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\begin{expandtable}
\begin{tabularx}{\linewidth}{|l|X|}
\hline
\multicolumn{2}{|c|}{Berühmte Zahlen} \\
\hline
\textsc{Fibonacci} &
$f(0) = 0 \quad
f(1) = 1 \quad
f(n+2) = f(n+1) + f(n)$ \\
\grayhline
\textsc{Catalan} &
$C_0 = 1 \qquad
C_n = \sum\limits_{k = 0}^{n - 1} C_kC_{n - 1 - k} =
\frac{1}{n + 1}\binom{2n}{n} = \frac{2(2n - 1)}{n+1} \cdot C_{n-1}$ \\
\grayhline
\textsc{Euler} I &
$\eulerI{n}{0} = \eulerI{n}{n-1} = 1 \qquad
\eulerI{n}{k} = (k+1) \eulerI{n-1}{k} + (n-k) \eulerI{n-1}{k-1} $ \\
\grayhline
\textsc{Euler} II &
$\eulerII{n}{0} = 1 \quad
\eulerII{n}{n} = 0 \quad$\\
& $\eulerII{n}{k} = (k+1) \eulerII{n-1}{k} + (2n-k-1) \eulerII{n-1}{k-1}$ \\
\grayhline
\textsc{Stirling} I &
$\stirlingI{0}{0} = 1 \qquad
\stirlingI{n}{0} = \stirlingI{0}{n} = 0 \qquad
\stirlingI{n}{k} = \stirlingI{n-1}{k-1} + (n-1) \stirlingI{n-1}{k}$ \\
\grayhline
\textsc{Stirling} II &
$\stirlingII{n}{1} = \stirlingII{n}{n} = 1 \qquad
\stirlingII{n}{k} = k \stirlingII{n-1}{k} + \stirlingII{n-1}{k-1} =
\frac{1}{k!} \sum\limits_{j=0}^{k} (-1)^{k-j}\binom{k}{j}j^n$\\
\grayhline
\textsc{Bell} &
$B_1 = 1 \qquad
B_n = \sum\limits_{k = 0}^{n - 1} B_k\binom{n-1}{k}
= \sum\limits_{k = 0}^{n}\stirlingII{n}{k}$\\
\grayhline
\textsc{Partitions} &
$p(0,0) = 1 \quad
p(n,k) = 0 \text{ für } k > n \text{ oder } n \leq 0 \text{ oder } k \leq 0$ \\
& $p(n,k) = p(n-k,k) + p(n-1,k-1)$\\
\grayhline
\textsc{Partitions} &
$f(0) = 1 \quad f(n) = 0~(n < 0)$ \\
& $f(n)=\sum\limits_{k=1}^\infty(-1)^{k-1}f(n - \frac{k(3k+1)}{2})+\sum\limits_{k=1}^\infty(-1)^{k-1}f(n - \frac{k(3k-1)}{2})$\\
\hline
\end{tabularx}
\end{expandtable}
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