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Matrix $A$ is 

\begin{array}
1 & 6 & 9 & 2 & 3 & 6 & 1 \\
1 & 5 & 6 & 2 & 2 & 2 & 0 \\
1 & 4 & 4 & 2 & 1 & 0 & 0 \\
1 & 3 & 3 & 1 & 0 & 0 & 0 \\
1 & 2 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}

According equation 3 and 4, we have

\begin{equation}
P_{ij} = A_{ij} * c_j / b_i
\end{equation}


\begin{eqnarray}
\nonumber
c_1 & = & exp(-12\beta) \\
\nonumber
c_2 & = & exp(-10\beta) - exp(-12\beta) \\
\nonumber
c_3 & = & exp(-8\beta) - 2exp(-10\beta)  + exp(-12\beta) \\
\nonumber
c_4 & = & exp(-8\beta) - 2exp(-10\beta) + exp(-12\beta) \\
\nonumber
c_5 & = & exp(-6\beta) - 3exp(-10\beta) + 2exp(-12\beta) \\
\nonumber
c_6 & = & exp(-4\beta) - 2exp(-6\beta) + 2exp(-10\beta) - exp(-12\beta) \\
\nonumber
c_7 & = & 0.5exp(-2\beta) - exp(-4\beta) + exp(-6\beta) - exp(-8\beta) +
2.5exp(-10\beta) - exp(-12\beta) \\
\nonumber
c_8 & = & 1 - 3exp(-2\beta) + 3exp(-4\beta) - 2exp(-6\beta) +
3exp(-8\beta) - 2exp(-10\beta) + exp(-12\beta)  \\
\nonumber
\end{eqnarray}

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