\begin{table}%t1a
\caption{\label{tabla 1a}Coefficients  $a_{r} (i,j)$ and $ b_{r}(i,j)$ of
  the power series for $r$, corresponding to a stratified medium with
  a generic  density distribution $F(z)^a$.}
\par
%\centerline
{
\begin{tabular}{c c l l}
\hline \hline
${i}$  &  ${j}$  &  $a_{r} {(i,j)}$     & $ b_{r}{(i,j)}$  \\ 
\hline
  1     &  1  &  1  &  $  {f}_{1}/2$  \\
\hline
  2 & 1 & $-  {f}_{1}^{2}+2   {f}_{2}$  &  $-2   {f}_{1}^{3}+12   {f}_{3} $ \\
    & 2 & $-  {f}_{1}^{2}- 2   {f}_{2} $  & $ -  {f}_{1}^{3}- 8   {f}_{1}
      {f}_{2} - 6   {f}_{3} $ \\
\hline 
  3 & 1 &   $4   {f}_{1}^{4}-12   {f}_{1}^{2}
      {f}_{2} + 48   {f}_{4} $  &  $5   {f}_{1}^{5}+ 52   {f}_{1}^{3}
      {f}_{2} -152   {f}_{1}   {f}_{2}^{2}-144   {f}_{1}^{2}
      {f}_{3} +180   {f}_{2}   {f}_{3} $  \\
   &&& $+ 264   {f}_{1}   {f}_{4} + 600   {f}_{5}$\\
    & 2 &  $ 3  {f}_{1}^{4}+ 22   {f}_{1}^{2}
      {f}_{2}-16   {f}_{2}^{2} -42   {f}_{1}   {f}_{3}-72   {f}_{4}$ &
    $ 4   {f}_{1}^{5}+ 104   {f}_{1}^{3}   {f}_{2}-360   {f}_{2}  {f}_{3}-528   {f}_{1}
      {f}_{4}- 480   {f}_{5}  $  \\
    & 3 &  $  {f}_{1}^{4}+ 22   {f}_{1}^{2}
      {f}_{2} +16   {f}_{2}^{2} + 42   {f}_{1}   {f}_{3} + 24   {f}_{4}$ &  $  {f}_{1}^{5}+ 52   {f}_{1}^{3}
      {f}_{2}+136    {f}_{1}
      {f}_{2}^{2}+192   {f}_{2}^{2}  {f}_{3} +180   {f}_{2}  {f}_{3}
      $  \\
      &&&$ + 264    {f}_{1}   {f}_{4}+ 120  {f}_{5}$\\
\hline 
4& 1&  $ -15   {f}_{1}^{6}- 54   {f}_{1}^{4}  {f}_{2}+ 464   {f}_{1}^{2}   {f}_{2}^{2}
-304   {f}_{2}^{3} + 300   {f}_{1}^{3}  {f}_{3} -1488   {f}_{1}   {f}_{2}  {f}_{3}
  $ & $ -14   {f}_{1}^{7}-
960   {f}_{1}^{5}  {f}_{2}+ 3072   {f}_{1}^{3}   {f}_{2}^{2} + 1404
  {f}_{1}^{4}  {f}_{3} - 11424   {f}_{2}^{2}  {f}_{3}   $ \\
&  & $+540   {f}_{3}^{2}-576   {f}_{1}^{2}  {f}_{4} + 1248
  {f}_{2}  {f}_{4}+1920   {f}_{1}  {f}_{5}+ 3600  {f}_{6}$ & $-7848   {f}_{1}   {f}_{3}^{2}
-14016   {f}_{1}   {f}_{2}  {f}_{4}
 + 32256    {f}_{3}  {f}_{4} -1920
  {f}_{1}^{2}  {f}_{5} $ \\
  &&&$+ 40320   {f}_{2}  {f}_{5}+63360
  {f}_{1}  {f}_{6}+70560  {f}_{7} $\\
 & 2& $ -11   {f}_{1}^{6}+ 322   {f}_{1}^{4}  {f}_{2} + 464   {f}_{1}^{2}
   {f}_{2}^{2}+304   {f}_{2}^{3}
 + 300   {f}_{1}^{3}  {f}_{3} +1488   {f}_{1}   {f}_{2}  {f}_{3}
 $ & $ -14   {f}_{1}^{7}-1440   {f}_{1}^{5}  {f}_{2}-1536   {f}_{1}^{3}
   {f}_{2}^{2} + 5888   {f}_{1}  {f}_{2}^{3}
-1836  {f}_{1}^{4}  {f}_{3} 
 $\\
 &  &  $-1620  {f}_{3}^{2}+576
   {f}_{1}^{2}  {f}_{4}-3744  {f}_{2}  {f}_{4}-5760
   {f}_{1}  {f}_{5}-6480   {f}_{6}$ &  $ +20880  {f}_{1}^{2}  {f}_{2}  {f}_{3} -1344   {f}_{2}   {f}_{3}
   -5688   {f}_{1}  {f}_{3}^{2}
 +8928    {f}_{1}^{3}   {f}_{4}  $\\
 &  & $  $ &  $ -15744   {f}_{1}   {f}_{2}  {f}_{4}-48384
   {f}_{3}   {f}_{4}-17760  {f}_{1}^{2}   {f}_{5}-60480   {f}_{2}
   {f}_{5}  $ \\
   &&&$-95040   {f}_{1}  {f}_{6}-70560   {f}_{6}$\\
 & 3&  $ -5   {f}_{1}^{6}- 342   {f}_{1}^{4}  {f}_{2} -720
   {f}_{1}^{2}  {f}_{2}^{2}+ 272   {f}_{2}^{3} -732   {f}_{1}^{3}  {f}_{3} +2304   {f}_{1}   {f}_{2}  {f}_{3}
$ &  $-6   {f}_{1}^{7}-
960   {f}_{1}^{5}  {f}_{2}-6144   {f}_{1}^{3}   {f}_{2}^{2} -5076
  {f}_{1}^{4}  {f}_{3} +14112   {f}_{2}^{2}  {f}_{3}  $\\
 &  & $ +1620  {f}_{3}^{2}+ 1824   {f}_{1}^{2}  {f}_{4}+3744  {f}_{2}  {f}_{4}+5760   {f}_{1}  {f}_{5}+ 3600  {f}_{6}$  &  $ +19224   {f}_{1}   {f}_{3}^{2}
+45504   {f}_{1}   {f}_{2}  {f}_{4} + 32256    {f}_{3}  {f}_{4} +37440
  {f}_{1}^{2}  {f}_{5} $ \\
  &&&$+ 40320   {f}_{2}  {f}_{5}+63360
  {f}_{1}  {f}_{6}+30240  {f}_{7}$\\
 & 4& $-   {f}_{1}^{6}- 114   {f}_{1}^{4}  {f}_{2} -720
   {f}_{1}^{2}  {f}_{2}^{2}- 272   {f}_{2}^{3} -732   {f}_{1}^{3}  {f}_{3} -2304   {f}_{1}   {f}_{2}  {f}_{3}
 $  &  $ -   {f}_{1}^{7}-240   {f}_{1}^{5}  {f}_{2}-3072   {f}_{1}^{3}
   {f}_{2}^{2}-3968   {f}_{1}  {f}_{2}^{3}
-2538  {f}_{1}^{4}  {f}_{3}  $ \\
 &  & $ -540  {f}_{3}^{2}- 1824   {f}_{1}^{2}  {f}_{4} -1248   {f}_{2}  {f}_{4} -1920  {f}_{1}  {f}_{5}-720  {f}_{6}$  &  $ -17928   {f}_{1}^{2}  {f}_{2}  {f}_{3} -7056   {f}_{2}   {f}_{3} -9612   {f}_{1}  {f}_{3}^{2}
 -10224    {f}_{1}^{3}   {f}_{4}  $ \\
 &  & $ $  &  $  -22752  {f}_{1}   {f}_{2}  {f}_{4}-8064
   {f}_{3}   {f}_{4}-18720  {f}_{1}^{2}   {f}_{5}-10080   {f}_{2}
   {f}_{5} $ \\
   &&&$-15840   {f}_{1}   {f}_{6}-5040   {f}_{7}$\\
\hline 
%\noalign{\smallskip}
%5& 1&  $ $ & $ $ \\
%\noalign{\smallskip}
% & 2& $ $ & $ $\\
%\noalign{\smallskip}
 %& 3&  $ $   &   $ $  \\
%\noalign{\smallskip}
% & 4& $ $  &  $ $ \\
%\noalign{\smallskip}
 %& 5& $ $  &  $ $ \\
%\noalign{\smallskip}
%\hline
\end{tabular}}
$^a$
 The terms ${  f_{1},
  f_{2},f_{3},~...}$  represent the coefficients of the Taylor expansion of
  $F(z)^{-1}$, whose numerical values are determined from the specific
  distribution function $F(z)$ that we choose.
\end{table}