&5372&714&68&-3264&-136&-51 \\
864&304&832&12601&43696&9162&2104&797&15424&204&24 \\
109&124&542&10924&46439&11427&3009&1297&28901&466&89 \\
-396&-3&316&9162&45708&12377&3492&1596&37748&657&144 \\
-686&-84&147&7364&42126&12222&3616&1722&42546&784&189 \\
-796&-126&28&5579&36316&11172&3444&1703&43876&854&224 \\
-761&-136&-48&3856&28901&9437&3039&1567&42319&874&249 \\
-616&-121&-88&2244&20504&7227&2464&1342&38456&851&264 \\
-396&-88&-99&792&11748&4752&1782&1056&32868&792&269 \\
-136&-44&-88&-451&3256&2222&1056&737&26136&704&264 \\
129&4&-62&-1436&-4349&-153&349&413&18841&594&249 \\
364&49&-28&-2114&-10444&-2163&-276&112&11564&469&224 \\
534&84&7&-2436&-14406&-3598&-756&-138&4886&336&189 \\
604&102&36&-2353&-15612&-4248&-1028&-309&-612&202&144 \\
539&96&52&-1816&-13439&-3903&-1029&-373&-4349&74&89 \\
304&59&48&-776&-7264&-2353&-696&-302&-5744&-41&24 \\
-136&-16&17&816&3536&612&4&-68&-4216&-136&-51 \\
-816&-136&-48&3009&19584&5202&1224&357&816&-204&-136 \\
\ \\
3876&1463&4389&74613&298452&74613&21318&10659&298452&6783&2261 \\
1.133&0.763&0.521&0.411&0.394&0.407&0.412&0.400&0.377&0.354&0.345 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrr}
\multicolumn{5}{c}{\text{QUARTIC}} \\
\multicolumn{5}{c}{N=7} \\
\ \\
0&1&2&3&4 \\
\ \\
25&456&25&-35&5 \\
-25&25&356&155&-30 \\
-5&-35&155&212&75 \\
15&10&-60&150&131 \\
7&20&-65&25&75 \\
-15&-19&70&-65&-30 \\
5&5&-19&20&5 \\
\ \\
7&462&462&462&231 \\
6.059&0.993&0.878&0.677&0.753 
\end{array}
\]
\[
\begin{array}{rrrrrr}
\multicolumn{6}{c}{N=9} \\
\ \\
0&1&2&3&4&5 \\
\ \\
50&1231&350&-250&-50&15 \\
-25&175&1412&1025&75&-55 \\
-25&-125&1025&1112&225&30 \\
0&-75&225&675&304&135 \\
18&45&-330&180&270&179 \\
15&81&-360&-105&135&135 \\
-5&5&37&-110&-35&30 \\
-20&-85&385&37&-120&-55 \\
10&35&-170&10&54&15 \\
\ \\
18&1287&2574&2574&858&429 \\
3.859&0.978&0.741&0.657&0.595&0.646 
\end{array}
\]
\[
\begin{array}{rrrrrrr}
\multicolumn{7}{c}{\text{QUARTIC}} \\
\multicolumn{7}{c}{N=11} \\
\ \\
0&1&2&3&4&5&6 \\
\ \\
75&131&30&-30&-45&-15&18 \\
-15&30&59&150&75&0&-45 \\
-35&-10&50&177&125&50&-10 \\
-20&-15&25&125&127&100&60 \\
4&-5&0&50&100&127&120 \\
20&6&-15&-10&60&120&143 \\
20&10&-16&-35&20&80&120 \\
5&5&-5&-23&-10&20&60 \\
-15&-5&10&10&-23&-35&-10 \\
-21&-10&15&30&-15&-48&-45 \\
15&6&-10&-15&15&30&18 \\
\ \\
33&143&143&429&429&429&429 \\
2.908&0.957&0.642&0.642&0.544&0.544&0.577 
\end{array}
\]
\[
\begin{array}{rrrrrrrr}
\multicolumn{8}{c}{N=13} \\
\ \\
0&1&2&3&4&5&6&7 \\
\ \\
275&2698&825&-825&-3795&-2915&-286&110 \\
0&825&1048&9900&6270&2255&-495&-198 \\
-100&-75&900&12428&10140&5745&640&-135 \\
-95&-345&570&10140&9992&7625&2230&110 \\
-41&-265&205&5745&7625&8042&3610&390 \\
20&-52&-90&1280&4460&7220&4339&600 \\
60&140&-252&-1890&1540&5460&4200&677 \\
65&220&-260&-3072&-470&3140&3200&600 \\
35&160&-135&-2245&-1283&715&1570&390 \\
-16&-5&60&-60&-990&-1283&-235&110 \\
-60&-177&220&2160&-60&-2245&-1536&-135 \\
-55&-195&198&2420&660&-1485&-1430&-198 \\
55&165&-195&-1947&-55&1760&1210&115 \\
\ \\
143&3094&3094&34034&34034&34034&17017&2436 \\
2.379&0.934&0.582&0.604&0.542&0.486&0.505&0.527 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrr}
\multicolumn{9}{c}{\text{QUARTIC}} \\
\multicolumn{9}{c}{N=15} \\
\ \\
0&1&2&3&4&5&6&7&8 \\
\ \\5005&9626&25025&25025&-50050&-1251250&-749749&-5005&2145 \\
715&3575&24482&260975&89375&1011725&177320&-46761&-2860 \\
-1265&275&20075&331466&147950&2240975&1038785&-2970&-2937 \\
-1705&-1100&13750&295900&151708&2698300&1732390&82940&-165 \\
-1249&-1250&7075&203725&122650&2614126&2187505&176235&3755 \\
-415&-749&1240&94435&78745&2187505&2365126&250875&7500 \\
405&-45&-2943&-2430&33930&1586115&2257875&289514&10125 \\
945&540&-5040&-67284&-1890&946260&1890000&283500&11063 \\
1065&810&-4995&-90495&-22842&372870&1317375&232875&10125 \\
751&695&-3130&-72385&-27085&-60499&627500&146375&7500 \\
115&251&-145&-23230&-16810&-311665&-60499&41430&3755 \\
-605&-340&2882&36740&1760&-369820&-595870&-55836&-165 \\
-1045&-770&4495&77341&18370&-255530&-796235&-110605&-2937 \\
-715&-605&2860&58435&18733&-20735&-447590&-79365&-2858 \\
1001&715&-4235&-70070&-15470&251251&695695&90090&2145 \\
\ \\
3003&11628&81396&1058148&529074&11639628&11639628&1293292&46191 \\
2.040&0.910&0.548&0.560&0.535&0.474&0.451&0.473&0.489 
\end{array}
\]
\[
\begin{array}{rrrrrrrrrr}
\multicolumn{10}{c}{N=17} \\
\ \\
0&1&2&3&4&5&6&7&8&9 \\
\ \\
1300&761&1300&260&-260&-2730&-4628&-2340&260&195 \\
325&325&1094&845&585&2210&1690&-403&-1755&-195 \\
-195&65&845&1042&975&4875&5915&2080&-1079&-260 \\
390&-65&585&975&1042&5850&8385&4615&1170&-117 \\
-372&-105&340&750&900&5648&9420&6810&4060&135 \\
-235&-89&130&455&645&4710&9322&8375&6845&415 \\
-55&-45&-31&160&355&3405&8375&9122&8965&660 \\
110&5&-135&-83&90&2030&6845&8965&10046&825 \\
220&45&-180&-240&-108&810&4980&7920&9900&883 \\
253&65&-170&-295&-215&-102&3010&6105&8525&825 \\
205&61&-115&-250&-225&-625&1147&3740&6105&660 \\
90&35&-31&-125&-150&-750&-415&1147&3010&415 \\
-60&-5&60&42&-20&-540&-1500&-1250&-204&135 \\
-195&-45&130&195&117&-130&-1950&-2925&-2795&-117 \\
-247&-65&145&260&195&273&-1625&-3250&-385&-195 \\
-130&-39&65&145&130&390&-403&-1495&-2210 \\
-195&260&65&-156&-260&-180&-130&1820&3172&3380 \\
\ \\
884&969&3876&3876&3876&25194&50388&50388&50388&4264 \\
1.804&0.886&0.531&0.518&0.518&0.473&0.430&0.425&0.447&0.450 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrrrr}
\multicolumn{11}{c}{\text{QUARTIC}} \\
\multicolumn{11}{c}{N=19} \\
\ \\
0&1&2&3&4&5&6&7&8&9&10 \\
\ \\
2550&3583&5100&25500&-17850&-47430&-50184&-36720&-15810&255&340 \\
850&1700&3949&49300&67150&40460&18870&2516&-8670&-680&-255 \\
-150&500&2900&57533&109450&89640&62360&33170&6466&-660&-420 \\
-625&-175&1975&54725&120266&109455&85515&55585&25445&-22&-290 \\
-729&-465&1190&44820&109455&108188&93060&70230&44795&945&18 \\
-595&-492&555&31180&85515&93060&89216&77700&61725&2000&405 \\
-335&-360&74&16585&55585&70230&77700&78716&74125&2950&790 \\
-40&-155&-255&3233&25445&44795&61725&74125&80566&3650&1110 \\
220&55&-440&-7260&-484&20790&44000&64900&80300&4003&1320 \\
396&220&-495&-13860&-19140&1188&26730&52140&73260&3960&1393 \\
460&308&-440&-16115&-28820&-12100&11616&37070&60060&3520&1320 \\
405&305&-301&-14155&-29180&-18225&-145&21041&41995&2730&1110 \\
245&215&-110&-8692&-21235&-17400&-7860&5530&21041&1685&790 \\
15&60&95&-1020&-7359&-10900&-11340&-7860&-145&528&405 \\
-229&-120&270&6985&8715&-1062&-10900&-17400&-18225&-550&18 \\
-410&-267&365&12865&21895&8715&-7359&-21235&-29180&-1310&-278 \\
-430&-305&324&13580&25730&13970&-2040&-17384&-28310&-1465&-420 \\
-170&-140&85&5508&12410&9180&3230&-3740&-10234&-680&-255 \\
510&340&-420&-15555&-27234&-12240&6120&21930&31110&1428&340 \\
\ \\
1938&4807&14421&245157&490314&490314&490314&490314&490314&22287&7441 \\
1.630&0.863&0.523&0.484&0.495&0.470&0.427&0.401&0.405&0.424&0.432 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrr}
\multicolumn{5}{c}{\text{QUINTIC}} \\
\multicolumn{5}{c}{N=7} \\
\ \\
0&1&2&3&4 \\
\ \\
36&923&1&-5&5 \\
-69&6&148&30&-30 \\
50&-15&15&233&75 \\
15&20&-20&100&131 \\
-48&-15&15&-75&75 \\
29&6&-6&30&-30 \\
-6&-1&1&-5&5 \\
\ \\
7&924&154&308&231 \\
15.625&0.999&0.980&0.870&0.753 
\end{array}
\]
\[
\begin{array}{rrrrrr}
\multicolumn{6}{c}{N=9} \\
\ \\
0&1&2&3&4&5 \\
\ \\
48&425&72&-54&32&15 \\
-57&18&1385&261&-213&-55 \\
-2&-27&522&400&582&30 \\
33&8&-213&291&905&135 \\
12&15&-220&60&540&179 \\
-23&-6&123&-101&-27&135 \\
-18&-13&-66&-202&30 \\
27&12&-187&93&123&-55 \\
-8&-3&48&-26&-24&15 \\
\ \\
12&429&1716&858&1716&429 \\
7.654&0.995&0.898&0.683&0.726&0.646 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrr}
\multicolumn{7}{c}{\text{QUINTIC}} \\
\multicolumn{7}{c}{N=11} \\
\ \\
0&1&2&3&4&5&6 \\
\ \\
108&557&27&-153&-4&6&18 \\
-81&54&184&666&3&-22&-45 \\
-46&-51&ill&719&38&13&-10 \\
24&-16&6&456&57&48&60 \\
48&24&-44&156&48&57&120 \\
20&24&-30&-40&20&40&143 \\
-24&-4&12&-96&-8&12&120 \\
-39&-24&34&-48&-18&-8&60 \\
-4&-9&9&29&-4&-8&-10 \\
45&26&-36&54&17&6&-45 \\
-18&-9&13&-27&-6&-1&18 \\
\ \\
33&572&286&1716&143&143&429 \\
5.078&0.987&0.802&0.647&0.631&0.631&0.577 
\end{array}
\]
\[
\begin{array}{rrrrrrrr}
\multicolumn{8}{c}{N=13} \\
\ \\
0&1&2&3&4&5&6&7 \\
\ \\
792&420&132&-429&-704&33&132&110 \\
-363&66&455&1881&1221&-352&-330&-198 \\
-398&-39&342&2030&2586&453&-50&-135 \\
-69&-32&ill&1293&3045&1314&405&110 \\
204&3&-64&453&2628&1680&720&390 \\
260&24&-120&-100&1620&1440&777&600 \\
120&20&-72&-270&440&780&600&677 \\
-90&0&20&-156&-480&40&300&600 \\
-216&-18&84&47&-816&-429&20&390 \\
-153&-18&69&147&-473&-408&-120&110 \\
78&3&-22&54&294&47&-78&-135 \\
253&24&-99&-121&759&462&55&-198 \\
-132&-11&48&33&-396&-198&0&195 \\
\ \\
286&442&884&4862&9724&4862&2431&2516 \\
3.849&0.975&0.717&0.646&0.560&0.588&0.565&0.514 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrr}
\multicolumn{9}{c}{\text{QUINTIC}} \\
\multicolumn{9}{c}{N=15} \\
\ \\
0&1&2&3&4&5&6&7&8 \\
\ \\
1716&11919&1287&-11583&-4147&-67353&5577&22165&2145 \\
-429&2574&2742&59202&7293&10582&-17160&-40326&-2860 \\
-814&-891&2277&67827&11913&227337&-605&-23925&-2937 \\
-429&-1276&1122&47652&12048&422532&26730&22660&-165 \\
96&-471&37&20667&9603&513417&47835&68055&3755 \\
425&390&-600&-550&6075&478350&55002&95250&7500 \\
450&775&-705&-10875&2575&340275&47625&98629&10125 \\
225&600&-400&-10680&-150&150200&30000&81000&11063 \\
-100&75&75&-3875&-1695&-29325&9125&50625&10125 \\
-345&-450&450&4050&-1975&-140730&-7500&18250&7500 \\
-366&-639&513&7983&-1203&-147947&-14073&-5865&3755 \\
-121&-324&198&5148&132&-52932&-8690&-14916&-165 \\
264&341&-327&-2937&1287&87813&4455&-8525&-2937 \\
429&726&-572&-8502&1287&146718&12870&4290&-2804 \\
-286&-429&363&4433&-1053&-91377&-6435&2145&2145 \\
\ \\
715&12920&6460&167960&41990&1847560&184756&369512&46245 \\
3.137&0.960&0.652&0.635&0.536&0.527&0.546&0.517&0.489 
\end{array}
\]
\[
\begin{array}{rrrrrrrrrr}
\multicolumn{10}{c}{N=17} \\
\ \\
0&1&2&3&4&5&6&7&8&9 \\
\ \\
3744&865&1872&-52&-832&-1794&-208&2236&6240&195 \\
-351&234&2825&403&1443&1391&-1391&-4407&-8515&-195 \\
-1534&-39&2418&486&2262&3939&1066&-2496&-7878&-260 \\
-1209&-104&1443&377&2201&5499&4511&2899&195&-117 \\
-348&-69&428&202&1692&5972&7276&8394&10100&135 \\
443&-6&-321&41&1041&5457&8511&12027&18255&415 \\
858&43&-678&-64&446&4197&8018&12994&22770&660 \\
825&60&-655&-101&15&2525&6085&11385&23117&825 \\
440&45&-360&-80&-216&810&3320&7920&19800&883 \\
-99&10&45&-25&-265&-597&485&3685&14025&825 \\
-558&-27&386&34&-186&-1417&-1670&-132&7370&660 \\
-733&-48&519&69&-51&-1497&-2573&-2505&1455&415 \\
-516&-41&372&62&68&-864&-1996&-2834&-2388&135 \\
39&-6&-13&13&117&221&-221&-1209&-3445&-117 \\
650&39&-438&-52&78&1209&1794&1326&-1950&510 \\
741&52&-507&-73&-13&1209&2249&2509&585&-195 \\
-624&-39&416&52&-48&-1066&-1664&-1404&1040&195 \\
\ \\
1768&969&7752&1292&7752&25194&33592&50388&100776&4969 \\
2.675&0.945&0.604&0.613&0.533&0.487&0.503&0.508&0.479&0.397 
\end{array}
\]

\newpage

\[
\begin{array}{rrrrrrrrrrr}
\multicolumn{11}{c}{\text{QUINTIC}} \\
\multicolumn{11}{c}{N=19} \\
\ \\
0&1&2&3&4&5&6&7&8&9&10 \\
\ \\
3672&8288&3978&-3978&-16796&-91494&-19890&5066&25704&1377&340 \\
102&2652&4697&135252&29614&78676&-1326&-18020&-36346&-1428&-255 \\
-1228&-78&3978&167888&46904&176046&33254&-6274&-33420&-1738&-420 \\
-1263&-988&2613&140712&46256&216996&68289&20256&1839&-660&-290 \\
-696&-897&1157&88023&36166&216475&93951&47689&46016&978&18 \\
-1&-390&-39&33254&22763&187902&105254&67442&83703&2594&405 \\
534&149&-795&-9411&10128&143067&101163&75361&106278&3819&790 \\
774&504&-1069&-33420&613&92032&83703&70852&110684&4464&1110 \\
704&594&-924&-38236&-4840&43032&57068&56012&98208&4487&1320 \\
396&440&-495&-27720&-6380&2376&26730&34760&73260&3960&1393 \\
-24&132&44&-8514&-4928&-25652&-1452&11968&42152&3036&1320 \\
-409&-204&513&11576&-1858&-38892&-22123&-7408&11877&1916&1110 \\
-624&-439&759&25197&1322&-37407&-31323&-19197&-11112&816&790 \\
-579&-474&689&27066&3289&-23582&-27378&-20882&-22123&-66&405 \\
-262&-273&303&15587&3224&-2223&-11791&-12469&-19446&-583&18 \\
228&104&-273&-5532&1131&19344&9867&2644&-5574&-672&-120 \\
648&468&-754&-25662&-1844&31174&27066&16798&11576&-387&-420 \\
578&468&-663&-25636&-3094&20604&23426&17204&17442&68&-255 \\
-612&-442&702&23868&1768&-27846&-24174&-14926&-10404&306&340 \\
\ \\
1938&9614&14421&490314&163438&980628&490314&326876&490314&22287&7599 \\
2.351&0.928&0.571&0.585&0.532&0.470&0.463&0.480&0.475&0.449&0.422 
\end{array}
\]
}

\end{document}

% </pre></body></html>
